Split (1)

train · 99.8k rows

text stringlengths 14 5.77M	meta dict	__index_level_0__ int64 0 9.97k ⌀
European Climate Technology Investment Is Closing the Gap With American Investment, According to New Data From Net Zero Insights and Alder & Co. Net Zero Insights, the leading market intelligence platform for climate tech in Europe and North America together with Alder & Co., a purpose-driven climate tech marketing agency, today announced year-end climate tech investment results for 2022 totaling a record-setting sum of $82 billion, a 20% increase compared to 2021. While the majority, $43.9 billion, came from the U.S., Europe's total funding reached $35.6B, representing a 33% year-over-year increase, compared to only 7% in the U.S. Europe's funding surge of 33% was driven by investments in energy, which saw an increase in funding by 81% ($18.5B), followed by transport ($9.4B), circular economy ($7.1B), and industry ($6B). The sectors showing the strongest growth in Europe are industry (+645%), GHG capture, removal and storage (+457%), and emissions control, reporting, and offsetting (+433%). Climate tech companies from the United Kingdom scored the highest funding ($8.1B), followed by Sweden ($7.7B) and Germany ($5.3B). Finland saw the biggest year-over-year growth of any European country driven by four mega-deals respectively across supply chain tracking, climate fintech, quantum computing, and circular electronics. "Unsurprisingly, energy was the strongest sector in Europe, with investors seeing a resurgence of renewable energy projects and technologies to strengthen Europe's energy independence," said Frederico Cristoforoni, co-founder, of Net Zero Insights. "Europe is also banking on hydrogen and decarbonizing heavy industry." In 2022 solutions to decarbonize the industry generated momentum as more investment poured into hardware. Even excluding the mega-round raised for H2 Green Steel of $4.54B, (the largest ever climate tech investment in Europe), the industry sector almost doubled year-over-year investment. H2 Green Steel was far from the only substantially large investment, as mega-rounds accounted for 6.6% of rounds in 2022, showing a growing maturity of the ecosystem. Other mega-rounds included Northvolt, TerraWatt Infrastructure, Flexport, and Enpal. While venture capital and overall funding slowed down in the U.S., the median deal size increased significantly from $2.4M to $5.5M, a growth of 132%. From a challenge perspective, GHG capture, removal, and storage was by far the fastest growing sector in the U.S. with a year-over-year growth rate of +1632%, translating to a total of $1.4B. The recently passed Inflation Reduction Act (IRA) was the single largest investment in climate and energy in American history at $369B. Included within are appropriations of $250B in loans: $11.7B for new loans, $100B for increasing existing loans, and $5B for a new loan program, the Energy Infrastructure Reinvestment (EIR). "We expect 2023 to be a big year for transatlantic partnerships in climate tech," said Melanie Adamson, chief marketing strategist, Alder & Co. "With European climate tech companies closing the funding gap vis-a-vis their North American counterparts, we're seeing companies seeking collaboration and expansion opportunities on both sides of the Atlantic." News Source: Businesswire Alder & Co. Climate Tech Investment Net Zero Insights Previous article A Cybersecurity Powerhouse Joins PlexTrac's Board of Directors Next article To Provide Narrative and Risk Intelligence to Leading Social and Media Technology Partners, Blackbird.AI Has Launched a Global Alliance Program	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	2,478
""" Tests for the API /nodes/ methods. """ import datetime import json import mock from oslo_config import cfg from oslo_utils import timeutils from oslo_utils import uuidutils import six from six.moves import http_client from six.moves.urllib import parse as urlparse from testtools import matchers from wsme import types as wtypes from ironic.api.controllers import base as api_base from ironic.api.controllers import v1 as api_v1 from ironic.api.controllers.v1 import node as api_node from ironic.api.controllers.v1 import notification_utils from ironic.api.controllers.v1 import utils as api_utils from ironic.api.controllers.v1 import versions from ironic.common import boot_devices from ironic.common import driver_factory from ironic.common import exception from ironic.common import states from ironic.conductor import rpcapi from ironic import objects from ironic.objects import fields as obj_fields from ironic.tests import base from ironic.tests.unit.api import base as test_api_base from ironic.tests.unit.api import utils as test_api_utils from ironic.tests.unit.objects import utils as obj_utils class TestNodeObject(base.TestCase): def test_node_init(self): node_dict = test_api_utils.node_post_data() del node_dict['instance_uuid'] node = api_node.Node(node_dict) self.assertEqual(wtypes.Unset, node.instance_uuid) class TestListNodes(test_api_base.BaseApiTest): def setUp(self): super(TestListNodes, self).setUp() self.chassis = obj_utils.create_test_chassis(self.context) p = mock.patch.object(rpcapi.ConductorAPI, 'get_topic_for') self.mock_gtf = p.start() self.mock_gtf.return_value = 'test-topic' self.addCleanup(p.stop) def _create_association_test_nodes(self): # create some unassociated nodes unassociated_nodes = [] for id in range(3): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) unassociated_nodes.append(node.uuid) # created some associated nodes associated_nodes = [] for id in range(4): node = obj_utils.create_test_node( self.context, uuid=uuidutils.generate_uuid(), instance_uuid=uuidutils.generate_uuid()) associated_nodes.append(node.uuid) return {'associated': associated_nodes, 'unassociated': unassociated_nodes} def test_empty(self): data = self.get_json('/nodes') self.assertEqual([], data['nodes']) def test_one(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) data = self.get_json( '/nodes', headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertIn('instance_uuid', data['nodes'][0]) self.assertIn('maintenance', data['nodes'][0]) self.assertIn('power_state', data['nodes'][0]) self.assertIn('provision_state', data['nodes'][0]) self.assertIn('uuid', data['nodes'][0]) self.assertEqual(node.uuid, data['nodes'][0]["uuid"]) self.assertNotIn('driver', data['nodes'][0]) self.assertNotIn('driver_info', data['nodes'][0]) self.assertNotIn('driver_internal_info', data['nodes'][0]) self.assertNotIn('extra', data['nodes'][0]) self.assertNotIn('properties', data['nodes'][0]) self.assertNotIn('chassis_uuid', data['nodes'][0]) self.assertNotIn('reservation', data['nodes'][0]) self.assertNotIn('console_enabled', data['nodes'][0]) self.assertNotIn('target_power_state', data['nodes'][0]) self.assertNotIn('target_provision_state', data['nodes'][0]) self.assertNotIn('provision_updated_at', data['nodes'][0]) self.assertNotIn('maintenance_reason', data['nodes'][0]) self.assertNotIn('clean_step', data['nodes'][0]) self.assertNotIn('raid_config', data['nodes'][0]) self.assertNotIn('target_raid_config', data['nodes'][0]) self.assertNotIn('network_interface', data['nodes'][0]) self.assertNotIn('resource_class', data['nodes'][0]) for field in api_utils.V31_FIELDS: self.assertNotIn(field, data['nodes'][0]) self.assertNotIn('storage_interface', data['nodes'][0]) # never expose the chassis_id self.assertNotIn('chassis_id', data['nodes'][0]) def test_get_one(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(node.uuid, data['uuid']) self.assertIn('driver', data) self.assertIn('driver_info', data) self.assertEqual('**', data['driver_info']['fake_password']) self.assertEqual('bar', data['driver_info']['foo']) self.assertIn('driver_internal_info', data) self.assertIn('instance_info', data) self.assertEqual('**', data['instance_info']['configdrive']) self.assertEqual('**', data['instance_info']['image_url']) self.assertEqual('bar', data['instance_info']['foo']) self.assertIn('extra', data) self.assertIn('properties', data) self.assertIn('chassis_uuid', data) self.assertIn('reservation', data) self.assertIn('maintenance_reason', data) self.assertIn('name', data) self.assertIn('inspection_finished_at', data) self.assertIn('inspection_started_at', data) self.assertIn('clean_step', data) self.assertIn('states', data) self.assertIn('network_interface', data) self.assertIn('resource_class', data) for field in api_utils.V31_FIELDS: self.assertIn(field, data) self.assertIn('storage_interface', data) # never expose the chassis_id self.assertNotIn('chassis_id', data) def test_node_states_field_hidden_in_lower_version(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: '1.8'}) self.assertNotIn('states', data) def test_node_interface_fields_hidden_in_lower_version(self): node = obj_utils.create_test_node(self.context) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: '1.30'}) for field in api_utils.V31_FIELDS: self.assertNotIn(field, data) def test_node_storage_interface_hidden_in_lower_version(self): node = obj_utils.create_test_node(self.context, storage_interface='cinder') data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: '1.32'}) self.assertNotIn('storage_interface', data) def test_get_one_custom_fields(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields = 'extra,instance_info' data = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MAX_VER)}) # We always append "links" self.assertItemsEqual(['extra', 'instance_info', 'links'], data) def test_get_collection_custom_fields(self): fields = 'uuid,instance_info' for i in range(3): obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), instance_uuid=uuidutils.generate_uuid()) data = self.get_json( '/nodes?fields=%s' % fields, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(3, len(data['nodes'])) for node in data['nodes']: # We always append "links" self.assertItemsEqual(['uuid', 'instance_info', 'links'], node) def test_get_custom_fields_invalid_fields(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields = 'uuid,spongebob' response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MAX_VER)}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertIn('spongebob', response.json['error_message']) def test_get_custom_fields_invalid_api_version(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields = 'uuid,extra' response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MIN_VER)}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_get_one_custom_fields_show_password(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id, driver_info={'fake_password': 'bar'}) fields = 'driver_info' data = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MAX_VER)}) # We always append "links" self.assertItemsEqual(['driver_info', 'links'], data) self.assertEqual('**', data['driver_info']['fake_password']) def test_get_network_interface_fields_invalid_api_version(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields = 'network_interface' response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MIN_VER)}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_get_network_interface_fields(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields = 'network_interface' response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertIn('network_interface', response) def test_get_all_interface_fields_invalid_api_version(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields_arg = ','.join(api_utils.V31_FIELDS) response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields_arg), headers={api_base.Version.string: str(api_v1.MIN_VER)}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_get_all_interface_fields(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields_arg = ','.join(api_utils.V31_FIELDS) response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields_arg), headers={api_base.Version.string: str(api_v1.MAX_VER)}) for field in api_utils.V31_FIELDS: self.assertIn(field, response) def test_get_storage_interface_fields_invalid_api_version(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields = 'storage_interface' response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MIN_VER)}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_get_storage_interface_fields(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) fields = 'storage_interface' response = self.get_json( '/nodes/%s?fields=%s' % (node.uuid, fields), headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertIn('storage_interface', response) def test_detail(self): node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id) data = self.get_json( '/nodes/detail', headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(node.uuid, data['nodes'][0]["uuid"]) self.assertIn('name', data['nodes'][0]) self.assertIn('driver', data['nodes'][0]) self.assertIn('driver_info', data['nodes'][0]) self.assertIn('extra', data['nodes'][0]) self.assertIn('properties', data['nodes'][0]) self.assertIn('chassis_uuid', data['nodes'][0]) self.assertIn('reservation', data['nodes'][0]) self.assertIn('maintenance', data['nodes'][0]) self.assertIn('console_enabled', data['nodes'][0]) self.assertIn('target_power_state', data['nodes'][0]) self.assertIn('target_provision_state', data['nodes'][0]) self.assertIn('provision_updated_at', data['nodes'][0]) self.assertIn('inspection_finished_at', data['nodes'][0]) self.assertIn('inspection_started_at', data['nodes'][0]) self.assertIn('raid_config', data['nodes'][0]) self.assertIn('target_raid_config', data['nodes'][0]) self.assertIn('network_interface', data['nodes'][0]) self.assertIn('resource_class', data['nodes'][0]) for field in api_utils.V31_FIELDS: self.assertIn(field, data['nodes'][0]) self.assertIn('storage_interface', data['nodes'][0]) # never expose the chassis_id self.assertNotIn('chassis_id', data['nodes'][0]) def test_detail_against_single(self): node = obj_utils.create_test_node(self.context) response = self.get_json('/nodes/%s/detail' % node.uuid, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_mask_available_state(self): node = obj_utils.create_test_node(self.context, provision_state=states.AVAILABLE) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: str(api_v1.MIN_VER)}) self.assertEqual(states.NOSTATE, data['provision_state']) data = self.get_json('/nodes/%s' % node.uuid, headers={api_base.Version.string: "1.2"}) self.assertEqual(states.AVAILABLE, data['provision_state']) def test_hide_fields_in_newer_versions_driver_internal(self): node = obj_utils.create_test_node(self.context, driver_internal_info={"foo": "bar"}) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: str(api_v1.MIN_VER)}) self.assertNotIn('driver_internal_info', data) data = self.get_json('/nodes/%s' % node.uuid, headers={api_base.Version.string: "1.3"}) self.assertEqual({"foo": "bar"}, data['driver_internal_info']) def test_hide_fields_in_newer_versions_name(self): node = obj_utils.create_test_node(self.context, name="fish") data = self.get_json('/nodes/%s' % node.uuid, headers={api_base.Version.string: "1.4"}) self.assertNotIn('name', data) data = self.get_json('/nodes/%s' % node.uuid, headers={api_base.Version.string: "1.5"}) self.assertEqual('fish', data['name']) def test_hide_fields_in_newer_versions_inspection(self): some_time = datetime.datetime(2015, 3, 18, 19, 20) node = obj_utils.create_test_node(self.context, inspection_started_at=some_time) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: str(api_v1.MIN_VER)}) self.assertNotIn('inspection_finished_at', data) self.assertNotIn('inspection_started_at', data) data = self.get_json('/nodes/%s' % node.uuid, headers={api_base.Version.string: "1.6"}) started = timeutils.parse_isotime( data['inspection_started_at']).replace(tzinfo=None) self.assertEqual(some_time, started) self.assertIsNone(data['inspection_finished_at']) def test_hide_fields_in_newer_versions_clean_step(self): node = obj_utils.create_test_node(self.context, clean_step={"foo": "bar"}) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: str(api_v1.MIN_VER)}) self.assertNotIn('clean_step', data) data = self.get_json('/nodes/%s' % node.uuid, headers={api_base.Version.string: "1.7"}) self.assertEqual({"foo": "bar"}, data['clean_step']) def test_hide_fields_in_newer_versions_network_interface(self): node = obj_utils.create_test_node(self.context, network_interface='flat') data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.19'}) self.assertNotIn('network_interface', data['nodes'][0]) new_data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.20'}) self.assertEqual(node.network_interface, new_data['nodes'][0]["network_interface"]) def test_hide_fields_in_newer_versions_resource_class(self): node = obj_utils.create_test_node(self.context, resource_class='foo') data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.20'}) self.assertNotIn('resource_class', data['nodes'][0]) new_data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.21'}) self.assertEqual(node.resource_class, new_data['nodes'][0]["resource_class"]) def test_hide_fields_in_newer_versions_interface_fields(self): node = obj_utils.create_test_node(self.context) data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.30'}) for field in api_utils.V31_FIELDS: self.assertNotIn(field, data['nodes'][0]) new_data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.31'}) for field in api_utils.V31_FIELDS: self.assertEqual(getattr(node, field), new_data['nodes'][0][field]) def test_hide_fields_in_newer_versions_volume(self): node = obj_utils.create_test_node(self.context) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: '1.31'}) self.assertNotIn('volume', data) data = self.get_json('/nodes/%s' % node.uuid, headers={api_base.Version.string: "1.32"}) self.assertIn('volume', data) def test_hide_fields_in_newer_versions_storage_interface(self): node = obj_utils.create_test_node(self.context, storage_interface='cinder') data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.32'}) self.assertNotIn('storage_interface', data['nodes'][0]) new_data = self.get_json( '/nodes/detail', headers={api_base.Version.string: '1.33'}) self.assertEqual(node.storage_interface, new_data['nodes'][0]["storage_interface"]) def test_many(self): nodes = [] for id in range(5): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) nodes.append(node.uuid) data = self.get_json('/nodes') self.assertEqual(len(nodes), len(data['nodes'])) uuids = [n['uuid'] for n in data['nodes']] self.assertEqual(sorted(nodes), sorted(uuids)) def test_many_have_names(self): nodes = [] node_names = [] for id in range(5): name = 'node-%s' % id node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), name=name) nodes.append(node.uuid) node_names.append(name) data = self.get_json('/nodes', headers={api_base.Version.string: "1.5"}) names = [n['name'] for n in data['nodes']] self.assertEqual(len(nodes), len(data['nodes'])) self.assertEqual(sorted(node_names), sorted(names)) def _test_links(self, public_url=None): cfg.CONF.set_override('public_endpoint', public_url, 'api') uuid = uuidutils.generate_uuid() obj_utils.create_test_node(self.context, uuid=uuid) data = self.get_json('/nodes/%s' % uuid) self.assertIn('links', data) self.assertEqual(2, len(data['links'])) self.assertIn(uuid, data['links'][0]['href']) for l in data['links']: bookmark = l['rel'] == 'bookmark' self.assertTrue(self.validate_link(l['href'], bookmark=bookmark)) if public_url is not None: expected = [{'href': '%s/v1/nodes/%s' % (public_url, uuid), 'rel': 'self'}, {'href': '%s/nodes/%s' % (public_url, uuid), 'rel': 'bookmark'}] for i in expected: self.assertIn(i, data['links']) def test_links(self): self._test_links() def test_links_public_url(self): self._test_links(public_url='http://foo') def test_collection_links(self): nodes = [] for id in range(5): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) nodes.append(node.uuid) data = self.get_json('/nodes/?limit=3') self.assertEqual(3, len(data['nodes'])) next_marker = data['nodes'][-1]['uuid'] self.assertIn(next_marker, data['next']) def test_collection_links_default_limit(self): cfg.CONF.set_override('max_limit', 3, 'api') nodes = [] for id in range(5): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) nodes.append(node.uuid) data = self.get_json('/nodes') self.assertEqual(3, len(data['nodes'])) next_marker = data['nodes'][-1]['uuid'] self.assertIn(next_marker, data['next']) def test_sort_key(self): nodes = [] for id in range(3): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) nodes.append(node.uuid) data = self.get_json('/nodes?sort_key=uuid') uuids = [n['uuid'] for n in data['nodes']] self.assertEqual(sorted(nodes), uuids) def test_sort_key_invalid(self): invalid_keys_list = ['foo', 'properties', 'driver_info', 'extra', 'instance_info', 'driver_internal_info', 'clean_step'] for invalid_key in invalid_keys_list: response = self.get_json('/nodes?sort_key=%s' % invalid_key, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertIn(invalid_key, response.json['error_message']) def _test_sort_key_allowed(self, detail=False): node_uuids = [] for id in range(3, 0, -1): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), resource_class='rc_%s' % id) node_uuids.append(node.uuid) node_uuids.reverse() headers = {'X-OpenStack-Ironic-API-Version': '1.21'} detail_str = '/detail' if detail else '' data = self.get_json('/nodes%s?sort_key=resource_class' % detail_str, headers=headers) data_uuids = [n['uuid'] for n in data['nodes']] self.assertEqual(node_uuids, data_uuids) def test_sort_key_allowed(self): self._test_sort_key_allowed() def test_detail_sort_key_allowed(self): self._test_sort_key_allowed(detail=True) def _test_sort_key_not_allowed(self, detail=False): headers = {'X-OpenStack-Ironic-API-Version': '1.20'} detail_str = '/detail' if detail else '' resp = self.get_json('/nodes%s?sort_key=resource_class' % detail_str, headers=headers, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, resp.status_int) self.assertEqual('application/json', resp.content_type) def test_sort_key_not_allowed(self): self._test_sort_key_not_allowed() def test_detail_sort_key_not_allowed(self): self._test_sort_key_not_allowed(detail=True) def test_ports_subresource_link(self): node = obj_utils.create_test_node(self.context) data = self.get_json('/nodes/%s' % node.uuid) self.assertIn('ports', data) def test_portgroups_subresource(self): node = obj_utils.create_test_node(self.context) headers = {'X-OpenStack-Ironic-API-Version': '1.24'} for id_ in range(2): obj_utils.create_test_portgroup(self.context, node_id=node.id, name="pg-%s" % id_, uuid=uuidutils.generate_uuid(), address='52:54:00:cf:2d:3%s' % id_) data = self.get_json('/nodes/%s/portgroups' % node.uuid, headers=headers) self.assertEqual(2, len(data['portgroups'])) self.assertNotIn('next', data) # Test collection pagination data = self.get_json('/nodes/%s/portgroups?limit=1' % node.uuid, headers=headers) self.assertEqual(1, len(data['portgroups'])) self.assertIn('next', data) def test_portgroups_subresource_link(self): node = obj_utils.create_test_node(self.context) data = self.get_json( '/nodes/%s' % node.uuid, headers={'X-OpenStack-Ironic-API-Version': '1.24'}) self.assertIn('portgroups', data.keys()) def test_portgroups_subresource_link_hidden_for_older_versions(self): node = obj_utils.create_test_node(self.context) data = self.get_json( '/nodes/%s' % node.uuid, headers={'X-OpenStack-Ironic-API-Version': '1.20'}) self.assertNotIn('portgroups', data.keys()) def test_portgroups_subresource_old_api_version(self): node = obj_utils.create_test_node(self.context) response = self.get_json( '/nodes/%s/portgroups' % node.uuid, expect_errors=True, headers={'X-OpenStack-Ironic-API-Version': '1.23'}) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_ports_subresource(self): node = obj_utils.create_test_node(self.context) for id_ in range(2): obj_utils.create_test_port(self.context, node_id=node.id, uuid=uuidutils.generate_uuid(), address='52:54:00:cf:2d:3%s' % id_) data = self.get_json('/nodes/%s/ports' % node.uuid) self.assertEqual(2, len(data['ports'])) self.assertNotIn('next', data) # Test collection pagination data = self.get_json('/nodes/%s/ports?limit=1' % node.uuid) self.assertEqual(1, len(data['ports'])) self.assertIn('next', data) def test_ports_subresource_noid(self): node = obj_utils.create_test_node(self.context) obj_utils.create_test_port(self.context, node_id=node.id) # No node id specified response = self.get_json('/nodes/ports', expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_ports_subresource_node_not_found(self): non_existent_uuid = 'eeeeeeee-cccc-aaaa-bbbb-cccccccccccc' response = self.get_json('/nodes/%s/ports' % non_existent_uuid, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_ports_subresource_invalid_ident(self): invalid_ident = '123~123' response = self.get_json('/nodes/%s/ports' % invalid_ident, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertIn('Expected a logical name or UUID', response.json['error_message']) def test_ports_subresource_via_portgroups_subres_not_allowed(self): node = obj_utils.create_test_node(self.context) pg = obj_utils.create_test_portgroup(self.context, node_id=node.id) response = self.get_json('/nodes/%s/portgroups/%s/ports' % ( node.uuid, pg.uuid), expect_errors=True, headers={api_base.Version.string: '1.24'}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_volume_subresource_link(self): node = obj_utils.create_test_node(self.context) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: '1.32'}) self.assertIn('volume', data) def test_volume_subresource(self): node = obj_utils.create_test_node(self.context) data = self.get_json('/nodes/%s/volume' % node.uuid, headers={api_base.Version.string: '1.32'}) self.assertIn('connectors', data) self.assertIn('targets', data) self.assertIn('/volume/connectors', data['connectors'][0]['href']) self.assertIn('/volume/connectors', data['connectors'][1]['href']) self.assertIn('/volume/targets', data['targets'][0]['href']) self.assertIn('/volume/targets', data['targets'][1]['href']) def test_volume_subresource_invalid_api_version(self): node = obj_utils.create_test_node(self.context) response = self.get_json('/nodes/%s/volume' % node.uuid, headers={api_base.Version.string: '1.31'}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_volume_connectors_subresource(self): node = obj_utils.create_test_node(self.context) for id_ in range(2): obj_utils.create_test_volume_connector( self.context, node_id=node.id, uuid=uuidutils.generate_uuid(), connector_id='test-connector_id-%s' % id_) data = self.get_json( '/nodes/%s/volume/connectors' % node.uuid, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(2, len(data['connectors'])) self.assertNotIn('next', data) # Test collection pagination data = self.get_json( '/nodes/%s/volume/connectors?limit=1' % node.uuid, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(1, len(data['connectors'])) self.assertIn('next', data) def test_volume_connectors_subresource_noid(self): node = obj_utils.create_test_node(self.context) obj_utils.create_test_volume_connector(self.context, node_id=node.id) # No node_id specified. response = self.get_json( '/nodes/volume/connectors', expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_volume_connectors_subresource_node_not_found(self): non_existent_uuid = 'eeeeeeee-cccc-aaaa-bbbb-cccccccccccc' response = self.get_json( '/nodes/%s/volume/connectors' % non_existent_uuid, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_volume_targets_subresource(self): node = obj_utils.create_test_node(self.context) for id_ in range(2): obj_utils.create_test_volume_target( self.context, node_id=node.id, uuid=uuidutils.generate_uuid(), boot_index=id_) data = self.get_json( '/nodes/%s/volume/targets' % node.uuid, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(2, len(data['targets'])) self.assertNotIn('next', data) # Test collection pagination data = self.get_json( '/nodes/%s/volume/targets?limit=1' % node.uuid, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(1, len(data['targets'])) self.assertIn('next', data) def test_volume_targets_subresource_noid(self): node = obj_utils.create_test_node(self.context) obj_utils.create_test_volume_target(self.context, node_id=node.id) # No node_id specified. response = self.get_json( '/nodes/volume/targets', expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_volume_targets_subresource_node_not_found(self): non_existent_uuid = 'eeeeeeee-cccc-aaaa-bbbb-cccccccccccc' response = self.get_json( '/nodes/%s/volume/targets' % non_existent_uuid, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.NOT_FOUND, response.status_int) @mock.patch.object(timeutils, 'utcnow') def _test_node_states(self, mock_utcnow, api_version=None): fake_state = 'fake-state' fake_error = 'fake-error' fake_config = '{"foo": "bar"}' test_time = datetime.datetime(2000, 1, 1, 0, 0) mock_utcnow.return_value = test_time node = obj_utils.create_test_node(self.context, power_state=fake_state, target_power_state=fake_state, provision_state=fake_state, target_provision_state=fake_state, provision_updated_at=test_time, raid_config=fake_config, target_raid_config=fake_config, last_error=fake_error) headers = {} if api_version: headers = {api_base.Version.string: api_version} data = self.get_json('/nodes/%s/states' % node.uuid, headers=headers) self.assertEqual(fake_state, data['power_state']) self.assertEqual(fake_state, data['target_power_state']) self.assertEqual(fake_state, data['provision_state']) self.assertEqual(fake_state, data['target_provision_state']) prov_up_at = timeutils.parse_isotime( data['provision_updated_at']).replace(tzinfo=None) self.assertEqual(test_time, prov_up_at) self.assertEqual(fake_error, data['last_error']) self.assertFalse(data['console_enabled']) return data def test_node_states(self): self._test_node_states() def test_node_states_raid(self): data = self._test_node_states(api_version="1.12") self.assertEqual({'foo': 'bar'}, data['raid_config']) self.assertEqual({'foo': 'bar'}, data['target_raid_config']) @mock.patch.object(timeutils, 'utcnow') def test_node_states_by_name(self, mock_utcnow): fake_state = 'fake-state' fake_error = 'fake-error' test_time = datetime.datetime(1971, 3, 9, 0, 0) mock_utcnow.return_value = test_time node = obj_utils.create_test_node(self.context, name='eggs', power_state=fake_state, target_power_state=fake_state, provision_state=fake_state, target_provision_state=fake_state, provision_updated_at=test_time, last_error=fake_error) data = self.get_json('/nodes/%s/states' % node.name, headers={api_base.Version.string: "1.5"}) self.assertEqual(fake_state, data['power_state']) self.assertEqual(fake_state, data['target_power_state']) self.assertEqual(fake_state, data['provision_state']) self.assertEqual(fake_state, data['target_provision_state']) prov_up_at = timeutils.parse_isotime( data['provision_updated_at']).replace(tzinfo=None) self.assertEqual(test_time, prov_up_at) self.assertEqual(fake_error, data['last_error']) self.assertFalse(data['console_enabled']) def test_node_by_instance_uuid(self): node = obj_utils.create_test_node( self.context, uuid=uuidutils.generate_uuid(), instance_uuid=uuidutils.generate_uuid()) instance_uuid = node.instance_uuid data = self.get_json('/nodes?instance_uuid=%s' % instance_uuid, headers={api_base.Version.string: "1.5"}) self.assertThat(data['nodes'], matchers.HasLength(1)) self.assertEqual(node['instance_uuid'], data['nodes'][0]["instance_uuid"]) def test_node_by_instance_uuid_wrong_uuid(self): obj_utils.create_test_node( self.context, uuid=uuidutils.generate_uuid(), instance_uuid=uuidutils.generate_uuid()) wrong_uuid = uuidutils.generate_uuid() data = self.get_json('/nodes?instance_uuid=%s' % wrong_uuid) self.assertThat(data['nodes'], matchers.HasLength(0)) def test_node_by_instance_uuid_invalid_uuid(self): response = self.get_json('/nodes?instance_uuid=fake', expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) def test_associated_nodes_insensitive(self): associated_nodes = (self ._create_association_test_nodes()['associated']) data = self.get_json('/nodes?associated=true') data1 = self.get_json('/nodes?associated=True') uuids = [n['uuid'] for n in data['nodes']] uuids1 = [n['uuid'] for n in data1['nodes']] self.assertEqual(sorted(associated_nodes), sorted(uuids1)) self.assertEqual(sorted(associated_nodes), sorted(uuids)) def test_associated_nodes_error(self): self._create_association_test_nodes() response = self.get_json('/nodes?associated=blah', expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_unassociated_nodes_insensitive(self): unassociated_nodes = ( self._create_association_test_nodes()['unassociated']) data = self.get_json('/nodes?associated=false') data1 = self.get_json('/nodes?associated=FALSE') uuids = [n['uuid'] for n in data['nodes']] uuids1 = [n['uuid'] for n in data1['nodes']] self.assertEqual(sorted(unassociated_nodes), sorted(uuids1)) self.assertEqual(sorted(unassociated_nodes), sorted(uuids)) def test_unassociated_nodes_with_limit(self): unassociated_nodes = ( self._create_association_test_nodes()['unassociated']) data = self.get_json('/nodes?associated=False&limit=2') self.assertThat(data['nodes'], matchers.HasLength(2)) self.assertIn(data['nodes'][0]['uuid'], unassociated_nodes) def test_next_link_with_association(self): self._create_association_test_nodes() data = self.get_json('/nodes/?limit=3&associated=True') self.assertThat(data['nodes'], matchers.HasLength(3)) self.assertIn('associated=True', data['next']) def test_detail_with_association_filter(self): associated_nodes = (self ._create_association_test_nodes()['associated']) data = self.get_json('/nodes/detail?associated=true') self.assertIn('driver', data['nodes'][0]) self.assertEqual(len(associated_nodes), len(data['nodes'])) def test_next_link_with_association_with_detail(self): self._create_association_test_nodes() data = self.get_json('/nodes/detail?limit=3&associated=true') self.assertThat(data['nodes'], matchers.HasLength(3)) self.assertIn('driver', data['nodes'][0]) self.assertIn('associated=True', data['next']) def test_detail_with_instance_uuid(self): node = obj_utils.create_test_node( self.context, uuid=uuidutils.generate_uuid(), instance_uuid=uuidutils.generate_uuid(), chassis_id=self.chassis.id) instance_uuid = node.instance_uuid data = self.get_json('/nodes/detail?instance_uuid=%s' % instance_uuid) self.assertEqual(node['instance_uuid'], data['nodes'][0]["instance_uuid"]) self.assertIn('driver', data['nodes'][0]) self.assertIn('driver_info', data['nodes'][0]) self.assertIn('extra', data['nodes'][0]) self.assertIn('properties', data['nodes'][0]) self.assertIn('chassis_uuid', data['nodes'][0]) # never expose the chassis_id self.assertNotIn('chassis_id', data['nodes'][0]) def test_maintenance_nodes(self): nodes = [] for id in range(5): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), maintenance=id % 2) nodes.append(node) data = self.get_json('/nodes?maintenance=true') uuids = [n['uuid'] for n in data['nodes']] test_uuids_1 = [n.uuid for n in nodes if n.maintenance] self.assertEqual(sorted(test_uuids_1), sorted(uuids)) data = self.get_json('/nodes?maintenance=false') uuids = [n['uuid'] for n in data['nodes']] test_uuids_0 = [n.uuid for n in nodes if not n.maintenance] self.assertEqual(sorted(test_uuids_0), sorted(uuids)) def test_maintenance_nodes_error(self): response = self.get_json('/nodes?associated=true&maintenance=blah', expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_maintenance_nodes_associated(self): self._create_association_test_nodes() node = obj_utils.create_test_node( self.context, instance_uuid=uuidutils.generate_uuid(), maintenance=True) data = self.get_json('/nodes?associated=true&maintenance=false') uuids = [n['uuid'] for n in data['nodes']] self.assertNotIn(node.uuid, uuids) data = self.get_json('/nodes?associated=true&maintenance=true') uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node.uuid, uuids) data = self.get_json('/nodes?associated=true&maintenance=TruE') uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node.uuid, uuids) def test_get_nodes_by_provision_state(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), provision_state=states.AVAILABLE) node1 = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), provision_state=states.DEPLOYING) data = self.get_json('/nodes?provision_state=available', headers={api_base.Version.string: "1.9"}) uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node.uuid, uuids) self.assertNotIn(node1.uuid, uuids) data = self.get_json('/nodes?provision_state=deploying', headers={api_base.Version.string: "1.9"}) uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node1.uuid, uuids) self.assertNotIn(node.uuid, uuids) def test_get_nodes_by_invalid_provision_state(self): response = self.get_json('/nodes?provision_state=test', headers={api_base.Version.string: "1.9"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_get_nodes_by_provision_state_not_allowed(self): response = self.get_json('/nodes?provision_state=test', headers={api_base.Version.string: "1.8"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) self.assertTrue(response.json['error_message']) def test_get_nodes_by_driver(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), driver='pxe_ssh') node1 = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), driver='fake') data = self.get_json('/nodes?driver=pxe_ssh', headers={api_base.Version.string: "1.16"}) uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node.uuid, uuids) self.assertNotIn(node1.uuid, uuids) data = self.get_json('/nodes?driver=fake', headers={api_base.Version.string: "1.16"}) uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node1.uuid, uuids) self.assertNotIn(node.uuid, uuids) def test_get_nodes_by_invalid_driver(self): data = self.get_json('/nodes?driver=test', headers={api_base.Version.string: "1.16"}) self.assertEqual(0, len(data['nodes'])) def test_get_nodes_by_driver_invalid_api_version(self): response = self.get_json( '/nodes?driver=fake', headers={api_base.Version.string: str(api_v1.MIN_VER)}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) self.assertTrue(response.json['error_message']) def _test_get_nodes_by_resource_class(self, detail=False): if detail: base_url = '/nodes/detail?resource_class=%s' else: base_url = '/nodes?resource_class=%s' node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), driver='fake', resource_class='foo') node1 = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), driver='fake', resource_class='bar') data = self.get_json(base_url % 'foo', headers={api_base.Version.string: "1.21"}) uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node.uuid, uuids) self.assertNotIn(node1.uuid, uuids) data = self.get_json(base_url % 'bar', headers={api_base.Version.string: "1.21"}) uuids = [n['uuid'] for n in data['nodes']] self.assertIn(node1.uuid, uuids) self.assertNotIn(node.uuid, uuids) def test_get_nodes_by_resource_class(self): self._test_get_nodes_by_resource_class(detail=False) def test_get_nodes_by_resource_class_detail(self): self._test_get_nodes_by_resource_class(detail=True) def _test_get_nodes_by_invalid_resource_class(self, detail=False): if detail: base_url = '/nodes/detail?resource_class=%s' else: base_url = '/nodes?resource_class=%s' data = self.get_json(base_url % 'test', headers={api_base.Version.string: "1.21"}) self.assertEqual(0, len(data['nodes'])) def test_get_nodes_by_invalid_resource_class(self): self._test_get_nodes_by_invalid_resource_class(detail=False) def test_get_nodes_by_invalid_resource_class_detail(self): self._test_get_nodes_by_invalid_resource_class(detail=True) def _test_get_nodes_by_resource_class_invalid_api_version(self, detail=False): if detail: base_url = '/nodes/detail?resource_class=%s' else: base_url = '/nodes?resource_class=%s' response = self.get_json( base_url % 'fake', headers={api_base.Version.string: str(api_v1.MIN_VER)}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) self.assertTrue(response.json['error_message']) def test_get_nodes_by_resource_class_invalid_api_version(self): self._test_get_nodes_by_resource_class_invalid_api_version( detail=False) def test_get_nodes_by_resource_class_invalid_api_version_detail(self): self._test_get_nodes_by_resource_class_invalid_api_version(detail=True) def test_get_console_information(self): node = obj_utils.create_test_node(self.context) expected_console_info = {'test': 'test-data'} expected_data = {'console_enabled': True, 'console_info': expected_console_info} with mock.patch.object(rpcapi.ConductorAPI, 'get_console_information') as mock_gci: mock_gci.return_value = expected_console_info data = self.get_json('/nodes/%s/states/console' % node.uuid) self.assertEqual(expected_data, data) mock_gci.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'get_console_information') def test_get_console_information_by_name(self, mock_gci): node = obj_utils.create_test_node(self.context, name='spam') expected_console_info = {'test': 'test-data'} expected_data = {'console_enabled': True, 'console_info': expected_console_info} mock_gci.return_value = expected_console_info data = self.get_json('/nodes/%s/states/console' % node.name, headers={api_base.Version.string: "1.5"}) self.assertEqual(expected_data, data) mock_gci.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') def test_get_console_information_console_disabled(self): node = obj_utils.create_test_node(self.context) expected_data = {'console_enabled': False, 'console_info': None} with mock.patch.object(rpcapi.ConductorAPI, 'get_console_information') as mock_gci: mock_gci.side_effect = ( exception.NodeConsoleNotEnabled(node=node.uuid)) data = self.get_json('/nodes/%s/states/console' % node.uuid) self.assertEqual(expected_data, data) mock_gci.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') def test_get_console_information_not_supported(self): node = obj_utils.create_test_node(self.context) with mock.patch.object(rpcapi.ConductorAPI, 'get_console_information') as mock_gci: mock_gci.side_effect = exception.UnsupportedDriverExtension( extension='console', driver='test-driver') ret = self.get_json('/nodes/%s/states/console' % node.uuid, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) mock_gci.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'get_boot_device') def test_get_boot_device(self, mock_gbd): node = obj_utils.create_test_node(self.context) expected_data = {'boot_device': boot_devices.PXE, 'persistent': True} mock_gbd.return_value = expected_data data = self.get_json('/nodes/%s/management/boot_device' % node.uuid) self.assertEqual(expected_data, data) mock_gbd.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'get_boot_device') def test_get_boot_device_by_name(self, mock_gbd): node = obj_utils.create_test_node(self.context, name='spam') expected_data = {'boot_device': boot_devices.PXE, 'persistent': True} mock_gbd.return_value = expected_data data = self.get_json('/nodes/%s/management/boot_device' % node.name, headers={api_base.Version.string: "1.5"}) self.assertEqual(expected_data, data) mock_gbd.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'get_boot_device') def test_get_boot_device_iface_not_supported(self, mock_gbd): node = obj_utils.create_test_node(self.context) mock_gbd.side_effect = exception.UnsupportedDriverExtension( extension='management', driver='test-driver') ret = self.get_json('/nodes/%s/management/boot_device' % node.uuid, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) mock_gbd.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'get_supported_boot_devices') def test_get_supported_boot_devices(self, mock_gsbd): mock_gsbd.return_value = [boot_devices.PXE] node = obj_utils.create_test_node(self.context) data = self.get_json('/nodes/%s/management/boot_device/supported' % node.uuid) expected_data = {'supported_boot_devices': [boot_devices.PXE]} self.assertEqual(expected_data, data) mock_gsbd.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'get_supported_boot_devices') def test_get_supported_boot_devices_by_name(self, mock_gsbd): mock_gsbd.return_value = [boot_devices.PXE] node = obj_utils.create_test_node(self.context, name='spam') data = self.get_json( '/nodes/%s/management/boot_device/supported' % node.name, headers={api_base.Version.string: "1.5"}) expected_data = {'supported_boot_devices': [boot_devices.PXE]} self.assertEqual(expected_data, data) mock_gsbd.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'get_supported_boot_devices') def test_get_supported_boot_devices_iface_not_supported(self, mock_gsbd): node = obj_utils.create_test_node(self.context) mock_gsbd.side_effect = exception.UnsupportedDriverExtension( extension='management', driver='test-driver') ret = self.get_json('/nodes/%s/management/boot_device/supported' % node.uuid, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) mock_gsbd.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'validate_driver_interfaces') def test_validate_by_uuid_using_deprecated_interface(self, mock_vdi): # Note(mrda): The 'node_uuid' interface is deprecated in favour # of the 'node' interface node = obj_utils.create_test_node(self.context) self.get_json('/nodes/validate?node_uuid=%s' % node.uuid) mock_vdi.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'validate_driver_interfaces') def test_validate_by_uuid(self, mock_vdi): node = obj_utils.create_test_node(self.context) self.get_json('/nodes/validate?node=%s' % node.uuid, headers={api_base.Version.string: "1.5"}) mock_vdi.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'validate_driver_interfaces') def test_validate_by_name_unsupported(self, mock_vdi): node = obj_utils.create_test_node(self.context, name='spam') ret = self.get_json('/nodes/validate?node=%s' % node.name, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, ret.status_code) self.assertFalse(mock_vdi.called) @mock.patch.object(rpcapi.ConductorAPI, 'validate_driver_interfaces') def test_validate_by_name(self, mock_vdi): node = obj_utils.create_test_node(self.context, name='spam') self.get_json('/nodes/validate?node=%s' % node.name, headers={api_base.Version.string: "1.5"}) # note that this should be node.uuid here as we get that from the # rpc_node lookup and pass that downwards mock_vdi.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') def test_ssh_creds_masked(self): driver_info = {"ssh_password": "password", "ssh_key_contents": "key"} node = obj_utils.create_test_node(self.context, chassis_id=self.chassis.id, driver_info=driver_info) data = self.get_json( '/nodes/%s' % node.uuid, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual("**", data["driver_info"]["ssh_password"]) self.assertEqual("**", data["driver_info"]["ssh_key_contents"]) class TestPatch(test_api_base.BaseApiTest): def setUp(self): super(TestPatch, self).setUp() self.chassis = obj_utils.create_test_chassis(self.context) self.node = obj_utils.create_test_node(self.context, name='node-57', chassis_id=self.chassis.id) self.node_no_name = obj_utils.create_test_node( self.context, uuid='deadbeef-0000-1111-2222-333333333333', chassis_id=self.chassis.id) p = mock.patch.object(rpcapi.ConductorAPI, 'get_topic_for') self.mock_gtf = p.start() self.mock_gtf.return_value = 'test-topic' self.addCleanup(p.stop) p = mock.patch.object(rpcapi.ConductorAPI, 'update_node') self.mock_update_node = p.start() self.addCleanup(p.stop) p = mock.patch.object(rpcapi.ConductorAPI, 'change_node_power_state') self.mock_cnps = p.start() self.addCleanup(p.stop) @mock.patch.object(notification_utils, '_emit_api_notification') def test_update_ok(self, mock_notify): self.mock_update_node.return_value = self.node (self .mock_update_node .return_value .updated_at) = "2013-12-03T06:20:41.184720+00:00" response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/instance_uuid', 'value': 'aaaaaaaa-1111-bbbb-2222-cccccccccccc', 'op': 'replace'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.assertEqual(self.mock_update_node.return_value.updated_at, timeutils.parse_isotime(response.json['updated_at'])) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') mock_notify.assert_has_calls([mock.call(mock.ANY, mock.ANY, 'update', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.START, chassis_uuid=self.chassis.uuid), mock.call(mock.ANY, mock.ANY, 'update', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.END, chassis_uuid=self.chassis.uuid)]) def test_update_by_name_unsupported(self): self.mock_update_node.return_value = self.node (self .mock_update_node .return_value .updated_at) = "2013-12-03T06:20:41.184720+00:00" response = self.patch_json( '/nodes/%s' % self.node.name, [{'path': '/instance_uuid', 'value': 'aaaaaaaa-1111-bbbb-2222-cccccccccccc', 'op': 'replace'}], expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_code) self.assertFalse(self.mock_update_node.called) def test_update_ok_by_name(self): self.mock_update_node.return_value = self.node (self .mock_update_node .return_value .updated_at) = "2013-12-03T06:20:41.184720+00:00" response = self.patch_json( '/nodes/%s' % self.node.name, [{'path': '/instance_uuid', 'value': 'aaaaaaaa-1111-bbbb-2222-cccccccccccc', 'op': 'replace'}], headers={api_base.Version.string: "1.5"}) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.assertEqual(self.mock_update_node.return_value.updated_at, timeutils.parse_isotime(response.json['updated_at'])) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') def test_update_state(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'power_state': 'new state'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) @mock.patch.object(notification_utils, '_emit_api_notification') def test_update_fails_bad_driver_info(self, mock_notify): fake_err = 'Fake Error Message' self.mock_update_node.side_effect = ( exception.InvalidParameterValue(fake_err)) response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/driver_info/this', 'value': 'foo', 'op': 'add'}, {'path': '/driver_info/that', 'value': 'bar', 'op': 'add'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') mock_notify.assert_has_calls([mock.call(mock.ANY, mock.ANY, 'update', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.START, chassis_uuid=self.chassis.uuid), mock.call(mock.ANY, mock.ANY, 'update', obj_fields.NotificationLevel.ERROR, obj_fields.NotificationStatus.ERROR, chassis_uuid=self.chassis.uuid)]) def test_update_fails_bad_driver(self): self.mock_gtf.side_effect = exception.NoValidHost('Fake Error') response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/driver', 'value': 'bad-driver', 'op': 'replace'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) def test_add_ok(self): self.mock_update_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') def test_add_root(self): self.mock_update_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/instance_uuid', 'value': 'aaaaaaaa-1111-bbbb-2222-cccccccccccc', 'op': 'add'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') def test_add_root_non_existent(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_remove_ok(self): self.mock_update_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/extra', 'op': 'remove'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') def test_remove_non_existent_property_fail(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/extra/non-existent', 'op': 'remove'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_update_allowed_in_power_transition(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), target_power_state=states.POWER_OFF) self.mock_update_node.return_value = node response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}]) self.assertEqual(http_client.OK, response.status_code) def test_update_allowed_in_maintenance(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), target_power_state=states.POWER_OFF, maintenance=True) self.mock_update_node.return_value = node response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/instance_uuid', 'op': 'remove'}]) self.assertEqual(http_client.OK, response.status_code) def test_add_state_in_deployfail(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid(), provision_state=states.DEPLOYFAIL, target_provision_state=states.ACTIVE) self.mock_update_node.return_value = node response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') def test_patch_ports_subresource_no_port_id(self): response = self.patch_json('/nodes/%s/ports' % self.node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) def test_patch_ports_subresource(self): response = self.patch_json( '/nodes/%s/ports/9bb50f13-0b8d-4ade-ad2d-d91fefdef9cc' % self.node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_patch_portgroups_subresource(self): response = self.patch_json( '/nodes/%s/portgroups/9bb50f13-0b8d-4ade-ad2d-d91fefdef9cc' % self.node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True, headers={'X-OpenStack-Ironic-API-Version': '1.24'}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_patch_volume_connectors_subresource_no_connector_id(self): response = self.patch_json( '/nodes/%s/volume/connectors' % self.node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.BAD_REQUEST, response.status_int) def test_patch_volume_connectors_subresource(self): connector = ( obj_utils.create_test_volume_connector(self.context, node_id=self.node.id)) response = self.patch_json( '/nodes/%s/volume/connectors/%s' % (self.node.uuid, connector.uuid), [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_patch_volume_targets_subresource(self): target = obj_utils.create_test_volume_target(self.context, node_id=self.node.id) response = self.patch_json( '/nodes/%s/volume/targets/%s' % (self.node.uuid, target.uuid), [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_remove_uuid(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/uuid', 'op': 'remove'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_add_state_in_cleaning(self): node = obj_utils.create_test_node( self.context, uuid=uuidutils.generate_uuid(), provision_state=states.CLEANING, target_provision_state=states.AVAILABLE) self.mock_update_node.return_value = node response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.CONFLICT, response.status_code) self.assertTrue(response.json['error_message']) def test_remove_mandatory_field(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/driver', 'op': 'remove'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_replace_chassis_uuid(self): self.mock_update_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_uuid', 'value': self.chassis.uuid, 'op': 'replace'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_add_chassis_uuid(self): self.mock_update_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_uuid', 'value': self.chassis.uuid, 'op': 'add'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_remove_chassis_uuid(self): self.mock_update_node.return_value = self.node headers = {api_base.Version.string: "1.25"} response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_uuid', 'op': 'remove'}], headers=headers) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_remove_chassis_uuid_invalid_api_version(self): self.mock_update_node.return_value = self.node headers = {api_base.Version.string: "1.24"} response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_uuid', 'op': 'remove'}], headers=headers, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) self.assertTrue(response.json['error_message']) @mock.patch("pecan.request") def test__update_changed_fields_remove_chassis_uuid(self, mock_pecan_req): mock_pecan_req.version.minor = versions.MINOR_MAX_VERSION controller = api_node.NodesController() node_dict = self.node.as_dict() del node_dict['chassis_id'] node_no_chassis = api_node.Node(node_dict) controller._update_changed_fields(node_no_chassis, self.node) self.assertIsNone(self.node.chassis_id) def test_add_chassis_id(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_id', 'value': '1', 'op': 'add'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_replace_chassis_id(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_id', 'value': '1', 'op': 'replace'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_remove_chassis_id(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_id', 'op': 'remove'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_replace_non_existent_chassis_uuid(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/chassis_uuid', 'value': 'eeeeeeee-dddd-cccc-bbbb-aaaaaaaaaaaa', 'op': 'replace'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_remove_internal_field(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/last_error', 'op': 'remove'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_replace_internal_field(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/power_state', 'op': 'replace', 'value': 'fake-state'}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_replace_maintenance(self): self.mock_update_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/maintenance', 'op': 'replace', 'value': 'true'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') def test_replace_maintenance_by_name(self): self.mock_update_node.return_value = self.node response = self.patch_json( '/nodes/%s' % self.node.name, [{'path': '/maintenance', 'op': 'replace', 'value': 'true'}], headers={api_base.Version.string: "1.5"}) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.mock_update_node.assert_called_once_with( mock.ANY, mock.ANY, 'test-topic') def test_replace_consoled_enabled(self): response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/console_enabled', 'op': 'replace', 'value': True}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_replace_provision_updated_at(self): test_time = '2000-01-01 00:00:00' response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/provision_updated_at', 'op': 'replace', 'value': test_time}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_patch_add_name_ok(self): self.mock_update_node.return_value = self.node_no_name test_name = 'guido-van-rossum' response = self.patch_json('/nodes/%s' % self.node_no_name.uuid, [{'path': '/name', 'op': 'add', 'value': test_name}], headers={api_base.Version.string: "1.5"}) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def _patch_add_name_invalid_or_reserved(self, name): self.mock_update_node.return_value = self.node_no_name response = self.patch_json('/nodes/%s' % self.node_no_name.uuid, [{'path': '/name', 'op': 'add', 'value': name}], headers={api_base.Version.string: "1.10"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_patch_add_name_invalid(self): self._patch_add_name_invalid_or_reserved('i am invalid') def test_patch_add_name_reserved(self): reserved_names = api_utils.get_controller_reserved_names( api_node.NodesController) for name in reserved_names: self._patch_add_name_invalid_or_reserved(name) def test_patch_add_name_empty_invalid(self): test_name = '' response = self.patch_json('/nodes/%s' % self.node_no_name.uuid, [{'path': '/name', 'op': 'add', 'value': test_name}], headers={api_base.Version.string: "1.5"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_patch_add_name_empty_not_acceptable(self): test_name = '' response = self.patch_json('/nodes/%s' % self.node_no_name.uuid, [{'path': '/name', 'op': 'add', 'value': test_name}], expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) self.assertTrue(response.json['error_message']) def test_patch_name_replace_ok(self): self.mock_update_node.return_value = self.node test_name = 'guido-van-rossum' response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/name', 'op': 'replace', 'value': test_name}], headers={api_base.Version.string: "1.5"}) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_patch_add_replace_invalid(self): self.mock_update_node.return_value = self.node_no_name test_name = 'Guido Van Error' response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/name', 'op': 'replace', 'value': test_name}], headers={api_base.Version.string: "1.5"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_patch_update_name_twice_both_invalid(self): test_name_1 = 'Windows ME' test_name_2 = 'Guido Van Error' response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/name', 'op': 'add', 'value': test_name_1}, {'path': '/name', 'op': 'replace', 'value': test_name_2}], headers={api_base.Version.string: "1.5"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertIn(test_name_1, response.json['error_message']) def test_patch_update_name_twice_second_invalid(self): test_name = 'Guido Van Error' response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/name', 'op': 'add', 'value': 'node-0'}, {'path': '/name', 'op': 'replace', 'value': test_name}], headers={api_base.Version.string: "1.5"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertIn(test_name, response.json['error_message']) def test_patch_duplicate_name(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) test_name = "this-is-my-node" self.mock_update_node.side_effect = exception.DuplicateName(test_name) response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/name', 'op': 'replace', 'value': test_name}], headers={api_base.Version.string: "1.5"}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.CONFLICT, response.status_code) self.assertTrue(response.json['error_message']) @mock.patch.object(api_node.NodesController, '_check_names_acceptable') def test_patch_name_remove_ok(self, cna_mock): self.mock_update_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/name', 'op': 'remove'}], headers={api_base.Version.string: "1.5"}) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) self.assertFalse(cna_mock.called) @mock.patch.object(api_utils, 'get_rpc_node') def test_patch_update_drive_console_enabled(self, mock_rpc_node): self.node.console_enabled = True mock_rpc_node.return_value = self.node response = self.patch_json('/nodes/%s' % self.node.uuid, [{'path': '/driver', 'value': 'foo', 'op': 'add'}], expect_errors=True) mock_rpc_node.assert_called_once_with(self.node.uuid) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.CONFLICT, response.status_code) self.assertTrue(response.json['error_message']) def test_update_in_UPDATE_ALLOWED_STATES(self): for state in states.UPDATE_ALLOWED_STATES: node = obj_utils.create_test_node( self.context, uuid=uuidutils.generate_uuid(), provision_state=state, target_provision_state=states.AVAILABLE) self.mock_update_node.return_value = node response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/extra/foo', 'value': 'bar', 'op': 'add'}]) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_update_network_interface(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node network_interface = 'flat' headers = {api_base.Version.string: str(api_v1.MAX_VER)} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/network_interface', 'value': network_interface, 'op': 'add'}], headers=headers) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_update_network_interface_old_api(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node network_interface = 'flat' headers = {api_base.Version.string: '1.15'} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/network_interface', 'value': network_interface, 'op': 'add'}], headers=headers, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) def test_update_resource_class(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node resource_class = 'foo' headers = {api_base.Version.string: '1.21'} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/resource_class', 'value': resource_class, 'op': 'add'}], headers=headers) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_update_resource_class_old_api(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node resource_class = 'foo' headers = {api_base.Version.string: '1.20'} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/resource_class', 'value': resource_class, 'op': 'add'}], headers=headers, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) def test_update_resource_class_max_length(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node resource_class = 'f' * 80 headers = {api_base.Version.string: '1.21'} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/resource_class', 'value': resource_class, 'op': 'add'}], headers=headers) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_update_resource_class_too_long(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node resource_class = 'f' * 81 headers = {api_base.Version.string: '1.21'} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/resource_class', 'value': resource_class, 'op': 'add'}], headers=headers, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_code) def test_update_interface_fields(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node headers = {api_base.Version.string: str(api_v1.MAX_VER)} for field in api_utils.V31_FIELDS: response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/%s' % field, 'value': 'fake', 'op': 'add'}], headers=headers) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_update_interface_fields_bad_version(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node headers = {api_base.Version.string: '1.30'} for field in api_utils.V31_FIELDS: response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/%s' % field, 'value': 'fake', 'op': 'add'}], headers=headers, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) def test_update_classic_driver_interface_fields(self): headers = {api_base.Version.string: '1.31'} self.mock_update_node.side_effect = ( exception.MustBeNone('error')) for field in api_utils.V31_FIELDS: node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/%s' % field, 'value': 'fake', 'op': 'add'}], headers=headers, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) def test_update_storage_interface(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node storage_interface = 'cinder' headers = {api_base.Version.string: str(api_v1.MAX_VER)} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/storage_interface', 'value': storage_interface, 'op': 'add'}], headers=headers) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.OK, response.status_code) def test_update_storage_interface_old_api(self): node = obj_utils.create_test_node(self.context, uuid=uuidutils.generate_uuid()) self.mock_update_node.return_value = node storage_interface = 'cinder' headers = {api_base.Version.string: '1.32'} response = self.patch_json('/nodes/%s' % node.uuid, [{'path': '/storage_interface', 'value': storage_interface, 'op': 'add'}], headers=headers, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_code) def _create_node_locally(node): driver_factory.check_and_update_node_interfaces(node) node.create() return node @mock.patch.object(rpcapi.ConductorAPI, 'create_node', lambda _api, _ctx, node, _topic: _create_node_locally(node)) class TestPost(test_api_base.BaseApiTest): def setUp(self): super(TestPost, self).setUp() self.config(enabled_drivers=['fake']) self.chassis = obj_utils.create_test_chassis(self.context) p = mock.patch.object(rpcapi.ConductorAPI, 'get_topic_for') self.mock_gtf = p.start() self.mock_gtf.return_value = 'test-topic' self.addCleanup(p.stop) @mock.patch.object(timeutils, 'utcnow') def _test_create_node(self, mock_utcnow, headers=None, remove_chassis_uuid=False, kwargs): headers = headers or {} ndict = test_api_utils.post_get_test_node(kwargs) if remove_chassis_uuid: del ndict['chassis_uuid'] test_time = datetime.datetime(2000, 1, 1, 0, 0) mock_utcnow.return_value = test_time response = self.post_json('/nodes', ndict, headers=headers) self.assertEqual(http_client.CREATED, response.status_int) result = self.get_json('/nodes/%s' % ndict['uuid'], headers=headers) self.assertEqual(ndict['uuid'], result['uuid']) self.assertFalse(result['updated_at']) return_created_at = timeutils.parse_isotime( result['created_at']).replace(tzinfo=None) self.assertEqual(test_time, return_created_at) # Check location header self.assertIsNotNone(response.location) expected_location = '/v1/nodes/%s' % ndict['uuid'] self.assertEqual(urlparse.urlparse(response.location).path, expected_location) return result @mock.patch.object(notification_utils.LOG, 'exception', autospec=True) @mock.patch.object(notification_utils.LOG, 'warning', autospec=True) def test_create_node(self, mock_warning, mock_exception): self._test_create_node() self.assertFalse(mock_warning.called) self.assertFalse(mock_exception.called) def test_create_node_chassis_uuid_always_in_response(self): result = self._test_create_node(chassis_uuid=None) self.assertIsNone(result['chassis_uuid']) result = self._test_create_node(uuid=uuidutils.generate_uuid(), remove_chassis_uuid=True) self.assertIsNone(result['chassis_uuid']) def test_create_node_invalid_chassis(self): ndict = test_api_utils.post_get_test_node(chassis_uuid=0) response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) def test_create_node_explicit_network_interface(self): headers = {api_base.Version.string: '1.20'} result = self._test_create_node(headers=headers, network_interface='neutron') self.assertEqual('neutron', result['network_interface']) def test_create_node_specify_interfaces(self): headers = {api_base.Version.string: '1.31'} for field in api_utils.V31_FIELDS: cfg.CONF.set_override('enabled_%ss' % field, ['fake']) for field in api_utils.V31_FIELDS: node = { 'uuid': uuidutils.generate_uuid(), field: 'fake', 'driver': 'fake-hardware' } result = self._test_create_node(headers=headers, node) self.assertEqual('fake', result[field]) def test_create_node_specify_interfaces_bad_version(self): headers = {api_base.Version.string: '1.30'} for field in api_utils.V31_FIELDS: ndict = test_api_utils.post_get_test_node({field: 'fake'}) response = self.post_json('/nodes', ndict, headers=headers, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_create_node_classic_driver_specify_interface(self): headers = {api_base.Version.string: '1.31'} for field in api_utils.V31_FIELDS: node = { 'uuid': uuidutils.generate_uuid(), field: 'fake', } ndict = test_api_utils.post_get_test_node(node) response = self.post_json('/nodes', ndict, headers=headers, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) def test_create_node_explicit_storage_interface(self): headers = {api_base.Version.string: '1.33'} result = self._test_create_node(headers=headers, storage_interface='cinder') self.assertEqual('cinder', result['storage_interface']) def test_create_node_name_empty_invalid(self): ndict = test_api_utils.post_get_test_node(name='') response = self.post_json('/nodes', ndict, headers={api_base.Version.string: "1.10"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) def test_create_node_name_empty_not_acceptable(self): ndict = test_api_utils.post_get_test_node(name='') response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) def test_create_node_reserved_name(self): reserved_names = api_utils.get_controller_reserved_names( api_node.NodesController) for name in reserved_names: ndict = test_api_utils.post_get_test_node(name=name) response = self.post_json( '/nodes', ndict, headers={api_base.Version.string: "1.10"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) def test_create_node_default_state_none(self): ndict = test_api_utils.post_get_test_node() response = self.post_json('/nodes', ndict, headers={api_base.Version.string: "1.10"}) self.assertEqual(http_client.CREATED, response.status_int) # default state remains NONE/AVAILABLE result = self.get_json('/nodes/%s' % ndict['uuid']) self.assertEqual(states.NOSTATE, result['provision_state']) result = self.get_json('/nodes/%s' % ndict['uuid'], headers={api_base.Version.string: "1.10"}) self.assertEqual(ndict['uuid'], result['uuid']) self.assertEqual(states.AVAILABLE, result['provision_state']) def test_create_node_default_state_enroll(self): ndict = test_api_utils.post_get_test_node() response = self.post_json('/nodes', ndict, headers={api_base.Version.string: "1.11"}) self.assertEqual(http_client.CREATED, response.status_int) # default state is ENROLL result = self.get_json('/nodes/%s' % ndict['uuid']) self.assertEqual(ndict['uuid'], result['uuid']) self.assertEqual(states.ENROLL, result['provision_state']) def test_create_node_doesnt_contain_id(self): # FIXME(comstud): I'd like to make this test not use the # dbapi, however, no matter what I do when trying to mock # Node.create(), the API fails to convert the objects.Node # into the API Node object correctly (it leaves all fields # as Unset). with mock.patch.object(self.dbapi, 'create_node', wraps=self.dbapi.create_node) as cn_mock: ndict = test_api_utils.post_get_test_node(extra={'foo': 123}) self.post_json('/nodes', ndict) result = self.get_json('/nodes/%s' % ndict['uuid']) self.assertEqual(ndict['extra'], result['extra']) cn_mock.assert_called_once_with(mock.ANY) # Check that 'id' is not in first arg of positional args self.assertNotIn('id', cn_mock.call_args[0][0]) def _test_jsontype_attributes(self, attr_name): kwargs = {attr_name: {'str': 'foo', 'int': 123, 'float': 0.1, 'bool': True, 'list': [1, 2], 'none': None, 'dict': {'cat': 'meow'}}} ndict = test_api_utils.post_get_test_node(kwargs) self.post_json('/nodes', ndict) result = self.get_json('/nodes/%s' % ndict['uuid']) self.assertEqual(ndict[attr_name], result[attr_name]) def test_create_node_valid_extra(self): self._test_jsontype_attributes('extra') def test_create_node_valid_properties(self): self._test_jsontype_attributes('properties') def test_create_node_valid_driver_info(self): self._test_jsontype_attributes('driver_info') def test_create_node_valid_instance_info(self): self._test_jsontype_attributes('instance_info') def _test_vendor_passthru_ok(self, mock_vendor, return_value=None, is_async=True): expected_status = http_client.ACCEPTED if is_async else http_client.OK expected_return_value = json.dumps(return_value) if six.PY3: expected_return_value = expected_return_value.encode('utf-8') node = obj_utils.create_test_node(self.context) info = {'foo': 'bar'} mock_vendor.return_value = {'return': return_value, 'async': is_async, 'attach': False} response = self.post_json('/nodes/%s/vendor_passthru/test' % node.uuid, info) mock_vendor.assert_called_once_with( mock.ANY, node.uuid, 'test', 'POST', info, 'test-topic') self.assertEqual(expected_return_value, response.body) self.assertEqual(expected_status, response.status_code) def _test_vendor_passthru_ok_by_name(self, mock_vendor, return_value=None, is_async=True): expected_status = http_client.ACCEPTED if is_async else http_client.OK expected_return_value = json.dumps(return_value) if six.PY3: expected_return_value = expected_return_value.encode('utf-8') node = obj_utils.create_test_node(self.context, name='node-109') info = {'foo': 'bar'} mock_vendor.return_value = {'return': return_value, 'async': is_async, 'attach': False} response = self.post_json('/nodes/%s/vendor_passthru/test' % node.name, info, headers={api_base.Version.string: "1.5"}) mock_vendor.assert_called_once_with( mock.ANY, node.uuid, 'test', 'POST', info, 'test-topic') self.assertEqual(expected_return_value, response.body) self.assertEqual(expected_status, response.status_code) @mock.patch.object(rpcapi.ConductorAPI, 'vendor_passthru') def test_vendor_passthru_async(self, mock_vendor): self._test_vendor_passthru_ok(mock_vendor) @mock.patch.object(rpcapi.ConductorAPI, 'vendor_passthru') def test_vendor_passthru_sync(self, mock_vendor): return_value = {'cat': 'meow'} self._test_vendor_passthru_ok(mock_vendor, return_value=return_value, is_async=False) @mock.patch.object(rpcapi.ConductorAPI, 'vendor_passthru') def test_vendor_passthru_put(self, mocked_vendor_passthru): node = obj_utils.create_test_node(self.context) return_value = {'return': None, 'async': True, 'attach': False} mocked_vendor_passthru.return_value = return_value response = self.put_json( '/nodes/%s/vendor_passthru/do_test' % node.uuid, {'test_key': 'test_value'}) self.assertEqual(http_client.ACCEPTED, response.status_int) self.assertEqual(return_value['return'], response.json) @mock.patch.object(rpcapi.ConductorAPI, 'vendor_passthru') def test_vendor_passthru_by_name(self, mock_vendor): self._test_vendor_passthru_ok_by_name(mock_vendor) @mock.patch.object(rpcapi.ConductorAPI, 'vendor_passthru') def test_vendor_passthru_get(self, mocked_vendor_passthru): node = obj_utils.create_test_node(self.context) return_value = {'return': 'foo', 'async': False, 'attach': False} mocked_vendor_passthru.return_value = return_value response = self.get_json( '/nodes/%s/vendor_passthru/do_test' % node.uuid) self.assertEqual(return_value['return'], response) @mock.patch.object(rpcapi.ConductorAPI, 'vendor_passthru') def test_vendor_passthru_delete(self, mock_vendor_passthru): node = obj_utils.create_test_node(self.context) return_value = {'return': None, 'async': True, 'attach': False} mock_vendor_passthru.return_value = return_value response = self.delete( '/nodes/%s/vendor_passthru/do_test' % node.uuid) self.assertEqual(http_client.ACCEPTED, response.status_int) self.assertEqual(return_value['return'], response.json) def test_vendor_passthru_no_such_method(self): node = obj_utils.create_test_node(self.context) uuid = node.uuid info = {'foo': 'bar'} with mock.patch.object( rpcapi.ConductorAPI, 'vendor_passthru') as mock_vendor: mock_vendor.side_effect = exception.UnsupportedDriverExtension( **{'driver': node.driver, 'node': uuid, 'extension': 'test'}) response = self.post_json('/nodes/%s/vendor_passthru/test' % uuid, info, expect_errors=True) mock_vendor.assert_called_once_with( mock.ANY, uuid, 'test', 'POST', info, 'test-topic') self.assertEqual(http_client.BAD_REQUEST, response.status_code) def test_vendor_passthru_without_method(self): node = obj_utils.create_test_node(self.context) response = self.post_json('/nodes/%s/vendor_passthru' % node.uuid, {'foo': 'bar'}, expect_errors=True) self.assertEqual('application/json', response.content_type, ) self.assertEqual(http_client.BAD_REQUEST, response.status_code) self.assertTrue(response.json['error_message']) def test_post_ports_subresource_no_node_id(self): node = obj_utils.create_test_node(self.context) pdict = test_api_utils.port_post_data(node_id=None) pdict['node_uuid'] = node.uuid response = self.post_json('/nodes/ports', pdict, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) def test_post_ports_subresource(self): node = obj_utils.create_test_node(self.context) pdict = test_api_utils.port_post_data(node_id=None) pdict['node_uuid'] = node.uuid response = self.post_json('/nodes/%s/ports' % node.uuid, pdict, expect_errors=True) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_post_portgroups_subresource(self): node = obj_utils.create_test_node(self.context) pgdict = test_api_utils.portgroup_post_data(node_id=None) pgdict['node_uuid'] = node.uuid response = self.post_json( '/nodes/%s/portgroups' % node.uuid, pgdict, expect_errors=True, headers={'X-OpenStack-Ironic-API-Version': '1.24'}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_post_volume_connectors_subresource_no_node_id(self): node = obj_utils.create_test_node(self.context) pdict = test_api_utils.volume_connector_post_data(node_id=None) pdict['node_uuid'] = node.uuid response = self.post_json( '/nodes/volume/connectors', pdict, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.NOT_FOUND, response.status_int) def test_post_volume_connectors_subresource(self): node = obj_utils.create_test_node(self.context) pdict = test_api_utils.volume_connector_post_data(node_id=None) pdict['node_uuid'] = node.uuid response = self.post_json( '/nodes/%s/volume/connectors' % node.uuid, pdict, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_post_volume_targets_subresource(self): node = obj_utils.create_test_node(self.context) pdict = test_api_utils.volume_target_post_data(node_id=None) pdict['node_uuid'] = node.uuid response = self.post_json( '/nodes/%s/volume/targets' % node.uuid, pdict, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_create_node_no_mandatory_field_driver(self): ndict = test_api_utils.post_get_test_node() del ndict['driver'] response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) def test_create_node_invalid_driver(self): ndict = test_api_utils.post_get_test_node() self.mock_gtf.side_effect = exception.NoValidHost('Fake Error') response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) def test_create_node_no_chassis_uuid(self): ndict = test_api_utils.post_get_test_node() del ndict['chassis_uuid'] response = self.post_json('/nodes', ndict) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.CREATED, response.status_int) # Check location header self.assertIsNotNone(response.location) expected_location = '/v1/nodes/%s' % ndict['uuid'] self.assertEqual(urlparse.urlparse(response.location).path, expected_location) @mock.patch.object(notification_utils, '_emit_api_notification') def test_create_node_with_chassis_uuid(self, mock_notify): ndict = test_api_utils.post_get_test_node( chassis_uuid=self.chassis.uuid) response = self.post_json('/nodes', ndict) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.CREATED, response.status_int) result = self.get_json('/nodes/%s' % ndict['uuid']) self.assertEqual(ndict['chassis_uuid'], result['chassis_uuid']) # Check location header self.assertIsNotNone(response.location) expected_location = '/v1/nodes/%s' % ndict['uuid'] self.assertEqual(urlparse.urlparse(response.location).path, expected_location) mock_notify.assert_has_calls([mock.call(mock.ANY, mock.ANY, 'create', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.START, chassis_uuid=self.chassis.uuid), mock.call(mock.ANY, mock.ANY, 'create', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.END, chassis_uuid=self.chassis.uuid)]) def test_create_node_chassis_uuid_not_found(self): ndict = test_api_utils.post_get_test_node( chassis_uuid='1a1a1a1a-2b2b-3c3c-4d4d-5e5e5e5e5e5e') response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertTrue(response.json['error_message']) def test_create_node_with_internal_field(self): ndict = test_api_utils.post_get_test_node() ndict['reservation'] = 'fake' response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_int) self.assertTrue(response.json['error_message']) @mock.patch.object(rpcapi.ConductorAPI, 'get_node_vendor_passthru_methods') def test_vendor_passthru_methods(self, get_methods_mock): return_value = {'foo': 'bar'} get_methods_mock.return_value = return_value node = obj_utils.create_test_node(self.context) path = '/nodes/%s/vendor_passthru/methods' % node.uuid data = self.get_json(path) self.assertEqual(return_value, data) get_methods_mock.assert_called_once_with(mock.ANY, node.uuid, topic=mock.ANY) # Now let's test the cache: Reset the mock get_methods_mock.reset_mock() # Call it again data = self.get_json(path) self.assertEqual(return_value, data) # Assert RPC method wasn't called this time self.assertFalse(get_methods_mock.called) def test_create_node_network_interface(self): ndict = test_api_utils.post_get_test_node( network_interface='flat') response = self.post_json('/nodes', ndict, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.CREATED, response.status_int) result = self.get_json('/nodes/%s' % ndict['uuid'], headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual('flat', result['network_interface']) def test_create_node_network_interface_old_api_version(self): ndict = test_api_utils.post_get_test_node( network_interface='flat') response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_create_node_invalid_network_interface(self): ndict = test_api_utils.post_get_test_node( network_interface='foo') response = self.post_json('/nodes', ndict, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_int) def test_create_node_resource_class(self): ndict = test_api_utils.post_get_test_node( resource_class='foo') response = self.post_json('/nodes', ndict, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.CREATED, response.status_int) result = self.get_json('/nodes/%s' % ndict['uuid'], headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual('foo', result['resource_class']) def test_create_node_resource_class_old_api_version(self): ndict = test_api_utils.post_get_test_node( resource_class='foo') response = self.post_json('/nodes', ndict, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_create_node_storage_interface_old_api_version(self): headers = {api_base.Version.string: '1.32'} ndict = test_api_utils.post_get_test_node(storage_interface='cinder') response = self.post_json('/nodes', ndict, headers=headers, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.NOT_ACCEPTABLE, response.status_int) def test_create_node_invalid_storage_interface(self): ndict = test_api_utils.post_get_test_node(storage_interface='foo') response = self.post_json('/nodes', ndict, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual('application/json', response.content_type) self.assertEqual(http_client.BAD_REQUEST, response.status_int) class TestDelete(test_api_base.BaseApiTest): def setUp(self): super(TestDelete, self).setUp() p = mock.patch.object(rpcapi.ConductorAPI, 'get_topic_for') self.mock_gtf = p.start() self.mock_gtf.return_value = 'test-topic' self.addCleanup(p.stop) @mock.patch.object(notification_utils, '_emit_api_notification') @mock.patch.object(rpcapi.ConductorAPI, 'destroy_node') def test_delete_node(self, mock_dn, mock_notify): node = obj_utils.create_test_node(self.context) self.delete('/nodes/%s' % node.uuid) mock_dn.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') mock_notify.assert_has_calls([mock.call(mock.ANY, mock.ANY, 'delete', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.START, chassis_uuid=None), mock.call(mock.ANY, mock.ANY, 'delete', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.END, chassis_uuid=None)]) @mock.patch.object(rpcapi.ConductorAPI, 'destroy_node') def test_delete_node_by_name_unsupported(self, mock_dn): node = obj_utils.create_test_node(self.context, name='foo') self.delete('/nodes/%s' % node.name, expect_errors=True) self.assertFalse(mock_dn.called) @mock.patch.object(rpcapi.ConductorAPI, 'destroy_node') def test_delete_node_by_name(self, mock_dn): node = obj_utils.create_test_node(self.context, name='foo') self.delete('/nodes/%s' % node.name, headers={api_base.Version.string: "1.5"}) mock_dn.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') @mock.patch.object(objects.Node, 'get_by_uuid') def test_delete_node_not_found(self, mock_gbu): node = obj_utils.get_test_node(self.context) mock_gbu.side_effect = exception.NodeNotFound(node=node.uuid) response = self.delete('/nodes/%s' % node.uuid, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) mock_gbu.assert_called_once_with(mock.ANY, node.uuid) @mock.patch.object(objects.Node, 'get_by_name') def test_delete_node_not_found_by_name_unsupported(self, mock_gbn): node = obj_utils.get_test_node(self.context, name='foo') mock_gbn.side_effect = exception.NodeNotFound(node=node.name) response = self.delete('/nodes/%s' % node.name, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_int) self.assertFalse(mock_gbn.called) @mock.patch.object(objects.Node, 'get_by_name') def test_delete_node_not_found_by_name(self, mock_gbn): node = obj_utils.get_test_node(self.context, name='foo') mock_gbn.side_effect = exception.NodeNotFound(node=node.name) response = self.delete('/nodes/%s' % node.name, headers={api_base.Version.string: "1.5"}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_int) self.assertEqual('application/json', response.content_type) self.assertTrue(response.json['error_message']) mock_gbn.assert_called_once_with(mock.ANY, node.name) def test_delete_ports_subresource_no_port_id(self): node = obj_utils.create_test_node(self.context) response = self.delete('/nodes/%s/ports' % node.uuid, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, response.status_int) def test_delete_ports_subresource(self): node = obj_utils.create_test_node(self.context) port = obj_utils.create_test_port(self.context, node_id=node.id) response = self.delete( '/nodes/%(node_uuid)s/ports/%(port_uuid)s' % {'node_uuid': node.uuid, 'port_uuid': port.uuid}, expect_errors=True) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_delete_portgroup_subresource(self): node = obj_utils.create_test_node(self.context) pg = obj_utils.create_test_portgroup(self.context, node_id=node.id) response = self.delete( '/nodes/%(node_uuid)s/portgroups/%(pg_uuid)s' % {'node_uuid': node.uuid, 'pg_uuid': pg.uuid}, expect_errors=True, headers={'X-OpenStack-Ironic-API-Version': '1.24'}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_delete_volume_connectors_subresource_no_connector_id(self): node = obj_utils.create_test_node(self.context) response = self.delete( '/nodes/%s/volume/connectors' % node.uuid, expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.BAD_REQUEST, response.status_int) def test_delete_volume_connectors_subresource(self): node = obj_utils.create_test_node(self.context) connector = obj_utils.create_test_volume_connector(self.context, node_id=node.id) response = self.delete( '/nodes/%s/volume/connectors/%s' % (node.uuid, connector.uuid), expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.FORBIDDEN, response.status_int) def test_delete_volume_targets_subresource(self): node = obj_utils.create_test_node(self.context) target = obj_utils.create_test_volume_target(self.context, node_id=node.id) response = self.delete( '/nodes/%s/volume/targets/%s' % (node.uuid, target.uuid), expect_errors=True, headers={api_base.Version.string: str(api_v1.MAX_VER)}) self.assertEqual(http_client.FORBIDDEN, response.status_int) @mock.patch.object(notification_utils, '_emit_api_notification') @mock.patch.object(rpcapi.ConductorAPI, 'destroy_node') def test_delete_associated(self, mock_dn, mock_notify): node = obj_utils.create_test_node( self.context, instance_uuid='aaaaaaaa-1111-bbbb-2222-cccccccccccc') mock_dn.side_effect = exception.NodeAssociated( node=node.uuid, instance=node.instance_uuid) response = self.delete('/nodes/%s' % node.uuid, expect_errors=True) self.assertEqual(http_client.CONFLICT, response.status_int) mock_dn.assert_called_once_with(mock.ANY, node.uuid, 'test-topic') mock_notify.assert_has_calls([mock.call(mock.ANY, mock.ANY, 'delete', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.START, chassis_uuid=None), mock.call(mock.ANY, mock.ANY, 'delete', obj_fields.NotificationLevel.ERROR, obj_fields.NotificationStatus.ERROR, chassis_uuid=None)]) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'update_node') def test_delete_node_maintenance_mode(self, mock_update, mock_get): node = obj_utils.create_test_node(self.context, maintenance=True, maintenance_reason='blah') mock_get.return_value = node response = self.delete('/nodes/%s/maintenance' % node.uuid) self.assertEqual(http_client.ACCEPTED, response.status_int) self.assertEqual(b'', response.body) self.assertFalse(node.maintenance) self.assertIsNone(node.maintenance_reason) mock_get.assert_called_once_with(mock.ANY, node.uuid) mock_update.assert_called_once_with(mock.ANY, mock.ANY, topic='test-topic') @mock.patch.object(objects.Node, 'get_by_name') @mock.patch.object(rpcapi.ConductorAPI, 'update_node') def test_delete_node_maintenance_mode_by_name(self, mock_update, mock_get): node = obj_utils.create_test_node(self.context, maintenance=True, maintenance_reason='blah', name='foo') mock_get.return_value = node response = self.delete('/nodes/%s/maintenance' % node.name, headers={api_base.Version.string: "1.5"}) self.assertEqual(http_client.ACCEPTED, response.status_int) self.assertEqual(b'', response.body) self.assertFalse(node.maintenance) self.assertIsNone(node.maintenance_reason) mock_get.assert_called_once_with(mock.ANY, node.name) mock_update.assert_called_once_with(mock.ANY, mock.ANY, topic='test-topic') class TestPut(test_api_base.BaseApiTest): def setUp(self): super(TestPut, self).setUp() self.node = obj_utils.create_test_node( self.context, provision_state=states.AVAILABLE, name='node-39') p = mock.patch.object(rpcapi.ConductorAPI, 'get_topic_for') self.mock_gtf = p.start() self.mock_gtf.return_value = 'test-topic' self.addCleanup(p.stop) p = mock.patch.object(rpcapi.ConductorAPI, 'change_node_power_state') self.mock_cnps = p.start() self.addCleanup(p.stop) p = mock.patch.object(rpcapi.ConductorAPI, 'do_node_deploy') self.mock_dnd = p.start() self.addCleanup(p.stop) p = mock.patch.object(rpcapi.ConductorAPI, 'do_node_tear_down') self.mock_dntd = p.start() self.addCleanup(p.stop) p = mock.patch.object(rpcapi.ConductorAPI, 'inspect_hardware') self.mock_dnih = p.start() self.addCleanup(p.stop) def _test_power_state_success(self, target_state, timeout, api_version): if timeout is None: body = {'target': target_state} else: body = {'target': target_state, 'timeout': timeout} if api_version is None: response = self.put_json( '/nodes/%s/states/power' % self.node.uuid, body) else: response = self.put_json( '/nodes/%s/states/power' % self.node.uuid, body, headers={api_base.Version.string: api_version}) self.assertEqual(http_client.ACCEPTED, response.status_code) self.assertEqual(b'', response.body) self.mock_cnps.assert_called_once_with(mock.ANY, self.node.uuid, target_state, timeout=timeout, topic='test-topic') # Check location header self.assertIsNotNone(response.location) expected_location = '/v1/nodes/%s/states' % self.node.uuid self.assertEqual(urlparse.urlparse(response.location).path, expected_location) def _test_power_state_failure(self, target_state, http_status_code, timeout, api_version): if timeout is None: body = {'target': target_state} else: body = {'target': target_state, 'timeout': timeout} if api_version is None: response = self.put_json( '/nodes/%s/states/power' % self.node.uuid, body, expect_errors=True) else: response = self.put_json( '/nodes/%s/states/power' % self.node.uuid, body, headers={api_base.Version.string: api_version}, expect_errors=True) self.assertEqual('application/json', response.content_type) self.assertEqual(http_status_code, response.status_code) self.assertTrue(response.json['error_message']) def test_power_state_power_on_no_timeout_no_ver(self): self._test_power_state_success(states.POWER_ON, None, None) def test_power_state_power_on_no_timeout_valid_soft_ver(self): self._test_power_state_success(states.POWER_ON, None, "1.27") def test_power_state_power_on_no_timeout_invalid_soft_ver(self): self._test_power_state_success(states.POWER_ON, None, "1.26") def test_power_state_power_on_valid_timeout_no_ver(self): self._test_power_state_failure( states.POWER_ON, http_client.NOT_ACCEPTABLE, 2, None) def test_power_state_power_on_valid_timeout_valid_soft_ver(self): self._test_power_state_success(states.POWER_ON, 2, "1.27") def test_power_state_power_on_valid_timeout_invalid_soft_ver(self): self._test_power_state_failure( states.POWER_ON, http_client.NOT_ACCEPTABLE, 2, "1.26") def test_power_state_power_on_invalid_timeout_no_ver(self): self._test_power_state_failure( states.POWER_ON, http_client.BAD_REQUEST, 0, None) def test_power_state_power_on_invalid_timeout_valid_soft_ver(self): self._test_power_state_failure( states.POWER_ON, http_client.BAD_REQUEST, 0, "1.27") def test_power_state_power_on_invalid_timeout_invalid_soft_ver(self): self._test_power_state_failure( states.POWER_ON, http_client.BAD_REQUEST, 0, "1.26") def test_power_state_soft_power_off_no_timeout_no_ver(self): self._test_power_state_failure( states.SOFT_POWER_OFF, http_client.NOT_ACCEPTABLE, None, None) def test_power_state_soft_power_off_no_timeout_valid_soft_ver(self): self._test_power_state_success(states.SOFT_POWER_OFF, None, "1.27") def test_power_state_soft_power_off_no_timeout_invalid_soft_ver(self): self._test_power_state_failure( states.SOFT_POWER_OFF, http_client.NOT_ACCEPTABLE, None, "1.26") def test_power_state_soft_power_off_valid_timeout_no_ver(self): self._test_power_state_failure( states.SOFT_POWER_OFF, http_client.NOT_ACCEPTABLE, 2, None) def test_power_state_soft_power_off_valid_timeout_valid_soft_ver(self): self._test_power_state_success(states.SOFT_POWER_OFF, 2, "1.27") def test_power_state_soft_power_off_valid_timeout_invalid_soft_ver(self): self._test_power_state_failure( states.SOFT_POWER_OFF, http_client.NOT_ACCEPTABLE, 2, "1.26") def test_power_state_soft_power_off_invalid_timeout_no_ver(self): self._test_power_state_failure( states.SOFT_POWER_OFF, http_client.NOT_ACCEPTABLE, 0, None) def test_power_state_soft_power_off_invalid_timeout_valid_soft_ver(self): self._test_power_state_failure( states.SOFT_POWER_OFF, http_client.BAD_REQUEST, 0, "1.27") def test_power_state_soft_power_off_invalid_timeout_invalid_soft_ver(self): self._test_power_state_failure( states.SOFT_POWER_OFF, http_client.NOT_ACCEPTABLE, 0, "1.26") def test_power_state_by_name_unsupported(self): response = self.put_json('/nodes/%s/states/power' % self.node.name, {'target': states.POWER_ON}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, response.status_code) def test_power_state_by_name(self): response = self.put_json('/nodes/%s/states/power' % self.node.name, {'target': states.POWER_ON}, headers={api_base.Version.string: "1.5"}) self.assertEqual(http_client.ACCEPTED, response.status_code) self.assertEqual(b'', response.body) self.mock_cnps.assert_called_once_with(mock.ANY, self.node.uuid, states.POWER_ON, timeout=None, topic='test-topic') # Check location header self.assertIsNotNone(response.location) expected_location = '/v1/nodes/%s/states' % self.node.name self.assertEqual(urlparse.urlparse(response.location).path, expected_location) def test_power_invalid_state_request(self): ret = self.put_json('/nodes/%s/states/power' % self.node.uuid, {'target': 'not-supported'}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) def test_power_change_when_being_cleaned(self): for state in (states.CLEANING, states.CLEANWAIT): self.node.provision_state = state self.node.save() ret = self.put_json('/nodes/%s/states/power' % self.node.uuid, {'target': states.POWER_OFF}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) def test_provision_invalid_state_request(self): ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': 'not-supported'}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) def test_provision_with_deploy(self): ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.ACTIVE}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) self.mock_dnd.assert_called_once_with( mock.ANY, self.node.uuid, False, None, 'test-topic') # Check location header self.assertIsNotNone(ret.location) expected_location = '/v1/nodes/%s/states' % self.node.uuid self.assertEqual(urlparse.urlparse(ret.location).path, expected_location) def test_provision_by_name_unsupported(self): ret = self.put_json('/nodes/%s/states/provision' % self.node.name, {'target': states.ACTIVE}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, ret.status_code) def test_provision_by_name(self): ret = self.put_json('/nodes/%s/states/provision' % self.node.name, {'target': states.ACTIVE}, headers={api_base.Version.string: "1.5"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) self.mock_dnd.assert_called_once_with( mock.ANY, self.node.uuid, False, None, 'test-topic') def test_provision_with_deploy_configdrive(self): ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.ACTIVE, 'configdrive': 'foo'}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) self.mock_dnd.assert_called_once_with( mock.ANY, self.node.uuid, False, 'foo', 'test-topic') # Check location header self.assertIsNotNone(ret.location) expected_location = '/v1/nodes/%s/states' % self.node.uuid self.assertEqual(urlparse.urlparse(ret.location).path, expected_location) def test_provision_with_configdrive_not_active(self): ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.DELETED, 'configdrive': 'foo'}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) def test_provision_with_tear_down(self): node = self.node node.provision_state = states.ACTIVE node.target_provision_state = states.NOSTATE node.save() ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': states.DELETED}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) self.mock_dntd.assert_called_once_with( mock.ANY, node.uuid, 'test-topic') # Check location header self.assertIsNotNone(ret.location) expected_location = '/v1/nodes/%s/states' % node.uuid self.assertEqual(urlparse.urlparse(ret.location).path, expected_location) def test_provision_already_in_progress(self): node = self.node node.provision_state = states.DEPLOYING node.target_provision_state = states.ACTIVE node.reservation = 'fake-host' node.save() ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': states.ACTIVE}, expect_errors=True) self.assertEqual(http_client.CONFLICT, ret.status_code) # Conflict self.assertFalse(self.mock_dnd.called) def test_provision_locked_with_correct_state(self): node = self.node node.provision_state = states.AVAILABLE node.target_provision_state = states.NOSTATE node.reservation = 'fake-host' node.save() self.mock_dnd.side_effect = exception.NodeLocked(node='', host='') ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': states.ACTIVE}, expect_errors=True) self.assertEqual(http_client.CONFLICT, ret.status_code) # Conflict self.assertTrue(self.mock_dnd.called) def test_provision_with_tear_down_in_progress_deploywait(self): node = self.node node.provision_state = states.DEPLOYWAIT node.target_provision_state = states.ACTIVE node.save() ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': states.DELETED}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) self.mock_dntd.assert_called_once_with( mock.ANY, node.uuid, 'test-topic') # Check location header self.assertIsNotNone(ret.location) expected_location = '/v1/nodes/%s/states' % node.uuid self.assertEqual(urlparse.urlparse(ret.location).path, expected_location) # NOTE(deva): this test asserts API funcionality which is not part of # the new-ironic-state-machine in Kilo. It is retained for backwards # compatibility with Juno. # TODO(deva): add a deprecation-warning to the REST result # and check for it here. def test_provision_with_deploy_after_deployfail(self): node = self.node node.provision_state = states.DEPLOYFAIL node.target_provision_state = states.ACTIVE node.save() ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': states.ACTIVE}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) self.mock_dnd.assert_called_once_with( mock.ANY, node.uuid, False, None, 'test-topic') # Check location header self.assertIsNotNone(ret.location) expected_location = '/v1/nodes/%s/states' % node.uuid self.assertEqual(expected_location, urlparse.urlparse(ret.location).path) def test_provision_already_in_state(self): self.node.provision_state = states.ACTIVE self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.ACTIVE}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_provide_from_manage(self, mock_dpa): self.node.provision_state = states.MANAGEABLE self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['provide']}, headers={api_base.Version.string: "1.4"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_dpa.assert_called_once_with(mock.ANY, self.node.uuid, states.VERBS['provide'], 'test-topic') def test_inspect_already_in_progress(self): node = self.node node.provision_state = states.INSPECTING node.target_provision_state = states.MANAGEABLE node.reservation = 'fake-host' node.save() ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': states.MANAGEABLE}, expect_errors=True) self.assertEqual(http_client.CONFLICT, ret.status_code) # Conflict def test_inspect_validation_failed_status_code(self): self.mock_dnih.side_effect = exception.InvalidParameterValue( err='Failed to validate inspection or power info.') node = self.node node.provision_state = states.MANAGEABLE node.reservation = 'fake-host' node.save() ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': 'inspect'}, headers={api_base.Version.string: "1.6"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) def test_inspect_validation_failed_missing_parameter_value(self): self.mock_dnih.side_effect = exception.MissingParameterValue( err='Failed to validate inspection or power info.') node = self.node node.provision_state = states.MANAGEABLE node.reservation = 'fake-host' node.save() ret = self.put_json('/nodes/%s/states/provision' % node.uuid, {'target': 'inspect'}, headers={api_base.Version.string: "1.6"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_manage_from_available(self, mock_dpa): self.node.provision_state = states.AVAILABLE self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['manage']}, headers={api_base.Version.string: "1.4"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_dpa.assert_called_once_with(mock.ANY, self.node.uuid, states.VERBS['manage'], 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_bad_requests_in_managed_state(self, mock_dpa): self.node.provision_state = states.MANAGEABLE self.node.save() for state in [states.ACTIVE, states.REBUILD, states.DELETED]: ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.ACTIVE}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertEqual(0, mock_dpa.call_count) @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_abort_cleanwait(self, mock_dpa): self.node.provision_state = states.CLEANWAIT self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['abort']}, headers={api_base.Version.string: "1.13"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_dpa.assert_called_once_with(mock.ANY, self.node.uuid, states.VERBS['abort'], 'test-topic') def test_abort_invalid_state(self): # "abort" is only valid for nodes in CLEANWAIT self.node.provision_state = states.CLEANING self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['abort']}, headers={api_base.Version.string: "1.13"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) def test_provision_with_cleansteps_not_clean(self): self.node.provision_state = states.MANAGEABLE self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['provide'], 'clean_steps': 'foo'}, headers={api_base.Version.string: "1.4"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) def test_clean_no_cleansteps(self): self.node.provision_state = states.MANAGEABLE self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['clean']}, headers={api_base.Version.string: "1.15"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) @mock.patch.object(rpcapi.ConductorAPI, 'do_node_clean') @mock.patch.object(api_node, '_check_clean_steps') def test_clean_check_steps_fail(self, mock_check, mock_rpcapi): self.node.provision_state = states.MANAGEABLE self.node.save() mock_check.side_effect = exception.InvalidParameterValue('bad') clean_steps = [{"step": "upgrade_firmware", "interface": "deploy"}] ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['clean'], 'clean_steps': clean_steps}, headers={api_base.Version.string: "1.15"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) mock_check.assert_called_once_with(clean_steps) self.assertFalse(mock_rpcapi.called) @mock.patch.object(rpcapi.ConductorAPI, 'do_node_clean') @mock.patch.object(api_node, '_check_clean_steps') def test_clean(self, mock_check, mock_rpcapi): self.node.provision_state = states.MANAGEABLE self.node.save() clean_steps = [{"step": "upgrade_firmware", "interface": "deploy"}] ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['clean'], 'clean_steps': clean_steps}, headers={api_base.Version.string: "1.15"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_check.assert_called_once_with(clean_steps) mock_rpcapi.assert_called_once_with(mock.ANY, self.node.uuid, clean_steps, 'test-topic') def test_adopt_raises_error_before_1_17(self): """Test that a lower API client cannot use the adopt verb""" ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['adopt']}, headers={api_base.Version.string: "1.16"}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, ret.status_code) @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_adopt_from_manage(self, mock_dpa): """Test that a node can be adopted from the manageable state""" self.node.provision_state = states.MANAGEABLE self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['adopt']}, headers={api_base.Version.string: "1.17"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_dpa.assert_called_once_with(mock.ANY, self.node.uuid, states.VERBS['adopt'], 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_adopt_from_adoptfail(self, mock_dpa): """Test that a node in ADOPTFAIL can be adopted""" self.node.provision_state = states.ADOPTFAIL self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['adopt']}, headers={api_base.Version.string: "1.17"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_dpa.assert_called_once_with(mock.ANY, self.node.uuid, states.VERBS['adopt'], 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_adopt_from_active_fails(self, mock_dpa): """Test that an ACTIVE node cannot be adopted""" self.node.provision_state = states.ACTIVE self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['adopt']}, headers={api_base.Version.string: "1.17"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertEqual(0, mock_dpa.call_count) @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_manage_from_adoptfail(self, mock_dpa): """Test that a node can be sent to MANAGEABLE from ADOPTFAIL""" self.node.provision_state = states.ADOPTFAIL self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.VERBS['manage']}, headers={api_base.Version.string: "1.17"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_dpa.assert_called_once_with(mock.ANY, self.node.uuid, states.VERBS['manage'], 'test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_bad_requests_in_adopting_state(self, mock_dpa): """Test that a node in ADOPTING fails with invalid requests Verify that an API request fails if the ACTIVE, REBUILD, or DELETED state is requested by an API client when the node is in ADOPTING state. """ self.node.provision_state = states.ADOPTING self.node.save() for state in [states.ACTIVE, states.REBUILD, states.DELETED]: ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': state}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertEqual(0, mock_dpa.call_count) @mock.patch.object(rpcapi.ConductorAPI, 'do_provisioning_action') def test_bad_requests_in_adoption_failed_state(self, mock_dpa): """Test that a node in ADOPTFAIL fails with invalid requests Verify that an API request fails if the ACTIVE, REBUILD, or DELETED state is requested by an API client when the node is in ADOPTFAIL state. """ self.node.provision_state = states.ADOPTFAIL self.node.save() for state in [states.ACTIVE, states.REBUILD, states.DELETED]: ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': state}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertEqual(0, mock_dpa.call_count) def test_set_console_mode_enabled(self): with mock.patch.object(rpcapi.ConductorAPI, 'set_console_mode') as mock_scm: ret = self.put_json('/nodes/%s/states/console' % self.node.uuid, {'enabled': "true"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_scm.assert_called_once_with(mock.ANY, self.node.uuid, True, 'test-topic') # Check location header self.assertIsNotNone(ret.location) expected_location = '/v1/nodes/%s/states/console' % self.node.uuid self.assertEqual(urlparse.urlparse(ret.location).path, expected_location) @mock.patch.object(rpcapi.ConductorAPI, 'set_console_mode') def test_set_console_by_name_unsupported(self, mock_scm): ret = self.put_json('/nodes/%s/states/console' % self.node.name, {'enabled': "true"}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, ret.status_code) @mock.patch.object(rpcapi.ConductorAPI, 'set_console_mode') def test_set_console_by_name(self, mock_scm): ret = self.put_json('/nodes/%s/states/console' % self.node.name, {'enabled': "true"}, headers={api_base.Version.string: "1.5"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_scm.assert_called_once_with(mock.ANY, self.node.uuid, True, 'test-topic') def test_set_console_mode_disabled(self): with mock.patch.object(rpcapi.ConductorAPI, 'set_console_mode') as mock_scm: ret = self.put_json('/nodes/%s/states/console' % self.node.uuid, {'enabled': "false"}) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) mock_scm.assert_called_once_with(mock.ANY, self.node.uuid, False, 'test-topic') # Check location header self.assertIsNotNone(ret.location) expected_location = '/v1/nodes/%s/states/console' % self.node.uuid self.assertEqual(urlparse.urlparse(ret.location).path, expected_location) def test_set_console_mode_bad_request(self): with mock.patch.object(rpcapi.ConductorAPI, 'set_console_mode') as mock_scm: ret = self.put_json('/nodes/%s/states/console' % self.node.uuid, {'enabled': "invalid-value"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) # assert set_console_mode wasn't called assert not mock_scm.called def test_set_console_mode_bad_request_missing_parameter(self): with mock.patch.object(rpcapi.ConductorAPI, 'set_console_mode') as mock_scm: ret = self.put_json('/nodes/%s/states/console' % self.node.uuid, {}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) # assert set_console_mode wasn't called assert not mock_scm.called def test_set_console_mode_console_not_supported(self): with mock.patch.object(rpcapi.ConductorAPI, 'set_console_mode') as mock_scm: mock_scm.side_effect = exception.UnsupportedDriverExtension( extension='console', driver='test-driver') ret = self.put_json('/nodes/%s/states/console' % self.node.uuid, {'enabled': "true"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) mock_scm.assert_called_once_with(mock.ANY, self.node.uuid, True, 'test-topic') def test_provision_node_in_maintenance_fail(self): self.node.maintenance = True self.node.save() ret = self.put_json('/nodes/%s/states/provision' % self.node.uuid, {'target': states.ACTIVE}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) @mock.patch.object(rpcapi.ConductorAPI, 'set_target_raid_config', autospec=True) def test_put_raid(self, set_raid_config_mock): raid_config = {'logical_disks': [{'size_gb': 100, 'raid_level': 1}]} ret = self.put_json( '/nodes/%s/states/raid' % self.node.uuid, raid_config, headers={api_base.Version.string: "1.12"}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) self.assertEqual(b'', ret.body) set_raid_config_mock.assert_called_once_with( mock.ANY, mock.ANY, self.node.uuid, raid_config, topic=mock.ANY) @mock.patch.object(rpcapi.ConductorAPI, 'set_target_raid_config', autospec=True) def test_put_raid_older_version(self, set_raid_config_mock): raid_config = {'logical_disks': [{'size_gb': 100, 'raid_level': 1}]} ret = self.put_json( '/nodes/%s/states/raid' % self.node.uuid, raid_config, headers={api_base.Version.string: "1.5"}, expect_errors=True) self.assertEqual(http_client.NOT_ACCEPTABLE, ret.status_code) self.assertFalse(set_raid_config_mock.called) @mock.patch.object(rpcapi.ConductorAPI, 'set_target_raid_config', autospec=True) def test_put_raid_iface_not_supported(self, set_raid_config_mock): raid_config = {'logical_disks': [{'size_gb': 100, 'raid_level': 1}]} set_raid_config_mock.side_effect = ( exception.UnsupportedDriverExtension(extension='raid', driver='fake')) ret = self.put_json( '/nodes/%s/states/raid' % self.node.uuid, raid_config, headers={api_base.Version.string: "1.12"}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, ret.status_code) self.assertTrue(ret.json['error_message']) set_raid_config_mock.assert_called_once_with( mock.ANY, mock.ANY, self.node.uuid, raid_config, topic=mock.ANY) @mock.patch.object(rpcapi.ConductorAPI, 'set_target_raid_config', autospec=True) def test_put_raid_invalid_parameter_value(self, set_raid_config_mock): raid_config = {'logical_disks': [{'size_gb': 100, 'raid_level': 1}]} set_raid_config_mock.side_effect = exception.InvalidParameterValue( 'foo') ret = self.put_json( '/nodes/%s/states/raid' % self.node.uuid, raid_config, headers={api_base.Version.string: "1.12"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) set_raid_config_mock.assert_called_once_with( mock.ANY, mock.ANY, self.node.uuid, raid_config, topic=mock.ANY) @mock.patch.object(rpcapi.ConductorAPI, 'set_boot_device') def test_set_boot_device(self, mock_sbd): device = boot_devices.PXE ret = self.put_json('/nodes/%s/management/boot_device' % self.node.uuid, {'boot_device': device}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) self.assertEqual(b'', ret.body) mock_sbd.assert_called_once_with(mock.ANY, self.node.uuid, device, persistent=False, topic='test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'set_boot_device') def test_set_boot_device_by_name(self, mock_sbd): device = boot_devices.PXE ret = self.put_json('/nodes/%s/management/boot_device' % self.node.name, {'boot_device': device}, headers={api_base.Version.string: "1.5"}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) self.assertEqual(b'', ret.body) mock_sbd.assert_called_once_with(mock.ANY, self.node.uuid, device, persistent=False, topic='test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'set_boot_device') def test_set_boot_device_not_supported(self, mock_sbd): mock_sbd.side_effect = exception.UnsupportedDriverExtension( extension='management', driver='test-driver') device = boot_devices.PXE ret = self.put_json('/nodes/%s/management/boot_device' % self.node.uuid, {'boot_device': device}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) mock_sbd.assert_called_once_with(mock.ANY, self.node.uuid, device, persistent=False, topic='test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'set_boot_device') def test_set_boot_device_persistent(self, mock_sbd): device = boot_devices.PXE ret = self.put_json('/nodes/%s/management/boot_device?persistent=True' % self.node.uuid, {'boot_device': device}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) self.assertEqual(b'', ret.body) mock_sbd.assert_called_once_with(mock.ANY, self.node.uuid, device, persistent=True, topic='test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'set_boot_device') def test_set_boot_device_persistent_invalid_value(self, mock_sbd): device = boot_devices.PXE ret = self.put_json('/nodes/%s/management/boot_device?persistent=blah' % self.node.uuid, {'boot_device': device}, expect_errors=True) self.assertEqual('application/json', ret.content_type) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) @mock.patch.object(rpcapi.ConductorAPI, 'inject_nmi') def test_inject_nmi(self, mock_inject_nmi): ret = self.put_json('/nodes/%s/management/inject_nmi' % self.node.uuid, {}, headers={api_base.Version.string: "1.29"}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) self.assertEqual(b'', ret.body) mock_inject_nmi.assert_called_once_with(mock.ANY, self.node.uuid, topic='test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'inject_nmi') def test_inject_nmi_not_allowed(self, mock_inject_nmi): ret = self.put_json('/nodes/%s/management/inject_nmi' % self.node.uuid, {}, headers={api_base.Version.string: "1.28"}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, ret.status_code) self.assertTrue(ret.json['error_message']) self.assertFalse(mock_inject_nmi.called) @mock.patch.object(rpcapi.ConductorAPI, 'inject_nmi') def test_inject_nmi_not_supported(self, mock_inject_nmi): mock_inject_nmi.side_effect = exception.UnsupportedDriverExtension( extension='management', driver='test-driver') ret = self.put_json('/nodes/%s/management/inject_nmi' % self.node.uuid, {}, headers={api_base.Version.string: "1.29"}, expect_errors=True) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) mock_inject_nmi.assert_called_once_with(mock.ANY, self.node.uuid, topic='test-topic') def _test_set_node_maintenance_mode(self, mock_update, mock_get, reason, node_ident, is_by_name=False): request_body = {} if reason: request_body['reason'] = reason self.node.maintenance = False mock_get.return_value = self.node if is_by_name: headers = {api_base.Version.string: "1.5"} else: headers = {} ret = self.put_json('/nodes/%s/maintenance' % node_ident, request_body, headers=headers) self.assertEqual(http_client.ACCEPTED, ret.status_code) self.assertEqual(b'', ret.body) self.assertTrue(self.node.maintenance) self.assertEqual(reason, self.node.maintenance_reason) mock_get.assert_called_once_with(mock.ANY, node_ident) mock_update.assert_called_once_with(mock.ANY, mock.ANY, topic='test-topic') @mock.patch.object(notification_utils, '_emit_api_notification') @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'update_node') def test_set_node_maintenance_mode(self, mock_update, mock_get, mock_notify): self._test_set_node_maintenance_mode(mock_update, mock_get, 'fake_reason', self.node.uuid) mock_notify.assert_has_calls([mock.call(mock.ANY, mock.ANY, 'maintenance_set', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.START), mock.call(mock.ANY, mock.ANY, 'maintenance_set', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.END)]) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'update_node') def test_set_node_maintenance_mode_no_reason(self, mock_update, mock_get): self._test_set_node_maintenance_mode(mock_update, mock_get, None, self.node.uuid) @mock.patch.object(objects.Node, 'get_by_name') @mock.patch.object(rpcapi.ConductorAPI, 'update_node') def test_set_node_maintenance_mode_by_name(self, mock_update, mock_get): self._test_set_node_maintenance_mode(mock_update, mock_get, 'fake_reason', self.node.name, is_by_name=True) @mock.patch.object(objects.Node, 'get_by_name') @mock.patch.object(rpcapi.ConductorAPI, 'update_node') def test_set_node_maintenance_mode_no_reason_by_name(self, mock_update, mock_get): self._test_set_node_maintenance_mode(mock_update, mock_get, None, self.node.name, is_by_name=True) @mock.patch.object(notification_utils, '_emit_api_notification') @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'update_node') def test_set_node_maintenance_mode_error(self, mock_update, mock_get, mock_notify): mock_get.return_value = self.node mock_update.side_effect = Exception() self.put_json('/nodes/%s/maintenance' % self.node.uuid, {'reason': 'fake'}, expect_errors=True) mock_notify.assert_has_calls([mock.call(mock.ANY, mock.ANY, 'maintenance_set', obj_fields.NotificationLevel.INFO, obj_fields.NotificationStatus.START), mock.call(mock.ANY, mock.ANY, 'maintenance_set', obj_fields.NotificationLevel.ERROR, obj_fields.NotificationStatus.ERROR)]) class TestCheckCleanSteps(base.TestCase): def test__check_clean_steps_not_list(self): clean_steps = {"step": "upgrade_firmware", "interface": "deploy"} self.assertRaisesRegex(exception.InvalidParameterValue, "not of type 'array'", api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_not_dict(self): clean_steps = ['clean step'] self.assertRaisesRegex(exception.InvalidParameterValue, "not of type 'object'", api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_key_invalid(self): clean_steps = [{"step": "upgrade_firmware", "interface": "deploy", "unknown": "upgrade_firmware"}] self.assertRaisesRegex(exception.InvalidParameterValue, 'unexpected', api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_missing_interface(self): clean_steps = [{"step": "upgrade_firmware"}] self.assertRaisesRegex(exception.InvalidParameterValue, 'interface', api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_missing_step_key(self): clean_steps = [{"interface": "deploy"}] self.assertRaisesRegex(exception.InvalidParameterValue, 'step', api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_missing_step_value(self): clean_steps = [{"step": None, "interface": "deploy"}] self.assertRaisesRegex(exception.InvalidParameterValue, "not of type 'string'", api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_min_length_step_value(self): clean_steps = [{"step": "", "interface": "deploy"}] self.assertRaisesRegex(exception.InvalidParameterValue, 'is too short', api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_interface_value_invalid(self): clean_steps = [{"step": "upgrade_firmware", "interface": "not"}] self.assertRaisesRegex(exception.InvalidParameterValue, 'is not one of', api_node._check_clean_steps, clean_steps) def test__check_clean_steps_step_args_value_invalid(self): clean_steps = [{"step": "upgrade_firmware", "interface": "deploy", "args": "invalid args"}] self.assertRaisesRegex(exception.InvalidParameterValue, 'args', api_node._check_clean_steps, clean_steps) def test__check_clean_steps_valid(self): clean_steps = [{"step": "upgrade_firmware", "interface": "deploy"}] api_node._check_clean_steps(clean_steps) step1 = {"step": "upgrade_firmware", "interface": "deploy", "args": {"arg1": "value1", "arg2": "value2"}} api_node._check_clean_steps([step1]) step2 = {"step": "configure raid", "interface": "raid"} api_node._check_clean_steps([step1, step2]) class TestAttachDetachVif(test_api_base.BaseApiTest): def setUp(self): super(TestAttachDetachVif, self).setUp() self.vif_version = "1.28" self.node = obj_utils.create_test_node( self.context, provision_state=states.AVAILABLE, name='node-39') p = mock.patch.object(rpcapi.ConductorAPI, 'get_topic_for') self.mock_gtf = p.start() self.mock_gtf.return_value = 'test-topic' self.addCleanup(p.stop) @mock.patch.object(objects.Node, 'get_by_uuid') def test_vif_subcontroller_old_version(self, mock_get): mock_get.return_value = self.node ret = self.get_json('/nodes/%s/vifs' % self.node.uuid, headers={api_base.Version.string: "1.26"}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, ret.status_code) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'vif_list') def test_vif_list(self, mock_list, mock_get): mock_get.return_value = self.node self.get_json('/nodes/%s/vifs' % self.node.uuid, headers={api_base.Version.string: self.vif_version}) mock_get.assert_called_once_with(mock.ANY, self.node.uuid) mock_list.assert_called_once_with(mock.ANY, self.node.uuid, topic='test-topic') @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'vif_attach') def test_vif_attach(self, mock_attach, mock_get): vif_id = uuidutils.generate_uuid() request_body = { 'id': vif_id } mock_get.return_value = self.node ret = self.post_json('/nodes/%s/vifs' % self.node.uuid, request_body, headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) mock_get.assert_called_once_with(mock.ANY, self.node.uuid) mock_attach.assert_called_once_with(mock.ANY, self.node.uuid, vif_info=request_body, topic='test-topic') @mock.patch.object(objects.Node, 'get_by_name') @mock.patch.object(rpcapi.ConductorAPI, 'vif_attach') def test_vif_attach_by_node_name(self, mock_attach, mock_get): vif_id = uuidutils.generate_uuid() request_body = { 'id': vif_id } mock_get.return_value = self.node ret = self.post_json('/nodes/%s/vifs' % self.node.name, request_body, headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) mock_get.assert_called_once_with(mock.ANY, self.node.name) mock_attach.assert_called_once_with(mock.ANY, self.node.uuid, vif_info=request_body, topic='test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'vif_attach') def test_vif_attach_node_not_found(self, mock_attach): vif_id = uuidutils.generate_uuid() request_body = { 'id': vif_id } ret = self.post_json('/nodes/doesntexist/vifs', request_body, expect_errors=True, headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.NOT_FOUND, ret.status_code) self.assertTrue(ret.json['error_message']) self.assertFalse(mock_attach.called) @mock.patch.object(objects.Node, 'get_by_name') @mock.patch.object(rpcapi.ConductorAPI, 'vif_attach') def test_vif_attach_conductor_unavailable(self, mock_attach, mock_get): vif_id = uuidutils.generate_uuid() request_body = { 'id': vif_id } mock_get.return_value = self.node self.mock_gtf.side_effect = exception.NoValidHost('boom') ret = self.post_json('/nodes/%s/vifs' % self.node.name, request_body, expect_errors=True, headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) self.assertFalse(mock_attach.called) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'vif_attach') def test_vif_attach_no_vif_id(self, mock_attach, mock_get): vif_id = uuidutils.generate_uuid() request_body = { 'bad_id': vif_id } mock_get.return_value = self.node ret = self.post_json('/nodes/%s/vifs' % self.node.uuid, request_body, expect_errors=True, headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'vif_attach') def test_vif_attach_invalid_vif_id(self, mock_attach, mock_get): request_body = { 'id': "invalid%id^" } mock_get.return_value = self.node ret = self.post_json('/nodes/%s/vifs' % self.node.uuid, request_body, expect_errors=True, headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.BAD_REQUEST, ret.status_code) self.assertTrue(ret.json['error_message']) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'vif_attach') def test_vif_attach_node_locked(self, mock_attach, mock_get): vif_id = uuidutils.generate_uuid() request_body = { 'id': vif_id } mock_get.return_value = self.node mock_attach.side_effect = exception.NodeLocked(node='', host='') ret = self.post_json('/nodes/%s/vifs' % self.node.uuid, request_body, expect_errors=True, headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.CONFLICT, ret.status_code) self.assertTrue(ret.json['error_message']) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'vif_detach') def test_vif_detach(self, mock_detach, mock_get): vif_id = uuidutils.generate_uuid() mock_get.return_value = self.node ret = self.delete('/nodes/%s/vifs/%s' % (self.node.uuid, vif_id), headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) mock_get.assert_called_once_with(mock.ANY, self.node.uuid) mock_detach.assert_called_once_with(mock.ANY, self.node.uuid, vif_id=vif_id, topic='test-topic') @mock.patch.object(objects.Node, 'get_by_name') @mock.patch.object(rpcapi.ConductorAPI, 'vif_detach') def test_vif_detach_by_node_name(self, mock_detach, mock_get): vif_id = uuidutils.generate_uuid() mock_get.return_value = self.node ret = self.delete('/nodes/%s/vifs/%s' % (self.node.name, vif_id), headers={api_base.Version.string: self.vif_version}) self.assertEqual(http_client.NO_CONTENT, ret.status_code) mock_get.assert_called_once_with(mock.ANY, self.node.name) mock_detach.assert_called_once_with(mock.ANY, self.node.uuid, vif_id=vif_id, topic='test-topic') @mock.patch.object(rpcapi.ConductorAPI, 'vif_detach') def test_vif_detach_node_not_found(self, mock_detach): vif_id = uuidutils.generate_uuid() ret = self.delete('/nodes/doesntexist/vifs/%s' % vif_id, headers={api_base.Version.string: self.vif_version}, expect_errors=True) self.assertEqual(http_client.NOT_FOUND, ret.status_code) self.assertTrue(ret.json['error_message']) self.assertFalse(mock_detach.called) @mock.patch.object(objects.Node, 'get_by_uuid') @mock.patch.object(rpcapi.ConductorAPI, 'vif_detach') def test_vif_detach_node_locked(self, mock_detach, mock_get): vif_id = uuidutils.generate_uuid() mock_get.return_value = self.node mock_detach.side_effect = exception.NodeLocked(node='', host='') ret = self.delete('/nodes/%s/vifs/%s' % (self.node.uuid, vif_id), headers={api_base.Version.string: self.vif_version}, expect_errors=True) self.assertEqual(http_client.CONFLICT, ret.status_code) self.assertTrue(ret.json['error_message'])	{ "redpajama_set_name": "RedPajamaGithub" }	7,642
package org.jeecgframework.tag.core.easyui; import javax.servlet.jsp.JspTagException; import javax.servlet.jsp.tagext.Tag; import javax.servlet.jsp.tagext.TagSupport; /** * * 类描述:列表默认操作项标签 * * 张代浩 * @date: 日期:2012-12-7 时间:上午10:17:45 * @version 1.0 */ public class DataGridDefOptTag extends TagSupport { protected String url; protected String title; private String exp;//判断链接是否显示的表达式 private String operationCode;//按钮的操作Code public int doStartTag() throws JspTagException { return EVAL_PAGE; } public int doEndTag() throws JspTagException { Tag t = findAncestorWithClass(this, DataGridTag.class); DataGridTag parent = (DataGridTag) t; parent.setDefUrl(url, title, exp,operationCode); return EVAL_PAGE; } public void setExp(String exp) { this.exp = exp; } public void setUrl(String url) { this.url = url; } public void setTitle(String title) { this.title = title; } public void setOperationCode(String operationCode) { this.operationCode = operationCode; } }	{ "redpajama_set_name": "RedPajamaGithub" }	4,271
\section{Introduction} \label{introduction} Visual pose estimation refers to the problem of estimating the camera pose with respect to the coordinate frame of a given reference 3D point cloud map. It is the foundation of visual Simultaneous Localization and Mapping (vSLAM) \cite{orb-slam,lsd-slam}, and Structure-from-Motion (SfM) \cite{SfM}, which are extremely important to applications such as autonomous driving \cite{auto-vehicle} and augmented reality \cite{AR} etc. A two-step approach ~\cite{SfM} is commonly used for visual pose estimation - (1) establish 2D-3D keypoint correspondences between the 2D image and 3D reference map, and (2) apply the Perspective-n-Point (PnP) \cite{p3p} algorithm to compute the camera pose with respect to the coordinate frame of the 3D reference map with at least three 2D-3D correspondences. The 3D point cloud of the reference map is usually built from a collection of images using SfM, and the associated keypoint descriptors, e.g. SIFT \cite{SIFT}, are stored with the map to facilitate the establishment of 2D-3D correspondences in visual pose estimation. It is eminent that the accuracy of visual pose estimation strongly depends on the quality of the 3D reference map. Unfortunately, it is hard to ensure the quality of the 3D point cloud reconstructed from SfM, since most images are taken by close-to-market photo-sensors that are noisy. Furthermore, absolute scale of the reconstructed 3D point cloud is not directly available from SfM and has to be obtained from other sources. The need to store keypoint descriptors from multiple views of the same keypoint also increases the memory consumption of the map. In contrast, a 3D point cloud map built from Lidars has the advantages of higher accuracy and absolute scale is directly observed. Despite the advantages, Lidars are seldom used to build the 3D reference map because of the lack of a descriptor that allows direct matching of the keypoints extracted from a 2D image and 3D point cloud. In this paper, we propose the 2D3D-MatchNet - a novel deep network approach to jointly learn the keypoint descriptors of the 2D and 3D keypoints extracted from an image and a point cloud. We use the existing detectors from SIFT~\cite{SIFT} and ISS~\cite{iss} to extract the keypoints of the image and point cloud, respectively. Similar to most deep learning methods, an image patch is used to represent an image keypoint, and a local point cloud volume is used to represent a 3D keypoint. We propose a triplet-like deep network to concurrently learn the keypoint descriptors of a given image patch and point cloud volume such that the distance in the descriptor space is small if the 2D and 3D keypoint are a matching pair, and large otherwise. The descriptors of the keypoints from both the image and point cloud are generated through our trained network during inference. The EPnP~\cite{epnp2009} algorithm is used to compute the camera pose based on the 2D-3D correspondences. We create our~\textit{Oxford 2D-3D Patches} dataset with 432,982 pairs of matching 2D-3D keypoints based on the Oxford dataset. We conduct extensive experiments on our dataset to verify that sufficient inliers for visual pose estimation can be found based on our learned feature vectors. Pose estimation succeeds without any prior on 2D-3D correspondences. \textbf{Contributions} (1) To the best of our knowledge, we are the first to propose a deep learning approach to learn the descriptors that allow direct matching of keypoints across a 2D image and 3D point cloud. (2) Our approach makes it possible for the use of Lidars to build more accurate 3D reference map for visual pose estimation. (3) We create a dataset with huge collection of 2D-3D image patch to 3D point cloud volume correspondences, which can be used to train and validate the network. \section{Related Work} \label{sec:relatedwork} \textbf{Traditional localization} Most existing work on visual pose estimation can be classified into two categories: (1) local structure based methods and (2) global appearance based methods. 2D-3D correspondences are first established from SIFT features given the query image and 3D scene model~\cite{efficientloc, worldwideloc} for local structure based methods. Each local feature pair votes for its own pose independently, without considering other pairs in the image. Then a minimal solver algorithm, e.g.~\cite{epnp2009}, combined with RANSAC iterations, is used for robust pose estimation. Global appearance based localization methods~\cite{global-loc, ulrich2000} aggregate all local features of the query image to a global descriptor and localize the query image by matching against its nearest neighbour in current image database as a retrieval problem. \textbf{Learnable localization} Deep learning is increasingly applied to the visual pose estimation since learned features are shown to be more robust against environmental changes, e.g. lighting and weather changes, compared with methods based on hand-crafted features such as SIFT~\cite{loc-lstm}. Existing deep network models solve the localization problem from different aspects. \cite{posenet, loc-lstm} learn to regress the 6D camera pose directly based on single image. \cite{sceneforest, regforest} learn to predict pixel-wise image to scene correspondences, followed by a RANSAC-like optimization algorithm for pose estimation. \cite{dsac} proposes to localize an image by learnable probabilistic 2D-3D correspondences and iterative refinement of camera pose. However, they cannot generalize to unseen environment due to the global pose estimation mechanism. \cite{semloc} proposes to learn robust 3D point cloud descriptors by fusing semantic and geometric information for long-term localization. \textbf{Deep similarity learning} Deep similarity learning is widely used to achieve the information retrieval task. Two common architectures of deep similarity learning are the Siamese network and the triplet network. The Siamese network learns the similarity relationship between a pair of inputs~\cite{Zagoruyko2015, Han2015}. Most existing work shows better results on the Triplet architecture~\cite{Hu2018, Liao2018, Vo2016, Guo2016, Schroff2015}. Liao $et.~al$~\cite{Liao2018} use the triplet network to re-identify a person's identity. Vo and Hays~\cite{Vo2016} conduct experiments on the ground-to-aerial image retrieval task using both Siamese architecture and Triplet architecture. They show that Triplet architecture performs better. The Triplet network outperforms the Siamese network because it can jointly pull the positive sample to the anchor while pushing the negative sample away. In our work, our proposed network is based on the Triplet network. \section{Approach} \label{sec:approach} In this section, we outline our pipeline for visual pose estimation with a 2D query image and 3D point cloud reference map built from Lidar scans. We first introduce the overview of our pipeline in section~\ref{subsec:overview}. In section~\ref{subsec:network}, we describe our novel 2D3D-MatchNet - a deep network to jointly extract the descriptors of the 2D and 3D keypoints from an image and a point cloud. The training loss is given in section~\ref{subsec:loss}. Finally, we discuss the pose estimation algorithm we use to compute the camera pose given at least three 2D-3D correspondences in section~\ref{subsec:pose_estimation}. \subsection{Overview} \label{subsec:overview} Given a query image $I$ and the 3D point cloud reference map $\mathit{M}$ of the scene, the objective of visual pose estimation is to compute the absolute camera pose $P=[R~\|~t]$ of the query image $I$ with respect to the coordinate frame of the 3D point cloud reference map $\mathit{M}$. Unlike existing visual pose methods which associate image-based descriptors, e.g. SIFT \cite{SIFT}, to each 3D point in the reference map, we propose the 2D3D-MatchNet - a deep network to jointly learn the descriptors directly from the 2D image and 3D point cloud. We first apply the SIFT detector on the query image $I$ to extract a set of 2D keypoints $U = \{u_1, ..., u_N~\|~u_n \in \mathcal{R}^2\}$, and the ISS detector \cite{iss} on the 3D point cloud of the reference map $\mathit{M}$ to extract a set of 3D keypoints $V = \{v_1, ..., v_M~\|~v_m \in \mathcal{R}^3\}$. Here, $N$ and $M$ are the total number of 2D and 3D keypoints extracted from the image $I$ and point cloud $\mathit{M}$, respectively. Given the set of 2D image patches centered around each 2D keypoint and 3D local point cloud volume centered around each 3D keypoint, our 2D3D-MatchNet learns the corresponding set of 2D and 3D descriptors denoted as $P = \{p_1, ..., p_N~\|~p_n \in \mathcal{R}^{D}\}$ and $Q = \{q_1, ..., q_M~\|~q_m \in \mathcal{R}^{D}\}$ for each corresponding 2D and 3D keypoint in $U$ and $V$. $D$ is the dimension of the descriptor. Furthermore, the descriptors $P$ and $Q$ learned from our network yield a much smaller similarity distance $d(p, q)$ between a matching pair of 2D-3D descriptors $(p, q)$ in comparison to the similarity distance $d(\bar{p},\bar{q})$ between a non-matching pair of 2D-3D descriptors $(\bar{p}, \bar{q})$, i.e. $d(p,q) \ll d(\bar{p},\bar{q})$, thus establishing the 2D-3D correspondences between $P$ and $Q$. Finally, the 2D-3D correspondences found from our 2D3D-MatchNet are used to estimate the absolute pose of the camera using a PnP algorithm. We run the PnP algorithm within RANSAC \cite{ransac} for robust estimation. \begin{figure} \centering \includegraphics[width=\linewidth]{network.pdf} \vspace{-0.4cm} \caption {Our triplet-like 2D3DMatch-Net.} \vspace{-0.1cm} \label {fig:network} \end{figure} \subsection{Our 2D3D-MatchNet: Network Architecture} \label{subsec:network} Our 2D3D-MatchNet is a triplet-like deep network that jointly learns the similarity between a given pair of image patch and local point cloud volume. The network consists of three branches as illustrated in Fig.~\ref{fig:network}. One of the branches learns the descriptor for the 2D image keypoint and the other two branches with shared weights learn the descriptor for the 3D point cloud keypoint. The inputs to the network are (1) image patches centered on the 2D image keypoints, and (2) local volume of point cloud within a fixed radius sphere centered on the 3D keypoints. Details on keypoints extraction and the definitions of image patch and point cloud sphere are given in Sec.~\ref{subsec:training_data}. The image patches and local volume of point clouds are fed into the network during training as tuples of anchor image patch, and positive and negative local volume point cloud. We denote the training tuple as $\{x^a_I, x^{+}_{\mathit{M}}, x^{-}_{\mathit{M}}\}$. Given a set of training tuples, our network learns the image descriptor function $G(x_I; \theta_I):x_I \mapsto p$ that maps an input image patch $x_I$ to its descriptor $p$, and the point cloud descriptor function $F(x_{\mathit{M}}; \theta_{\mathit{M}}):x_{\mathit{M}} \mapsto q$ that maps an input local point cloud volume $x_{\mathit{M}}$ to its descriptor $q$. $\theta_I$ and $\theta_{\mathit{M}}$ are the weights of the network learned during training. More specifically: \textbf{Image Descriptor Function} We design $G(x_I; \theta_I)$ as a convolutional neural network followed by several fully connected layers to extract the descriptor vector of an image patch. We use the well-known VGG network~\cite{VGG} as the basis of our image descriptor function network. We make use of the first four convolution block (conv1 $\sim$ conv4) of VGG16 to make the network fit better for image patch with small size. Global average pooling is applied on the feature maps from conv4. Compared to the max pooling layer which is widely used after convolutional layers, global average pooling has the advantages of reducing the number of parameters and avoiding over-fitting. Two fully connected layers are concatenated at the end of the network to achieve the desired output descriptor dimension. The output descriptor vector is L2-normalized before feeding into the loss function. \textbf{Point Cloud Descriptor Function} We use the state-of-art PointNet~\cite{Qi2017} as our point cloud descriptor function $F(x_{\mathit{M}}; \theta_{\mathit{M}})$ to extract the descriptor vector of a local point cloud volume. The dimension of the last fully connected layer is changed to fit our feature dimension. The softmax layer in the end is replaced by a L2-normalization. \textbf{Loss Function for Training} \label{subsec:loss} Our network is trained using the Triplet loss, so that the similarity distance $d_{pos} = d(G(x_I^a; \theta_I), F(x_{\mathit{M}}^{+}; \theta_{\mathit{M}}))$ between the matching anchor $x_I^a$ and positive $x_{\mathit{M}^{+}}$ pair is small and much lesser than the similarity distance $d_{neg} = d(G(x_I^a; \theta_I), F(x_{\mathit{M}}^{-}; \theta_{\mathit{M}}))$ between the non-matching anchor $x_I^a$ and negative $x_{\mathit{M}^{-}}$ pair, i.e. $d(G(x_I^a; \theta_I), F(x_{\mathit{M}}^{+}; \theta_{\mathit{M}})) \ll d(G(x_I^a; \theta_I), F(x_{\mathit{M}}^{-}; \theta_{\mathit{M}}))$. Specifically, Our network is trained with the weighted soft-margin triplet loss~\cite{Hu2018}: \begin{equation} \mathcal{L} = ln(1 + e^{\alpha d}), \end{equation} where $d = d_{pos} - d_{neg}$. We use this loss because it allows the deep network to converge faster and increase the retrieval accuracy~\cite{Hu2018}. In contrast to the basic Triplet loss~\cite{Schroff2015, Chechik2010}, it also avoids the need of selecting an optimal margin. In our experiment, we set $\alpha = 5$. We use the Euclidean distance between two vectors as the similarity distance $d(.,.)$ in this work. During inference, we check the distance $d(G(x_I; \theta_I), F(x_{\mathit{M}}; \theta_{\mathit{M}}))$ between the descriptors of a given pair of image patch $x_I$ and local point cloud volume $x_{\mathit{M}}$. $x_I$ and $x_{\mathit{M}}$ are deemed matching pair if the distance is lesser than a threshold, and non-matching pair otherwise. \subsection{Pose Estimation} \label{subsec:pose_estimation} The pose of the camera is computed from the putative set of 2D-3D correspondences obtained from our 2D3DMatch-Net. Specifically, we obtain the 2D keypoints of the 2D query image with the SIFT detector, and the 3D keypoints of the 3D point cloud with the ISS detector. We compute the 2D and 3D keypoint descriptors with our network from the image patches and local point cloud volume extracted around the keypoints. The similarity distance is computed for every pair of 2D and 3D keypoints, and we find the top $K$ closest 3D point cloud keypoints for every 2D image keypoint. Finally, we apply the EPnP algorithm~\cite{epnp2009} to estimate the camera pose with all the putative 2D-3D correspondences. The EPnP algorithm is ran within RANSAC for robust estimation to eliminate outliers. \section{Dataset} \label{sec:dataset} In this section, we present the creation of our benchmark dataset -- \textit{Oxford 2D-3D Patches Dataset}. The dataset contains a total of 432,982 image patch to pointcloud pairs, which allows sufficient training and evaluation for the 2D-3D feature matching task. \subsection{The Oxford 2D-3D Patches Dataset} Our Oxford 2D-3D Patches dataset is created based on the Oxford RobotCar Dataset~\cite{RobotCarDataset}. The Oxford RobotCar Dataset collects data from different kinds of sensors, including cameras, Lidar and GPS/INS, for over one year. We use the images from the two (left and right) Point Grey Grasshopper2 monocular cameras, the laser scans from the front SICK LMS-151 2D Lidar, and the GPS/INS data from the NovAtel SPAN-CPT ALIGN inertial and GPS navigation system. Ignoring the traversals collected with poor GPS, night and rain, we get 36 traversals for over a year with sufficiently challenging lighting, weather and traffic conditions. We synchronize the images from the left and right cameras, and 2D laser scans from the Lidar with the timestamps, and get their global poses using the GPS/INS data. We remove camera and Lidar frames with small motion. To simplify point cloud processing and reduce the detrimental effects of GPS jumps over long distance, we split each traversal into disjoint submaps at every 60m interval. Each submap contains the corresponding sets of left and right cameras and Lidar frames. A visualization of the reconstructed point cloud map from Lidar scans is illustrated in Fig.~\ref{fig:pointcloud_map}. \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{pointcloud_map.pdf} \vspace{-0.3cm} \caption {Reconstructed point cloud map from Lidar. Different colors represent different submaps. Two zooming in examples of point cloud are shown, the top one indicates a 60m submap, the bottom shows the last unseen 10\% of the path for testing.} \label {fig:pointcloud_map} \end{figure} \subsection{Training Data Generation} \label{subsec:training_data} \textbf{Keypoint Detection} We build a point cloud based reference map from the laser scans for every submap, where the coordinates of the first laser scan is used as the reference frame. We detect the ground plane and remove all points lying on it. This is because the flat ground plane is unlikely to contain any good 3D keypoint and descriptor. The ISS keypoint detector is applied on the remaining point cloud to extract all 3D keypoints. We apply the SIFT detector on every image to extract all 2D keypoints. \textbf{2D-3D Correspondences} To establish the 2D-3D correspondences, we project each ISS keypoint to all images within its view and find the nearest neighbour of SIFT keypoint in each image. To increase the confidence of the correspondences, we require the distance of the projected nearest neighbour to be smaller than 3 pixels and each ISS point must have at least SIFT correspondences in three different views within each submap. The ISS points and their corresponding SIFT points that satisfy these requirements are reserved for further processing. \begin{figure} \centering \includegraphics[width=\linewidth]{dataset.pdf} \vspace{-0.3cm} \caption {Four examples of our dataset. The first image of each example shows the ISS volume. The other three are some corresponding SIFT patches across multiple frames with different scale, viewpoint and lighting.} \label {fig:dataset} \end{figure} \textbf{ISS Volume and SIFT Patch Extraction} We remove all ISS keypoints that are within 4m from a selected ISS keypoint in each submap, and remove all SIFT keypoints within 32 pixels from a selected SIFT keypoint in each image. We find all 3D points that are within 1m radius from each selected ISS keypoint, and discard those ISS keypoints with less than 100 neighboring 3D points. We discard SIFT keypoints with scale larger than a threshold value, since larger scale results in smaller patch size. In our experiments, we set this threshold value as 4 and the patch size at the basic scale as $256\times256$. We vary the extracted patch size with the scale of the extracted SIFT keypoints for better scale invariance. In our experiments, we extract the ISS volume and its corresponding SIFT patch if the number of points within the ISS volume is larger than 100 and the SIFT patch is at suitable scale. We discard both the ISS volume and SIFT patch otherwise. Fig.~\ref{fig:dataset} shows the visualization of several examples of the local ISS point cloud volumes and their corresponding image patches with different scales, viewpoints and lightings. \textbf{Data Pre-processing} Before training, we rescale all the SIFT patches with different scales to the same size, i.e. 128$\times$128, and zero-center by mean subtraction. We subtract each point within each ISS point cloud volume with the associated ISS keypoint, thus achieving zero-center and unit norm sphere. Additionally, we pad the number of points to 1024 for each local volume in our experiments. \subsection{Testing Data Generation} \label{subsec:testing_data} Our objective during inference is to localize a query image based on the 2D-3D matching of the descriptors from the keypoints extracted from the image and point cloud. We test our trained network with reference submaps and images that are not used in training the network. We use the GPS/INS poses of the images as the ground truth pose for verification. The ground truth 2D-3D correspondences are computed as follows: (1) We detect all ISS keypoints from the point cloud of each submap and retain keypoints with more than 100 neighboring 3D points within 1m radius. (2) We detect SIFT keypoints on each image and extract the corresponding patches with scale smaller than the threshold value, i.e. 4 as mentioned above. (3) Each ISS keypoint is projected to all images within its view and the nearest SIFT keypoint with a distance smaller than 3 pixels is selected as the correspondence. We discard an ISS to SIFT keypoint correspondence if a nearest SIFT within 3 pixels is found in less than 3 image views. \section{Experiments} \label{sec:result} In this section, we first outline the training process of our 2D3D-MatchNet that jointly learns both 2D image and 3D point cloud descriptors. Next, we describe the camera pose estimation given a query image. Then we evaluate our results and compare with different methods. Finally, we discuss and analyze the localization results of the proposed methods. \subsection{Network Training} \textbf{Data splitting and evaluation metric} \hspace{0.1cm} As mentioned in Sec.~\ref{subsec:training_data}, we split each traversal into a set of disjoint 60m submaps. We leave one full traversal for testing. For the remaining 35 traversals, we use the first 90\% submaps of each traversal for training and leave the remaining 10\% unseen for testing as in Fig.~\ref{fig:pointcloud_map}. We evaluate the accuracy of the estimated pose by computing the position error and the rotational error from the ground truth poses. Similar to~\cite{sattler2018benchmarking}, we define the pose precision threshold as (10m, 45\degree). We measure the percentage of query images localized within this range and report the average position error and rotation error. We choose the threshold values to satisfy the high requirements for autonomous driving. \textbf{Network Training} \hspace{0.1cm} Our network is implemented in Tensorflow~\cite{Tensorflow} with $2\times$ Nvidia Titan X GPUs. We train the whole network in an end-to-end manner. For each triplet input, we choose an image patch as the anchor, and its corresponding 3D point cloud volume as positive sample. The negative point cloud volume is randomly sampled from the rest of the point clouds. We initialize the image descriptor network branch with VGG model pre-trained on ImageNet~\cite{Deng2009}. Both descriptor extraction networks are optimized with Adam optimizer and the initial learning rate is $6\times10^{-5}$. The total training time is around two days. We also explore the effect of the output feature dimension $D$ on the performance of localization. In our experiments, we train and test with different $D$ values, i.e. $D \in \{64, 128, 256\}$. The localization results from different descriptor dimensions are presented and discussed below. \begin{table}[h] \centering \centering \small \caption{The localization results on different traversals for over one year} \label{tab:full_diff_run_result} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline Date and Time & \begin{tabular}[c]{@{}c@{}}2014.06.26 \\ 09:53:12\end{tabular} & \begin{tabular}[c]{@{}c@{}}2014.07.14 \\ 15:16:36\end{tabular} & \begin{tabular}[c]{@{}c@{}}2015.02.03 \\ 08:45:10\end{tabular} & \begin{tabular}[c]{@{}c@{}}2015.04.24 \\ 08:15:07\end{tabular} & \begin{tabular}[c]{@{}c@{}}2015.06.09 \\ 15:06:29\end{tabular} & \begin{tabular}[c]{@{}c@{}}2015.07.14 \\ 16:17:39\end{tabular} & \begin{tabular}[c]{@{}c@{}}2015.08.13 \\ 16:02:58\end{tabular} \\ \hline Test submaps & 32 & 5 & 5 & 5 & 3 & 3 & 4 \\ \hline Test frames & 7095 & 866 & 666 & 898 & 548 & 624 & 536 \\ \hline Average T error & 1.41 & 1.34 & 1.88 & 1.67 & 1.68 & 1.44 & 1.71 \\ \hline Average R error & 6.40 & 6.62 & 7.33 & 7.24 & 7.24 & 7.17 & 7.44 \\ \hline \end{tabular} \end{table} \begin{figure} \centering \includegraphics[width=0.9\linewidth]{top_k_recall.png} \vspace{-0.3cm} \caption {The recall in top $K$.} \label {fig:top_k_recall} \end{figure} \subsection{Results} We test on one full traversal and the last unseen 10\% from another six traversals. As mentioned in \ref{subsec:testing_data}, we reconstruct the point cloud from GPS/INS data for each submap. We remove the redundant points from the ground plane. Next, we detect all the 3D keypoints and infer the corresponding descriptors from the point cloud descriptor network. All descriptors of the point cloud keypoints are stored in the database. \textbf{Network results} Given a query image, we extract the 2D SIFT keypoints and feed all the corresponding image patches into the image descriptor network to get the descriptors of the query image. For each image descriptor, we find its top $K$ nearest point descriptor from our database thus establishing the 2D-3D correspondences. In Fig.~\ref{fig:top_k_recall}, it shows the recall from top-1 to top-6. The selection of $K$ can largely effect the localization results. With a larger $K$, we have more point feature candidates for each image feature. Consequently, the RANSAC algorithm is more likely to find the correct match. On the other hand, a larger $K$ unfavorably increases the number of iterations of RANSAC exponentially. Considering the trade-off, we choose $K=5$ for our experiments. \textbf{Camera pose estimation} Finally, we solve the camera pose using the EPnP algorithm~\cite{epnp2009}. For all images in each test submap, the localization results are presented in Tab.~\ref{tab:full_diff_run_result}. The unit of T and R error are meter and degree. The first result (@2014.06.26) reports the localization results on most submaps of the full run, except for those with bad point cloud maps due to GPS inaccuracy. We denote it as $full\_test$. Others show the localization results on several different test submaps of each traversal across different times over one year. We denote them as $submap\_test$. The ratio of successfully localized frames are shown in Fig.~\ref{fig:curve}. A qualitative visualization of the localization is presented in Fig.\ref{fig:submap_visual}. In these results, the output feature dimension $D$ is set 128. \begin{figure} \centering \includegraphics[width=\linewidth]{comparison_result.pdf} \vspace{-0.3cm} \caption {The localization result on ORB-SLAM2. ORB-SLAM2 result: The red points are the failure position. An image of the failure case is shown.} \label {fig:comparison} \end{figure} \subsection{Evaluation and Comparison} For evaluation, we choose 8500 images collected in the overcast afternoon of July 14, 2015 with frame rate 16$Hz$. The path is around 3km with partial overlaps for loop closure detection, covering an spatial area of 845$m$ $\times$ 617$m$. We test the performance of two well-known algorithms for the task of visual localization, i.e., ORB-SLAM2~\cite{orb-slam2} (traditional method) and PoseNet~\cite{posenet} (deep learning method). We use ORB-SLAM2 algorithm to build the point cloud map of the testing area. However, it cannot succeed to build the whole area. Partial mapping result is shown in Fig.~\ref{fig:comparison}-(b). The localization result using the point cloud map by ORB-SLAM2 is shown in Fig.~\ref{fig:curve}-(a). The large error is due to the inaccurate mapping. We observe that the map is unreliable when the images are captured near trees or at the turns. We train the PoseNet for visual localization. However, the localization error is huge. We argue that the PoseNet is not suitable in a large area with training data captured by cameras on a running vehicle, which do not provide rich variance on the angle and viewpoint. Furthermore, this method can not generalize to the unseen test sequences. The existing algorithms fail to localize images within large-scale urban environments. In the next section, we show that our algorithm can successfully localize more than 40\% of the images throughout the whole testing area. \begin{table}[h] \centering \small \caption{Localization results on different output descriptor dimension $D$ on 2015-02-13, 09:16:26} \label{tab:diff_feature_dim} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|} \hline \begin{tabular}[c]{@{}c@{}}$D$\end{tabular} & \begin{tabular}[c]{@{}c@{}}Test \\ frames\end{tabular} & \begin{tabular}[c]{@{}c@{}}Success \\ frames\end{tabular} & \begin{tabular}[c]{@{}c@{}}Average \\ inliers\end{tabular} & \begin{tabular}[c]{@{}c@{}}Average\\ T error\end{tabular} & \begin{tabular}[c]{@{}c@{}}Average \\ R error\end{tabular} \\ \hline 64 & & 182 & 9 & 1.18 & 6.00 \\ \cline{1-1} \cline{3-6} 128 & 625 & 187 & 10 & 1.14 & 6.10 \\ \cline{1-1} \cline{3-6} 256 & & 179 & 9 & 0.99 & 5.31 \\ \hline \end{tabular} \end{table} \begin{figure} \centering \includegraphics[width=\linewidth]{curve.pdf} \vspace{-0.6cm} \caption {The curves of ratio of successfully localized frames w.r.t error thresholds. (a) full\_test: results of testing on the $full\_test$ with both our method and ORB-SLAM2~\cite{orb-slam2}. (b) submap\_test: average results of testing on the $submap\_test$.} \label {fig:curve} \end{figure} \subsection{Analysis and Discussion} \textbf{Generalization} As can be seen in the Tab.~\ref{tab:full_diff_run_result}, the results of $submap\_test$ are slightly worse than the result of $full\_test$ since the testing area of $submap\_test$ is totally unseen. However, the results in unseen area are close to the results of seen area. It shows the good generalization of our proposed network. \textbf{Output Feature Dimension $D$} We investigate the effect of the output feature dimension $D$. We test three submaps from another traversal on 2015, Feb 13, at 09:16:26. The localization result is shown in Tab.~\ref{tab:diff_feature_dim}. As we can see, the feature dimension $D$ of 128 successfully localize more images than the other two. A higher feature dimension can better represent the image and point cloud, but on the other hand, it may also cause over-fitting since our patch size and point cloud volume are small. Considering the trade-off between accuracy and inference efficiency, we choose $D$ as 128 in our experiments. From the localization results, we show that our proposed method is able to estimate camera pose from a point cloud based reference map directly through 2D image to 3D point cloud descriptor matching using deep learning. There are two main cases where the localization is likely to fail. (1) The scene contains many trees: 3D points from trees are quite likely to be detected as key points due to their strong gradients, i.e. irregularity in shape. However, the SIFT keypoints on trees do not contain discriminative information. Consequently, wrong matches arose from patches and point clouds on trees. (2) The scene is dominated by flat building walls: buildings are always full of texture seen from image, and thus create many meaningful patches. However, points on smooth wall are less likely to be detected as keypoints. This leads to low 3D keypoints and descriptor candidates, which decreases localization performance. \begin{figure} \centering \includegraphics[width=\linewidth]{path.pdf} \vspace{-0.6cm} \caption {A qualitative visualization of our camera pose estimation. Red camera: our estimated camera pose. Yellow camera: the ground-truth camera pose. Purple line: some predicted 2D-3D correspondences between 2D SIFT patches and 3D ISS volumes.} \label {fig:submap_visual} \end{figure} \section{Conclusion} \label{sec:conclusion} We presented a novel method for camera pose estimation given a 3D point cloud reference map of the outdoor environment. Instead of the association of local image descriptors to points in the reference map, we proposed to jointly learn the image and point cloud descriptors directly through our deep network model, thus obtaining the 2D-3D correspondences and estimating the camera pose with the EPnP algorithm. We demonstrated that our network is able to map cross-domain inputs (i.e. image and point cloud) to a discriminative descriptor space where their similarity / dis-similarity can be easily identified. Our method achieved considerable localization results with average translation and rotation errors of 1.41m and 6.40 degree on the standard Oxford RobotCar dataset. In future work, we aim at an end-to-end network for camera pose estimation by incorporating the hand-crafted key point selection and the RANSAC algorithm into the network. Furthermore, we will enforce temporal consistencies on multiple continuous frames to help improve the localization accuracy. \section{ACKNOWLEDGMENT} This research was supported in parts by the National Research Foundation (NRF) Singapore through the Singapore-MIT Alliance for Research and Technology's (FM IRG) research programme and a Singapore MOE Tier 1 grant R-252-000-637-112. We are grateful for the support. \clearpage \bibliographystyle{IEEEtran}	{ "redpajama_set_name": "RedPajamaArXiv" }	6,876
CHICAGO — The Chicago police superintendent recommends an officer be fired for shooting and killing a teenager and his neighbor. Chicago Police Supt. Eddie Johnson is accusing officer Robert Rialmo of violating multiple rules in charges he filed with the Police Board. Rialmo opened fire on 19-year-old Quintonio LeGrier and 55-year-old Bettie Jones as he responded to a disturbance call in December 2015. A jury awarded more than $1 million to LeGrier's family in a wrongful death lawsuit, but a judge reversed it, noting jurors also found Rialmo feared for his life. The City Council also agreed to pay the Jones family $16 million.	{ "redpajama_set_name": "RedPajamaC4" }	7,966
{"url":"https:\/\/db0nus869y26v.cloudfront.net\/en\/Frederick_Blackman","text":"Frederick Blackman\nBorn\nFrederick Frost Blackman\n\n25 July 1866\nDied25 January 1947 (aged\u00a080)\nResting placeParish of the Ascension Burial Ground\nNationalityBritish\nAlma\u00a0materUniversity of Cambridge\nSpouse(s)Elsie\nAwardsRoyal Medal\nScientific career\nFieldsBotany\n\nFrederick Frost Blackman FRS[1] (25 July 1866 \u2013 30 January 1947) was a British plant physiologist.[2]\n\nFrederick Blackman was born in Lambeth, London to a doctor. He studied medicine at St. Bartholomew's Hospital, graduating MA. In the subsequent years, he studied natural sciences at the University of Cambridge and was awarded DSc.\n\nHe conducted research on plant physiology, in particular photosynthesis, in Cambridge until his retirement in 1936. Gabrielle Matthaei was his assistant until 1905. He was elected in May 1906 a Fellow of the Royal Society,[1] his candidature citation reading \"Fellow of St John's College, Cambridge. Ex-Lecturer and now Reader in Botany in the University.\" He has made distinguished investigations in plant physiology. In 1921 he was awarded the Royal Medal and in 1923 delivered the Croonian lecture.\n\nHe was buried at the Parish of the Ascension Burial Ground in Cambridge, with his wife Elsie (1882 - 1967).\n\n## Blackman's law of limiting factors\n\nBlackman proposed the law of limiting factors in 1905. According to this law, when a process depends on a number of factors, its rate is limited by the pace of the slowest factor. Blackman's law is illustrated by ${\\displaystyle {\\ce {C O2))}$ concentration as a limiting factor in the rate of oxygen production in photosynthesis:\n\nSuppose a leaf is exposed to a certain light intensity which can use 5\u00a0mg. of ${\\displaystyle {\\ce {C O2))}$ per hour in photosynthesis. If only 1\u00a0mg. of ${\\displaystyle {\\ce {C O2))}$ enters the leaf in an hour, the rate of photosynthesis is limited due to ${\\displaystyle {\\ce {C O2))}$ factor. But as the concentration of the ${\\displaystyle {\\ce {C O2))}$ increases from 1 to 5\u00a0mg.\/hour the rate of photosynthesis is also increased.\n\n## Works\n\n\"Experimental researches in vegetable assimilation and respiration\":\n\nThe standard author abbreviation F.F.Blackman is used to indicate this person as the author when citing a botanical name.[3]","date":"2022-12-08 13:40:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 5, \"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\": 0.2958781123161316, \"perplexity\": 4063.2106352830356}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711336.41\/warc\/CC-MAIN-20221208114402-20221208144402-00111.warc.gz\"}"}	null	null
Root Cause of Asthma Attack, Respiratory Natural Treatment, Prevention There is overwhelming evidence that asthma is closely connected with lifestyle, and especially diet. This is strongly suggested by the fact that asthma is very common in Western industrialized countries but rare in poorer Asian and African countries. However, when these poorer populations exchange their traditional foods for Western food then asthma rates begin to climb. "Conventional medicine seeks to reduce this inflammation by using anti-inflammatory medications, or use pharmaceutical inhalants to temporarily dilate the passageway. There are two limitations to this approach. Firstly, every drug has the risk of side effects. Secondly, the drug approach does not offer a long term solution. In my opinion, there are natural approaches that work to treat the cause of asthma and offer long term success." Mark Stengler ND Also See: Vaccine, Immunization, and the Adverse Effects to Our Health Beware of Conventional Drugs used for Treating Asthma Food Allergies and Chemical Sensitivities Trigger Asthma Asthma and Allergies in Children - Dr. Doris Rapp Studies Confirm Unvaccinated Children Have Less Asthma Natural Remedies for the Relief of Asthma Dr. John Mills MD - "Conventional drugs used for treating asthma, particularly steroids, can impair immune function and lead to more serious health problems. "Doctors tell you that steroids (cortisone, prednisone) only cause side effects after many years. But new research shows that permanent damage is immediate and devastating. Studies show that steroids cause permanent, debilitating effects after a single dosage. Steroids are probably the most sleazy of modern day medications" John Mills, former professor of medicine at the University of California, San Francisco and chief of infectious diseases at San Francisco General Hospital. Dr. Lisa Landymore-Lim wrote all about this in her book " Poisonous Prescriptions"asking, 'Do Antibiotics Cause Asthma and Diabetes?' We are now even beginning to question the role of antibiotics as a cause of cancer since they do lead to pathogen overgrowth especially in the area of yeast and fungi. Chris Woollams writes, "It is estimated that 70 per cent of the British population have a yeast infection. The primary cause of this is our love of antibiotics. In the past few years heavy antibiotic use has been linked to the inflammatory bowel disorder, Crohn's disease, and to children developing allergies such as hayfever and asthma. An association between antibiotic exposure and asthmais accepted both by the medical profession and the Department of Social Security in the UK and the Health Department in Australia. However, general practitioners and the general public are either apparently unaware of this association, or have not drawn from it what I consider to be a logical conclusion The FDA has recently called into question the safety of a popular allergy and asthma drug, Singulair, after reports linked it to suicidal behavior and mood changes. Over the course of the past year, the drug's manufacturer, Merck, changed the product labeling 4 times to include various warnings of side effects, including tremors, anxiousness, depression, and suicidal behavior. The drug currently pulls in $4.3 billion dollars in sales annually. Asthma are abnormal inflammatory responses of the immune system to dust, pollen, a food or some other substance. Those that involve an antibody called immunoglobulin E (IgE) occur immediately or within an hour. Reactions may include coughing, sneezing, runny nose, hives, diarrhoea, facial swelling, shortness of breath, a swollen tongue, difficulty swallowing, lowered blood pressure, excessive perspiration, fainting, anaphylactic shock or even death. Kenneth A. Bock, MD - "Allergies are adverse immune reactions to every day substances that most people can tolerate. "Allergens," as they're called, can include inhalants such as pollens (e.g., ragweed), dust, cat hair or mold, Allergens also are found in foods including wheat, peanut or egg. These two types of allergens-food and inhalant-can join forces to increase your suffering. Study links asthma with indoor swimming pools - Researchers compared the rates of asthma in 13 and 14-year-old children in 21 European countries and the number of chlorinated swimming pools per 100,000 people. They found that after taking into account other factors such as climate, childhood asthma and wheeze increased by 2-3 percent for every indoor swimming pool, according the research published in the journal Occupational and Environmental Medicine. According to Dr. Bock MD, deadly modern toxins, nutritional deficiencies, metabolic imbalances, genetic vulnerabilities and assaults on the immune and gastrointestinal systems trigger most of the asthma and allergy symptoms. Asthma is notorious for being triggered by inhaled allergens but this does not mean that such triggers are a basic cause of asthma. The main problem is that there is an underlying chronic inflammation and oversensitivity of the airways which then reacts indiscriminately to a wide range of inhaled irritants. One common cause of a chronic inflammatory setting is the presence of hidden food allergies. 'Hidden' means that people are not aware of the allergy because usually the body does not react to an allergenic food to which it is exposed every day. Most asthmatics have been shown to have such hidden food allergies. The most common foods to which asthmatics react are cows' milk and cheese, gluten, eggs, nuts, and seafood. Drinking unpasteurized (raw) milk can protect children against asthma and hayfever, according to a study of nearly 15,000 children published in the May issue of Clinical and Experimental Allergy. The same applies to chemical sensitivities. The most common reactions for asthmatics are caused by sulfur dioxide and sulfites (codes 221 to 224), by monosodium glutamate or MSG (621 to 623), the yellow food dye tartrazine (code 102), and also salicylates such as aspirin. MSG is not necessarily declared as such on a food label, it may just be called hydrolyzed vegetable protein, vegetable or Thai seasoning or natural flavoring. All food additives are potentially dangerous and best avoided, except a few such as vitamin C or citric acid. Medical researchers think mainly in terms of inhaled allergens, but in response to a study showing that cases of wheezing disorders in preschool children in the UK doubled between 1990 and 1998 (The Lancet (Vol 357, p 1821), even they admit that there must be other unknown factors present to explain this dramatic increase. The immune system plays a central role in the health of the individual. It protects us from disease by recognizing and eliminating or removing foreign material from our bodies. Exposure to a variety of chemicals can effect this system adversely; putting us at risk for illness and disease. Allergy is caused by an oversensitive immune system, which leads to a misdirected immune response. The immune system normally protects the body against harmful substances, such as bacteria and viruses. In contrast, an allergic reaction is when the immune system reacts to substances (allergens) that are generally harmless and in most people do not cause an immune response. Asthma and Allergies in Children - Dr. Doris Rapp M.D Dr. Rapp is board certified in pediatrics and pediatric allergy. She was a Clinical Assistant Professor of Pediatrics at the State University of New York at Buffalo until January 1996. She practiced traditional allergy for 18 years and then, in 1975, began incorporating the principles of environmental medicine in her pediatric allergy practice. She is a certified specialist in Environmental Medicine. She has also produced numerous educational videos and audiotapes for the public, educators and physicians. These demonstrate the dramatic physical and behavioral changes in both children and adults that can be produced using a more precise method of allergy testing called Provocation/Neutralization. It has been estimated that about 90 million Americans suffer from food allergies especially gluten sensitivity. According to a new study in the Journal of Allergy and Clinical Immunology, there is a strong correlation between gluten intolerance and asthma- celiac disease confers a 1.6-fold increased risk of asthma. Wheat and gluten are often a problem for those with asthma and hay fever. If you have any problem with mucus, avoid all gluten products initially together with all foods containing lactose. About 1.5 million Americans are thought to suffer from it. In her breakthrough book, Is This Your Child? - Dr. Rapp identifies the major symptoms of potentially unrecognized allergies in children and adults, suggesting possible sensitivities to dust, mold, pollen, foods or chemicals. Allergies are much more than high fever, asthma and itchy skin. It is possible to identify allergies by simply looking at someone. At times it is surprisingly easy to find and eliminate the cause. The typical clues of allergies and environmental illness can include any combination of the following: Rubbing Nose Upwards; Eye Wrinkles; Dark Eye Circles Sudden Aggression; Scarlet Earlobes A Spacey Look; Extreme Activity Changes Wiggly Legs; Red Cheeks A Mottled Tongue. Cow's milk allergy is the most common food allergy in young children. Fortunately, most babies outgrow milk allergies by their second or third year. In the meantime, parents of babies with milk allergies can be reassured that - although there is no treatment that can cure milk allergies - symptoms can be controlled through a dairy-free diet. Today, supermarket milk is a brew of hormones, chemicals, DDT, fungicides, defoliants and radioactive fallout, produced by artificially inseminated creatures forced to stand around in muddy feed lots all day long. Some Other Contributing Factors A headline in New Scientist (19 July 2001) says "Margarine linked to dramatic asthma rise". This was a study of children in two rural Australian towns. Toddlers who consume large amounts of margarine and foods fried in vegetable oil may be twice as likely to develop asthma as others who eat less of these foods. This confirms the well known fact that (omega-6) linoleic acid increases inflammatory tendencies; this applies generally to polyunsaturated seed oils. Finnish researchers came to a similar conclusion. They found that children who eventually developed allergies ate less butter and more margarine compared with children who did not develop allergies. Of course, health authorities have been urging us for decades to consume more polyunsaturated fats and less saturated fat thereby increasing the severity of asthma. A study in the British Medical Journal (September 25, 1999; 319, 815-819) shows that giving babies other milk than breast milk before the age of four months greatly increases the risk of asthma and allergies. Children with low birth weight of less than 1 kg in the US had an asthma rate of 21% compared to 9% for children with higher birth weight. Another New Scientist headline reads "Weekly swimming linked to lung damage" (28 September 2001). This article reports that children who use chlorinated swimming pools every week get lung damage just like smokers. Also lifeguards who work in indoor pools have an increased incidence of asthma. A surprising Japanese study found that school children who ate more fish had also higher rates of asthma (Preventive Medicine February 2002;34:221-225). As we know from other studies that fish oil and even consumption of oily fish reduce inflammations and asthma, the conclusion is that in this case the high rates of mercury in Japanese coastal waters are the cause of such fish causing increased asthma. There are also various reports of vaccinations causing asthma. One such case is described under the title: "A case of asthma after vaccination against smallpox." (Ekbom, K. .Acta Med Scand Suppl. 1966; 464:170-1). Finish researchers found that mothers can prevent eczema and asthma in their children by taking Probiotics (acidophilus-bifido bacteria) while they are pregnant and breastfeeding. Babies normally get their mother's bacteria as they travel down the birth canal, but modern medicine is preventing this. Babies born by caesarian section are inoculated with hospital bacteria such as Streptococci and Clostridia. In an article entitled "The Dark Side of Immunizations?," Science News reviews new reports by researchers that show that vaccinated children have a higher incidence of asthma and diabetes than do unvaccinated children . Science News reports that a study by researchers at the Wellington School of Medicine in New Zealand found that unvaccinated New Zealand children report fewer cases of asthma than vaccinated children. Another study by New Zealand researchers published in the November 1997 Epidemiology analyzed the health of 1,265 people born in 1977. Of these, 23 didn't get any childhood vaccinations and none of them suffered childhood asthma. Among the 1,242 who got polio and DPT shots, more than 23 percent later had episodes of asthma. Science News adds that a 1994 survey of 446 British children, most of them eight years old, showed that 91 received no vaccinations in early childhood. Only one child out of 91 got asthma. About 11 percent of the other 355 children who had been vaccinated with pertussis and other vaccines had asthma. Several asthma and allergy researchers have found results similar to the earlier described relationship between immunization and IDDM. A group from New Zealand (Kemp, Pearce, Fitzharriset al.1997) found that asthma and allergies were more common in children that received pertussis vaccine than in those that were not immunized. Similar results have been suggested by others (Odent, Culpin & Kimmel, 1994) . More recently Dr. Julian Hopkin presented data at the British Thoracic Society meeting in 1997 which linked asthma to immunization. He has also published data that early immunization with BCG in Japan is associated with a decreased risk of IDDM (Shirakawa, Enomoto, Shimazuet al.1997). Athma, Allergies and Vaccines Several asthma and allergy researchers have found results similar to the earlier described relationship between immunization and IDDM. A group from New Zealand (Kemp, Pearce, Fitzharriset al.1997) found that asthma and allergies were more common in children that received pertussis vaccine than in those that were not immunized. Similar results have been suggested by others (Odent, Culpin and Kimmel, 1994) . More recently Dr. Julian Hopkin presented data at the British Thoracic Society meeting in 1997 which linked asthma to immunization. He has also published data that early immunization with BCG in Japan is associated with a decreased risk of IDDM (Shirakawa, Enomoto, Shimazuet al.1997). The culprit behind asthma and allergies: vaccination A summary of published medical research findings pertaining to the atopic effect of vaccination. Appears here as published in New Vegetarian and Natural Health Magazine, Winter 2000 issue. Can vaccines cause immune dysfunction resulting in allergies, asthma and anaphylaxis? Several asthma and allergy researchers have found results similar to the earlier described relationship between immunization and IDDM. A group from New Zealand (Kemp, Pearce, Fitzharriset al.1997) found that asthma and allergies were more common in children that received pertussis vaccine than in those that were not immunized. Similar results have been suggested by others (Odent, Culpin and Kimmel, 1994) . More recently Dr. Julian Hopkin presented data at the British Thoracic Society meeting in 1997 which linked asthma to immunization. He has also published data that early immunization with BCG in Japan is associated with a decreased risk of IDDM (Shirakawa, Enomoto, Shimazuet al.1997). The Institute of Medicine's Immunization Safety Review Committee held a public meeting in Seattle, Washington on November 12, 2001 to review the "Possible association between multiple immunizations in newborns and infants and immune system dysfunction." The Institute of Medicine Report stated... "The committee looked at five studies examining multiple vaccinations and their potential to cause allergic diseases, which reflect a hypersensitivity of the immune system to relatively harmless agents in the environment, like pollens, dust mites, insect venom, and specific foods. Some, but not all, of these studies suggested that certain vaccines increase the risk of developing allergic disorders. Methodological weaknesses and inconsistent findings among the studies, however, led the committee to conclude that there is inadequate evidence to either accept or reject a causal relationship between multiple immunizations and increased risk of allergic diseases, particularly asthma." Why the "surge" in anaphylactic children entering school a decade ago? These children were among the first to receive an additional vaccination, Hib meningitis. Is it possible that the Pertussis and Hib vaccine, both shown below to cause allergic responses, are creating a hypersensitive immune system in some children? Has any study looked into what happens to atopy incidence and IgE levels when 5 vaccines are given concurrently in infants? Possible association between multiple immunizations in newborns and infants and immune system dysfunction The exact numbers of children affected by anaphylaxis are difficult to pinpoint. A study in Arch Intern Med 2001 Jan 8;161(1):15-2, Anaphylaxis in the United States: an investigation into its epidemiology, concluded with "The occurrence of anaphylaxis in the US is not as rare as is generally believed. On the basis of our figures, the problem of anaphylaxis may, in fact, affect 1.21% (1.9 million) to 15.04% (40.9 million) of the US population."PMID 11146694 Pediatr Res 1987 Sep;22(3):262-7 Murine responses to immunization with pertussis toxin and bovine serum albumin: I. Mortality observed after bovine albumin challenge is due to an anaphylactic reaction..........the results of our experiments have established that the disease induced by co immunizing mice with Ptx and BSA is due to an immediate type hypersensitivity. Natural Remedies for the Relief of Asthma and Allergies Increasing glutathione levels in the lungs may be beneficial in lung disease. Although there are few studies using glutathione (GSH) enhancement therapies or nebulized (aerosolized and inhaled) glutathione in lung diseases, there is good data (mostly in animal models) suggesting increasing the potent antioxidant and free-radical scavenger glutathione would be beneficial in certain degenerative lung conditions associated with increased free radical load including asthma, cystic fibrosis, bronchiectasis and bronchiolitis obliterans. The Homeopathic Treatment of Asthma Homeopaths have a long history of successful allergy treatment, and they have made some important contributions to present understanding of allergies. Sir William Osler, considered the Father of Modern Medicine, was known to say, "Asthmatics don't die, they just 'pant into old age.'" However, new research on the homeopathic treatment of asthma that has been published in The Lancet (December 10, 1994) suggests that relief is in sight for asthma sufferers. Research conducted by professors at the University of Glasgow, Europe's largest medical school, indicates that those patients given an exceedingly small homeopathic doses of whatever substance to which they are most allergic can provide significant relief within the first week of treatment. The authors called this unique method of individualizing medicines "homeopathic immunotherapy." This double-blind, placebo-controlled trial showed that over 80% patients given a homeopathic remedy improved, while only 38% of patients given a placebo experienced a similar degree of relief. The patients were assessed by a homeopathic physician and a conventional physician. When the patients and doctors were asked if they felt the patient received the homeopathic medicine or the placebo, both the patients and the doctors tended to guess correctly. How Can Homeopathy Help Asthma? Like with Traditional Chinese Medicine, each individual is analyzed for their specific symptoms and an appropriate therapy is chosen, not for the disease, but for the person displaying signs of health out of balance. This is a very important distinction, and, very generally speaking, one of the main differences between conventional and "complementary" approaches to health care. Please refer to the essay on Homeopathy in the Introduction to Modalities section. American Association of Naturopathic Physicians "Homeopathic remedies are prescribed on an individual basis so that specific imbalances can be focused on and resolved. They work similar in action to a vaccine- in that a small amount of a substance stimulates the healing mechanisms of the immune system. The beauty of homeopathic treatment is that there are no worries about toxic side effects, relatively inexpensive, and works to repair the immune system and other imbalances in the body. Recent clinical studies have confirmed the effectiveness of homeopathic medicine in the treatment of asthma." Mark Stengler ND Treatment of Allergies with Homeopathy Dana Ullman, M.P.H. Solid research have proven the effectiveness of homeopathic medicines in hayfever. Dr. David Taylor-Reilly, a professor and homeopath at the University of Glasgow in Scotland, published an important study in the Lancet (October 18, 1986) which showed that homeopathically prepared doses of 12 common flowers were very effective in reducing hayfever symptoms when compared with patients given a placebo. This same researcher published in the Lancet (12-10-94) another high caliber study on the homeopathic treatment of asthma. This double-blind, placebo-controlled randomized trial performed conventional allergy testing to determine what substance asthma sufferers were most allergic. Then, half of the subjects were given a homeopathic preparation of this substance, while the other half were given a placebo. Those people given the homeopathic medicine experienced a very significant improvement in their symptoms of asthma. Asthma and Magnesium Deficiency A population-based clinical study of over 2,500 children aged 11 - 19 years found that low dietary magnesium intake may be associated with a risk of developing asthma. The same was found in a group of over 2,600 adults aged 18 - 70. In addition, some clinical studies suggest that intravenous and inhaled magnesium can help treat acute attacks of asthma in children aged 6 - 18 as well as adults. However, evidence from other clinical studies report that long-term oral magnesium supplementation does not lead to improved control in adult asthma. Glyconutrients - Phytoplankton Nutritional Support for Asthma Major universities are involved in the study of glyconutrients. Academic institutions throughout the United States now have laboratories focusing on glycobiology research, including the University of California, University of Oklahoma, San Francisco State University, University of Wisconsin, The Burnham Institute, The Scripps Research Institute, and the Massachusetts Institute of Technology. The focus of their research involves a wide range of possible uses including cancer treatments, anti-inflammatory and anti-fungal agents as well as treatments for influenza, asthma, and other infections. "Chelsea is 8 years old and has had severe food allergies and asthma her entire life. Last week she went to the doctor to refill her inhalant prescription for her asthma. He asked her to take deep breaths, and by the fifth one she needed the inhaler because she was wheezing so badly. The day after the visit, Chelsea began to take half an ounce of Marine Phytoplankton daily added to her juice. After a week, she was able to endure over an hour of exercise on a trampoline without the use of her asthma medication, which would not have been possible a week prior. She visited the doctor again today and he said he could not explain why she was no longer wheezing and has reduced her need for the inhaler from several times a day to two times a day. Because of Marine Phytoplankton, her moods have significantly improved and she seems to be feeling much happier!" V. G. Asthma and Emphysema population-based study of over 2,500 children aged 11 to 19 years found that low dietary magnesium intake may be associated with a risk of developing asthma. The same was found in a group of over 2,600 adults aged 18 to 70. In addition, some studies suggest that intravenous magnesium can help treat acute attacks of asthma in children aged 6 to 18 as well as adults. This may also be true for those with emphysema (also known as chronic obstructive pulmonary disease or COPD). A doctor will determine if this is necessary and appropriate in a hospital setting. Pollen is also a remedy for hay fever and allergies. However it must be taken at least six weeks before the season begins and then continued throughout the season if it going to work. Bee pollen has been effectively used down through the ages to rid allergy sufferers of their afflictions. Bee Pollen Treats Allergies This technique, called desensitization, was developed at St. Mary's Hospital Medical School in London soon after the turn of the century. The treatment consists of administering small amounts of the allergen to stimulate the patient's own immune system to produce antibodies that will eliminate the allergic reaction. It works rather like a vaccination does against childhood diseases. Desensitization is based on the premise that the administration of the allergen will cause the body to produce antibodies that will cancel out the effects of the offending substance when the patient is again exposed to it.	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	3,540
Angel Eyes 2001 2001102 min76 views A story about a seemingly unlikely couple who cross paths under life-threatening circumstances as though they are destined not only to meet but to save each other's lives. Not once, but twice. DramaRomanceThriller IMDb: 5.0201479 min71 views A single mom and her two boys help take care of their grandmother with mystical powers Paul Blart: Mall Cop Mild-mannered Paul Blart (Kevin James) has always had huge dreams of becoming a State Trooper. Until then, he patrols the local mall as a security guard. With his closely cropped moustache, personal transporter and gung-ho attitude, only Blart seems to take his job seriously. All that changes when a team of thugs raids the mall and takes hostages. Untrained, unarmed ... ActionAdventureComedyFamily Even though he's 35, Alex acts more like he's 13, spending his days as the world's oldest video game tester and his evenings developing the next big Xbox game. But he gets kicked out of his apartment and is forced to move in with his grandmother. IMDb: 7.71997139 min108 views New York City. Melvin Udall, a cranky, bigoted, obsessive-compulsive writer, finds his life turned upside down when neighboring gay artist Simon is hospitalized and his dog is entrusted to Melvin. In addition, Carol, the only waitress who will tolerate him, must leave work to care for her sick son, making it impossible for Melvin to eat breakfast.	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	5,960
Q: Kohana Change Password Error ORM Driver Okay so im attempt to allow users to change their passwords via an admin panel with this code but i keep getting ORM_Validation_Exception [ 0 ]: Failed to validate array ~ MODPATH/orm/classes/kohana/orm.php [ 1174 ] Anyone have any solutions? heres the controller function and model functions and view respectively Controller: public function action_edituser() { $this->auto_render = false; $edit = Model::factory('manageusers'); $post = $_POST; $edit->editUser($post); } Model: public function editUser($array) { try { $user = ORM::factory('user') ->where('id', '=', $array['id']) ->find() ->update_user($array, array( 'username', 'password', 'email', )); return $this->response('success', 'User Edited'); } catch (Database_Exception $e) { $this->response('error', $e->getMessage()); } } View: <div class="modal" id="editmodal" style="display: none;"> <div class="modal-header"> <a class="close" data-dismiss="modal">×</a> <h3>Edit User</h3> </div> <div class="modal-body"> <?php echo Form::open('admin/manageusers/edituser', array('data-function'=> '/admin/manageusers/updategrid', 'data-element' => 'tbody')); echo Form::label('username', 'Username'); echo Form::input('username', null, array('id' => 'username')); echo Form::label('email', 'Email'); echo Form::input('email', null, array('id' => 'email')); echo HTML::anchor('#', 'Change Password', array('id' => 'changepass', 'style' => 'display: block;')); echo '<div id="passwordchange" style="display: none;">'; echo Form::label('password', 'Password'); echo Form::password('password', null, array('id' => 'password')); echo Form::label('password_confirm', 'Confirm Password'); echo Form::password('password_confirm'); echo '</div>'; echo Form::hidden('id', null, array('id' => 'id')); ?> </div> <div class="modal-footer"> <a href="#" class="btn" data-dismiss="modal">Close</a> <input type="submit" class="btn btn-success" value="Save changes" /> </div> <?php echo Form::close(); ?> </div> A: The problem is that there are validation errors, meaning your input does not match the rules of your model. You need to catch ORM_Validation_Exceptions, and figure out what causes the error. public function editUser($array) { try { ... } catch (Database_Exception $e) { ... } catch (ORM_Validation_Exception $e) { var_dump($e->errors()); } }	{ "redpajama_set_name": "RedPajamaStackExchange" }	6,434
\section{Introduction} Quantum walks constitute a promising route to the development of algorithms for quantum information processing. Successful applications of quantum walks include a variety of search algorithms \cite{Shenvi2003,Childs2004a,Ambainis2004a,Childs2004b}, graph hitting problems \cite{Childs2002,Childs2003,Kempe03a}, Boolean function evaluation \cite{Farhi2008, Childs2009a}, among others. Quantum walks are in fact a universal paradigm for quantum computation \cite{Childs2009, Lovett2010}. The simplest and most dramatic improvement demonstrated by quantum walks, when compared with the corresponding classical random walks, is the hitting probability of a walk on certain graphs. Two graphs in particular have demonstrated an exponential separation between the quantum and classical walks: the glued binary trees \cite{Childs2002} and the hypercube \cite{ Kempe03a}. An example of the former is shown in Fig. \ref{fig1}; this graph is formed by connecting the leaves of two binary trees of depth $d$. Due to the graph's symmetry, the quantum walk can be restricted to a subspace of the total (exponentially large) Hilbert space, known as the ``column space'' (also shown in Fig. 1). It has been argued that this symmetry is the heart of the quantum speed-up \cite{Krovi2007}. A quantum walk algorithm on a slight modification of this graph, to be described below, was proven to have an exponential speedup over any classical algorithm \cite{Childs2003}. Keating {{\it et al.}} have argued that physical implementations of these walks will not be able to maintain this quantum speed-up for large graphs \cite{Keating07}. Indeed, one expects that physical systems will be subject to decoherence and disorder. While one can always argue that future quantum computers will be protected from these effects by error correction, it is more likely that near term demonstrations of quantum walks will use a physical network. In fact, many of the recent experiments on quantum walks involve an encoding of the degrees of freedom that is not, strictly speaking, computationally useful \cite{Kendon11}. Nevertheless, these experiments demonstrate the dynamical speed-up characteristic of quantum walks. Determining when disorder or decoherence will limit the observability of this speed-up is an important theoretical problem \cite{Kendon07}. This problem may already have been encountered by nature. Recent evidence has shown that photosynthetic complexes operate using a type of quantum walk \cite{AGuzik08}, where the interplay of decoherence with disorder plays a crucial role in their energy harvesting efficiency. This phenomenon, known as dephasing or environmentally assisted transport, has relations to earlier studies of phonon effects on electron transport in disordered solids \cite{Jonson79}. Understanding and potentially reverse-engineering this efficiency is a topic of great scientific and practical importance. As the complexes and potential technologies under study can be far from the thermodynamic limit, examining quantum transport on graphs of modest size may reveal new surprises. In this paper, we examine the continuous-time quantum walk on the glued trees graph with static disorder, corresponding to the usual Anderson model with disordered on-site energies \cite{Anderson58}. Previous authors \cite{Keating07} introduced disorder through an effective one-dimensional representation (the column space). Using the fact that all eigenstates are localized in one-dimension with arbitrary disorder, they concluded that Anderson localization would cause an exponential suppression of the hitting probability. From a physical perspective, however, this model has some flaws. First, their model introduces a highly correlated form of disorder, in that all of the on-site energies on a given column are the same Second, the glued trees graph is quite similar to a Cayley tree, whose dimensionality is formally infinite, as far as Anderson localization is concerned \cite{Evers2008}. The Cayley tree exhibits a localization transition at large disorder, and thus significant speed-up may still be possible for weak disorder. We have performed a numerical study of this problem for the full Hilbert space of the glued trees graph and modifications thereof. Analysis of the eigenvalues and eigenvectors agree with previous studies of localization the Cayley tree. We pay special attention to the less well studied regime of weak disorder, using dynamical simulations to find a type of quantum decay of the quantum walk from the column space. The resulting hitting probability is well simulated by a model with local decay from each column state. We further consider the crossover from wavelike to diffusive transport for intermediate levels of disorder. A scattering theory analysis of this regime indicates a transition from the quantum walk to a classical random walk. Our results augment the many results for one-dimensional quantum walks with disorder. These include theoretical studies for the continuous-time quantum walk \cite{Yin2008} and the discrete-time quantum walk \cite{Ahlbrecht2011}, and the many recent experiments on atoms \cite{Karski2009}, ions \cite{Schmitz2009,Zahringer2010}, and photons \cite{Broome2010, Schreiber2011}. Higher-dimensional quantum walks could be realized in these or other systems, such as networks of superconducting circuits \cite{Strauch2008, Chudzicki10}. In particular, our simulations show that the effects of disorder on quantum transport can be identified in systems of 10-100 lattice sites. This paper is organized as follows. In Section II we review the known results for Anderson localization on the Cayley tree, and in Section III numerically study the phase diagram for the two types of glued trees, and find evidence of the localization transition. In Section IV we study the time-dependence of the quantum walk with disorder, and introduce the local decay model for weak disorder. In Section V we consider a scattering theory model for transport through the glued trees graph, and find evidence for the transition to classical random walk. We conclude in Section VI, comparing our results with the hypercube and highlighting the major open questions. Results for the quantum scattering and (classical) diffusive transport through the glued trees are found in the Appendices. \begin{figure}[t] \begin{center} \includegraphics[width=3.5in]{fig1} \caption{Glued binary trees graph $G_4$, showing the reduction to the column space for both the classical (top) and quantum (bottom) random walks. } \label{fig1} \end{center} \end{figure} \section{Localization and Diffusion on the Binary Tree} Anderson localization is the phenomenon that, for a sufficiently large amount of disorder, the eigenstates of a quantum system become exponentially localized about the nodes in a graph \cite{Anderson58}. Localization transitions are a rich field of study \cite{Evers2008}, for which both symmetry and dimensionality play key roles. The quantum walk we consider corresponds to the simplest such model, a tight-binding Hamiltonian with random on-site energies \begin{equation} \mathcal{H} = - \gamma \sum_{\langle j,k \rangle} \left(c_j^{\dagger} c_k + c_k^{\dagger} c_j \right) + \sum_{j} \epsilon_j c_j^{\dagger} c_j, \label{andersonmodel} \end{equation} where $\gamma$ is a hopping rate, $\epsilon_j$ is a set of random variables, uniformly distributed in the range $[-W/2,W/2]$, and $c_j^{\dagger}$ is the creation operator for an excitation at site $j$, and the sum is over neighboring sites. This model, originally inspired by electron transport in solids, can be used for many physical systems, such as quantum state transfer of a single excitation on a qubit network \cite{Bose2003,Bose2008,Strauch2008} or exciton transfer in a photosynthetic complex \cite{AGuzik08}. Much is known about the Anderson model given by Eq. \ref{andersonmodel} \cite{Evers2008}. For example, for a one-dimensional infinite lattice, localization occurs for arbitrarily small amounts of disorder \cite{Borland63}. It was argued that this property was generic to quantum walks \cite{Keating07}. However, it is known that for systems with dimension greater than two, localization requires a sufficiently large amount of disorder, i.e. there is a localization transition as the disorder is increased \cite{Abrahams79}. At first glance, the glued trees graph of Fig. 1 might appear to be a subset of a two-dimensional system, and thus one might expect localization for even small amounts of disorder. In fact, the infinite binary tree, or more generally the infinite Cayley tree (also known as the Bethe lattice \cite{Ostilli12}) has been used as a model for an infinite-dimensional system. This has been studied extensively, and exhibits a localization transition found numerically \cite {Jonson79,Miller94,Monthus2009,Biroli2010} and by an analytical mean field calculation \cite{AbC73,AbC74}. For this system, there is a mobility edge in the energy spectrum, such that eigenstates inside the mobility edge are extended, while those outside are localized. For sufficiently large values of disorder, there is a localization transition beyond which all eigenstates are localized. For a binary tree (or Cayley tree with $K=1$), this transition occurs for $W_c \approx 17$ \cite{AbC74,Jonson79,Miller94,Monthus2009,Biroli2010}. The description of localization given above, in which the eigenstates of the system exhibit a transition from extended to localized states, can be called eigenstate localization. There are two other ways to identify localization that we will encounter in this paper. The first is dynamical localization, in which the spreading of a wavepacket shows a saturation as a function of time. The second is spectral localization, in which the eigenvalues of the system move from an absolutely continuous spectrum (corresponding to extended states) to a pure point spectrum (corresponding to localized states). These other indicators of localization, which have also been studied extensively, we now briefly summarize. The spectral properties of the system were studied numerically \cite{Sade2003,Biroli2010}, and found to exhibit a transition in agreement with the self-consistent approaches described above. These numerical studies are sensitive to the handling of the boundary of the tree \cite{Aizenman2006}, a fact to which we will return in Section III. While there are some subtleties regarding the the phase diagram at weak disorder \cite{Aizenman2011a,Aizenman2011b}, the existence of a localization transition for large disorder is well established. For small disorder, the existence of extended eigenstates has been proven \cite{Klein98}. It has also been proven that, for small disorder, states that are initially localized spread ballistically \cite{Klein95}. The ballistic spreading does not preclude classical behavior, however. The random walk on the Bethe lattice has been studied, and can be mapped onto an asymmetric random walk on a one-dimensional half-line \cite{Hughes82}, as indicated in Fig. \ref{fig1} ({\it e.g.} the left half). A classical walker is twice as likely to move right as left, and this asymmetry leads to the peculiar fact that the classical walk also spreads ballistically. Exact results and limits have been established for this and other properties of the classical walk \cite{Cassi89, Monthus96}. Note that the asymmetry is linked to the exponential growth of sites away from the origin, such that one often calls the Bethe lattice a graph of infinite dimensionality. \section{Quantum Walk Eigenstates} The continuous-time quantum walk \cite{Farhi98,Mulken2011} is precisely the quantum dynamics of a single particle moving on a graph. This is given by the Schr{\"o}dinger equation \begin{equation} i \frac{d \psi_j}{d t} = - \gamma \sum_{k} A_{jk} \psi_k, \end{equation} where $A_{jk}$ is the adjacency matrix for the graph and $\gamma$ is hopping rate. One could also use the Laplacian matrix, or introduce potentials to implement search algorithms \cite{Childs2004a,Childs2004b}, but here we consider the addition of static disorder, such that \begin{equation} i \frac{d \psi_j}{d t} = - \gamma \sum_{k} A_{jk} \psi_k + \epsilon_j \psi_j, \end{equation} where the on-site energies $\epsilon_j$ are i.i.d. random variables uniformly distributed in the range $[-W/2,W/2]$. In this section we consider the nature of the eigenstates of the Hamiltonian of the quantum walk with disorder, namely $\mathcal{H} \|\Psi\rangle = E \|\Psi\rangle$, with $\mathcal{H} = \mathcal{H}_0 + \mathcal{H}'$, with the unperturbed Hamiltonian given by \begin{equation} \mathcal{H}_0 = -\gamma \sum_{j,k=1}^{N_d} A_{jk} \|j\rangle \langle k\|, \end{equation} and diagonal static disorder \begin{equation} \mathcal{H}' = \sum_{j=1}^{N_d} \epsilon_j \|j\rangle \langle j\|, \label{hprime} \end{equation} where $N_d = 3 \times 2^d -2$ is the number of vertices for the glued trees graph $G_d$. We begin by analyzing $\mathcal{H}_0$, and follow by studying the eigenvalues and eigenstates of $\mathcal{H}$. \subsection{Eigenstates without disorder} As described above, there is a great deal of symmetry in the glued trees graph $G_d$, and a great deal of structure in the eigenstates and eigenvalues of the system. We begin by providing a notation for the graph. We consider a labeling along the ``columns'' and ``rows'' of the graph, of the form $(j,n)$, where $j = 0 \to 2d$ indicates the distance from the left root along the graph, and $n = 0 \to N_{j,d}-1$ is the location within a given column. Here $N_{j,d}$ is the number of sites in a given column $j$, given by $N_{j,d} = 2^j$ for $j \le d$, and $N_{j,d} = 2^{2d-j}$ for $j > d$. The coordinates $(j,n)$ can be combined into a single coordinate $v$ by the following rule \begin{equation} v = \left\{ \begin{array}{ll} 2^j + n & \mbox{for} \ 0 \le j \le d, \\ 3 \times 2^d - 2^{2d + 1 - j} + n & \mbox{for} \ d < j \le 2 d. \end{array} \right. \end{equation} Note that $v$ ranges from $1 \to 3 \times 2^d -2$. The adjacency matrix elements $A_{v,w}$ are equal to one if vertices $v$ and $w$ are connected, and zero otherwise. This can be given in terms of the coordinates $(j,n)$ by observing that $(j,n)$ is connected to \begin{equation} (j-1, \lfloor n/2 \rfloor), (j+1,2n), (j+1,2n+1) \ \mbox{for} \ j \le d, \end{equation} and \begin{equation} (j+1, \lfloor n/2 \rfloor), (j-1,2n), (j-1,2n+1) \ \mbox{for} \ j > d. \end{equation} What is most important is that a given vertex is symmetrically coupled to those sites one step further from the left root (or closer to the right root). This allows us to express the eigenstates of the system in terms of ``column-states'' that are equal superpositions of states on a given column $j$. This column-space reduction is well-known for its utility in the analysis of the quantum walk \cite{Childs2002,Childs2003, Krovi2007}. Letting $\|j,n\rangle$ denote the Hilbert-space vector associated with vertex $(j,n)$, we define the column-space vector $\|\mbox{col} \ j \rangle$ by \begin{equation} \|\mbox{col} \ j \rangle = \frac{1}{\sqrt{N_{j,d}}} \sum_{n=0}^{N_{j,d}-1} \|j,n\rangle. \end{equation} By the symmetry noted above, the Hamiltonian $\mathcal{H}_0 = -\gamma A$ acts on the column-space states as \begin{equation} \mathcal{H}_0 \|\mbox{col} \ j \rangle = -\sqrt{2} \gamma \| \mbox{col} \ j-1 \rangle - \sqrt{2} \gamma \| \mbox{col} \ j+1\rangle, \end{equation} where one factor of $\sqrt{2}$ is due to the number of connections to a neighboring column, and the other due to the normalization of the column states. Hence, we can reduce the dynamics to a quantum walk on a finite line, whose eigenstates are equally well-known \begin{equation} \ket{\Psi_{k,d}} = \frac{1}{\sqrt{d+1}}\sum_{j=0}^{2d} \,\sin\!\left(\frac{k\,(j+1)\,\pi}{2(d+1)}\right)\ket{ \mbox{col} \ j}, \label{ColSpEs} \end{equation} with energies \begin{equation} E_{k,d} = - 2\sqrt{2}\,\gamma \,\cos\!\left(\frac{k\,\pi}{2(d+1)}\right), \label{ColspEn} \end{equation} and $k = 1 \to 2d+1$. We have annotated the states by the depth of the graph, as there are in fact many more eigenstates for $G_d$, whose enumeration we now consider. The glued trees graph is self-similar, in that $G_d$ contains $2^\nu$ subgraphs, each equivalent to $G_{d-\nu}$. These subgraphs can be grouped into $2^{\nu-1}$ pairs, each pair formed by removing the roots of a larger graph equivalent to $G_{d-\nu+1}$. That is, we can repeatedly split the glued trees graph by removing the left and right roots, such that $G_d$ contains 2 copies of $G_{d-1}$, 4 copies of $G_{d-2}$, and so forth, formed by removing the roots until we are left with $2^d$ copies of $G_0$ (the isolated vertices at the center of the graph). Some representative subgraphs of $G_4$ are shown in Fig. \ref{subgraphs}. On their own, each subgraph would have eigenstates of the form of Eq. (\ref{ColSpEs}). To find how these subgraphs contribute to the spectrum of $G_d$, we observe that an equal but opposite-signed superposition over two paired subgraph eigenstates is an eigenstate of $G_d$. This occurs because the components with opposite phase, when acted upon by $\mathcal{H}_0$, will interfere destructively on the two nodes to which they are connected on the larger graph. By this pairing, we can thus construct the complete set of eigenstates for $G_d$. \begin{figure}[t] \begin{center} \includegraphics[width=3in]{fig2} \caption{(Color online) Glued binary trees graph $G_4$, with several highlighted subgraphs $G_3$ (top, in gray) and two copies of $G_2$ (middle, in red and bottom, in blue). } \label{subgraphs} \end{center} \end{figure} To see this more clearly, we define a set of ``sub-column'' states whose elements combine the paired subgraphs described above. These are given by \begin{equation} \|\mbox{scol} \ j; \alpha, \nu \rangle = \sum_{n= 2 \alpha N_{d-\nu}}^{ (2 \alpha +1) N_{d-\nu} - 1 } \frac{ \|j+\nu,n\rangle - \|j+\nu, n + N_{j,d-\nu} \rangle}{\sqrt{2 N_{j,d-\nu}}}, \end{equation} where $j = 0 \to 2(d-\nu)$ indicates the column in $G_{d-\nu}$, $\alpha = 0 \to 2^{\nu-1}-1$ labels the paired subgraphs, and $\nu = 1 \to d$ indicates the depth of the subgraphs' left root. Using these states, the remaining eigenstates of $G_d$ can be obtained by using the eigenstates $\|\Psi_{k,d-\nu}\rangle$ from Eq. (\ref{ColSpEs}) with $\|\mbox{col} \ j \rangle$ replaced by $\|\mbox{scol} \ j; \alpha, \nu\rangle$ and eigenvalues $E_{k,\,d-\nu}$ from Eq. (\ref{ColspEn}), where $k = 1 \to 2(d-\nu)+1$. Defining \begin{equation} \sigma_d = \{ E_{k,d}, k = 1 \to 2 d \}, \end{equation} the total spectrum can then be written as \begin{equation} \sigma = \sigma_d + \sum_{\nu=1}^{d} 2^{\nu-1} \sigma_{d-\nu}. \end{equation} This spectrum exhibits a very large degeneracy, especially for $E=0$, which has a multiplicity of $2^d$ (one from each copy of $\sigma_{d-\nu}$, and half from the $\sigma_0$). This large degeneracy (also observed in \cite{Aizenman2006}) occurs for trees that are both glued and unglued and can make numerical analysis of the disordered system problematic, as will be described below. \subsection{Eigenstates with disorder} The introduction of static disorder changes both the eigenvalues and eigenvectors of the system. For the Cayley tree, early studies used a self-consistent approach \cite{AbC74,Miller94} to find the localization transition and a phase diagram between extended and localized eigenstates. Recent numerical studies \cite{Monthus2009,Biroli2010} have confirmed these earlier results, which we now summarize. The eigenvalue spectrum has been studied for a single Cayley tree numerically by \cite{Sade2003} through the use of spectral statistics. A transition of the distribution of the energy level spacings from Wigner to Poisson (indicative of a localization transition) was observed when the tree was modified so that the leaves are randomly connected to each other. This random connection presumably reduces the prevalence of the zero eigenvalue described above, which would otherwise lead to a Poisson distribution for the spectral statistics \cite{Aizenman2006}. We performed a similar analysis for the glued trees graph, with the leaves connected as in Fig. \ref{fig1}, which we call a simple glued trees (SGT) graph, or interrupted by a random cycle (as in \cite{Childs2003}), which we call a modified glued trees (MGT) graph. An example of the MGT (with a regular connection \cite{Douglas09}) is shown in Fig. 9. A Wigner to Poisson transition was observed for the MGT, while the SGT exhibited a spectrum that was always far from Wigner. For the remainder of this section, our results were obtained using the MGT. The localization of the eigenstates and the localization transition can be found by studying a simple property of the eigenstates, namely the inverse participation ratio (IPR) $I_2$: \begin{equation} I_2(\psi) = \sum_{j} \|\psi_j\|^4, \end{equation} where we assume that the states are normalized with $I_1(\psi) = \sum_{j} \|\psi_j\|^2 = 1$. This quantity has the property that an eigenstate localized to a single site has $I_2(\psi) = 1$, while an eigenstate extended over $N$ sites has $I_2(\psi) = 1/N$. We further define an averaged value of this quantity \begin{equation} I_2(E) = \frac{1}{N(E;\Delta E)} \sum_{ \|\langle \psi \| \ham \| \psi \rangle - E\| < \Delta E} I_2(\psi), \end{equation} where $N(E;\Delta E)$ is the number of eigenstates found to have eigenvalue $E_j$ with in the range $E-\Delta E < E_j < E+\Delta E$, leaving the dependence on $\Delta E$ implicit.. By calculating $I_2(E)$ the eigenstates of the system as a function of energy and disorder, the phase diagram and localization transition can be visualized. For the SGT, we again encounter a difficulty in that there are a large number of states with an $I_2(\psi) = 1/2$, associated with the $2^{d-1}$ states in $\sigma_0$ (with $E=0$). However, by using the MGT, we can calculate meaningful results for $I_2(E)$ for various disorder strengths $W$ to obtain the diagram in \ref{IPRfig1}. Here we have let $d = 8$ and set $\Delta E = 0.15 \gamma$ and averaged over 500 realizations of $\mathcal{H}$. We observe a gradual movement of extended states (with small $I_2$) to localized states (with $I_2 > 1/2$) as disorder is increased. Also shown are the expected results for an infinite Bethe lattice, obtained using the self-consistent method of Miller and Derrida \cite{Miller94} To obtain an estimate of the localization transition, we fix our attention to states near $E=0$, and repeat the calculation of the averaged IPR for graphs of various sizes. As expected, $I_2(0)$ exhibits a small size-dependent value for $W=0$, which increases to approximately 0.5 for large $W$. At a certain value, the averaged IPRs for all of the graphs coalesce, from which we estimate $W_c \approx 17$, in agreement with recent results for the Cayley tree \cite{Monthus2009,Biroli2010}. \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig3} \caption{Inverse participation ratio $I_2(E)$ as a function of energy and disorder, for $d=8$. For each value of disorder the IPR was averaged over 500 realizations of $\mathcal{H}$ with $\Delta E = 0.15$ (with $\gamma = 1$). Also shown is the mobility edge from the self-consistent theory, predicting a localization transition with $W_c \approx 17$.} \label{IPRfig1} \end{center} \end{figure} \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig4} \caption{(Color online) Inverse participation ratio $I_2(E)$ at the band center ($E=0$) as a function of disorder for various depths. For each value of disorder the IPR was averaged over the middle 100 eigenvalues and a number of realizations given by $500, 250, 125, \mbox{and} \ 50$ for $d = 5, 6, 7, \mbox{and} \ 8$, respectively. The localization transition occurs when the curves collapse near $W \approx 17$.} \label{IPRfig2} \end{center} \end{figure} \section{Quantum Walk Dynamics} The dynamics of a quantum walk on the glued trees graph with disorder has been studied using the one-dimensional column-space model by Keating {{\it et al.}} \cite{Keating07}. In the previous section, however, we have seen that the eigenstates of the MGT undergo a localization transition at large disorder. This leaves open the possibility that a dynamical speed-up can be observed for small disorder. To explore this possibility, we again turn to numerical studies. There are three issues to be studied: first, what is the probability to hit the right root should the quantum walk begin at the left root? Second, how does this probability decay as a function of disorder and size, and by what mechanism? Finally, if the disorder quantum walk does not hit the right root, how far does it get? The first question can be answered by calculating the average of \begin{equation} p_{\subs{hit}}(t) = \|\langle \mbox{col} \ 2d \| e^{-i \mathcal{H} t} \|\mbox{col} \ 0\rangle\|^2 \end{equation} for instances of the SGT Hamiltonian with disorder strength $W$ and for various sizes $d$. This corresponds to the single-shot measurement procedure \cite{Kempe03a} for the quantum walk, and a representative set of $p_{\subs{hit}}(t)$ as a function of time and values of $W$ are shown in Fig. \ref{phit}. The first thing to observe is the oscillatory structure, due to the Bessel function structure for $p_{\subs{hit}}(t)$ \cite{Childs2002,Bose2003}, from which we can see that probability is maximized at the hitting time \begin{equation} t_{\subs{hit}} \approx \frac{1}{2\sqrt{2} \gamma} \left[ 2 d + 1 + 1.0188 \left(d + \frac{1}{2}\right)^{1/3} \right], \end{equation} with a value $p_d \sim d^{-2/3}$ for large $d$. Second, as disorder increases the maximum of the hitting probability is seen to decrease. This hitting probability can be approximated by \begin{equation} p_{\subs{hit}}(t_{\subs{hit}}) \approx p_d \exp \left[ - \frac{1}{16} (d - \frac{1}{2}) W^2 \right], \label{phitapprox} \end{equation} The exponential decay of $p_{\subs{hit}}/p_d$ is shown in the inset to Fig. \ref{phit}. \begin{figure}[t] \begin{center} \includegraphics[width=3.5in]{fig5} \caption{(Color online) Hitting probability $p_{\subs{hit}}(t)$ as a function of time for the quantum walk on the glued trees graph with $d=15$ and various values of disorder. The symbols were calculated using exact numerical simulations with (from top to bottom) $W = 0, 0.2, 0.4, 0.6, 0.8, 1.0, \mbox{and} \ 1.2$, with 10 realizations of $\mathcal{H}$ for each value of $W$. The solid curves were calculated using the locay decay model (see text). The inset shows the hitting probability $p{\subs{hit}}(t_{\subs{hit}})/p_d$ as a function of $d$ with symbols from numerical simulations and the curve from the approximation of Eq. (\ref{phitapprox}) (see text). } \label{phit} \end{center} \end{figure} By what mechanism does this decay occur? It is not associated with eigenstate localization, as the eigenstates of the system are well within the extended regime in the phase diagram of Fig. \ref{IPRfig1}. To understand this, we turn to another quantity of interest, the column-space probability \begin{equation} p_{\subs{col}}(t) = \sum_{j=0}^{2d} \; \left\| \,\braket{\mbox{col }j}{\psi(t)} \,\right\|^2. \end{equation} Quantum walks on the glued trees graph that are initially in a column space state $\| \mbox{col} \ j_0 \rangle$ will remain in the column space in the absence of disorder. Once static disorder is introduced, the eigenstates lying within the column space become coupled to the other eigenstates of $G_d$. These are associated with the subgraphs of $G_d$, and have zero amplitude on the graph's two roots. The resulting decay of $p_{\subs{col}}(t)$, shown in Fig. \ref{pcol} leads to the decay of $p_{\subs{hit}}(t)$. \begin{figure}[b] \begin{center} \includegraphics[width=3.5in]{fig6} \caption{(Color online) Column space probability $p_{\subs{col}}(t)$ as a function of time for the quantum walk on the glued trees graph with $d=15$ and various values of disorder. The symbols were calculated using exact numerical simulations with (from top to bottom) $W = 0, 0.2, 0.4, 0.6, 0.8, 1.0, \mbox{and} \ 1.2$, with 10 realizations of $\mathcal{H}$ for each value of $W$. The error bars indicate the standard deviation of each average. The solid curves were calculated using the local decay model (see text). } \label{pcol} \end{center} \end{figure} We can provide an analytical estimate of this decay through perturbation theory. Assuming the walk begins in a column state $\kc{j_0}$, taking the second order expansion of $p_{\subs{col}}(t)$ gives \begin{eqnarray} p_{\subs{col}}(t) &=& \sum_{j=0}^{2d} \; \left\| \,\bc{j} \exp(-i\,\ham\,t)\kc{j_0} \,\right\|^2 \nonumber \\ &\approx& \sum_{j=0}^{2d} \; \|\braket{\mbox{col $j$}}{\mbox{col $j_0$}}\|^2 + t^2 \|\bc{j}\ham\kc{j_0}\|^2 \nonumber \\ & & \qquad - t^2 \left( \braket{\mbox{col $j$}}{\mbox{col $j_0$}} \bc{j_0}\ham^2\kc{j} \right), \nonumber \\ \end{eqnarray} where the Hamiltonian is given by $\ham = \ham_0 + \ham'$. It is straightforward to show that $\ham_0$ contributes nothing to the quadratic term, and for $\ham'$ given by Eq. (\ref{hprime}) we find \begin{equation} p_{\subs{col}}(t) = 1 - \frac{t^2}{N_{j_0,d}^2} \left[ (N_{j_0,d} - 1) \sum_i {\epsilon_i}^2 - \sum_{i \ne j} \epsilon_i \, \epsilon_j \right]. \end{equation} Averaging $p_{\subs{col}}(t)$ over the disorder we find \begin{equation} \langle p_{col}(t) \rangle = 1 - \frac{1}{12} t^2 W^2 \left( 1 -\frac{1}{N_{j_0,d}} \right), \end{equation} where we have used $\langle \epsilon_i \epsilon_j \rangle = \frac{1}{12} W^2 \delta_{ij}$. This result for the short time decay applies to any graph for which the quantum walk can be projected onto a column space. On longer timescales, we conjecture that the decay will have an exponential character, while retaining the position dependence. Hence, extrapolating from the short time result, we postulate a model of ``local (exponential) decay'', in which the walk evolution is computed as in the ideal column space representation, but the probability at each site $j$ is allowed to decay: \begin{equation} p_{j}(t) = p_0 \exp\left[ - \frac{t}{12} \frac{W^2}{\gamma} \left(1-\frac{1}{N_{j,d}} \right) \right]. \label{loc_gamma} \end{equation} This is equivalent to applying the mapping \begin{equation} \ham \quad\mapsto \quad \hcol - \, i\Gamma/2, \label{ldm} \end{equation} where $\hcol$ is the projection of the graph Hamiltonian onto the column space, and $\Gamma$ is given by \begin{equation} \Gamma = \frac{1}{12} \frac{W^2}{\gamma}\, \sum_{j=0}^{2d}\left(1-\frac{1}{N_{j,d}} \right) \kc{j}\bc{j}, \end{equation} with $\gamma$ denoting the unit time hopping probability from $\mathcal{H}_0$. This expression for $\Gamma$ can be anticipated by applying Fermi's Golden Rule to $\mathcal{H}'$ in the eigenstate basis, but the presence of a large discrete component to the spectrum ($\sigma_0$) prevents us from making a precise derivation. Numerical evidence, however, suggests that this is in fact the correct mechanism. Along with the exact numerical results for $p_{\subs{hit}}$ and $p_{\subs{col}}$ in Figs. \ref{phit} and \ref{pcol}, we have included results from the local-decay model calculated in the (exponentially smaller) column-space representation for $\mathcal{H}_0$. We observe that the local decay model predictions (solid lines) agree well with the results of simulations (points), to within one standard deviation. Our model accurately reproduces many features of the exact simulations, such as the variation in the decay rate of $p_{\subs{col}}$ and the oscillations in $p_{\subs{hit}}$. Furthermore, if we use the column space probability to predict the hitting probability at the target node (opposite root), the agreement is quite good. This supports the claim that the disorder-induced reduction in quantum transport is primarily explained by decay from the column space. So far we have looked at the probability at the right root and the column space, but a more global characterization of the walk propagation can be found by analyzing the average depth reached by the quantum walk, \begin{equation} r(t) = \bra{\psi(t)} \;\hat{r}\; \ket{\psi(t)}, \end{equation} where the column position operator $\hat{r}$ is defined as \begin{equation} \hat{r} = \sum_{j=0}^{2d} \sum_{n=0}^{N_{j,d}-1} j \|j,n\rangle \langle j,n\| \end{equation} with the property $\hat{r} \kc{j} = j \kc{j}$. The value of $r(t)$ thus gives a snapshot of the expected position of the walk along the graph at any one time. This is displayed in Fig. \ref{rtime} as a function of time and disorder. In the absence of disorder, the average depth shows an oscillatory character consistent with ballistic propagation of the wavepacket and reflections at the two ends of the graph. Disorder-induced decay from the column space causes the amplitude of the oscillations to decay faster than in the ideal case, ultimately causing a ``damping'' of the oscillations. Therefore, the walk has a reduced probability of traversing the graph, and substantial probability is instead deposited in the center of the graph, where the concentration of nodes is highest. \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig7} \caption{(Color online) Average distance $r(t)$ for $d=15$ as a function of time and disorder $W$. For each $W$, the simulations were averaged over $10$ realizations of $\mathcal{H}$. The quantum oscillations decay in time for small disorder, with a critical damping near $W \approx 2$, indicating a type of quantum-to-classical transition.} \label{rtime} \end{center} \end{figure} In Fig. \ref{rmax} we plot the maximum value of $r(t)$ in the range $t < 3 t_{\subs{hit}}$ against the strength of disorder, illustrating the localization transition on the glued trees graph. We see that for small disorder ($W < 2$), the maximum value of $r$ is high, corresponding to the quantum walk hitting the right root, for which $r = 2d$. As disorder increases, the curve falls, and begins to level off at the graph center ($r \approx d$ for $2 < W < 4$). For larger amounts of disorder ($W > 4$), the curves continue to decrease, converging on a single value around $W = 16$, precisely when we expect all eigenstates to be localized. \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig8} \caption{(Color online) Maximum average distance $r(t)$ as a function of disorder $W$ for various depths. For each value of disorder the maximum of $r(t)$ was averaged over $1000, 100$, and $10$ realizations for $d=5, 10$, and $15$, respectively. For intermediate disorder ($2 < W < 4$), there is a quantum-to-classical transition, while for large disorder ($W>15$) a localization transition is observed.} \end{center} \label{rmax} \end{figure} This analysis of the quantum walk dynamics strongly suggests a type of quantum-to-classical or ``wavelike-to-diffusive'' crossover at weak disorder. Note that this not a ``ballistic-to-diffusive'' crossover, and does not conflict the ballistic spreading of wavepackets found by Klein \cite{Klein95} for the Bethe lattice at weak disorder, as classical diffusion also leads to ballistic spreading ($ r(t) \sim t$). It does, however, suggest that there may be a length scale (the mean-free-path) that limits the size of graphs for which a speedup could occur in the presence of disorder. That is, the exponential decay of the hitting probability (in $d$) seen in Fig. \ref{phit} may be interpreted as the classical probability for a walker to traverse $d$ sites given a mean-free-path of order $\ell \sim 1/W^2$. To understand this crossover in more detail we extend our analysis of the quantum walk to a transport model. \section{Quantum Walk Transport} To identify the quantum-to-classical transition, we consider the quantum transmission through the glued trees graph, subject to disorder. To transform the quantum walk into a transmission problem, we attach ``tails'' to the input and output nodes, and look at the transmission coefficient through the graph for a wavefunction of the form \begin{equation} \Psi(n) = \left\{ \begin{array}{ll} e^{i k n} + \mathcal{R} e^{-i k n} & \mbox{for} \ n < 0 \\ \mathcal{T} e^{i k n} & \mbox{for} \ n > 2d+1 \end{array} \right. \end{equation} This represents an ingoing wave that is reflected and transmitted through the graph, as illustrated in Fig. \ref{scatterfig}. This type of quantum walk was been used to develop a quantum algorithm for NAND-tree evaluation \cite{Farhi2008}, and has been generally analyzed in \cite{Varbonov2009}. We consider the transmission probability $T = \|\mathcal{T}\|^2$ as a function of depth and disorder. \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig9} \caption{Scattering approach to the quantum walk, in which an incident wavepacket is transmitted and reflected along the ``tails'' connected to the modified glued trees graph.} \label{scatterfig} \end{center} \end{figure} Using a standard analysis for transmission in tight-binding lattices \cite{Sadreev2003}, we find the transmission amplitude \begin{equation} \mathcal{T} = \langle \mbox{col} \ 0 \| \frac{2 i \sin k}{\tilde{\mathcal{H}} - 2 \cos k} \| \mbox{col} \ 2 d+1\rangle \end{equation} where \begin{equation} \tilde{\mathcal{H}} = \mathcal{H} + e^{i k} \left( \| \mbox{col} \ 0\rangle \langle \mbox{col} \ 0\| + \|\mbox{col} \ 2d+1 \rangle \langle \mbox{col} \ 2d+1\| \right) \end{equation} is an effective Hamiltonian for the MGT graph alone (note that this graph has $2d+1$ columns). This quantity can be calculated by diagonalizing the non-Hermitian Hamiltonian $\tilde{\mathcal{H}}$, and forming the appropriate matrix elements in $\mathcal{T}$. The resulting transmission probability $T = \|\mathcal{T}\|^2$ is shown as a function of momentum $k$ and disorder $W$ in Figs. \ref{transfig1} and \ref{transd6}, for depths $d=5$ and $d=6$, respectively. For small disorder, there is a size-dependent oscillatory structure as a function of $k$ due to resonances, much like those found in \cite{Sadreev2003} (and briefly described in Appendix A). These oscillations in $T$ disappear when the disorder strength $W \approx 2$, after which the transmission decays monotonically. Figure \ref{transmission_fit} shows $T$ for $k=\pi/2$ as a function of disorder for many graph sizes, which all exhibit the same behavior for $W > 2$, indicative of a transition in $T$. \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig10} \caption{Transmission probability $T$ as a function of momentum $k$ and disorder $W$ for a modified glued trees graph of depth $d=5$. For each value of momentum and disorder, the transmission was averaged over 250 realizations of $\tilde{\mathcal{H}}$. } \label{transfig1} \end{center} \end{figure} \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig11} \caption{Transmission probability $T$ as a function of momentum $k$ and disorder $W$ for a modified glued trees graph of depth $d=6$. For each value of momentum and disorder, the transmission was averaged over 100 realizations of $\tilde{\mathcal{H}}$.} \label{transd6} \end{center} \end{figure} The decay of the transmission probability with disorder can be understood using a classical model, described in the Appendix B. This model uses a diffusion constant proportional to the mean-free-path $\lambda \sim \ell \sim W^{-2}$ and leads to a classical transmission probability of the form \begin{equation} T_c = \frac{T_0}{1 + c (W/\gamma)^2}, \label{T_c} \end{equation} where the coefficients $T_0$ and $c$ presumably depend on the exact mapping of the disordered quantum walk to the diffusion equation, such as the method of \cite{Amir2009} or the results of \cite{Erdos2005}. This expression for Eq. (\ref{T_c}), fit using $T_0 = 0.8$ and $c = 0.2$, is shown in Fig. \ref{transmission_fit}. This agreement, for intermediate values of disorder, provides confirmation of the quantum-to-classical crossover observed in the dynamical studies of the previous section. \begin{figure} \begin{center} \includegraphics[width = 3.5in]{fig12} \caption{(Color online) Transmission probability $T$ for $k=\pi/2$ as function of disorder and for various depths. The various symbols are $d = 7$ (blue circles), $d=8$ (red upward triangles), $d=9$ (blue squares), and $d=10$ (red downward triangles), averaged over $500, 200, 100, \mbox{and} \ 50$ realizations of $\tilde{\mathcal{H}}$, respectively. The solid black curve is the transmission probability for the classical random walk $T_c$ (see text).} \label{transmission_fit} \end{center} \end{figure} \section{Conclusion} In this paper, we have carried out an investigation of the effects of diagonal disorder on quantum walks on the glued trees graph. While disorder does lead to localization in the strong disorder limit, we find the primary effect in the case of small disorder to be quantum decay out of the column space. The quantum decay can be accurately modeled in the column space by a non-unitary mapping that enforces position dependent decay of the probability. This local decay model is efficient to compute, owing to the exponential reduction in size of the representation, yet it allows prediction of the end-to-end hitting probability, and should be extendable to other graphs with similar symmetries. One such graph is the hypercube. This problem had been previously studied for quantum state transfer \cite{Strauch2008}, where the effects of off-diagonal disorder were emphasized. Numerical simulations for diagonal disorder, however, provide very similar results to those found in Sec. IV, with one important difference. The hitting time for the hypercube is independent of the dimension $d$ (which is analogous to the depth of the glued binary trees graph). The local decay model then predicts that the hitting probability should decay as $e^{-W^2/12}$, a result borne out by simulations. Thus, for the hypercube, quantum transport outperforms classical transport for small $W$. Such a result is also possible for the glued trees graph. The exponential suppression of the hitting probability occurs for the specific case of a state initially localized to the left root of the graph. By using a graph with ``tails'', the results of Sec. V show that appreciable transport is possible for $W<2$, provided there is no disorder in the tails and an appropriate initial state can be found. In addition, alternative measurement strategies \cite{Varbonov2008}, or the inclusion of traps \cite{Mulken2011} could provide opportunities for speedup. Exploring this possibility could provide additional context for understanding environmentally assisted quantum transport \cite{AGuzik08}. In summary, we have performed an analysis of the effect of static disorder on a quantum walk on the glued trees graph. For small disorder, we find that the dominant effect is a type of quantum decay, and not quantum localization. For intermediate disorder, there is a crossover to diffusive transport, while a localization transition is observed at large disorder, in agreement with Anderson localization on the Cayley tree. Our results suggest that intermediate disorder will inhibit any quantum speedup, but also that large speedups are possible for quantum walks on complex networks with small disorder. \begin{acknowledgments} We thank A. Aspuru-Guzik, S. M. Girvin, T. Kottos, and S. Lloyd for helpful discussions. FWS was supported by the Research Corporation for Science Advancement. \end{acknowledgments}	{ "redpajama_set_name": "RedPajamaArXiv" }	2,119
Q: arabic language not inserting in mysql database I am using MySQL database to store Arabic records, issue is when i am trying to insert records using MySQL insert query it will convert like this in database سعود instead of Arabic word. After google it, i found that need to set collation. So i set it to utf8_general_ci , utf8mb4_unicode_ci and many other. Also i tried this before insert query mysql_query("SET NAMES utf8;"); mysql_query("SET CHARACTER_SET utf8;"); But not worked for me, Still it inserting سعود in database. Where i am doing wrong? A: After establish your connection, write this MySQL lines. mysql_query("SET NAMES cp1256"); mysql_query("set characer set cp1256"); Can't insert arabic text into mysql database using mysql prompt dba.stackexchange.com A: Can u try with utf8_encode() at time of insertion and utf8_decode() function for view. A: For Arabic characters to store properly in database, you have to keep the Database charset to utf8, and database collation to utf8_general_ci. This way all your tables will have a charset of utf8. You can also set those charset for individual tables. The code mysql_query("SET NAMES utf8"); mysql_query("set characer set utf8"); wont work, because it sets the charset of the mysql database connection, not the database itself. A: First of all, usage of Mysql_* is deprecated, and no longer maintained. You should convert to MySQLi or PDO for security reasons. Here's a few pointers: * ALL attributes must be set to UTF-8 or UTF-8 w/o BOM (collation is NOT the same as charset) Save the document as UTF-8 or UTF-8 w/o BOM (If you're using Notepad++, it's Format -> Convert to UFT-8 or UTF-8 w/o BOM * Note that even though they both are UTF-8, they behave somewhat different The header in both PHP and HTML should be set to UTF-8 * HTML: <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> PHP: header('Content-Type: text/html; charset=utf-8'); Upon connecting to the database, set the charset to UTF-8 there as well, like this: mysql_set_charset("utf8"); (directly after connecting, this is for MySQL, there are similar ones for MySQLi_ and PDO). *Also make sure your database and tables are set to UTF-8, you can do that like this: ALTER DATABASE databasename CHARACTER SET utf8 COLLATE utf8_unicode_ci; ALTER TABLE tablename CONVERT TO CHARACTER SET utf8 COLLATE utf8_unicode_ci; Remember that EVERYTHING that can be set to a specific charset, needs to be set to UFT-8 (If you're using another charset, everything needs to be set to that), otherwise you will get some weird characters in your database. This will also work for English characters, so don't worry about that. Hope this helped!	{ "redpajama_set_name": "RedPajamaStackExchange" }	4,822
{"url":"https:\/\/mitgcm.readthedocs.io\/en\/latest\/overview\/kinematic_bound.html","text":"# 1.3.1. Kinematic Boundary conditions\u00b6\n\n## 1.3.1.1. Vertical\u00b6\n\nat fixed and moving $$r$$ surfaces we set (see Figure 1.18):\n\n(1.7)$\\dot{r}=0 \\text{ at } r=R_{\\rm fixed}(x, y)\\text{ (ocean bottom, top of the atmosphere)}$\n(1.8)$\\dot{r}=\\frac{Dr}{Dt} \\text{ at } r=R_{\\rm moving}(x, y)\\text{ (ocean surface, bottom of the atmosphere)}$\n\nHere\n\n$R_{\\rm moving}=R_{o} + \\eta$\n\nwhere $$R_{o}(x,y)$$ is the \u2018$$r-$$value\u2019 (height or pressure, depending on whether we are in the atmosphere or ocean) of the \u2018moving surface\u2019 in the resting fluid and $$\\eta$$ is the departure from $$R_{o}(x,y)$$ in the presence of motion.\n\n## 1.3.1.2. Horizontal\u00b6\n\n(1.9)$\\vec{\\mathbf{v}}\\cdot \\vec{\\mathbf{n}}=0$\n\nwhere $$\\vec{\\mathbf{n}}$$ is the normal to a solid boundary.","date":"2022-06-30 07:19:37","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\": 0.7185847163200378, \"perplexity\": 2313.4161303437622}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"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-2022-27\/segments\/1656103669266.42\/warc\/CC-MAIN-20220630062154-20220630092154-00061.warc.gz\"}"}	null	null
We create each color and cut with love. Each client's wants and needs are taken into careful consideration with the help of our stylist's expertise and creativity. Creating a look our client loves is what makes us happy!	{ "redpajama_set_name": "RedPajamaC4" }	5,953
{"url":"http:\/\/googology.wikia.com\/wiki\/Megoexplodaination","text":"## FANDOM\n\n10,828 Pages\n\nMegoexplodaination refers to the function $$\\{a,b,1,3,2,2\\}$$, using BEAF.[1] The term was coined by Aarex Tiaokhiao.\n\nMegoexplodainational growth rate is equivalent to $$f_{\\omega^3+\\omega^2+\\omega2+1}(n)$$ in the fast-growing hierarchy.","date":"2017-08-19 20:31:53","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\": 0.9292948842048645, \"perplexity\": 1719.8713257455433}, \"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-2017-34\/segments\/1502886105922.73\/warc\/CC-MAIN-20170819201404-20170819221404-00668.warc.gz\"}"}	null	null
<?php namespace Oro\Bundle\DataGridBundle\Extension\FieldAcl; use Oro\Bundle\DataGridBundle\Extension\Formatter\Property\PropertyInterface; use Symfony\Component\Config\Definition\Builder\TreeBuilder; use Symfony\Component\Config\Definition\ConfigurationInterface; class Configuration implements ConfigurationInterface { const FIELDS_ACL = '[fields_acl]'; const COLUMNS_PATH = '[fields_acl][columns]'; /** * {@inheritDoc} */ public function getConfigTreeBuilder() { $builder = new TreeBuilder('fields_acl'); $builder->getRootNode() ->children() ->arrayNode('columns') ->useAttributeAsKey('name') ->prototype('array') ->treatFalseLike([PropertyInterface::DISABLED_KEY => true]) ->treatTrueLike([PropertyInterface::DISABLED_KEY => false]) ->treatNullLike([PropertyInterface::DISABLED_KEY => false]) ->children() ->scalarNode(PropertyInterface::DATA_NAME_KEY)->end() ->booleanNode(PropertyInterface::DISABLED_KEY)->defaultFalse()->end() ->end() ->end() ->end() ->end(); return $builder; } }	{ "redpajama_set_name": "RedPajamaGithub" }	4,418
Q: Four Errors In C++ Code, I Can't Find The Solutions To I was looking at some code I'm working on, and there are 3-4 errors that I have tried for about a week to get rid of, and I just can't do it! I'm kind of new to programming, so if you could answer in stupid form, that would be great! Here is the code. #include <iostream> #include <string> using namespace std; int main() { string password; int choice; cout << "Command Line Multi-Tool" << endl; cout << endl; cout << "plase enter your password: " << endl; cin >> password; if (password == "creeper1") { cout << endl; cout << "Main Menu" << endl; cout << "1. Class Schedule" << endl; cout << "2. School Info" << endl; cout << "3. Exit" << endl; cin >> choice; } else { cout << "Incorrect, Access Denied" << endl; return(0); } } else (password == "admin1"){ cout << "/adminLogin=='1'/" << endl; cout << endl; cout << "Menu::Main" << endl; } return(0); } } And here is the error log. /Users/student/Documents/TO BE FILED/Tuesday/main.cpp:31:0 /Users/student/Documents/TO BE FILED/Tuesday/main.cpp:31: error: expected unqualified-id before 'else' /Users/student/Documents/TO BE FILED/Tuesday/main.cpp:36:0 /Users/student/Documents/TO BE FILED/Tuesday/main.cpp:36: error: expected unqualified-id before 'return' /Users/student/Documents/TO BE FILED/Tuesday/main.cpp:36:0 /Users/student/Documents/TO BE FILED/Tuesday/main.cpp:36: error: expected declaration before '}' token Again thanks so much! A: You have one if but 2 else brances in your code. Decide which one you want and lose the other one. Looking at the code you probably want if (password == "creeper1") { cout << endl; cout << "Main Menu" << endl; cout << "1. Class Schedule" << endl; cout << "2. School Info" << endl; cout << "3. Exit" << endl; cin >> choice; } else if (password == "admin1") // Your processing code here } else { cout << "Incorrect, Access Denied" << endl; return(0); } A: Unbalanced else,i.e. without a corresponding if. Perhaps you wanted something like: if (password == "creeper1") { } else if (password == "admin1") { } else { } A: You can't use an else if after an else, that block will never be executed.Also, the right syntax is return 0, not return(0). You also include extra braces, but if you indent the code in the right way you see when a block ends and starts, so you could rarely make mistakes like adding an extra brace: #include <iostream> #include <string> using namespace std; int main() { string password; int choice; cout << "Command Line Multi-Tool" << endl; cout << endl; cout << "plase enter your password: " << endl; cin >> password; if (password == "creeper1") { cout << endl; cout << "Main Menu" << endl; cout << "1. Class Schedule" << endl; cout << "2. School Info" << endl; cout << "3. Exit" << endl; cin >> choice; } else if(password == "admin1") { cout << "/adminLogin=='1'/" << endl; cout << endl; cout << "Menu::Main" << endl; } else { cout << "Incorrect, Access Denied" << endl; return 0; } return 0; } This is the code with all the syntax errors fixed.About semantics I don't know if it works, but now you can compile and execute it, to test the code. A: becasue else cannot comprise judgement the correct usage is like this: if(password == "creeper1"){ } else if(password == "admin1"){ } else{ }	{ "redpajama_set_name": "RedPajamaStackExchange" }	2,025
Special incentive rates for exhibitors who participate in the exhibition twice a year: in February and September. The list of extra furniture items which are ordered and paid for separately to "ExpoConsta" ZAO is available in the section Furniture on this page. Only equipped space is provided if you rent under 12 sq m.	{ "redpajama_set_name": "RedPajamaC4" }	4,661
\section{Introduction} The dynamics of polymer barrier crossing has attracted considerable attention in recent years, not only for basic understanding of numerous biological processes, but also for many practical applications. Biopolymers often need to surmount an entropic or energetic barrier for biological functions and in biotechnological applications, such as gene therapy, protein translocation, etc~\cite{Albert, Kasianowicz, SungPark}. Han \textit{et al.}~\cite{Han} studied the transport of double-stranded DNA (dsDNA) molecules through a fabricated channel of alternating thickness, driven by electric field. Within the relatively wide channels of one micron thick, the DNA are trapped in a coiled conformation. When an electric field drives the DNA into narrow channels, the chain, being severely confined and suffering a free energy barrier, becomes stretched. They found counterintuitively that the longer DNA molecules move faster than the shorter ones. The authors explained it is attributable to the fact that longer DNAs have larger contact area with the thin region and thus higher probability to escape the free energy barrier. The origin of the free energy barrier is the competition between the electric potential energy and the confinement entropy characteristic of the polymer. A similar effect has been reported for polymer translocation through long nanoscale channels \cite{Muthukumar03}. This transport process can be viewed as an extension of the famous Kramers problem~\cite{Kramers} of a Brownian particle crossing an activation energy barrier. As an interconnected many-particle system, the long chain polymer manifests cooperative dynamics in the presence of internal noise and external fields. Depending upon various length scales such as the chain contour length, radius of gyration, stretching and bending stiffnesses and other relevant parameters, many features emerge. For contour length much smaller than the width of a metastable potential as shown in Fig.~\ref{fig:potential}, Park and Sung~\cite{ParkSung} developed multi-dimension generalization of the Kramers rate. They found that the rate of polymer crossing the barrier is enhanced due to its flexibility. In particular, the flexibility enables the chain coiled in the well to stretch at the barrier top, which significantly lowers the activation energy, and enhances the barrier crossing rate. Also similar features were found in a double well potential, both for the flexible~\cite{SLee} and semiflexible ring polymers \cite{KLee}. In the opposite regime where the contour length is much larger than the potential width, Sebastian \textit{et al.}~\cite{Sebastian} suggested that the flexible polymers cross the barrier by excitation and motion of a kink and an anti-kink pair along the contour. Kraikivski \textit{et al.}~\cite{Kraikivski} studied a similar mechanism for semiflexible polymers. Because the kinks are a local property of the chain, activation energies are independent of the contour length and the barrier crossing rate monotonically decreases with the polymer contour length. For the intermediate cases of the semiflexible and self-avoiding polymers where the radii of gyration are comparable to the width of the potential barrier, crossing dynamics has not yet been studied either analytically or numerically. Computer simulation is a useful tool to study complex systems, complementing analytical methods. Since the rate of polymer crossing over the barrier can become very small for high barriers or long chains, conventional simulations using either molecular dynamics or Brownian dynamics are impractical because of long computing time required. To overcome this limitation, Voter~\cite{Voter} proposed the so-called hyperdynamics (HD) simulation method of accelerating the rate by raising the well bottom with an appropriate correction factor for the cases where the transition state theory (TST) is valid. Recently, two groups (\cite{Chen}, \cite{nummela}) introduced the path integral hyperdynamics (PIHD) method, for the Langevin dynamics of a single Brownian particle. Unlike HD, PIHD allows an {\it exact} correction of accelerated dynamics without the TST assumption. In the present work we extend the PIHD method to a chain of many-particle systems, namely, polymers, where the internal degrees of freedom contribute significantly to the total free energy of the system. We examine flexible, semiflexible, and self-avoiding polymers escaping a metastable well using the PIHD method. We consider the polymer's radius of gyration $R_g$ that is either smaller or comparable to the width of the potential well. We focus on the dependence of the escape rates on the contour length $L$ (or bead number $N$), chain stiffness, and excluded volume effect. We study how the chain's conformation and its transition affects the crossing rates. The outline of the paper is as follows. In the next section, we first recapitulate the path integral hyperdynamics method for the case of a single particle and then extend the method to polymer chains. In Sec.~\ref{sec:models}, we describe our simulation models and methods, whose results are discussed in Sec.~\ref{sec:results}. Finally, we conclude and summarize our results in Sec.~\ref{sec:conclusions}. \section{The path integral hyperdynamics method} \label{sec:PIHD} \subsection{The single-particle case} A Brownian particle moving subject to a potential $V(\vec{r})$ is described by the Langevin equation, \begin{eqnarray} m\ddot{\vec{r}}(t)+\zeta\ \dot{\vec{r}}(t)+ \nabla V(\vec{r}) =\vec{\xi }(t), \end{eqnarray} where $m$ is the mass, $\zeta$ is the friction coefficient and $\vec{\xi}$ is Gaussian white random force with $\langle \vec{\xi}(t)\rangle =0$ and $\langle \xi_{p}(t)\xi_{q}(0)\rangle=2\zeta k_{B} T\delta_{p,q}\delta (t) $. Here, $\langle \cdots \rangle$ denotes the ensemble average, $p$ and $q$ are cartesian coordinate indices, $k_{B}$ is the Boltzmann constant, and $T$ is the absolute temperature. Hereafter we drop the vector notation for brevity. The probability density of finding the particle at $r_{f}$ at time $t$ given an initial position $r_{0}$ at time $t_{0}$ is \begin{eqnarray} P(r_{0}, t_{0}\|r_{f}, t)= C\int[Dr]\exp \lbrace-\beta I[r(t)]\rbrace, \end{eqnarray} where $C$ is a normalization constant, $\beta = (k_{B}T)^{-1}$, and $[Dr]$ represents the path integral over all possible trajectories $r(t)$, and the effective action is given by \begin{eqnarray} I[r(t)]=\frac{1}{4\zeta}\int_{t_{0}}^{t} dt'[m \ddot{r }(t')+\zeta\ \dot{r}(t')+ \nabla V(r) ]^2. \end{eqnarray} In a system with an energy barrier much larger than the thermal energy $k_\mathrm{B}T$, the probability of the particle crossing the barrier is very small. To make such transition events more frequent, a bias potential $V_\mathrm{bias}(r)$ is added to the actual potential $V(r)$. In the boosted potential, $V_\mathrm{b}(r) \equiv V(r)+V_\mathrm{bias}(r)$, the particle obeys the Langevin equation \begin{eqnarray} m \ddot r(t)+\zeta\ \dot{r}(t)+ \nabla V_\mathrm{b}(r) =\xi (t). \end{eqnarray} Obviously, this leads to dynamics and transition probabilities that are different from the those given by Eqs. (1) and (2). However, as shown by \cite{Chen} and \cite{nummela}, it is possible to exactly recover the original probability density of Eq.~(2) from the biased dynamics by writing \begin{eqnarray} P(r_{0}, t_{0}\|r_{f}, t)= C\int[Dr]\exp (-\beta I_\mathrm{b}[r(t)])\exp (-\beta I_{\xi}[r(t)]), \end{eqnarray} where the effective action can now be written in two parts: the action in the boosted potential ($I_\mathrm{b}$) and the correction factor \begin{eqnarray} I_{\xi}(t)=\frac{1}{4\zeta} \int_{t_{0}}^{t}dt'\nabla V_\mathrm{bias}(r(t'))[\nabla V_\mathrm{bias}(r(t'))-2\xi(t')]. \end{eqnarray} To calculate the transition rates from the transition probability density, it is convenient to use the transition path sampling method~\cite{Chandler}. Here, the sampling is done over all dynamical paths $r(t)$ starting from a pre-transition state $1$ ($x_{1}<x_{c}$) at time $t=t_{0}$ to state $2$ located at $x_{2}>x_{c}$ at time $t$, where $x_{c}$ represents a certain transition state. The phenomenological rate constant $\mathcal{R}$ is then given by the relation $\mathcal{R}(t)=dP_{1\rightarrow 2}/dt= \mathcal{R}\exp(-t/t_{r})$, where $P_{1\rightarrow 2}(t)$ is the transition probability \begin{eqnarray} P_{1\rightarrow 2}(t)=\int_{x_{f}\geq x_{c}} dr_{f} \int_{x_{0}\leq x_{c}} dr_{0}P(r_{0})P(r_{0},t_{0}\|r_{f},t), \end{eqnarray} and $t_{r}$ is the transition time. The first and the second integrals are calculated over all accessible post-transition and all pre-transition states given by the initial quasiequilibrium distribution $P(r_{0})$ of the particle. For barriers larger than thermal energy, the rate reaches a well defined plateau after an initial transient period. Sampling all the events that have started with the initial configurations and crossed the transition state \textit{under the boosted potential,} $P_{1\rightarrow 2}(t)$ is given by \begin{eqnarray} P_{1\rightarrow 2}(t)=\frac{1}{n} \sum_{\xi} \exp(-\beta I_{\xi}(t)), \label{eq:crossing_prob} \end{eqnarray} where $n$ is number of all the paths and the summation is over the crossing paths only. \subsection{Extension to polymer chains with internal degrees of freedom} To date, the PIHD method has been demonstrated to work for systems where there are no internal degrees of freedom \cite{Chen,Khandkar09}. It is clear from the PIHD formalism that the external bias potential affects the evolution of these internal degrees of freedom in a many-particle system and its entropy changes. Thus, any transition rates that are influenced by an entropic contribution to the free energy barrier are not necessarily correctly described by the formalism. In this section we extend the PIHD method for systems with internal degrees of freedom. In particular, we consider the case of polymer chains consisting of $N$ beads and interacting with each other via a potential $U$. The position $r_{i}$ of the $i$th bead of the polymer can be described by the Langevin equation \begin{eqnarray} m \ddot r_{i}(t)+\zeta\ \dot{r_{i}}(t)+ \nabla_{i} \Phi(r_{i}) =\xi_{i}(t), \end{eqnarray} where $\Phi(r_{i})=V(r_{i})+U$. For the center-of-mass (CM) coordinate, $R(t)=(1/N)\sum_{i} r_{i}(t)$, which we regard as the reaction coordinate of the barrier crossing dynamics, we have \begin{eqnarray} M \ddot{R }(t)+N\zeta \dot{R}(t)+ \sum_{i}\nabla_{i} V(r_{i}) = \Xi(t), \end{eqnarray} where $M$ is total mass ($Nm$) and $\Xi(t)=\sum_{i}\xi_{i} (t)$ is a Gaussian random force that satisfies $\langle\Xi(t)\rangle=0$ and $ \langle\Xi_{p}(t)\Xi_{q}(0)\rangle=2N\zeta k_{B} T\delta_{p,q}\delta (t) $. Applying bias potentials to all beads, we have \begin{eqnarray} M \ddot{R }(t)+N\zeta \dot{R}(t)+ \sum_{i}\nabla_{i} V(r_{i})+\sum_{i}\nabla_{i} V_\mathrm{bias}(r_{i}) =\Xi (t). \end{eqnarray} We calculate the transition rate in the same way as in the single-particle case. However, now we define that the final state is reached whenever the polymer's center of mass has crossed the potential barrier. The probability is given by Eq.~(\ref{eq:crossing_prob}), except that now the correction factor is \begin{eqnarray} I_{\Xi}(t)=\frac{1}{4N\zeta} \int_{t_{0}}^{t}dt'\sum_{i}\nabla_{i}V_\mathrm{bias}(r_{i}(t')) [\sum_{i}\nabla_{i}V_\mathrm{bias}(r_{i}(t'))-2\Xi(t')]. \end{eqnarray} In the polymer barrier crossing problem, the choice of bias potential is a critical one. Because the polymer conformation has a significant effect on the crossing rate through its entropic contribution to the free energy barrier, in particular for long chains~\cite{ParkSung, SLee, KLee}, the bias potential should be chosen in such a way that it does not affect the polymer conformation. With a properly chosen bias, only the energetic part of the barrier is changed, while the entropic part remains unchanged. In the absence of bias potential, the conformational free energy of the chain with the CM position given as $R$ is \begin{eqnarray} F_{0}(R)=-k_{B}T\ln Z_{0}(R)\label{eq:free_energy}, \end{eqnarray} where % \begin{eqnarray} Z_{0}(R)=\int \prod_{i}dr_{i}\delta(\frac{1}{N}\sum_{i} r_{i}-R)\exp(-\beta (\sum_{i}V(r_{i})+U)). \end{eqnarray} Here $\int \prod_{i} dr_{i}$ stands for integration over all possible internal configurations of the polymer. When we add a bias potential $V_\mathrm{bias}(r_{i})$ to each polymer segment, the free energy is given as \begin{eqnarray}F(R)=-k_{B}T\ln Z(R), \end{eqnarray} where \begin{eqnarray}Z(R)=\int \prod_{i}dr_{i}\delta(\frac{1}{N}\sum_{i} r_{i}-R)\exp(-\beta (\sum_{i}V(r_{i})+U)-\beta \sum_{i}V_\mathrm{bias}(r_{i})). \end{eqnarray} For an arbitrary choice of bias $V_\mathrm{bias}(r_{i})$, $F(R)-F_{0}(R)\neq \sum_{i}V_\mathrm{bias}(r_{i})$, which means that adding the bias changes the entropic part of the activation barrier. However, if we choose a uniform force bias $V_\mathrm{bias}(r)=-br$, we have \begin{eqnarray} F(R)-F_{0}(R)= \sum_{i}V_\mathrm{bias}(r_{i}). \end{eqnarray} That is, this particular choice of the bias potential does not change the entropy or the polymer's conformation. By choosing a uniform force bias, the crossing probability is given by Eq.~(\ref{eq:crossing_prob}), with the correction factor $I_{\Xi}(t)$ determined by \begin{eqnarray} I_{\Xi}(t)=\frac{b}{4\zeta} \int_{t_{0}}^{t}dt'(bN+2\Xi(t')). \end{eqnarray} We note that in Ref.~\cite{nummela} the PIHD method was used to study pulled biopolymers without considering the influence of bias to the free energy. \section{Polymer models and simulation methods} \label{sec:models} In this paper, we consider three different polymer models. In all cases, the polymers are modeled as bead-spring chains, with the interaction potential $U$ chosen differently for the three models. In the simplest model of a flexible polymer, we include the stretching energy only, {\it i.e.}, $U=U_s$, where \begin{eqnarray} U_{s}=\sum_{i=1}^{N-1} \frac{1}{2} k\left(\|r_{i}-r_{i+1}\|-l_{0}\right)^2. \end{eqnarray} Here $l_{0}$ is the natural bond length and $k$ is the spring constant. In the second case, we also include the bending energy such that $U=U_{s}+U_{b}$. Here \begin{eqnarray} U_{b} = \sum_{i=2}^{N-1} \frac{1}{2} \kappa \left(r_{i-1}-2r_{i}+r_{i+1}\right)^2, \end{eqnarray} where $\kappa$ is the bending stiffness. As the third model, we consider the self-avoiding FENE chain, with the interaction potential given by the finite extension nonlinear elastic (FENE) potential and the short-range repulsive Lennard-Jones (LJ) potential: $U=U_\mathrm{F}+U_\mathrm{LJ}$. The FENE potential is defined between neighboring monomers as \begin{equation} U_\mathrm{F}=-\sum_{i=1}^{N-1}\frac{1}{2}k_\mathrm{F}R_0^2\ln\left(1-\left(r_{i}-r_{i+1}\right)^2/R_0^2\right), \end{equation} where $R_0$ is the maximum allowed separation between connected monomers. The LJ potential is defined as \begin{equation} U_\mathrm{LJ}=\sum_{i<j}^{N}4\epsilon\left[\left(\sigma/r_{ij}\right)^{12} -\left(\sigma/r_{ij} \right)^6\right], \end{equation} for $r_{ij}\leq 2^{1/6}\sigma$ and 0 for $r_{ij} > 2^{1/6}\sigma$. Here $r_{ij}=\|r_{i}-r_{j}\|$ is the separation of the monomers, $\sigma$ is the diameter of the monomer and $\epsilon$ is the depth of the potential. \begin{figure} \includegraphics[width=8.4cm] {potentialB.eps} \caption[0]{Actual potential $V(x)$ (solid line) and the boosted potential $V_{b}(x)$ (dashed line) we study.\label{fig:potential}} \end{figure} We consider a two-dimensional space with an external metastable potential that depends on $x$ as in Fig.~\ref{fig:potential} while $y$ represents the coordinate lateral to the potential force. The external potential is a one-dimensional piecewise-harmonic potential, defined by the equations \begin{eqnarray} V(x)&=& \frac{1}{2} \omega_{0}^2 x^2~\mathrm{for}~x< x_{0};\label{eq:extpot1}\\ V(x)&=& V_{B}-\frac{1}{2} \omega_{B}^2 (x-d_{0})^2~\mathrm{for}~x>x_{0},\label{eq:extpot2} \end{eqnarray} where $\omega_{0}^2$ and $\omega_{B}^2$ are the curvatures at the potential well and barrier, respectively, and $V_{B}$ is the potential barrier energy per segment. The position of the barrier is $d_{0}$ and $x_{0}$ is the crossover point between the piecewise harmonic potential. The position of each monomer as a function of time is given by the Langevin equation \begin{eqnarray} m \ddot r_{i}(t)+\zeta\ \dot{r_{i}}(t)+ \nabla_{i} \left[ \Phi(r_{i})-br_i \right] =\xi_{i}(t),\label{eq:single_langevin} \end{eqnarray} where $-br_i$ is the bias potential. The Langevin equations~(\ref{eq:single_langevin}) are integrated in time by the method described by Ermak and Buckholz~\cite{Ermak, Allen}. Initially, the system is equilibrated without the bias potential, i.e., $b=0$. Then the bias is switched on to expedite barrier crossing. The transition state of the chain crossing is $X=X_{c}$, where $X_{c}$ is CM position where CM free energy $F(X)$ is the maximum. Due to asymmetry of the potential $V(x)$, the $X_{c}$ is different from $d_{0}$~\cite{Sebastian06}, and found to be smaller than $d_{0}$. Therefore we can choose $X=d_{0}+2l_{0}$ ($X=d_{0}+2\sigma$ for the FENE chain) such that, whenever CM reach these positions the crossing occurs with very few chains recrossing back to the well. For the first two cases, we use the parameters $l_{0}$, $m_{0}$ and $(k_{B}T/1.2)$ to fix the length, mass and energy scales, respectively. We consider one bead as three bases of dsDNA, for which $l_{0}=1.02$ nm and $m_{0}\approx 1870~\mathrm{amu}$ and the characteristic time is $ t_{0}=\sqrt{\frac{1.2m_{0}l_{0}^2}{k_{B}T}}=30.9$ ps. For the FENE chain, we fix the parameters $\sigma$, $m$ and $\epsilon$. The time scale is then given by $t_\mathrm{LJ}=\left(m\sigma^2/\epsilon \right)^{1/2}$. The dimensionless parameters in all of our simulations are $m =1$, $\zeta=0.7$, $k_{B}T=1.2$ and, in addition for the FENE chain, $k_\mathrm{F}=15$ and $R_0=2$. For the spring constant $k$ and bending stiffness $\kappa$ we use various values, as indicated in Sec.~\ref{sec:results}. For a given $\kappa$ the persistence length is $2\kappa/k_{B}T$ in 2D. The dimensionless curvature of potential well and barrier are set to be $\frac{13}{9} \times 10^{-3}$ and $3.2\times 10^{-2}$, which correspond to $1.157\times 10^{-3} k_{B}T/{\rm nm}^2$ and $2.56\times 10^{-2} k_{B}T/{\rm nm}^2$, respectively. We choose $V_{B}=0.3 k_{B}T$, $x_{0}=12l_{0}$ ($x_0=12\sigma$ for the FENE chain) and $d_{0}=16l_{0}$ ($d_0=16\sigma$). \section{Results and Discussion} \label{sec:results} \subsection{Flexible polymer chains} For a flexible polymer, we first study its crossing rate as a function of polymer length $N$ for two values of spring constant $k=15,~30$. Figure~\ref{fig:rate_flexible} shows that the crossing rate decreases with $N$, but is still much larger than the Kramers rate~\cite{Kramers} $\mathcal{R}_{0}=\omega_{0}\omega_{B}/(2\pi\zeta)\exp(-\beta NV_{B})$ that the polymer has in the globular limit ($k\rightarrow \infty, l_{0}\rightarrow0$). The enhancement of the rates over this limit, larger for smaller $k$, is due to the chain flexibility that induces an entropy increase and conformational change in surpassing the potential barrier~\cite{ParkSung, SLee}. \begin{figure} \includegraphics[width=8.5cm] {flexible_rateC.eps} \caption[0]{Rate of flexible polymer barrier crossing for $k=15$, $k=30$ and the globular limit ($\mathcal{R}_0$) as a function of the chain length $N$. \label{fig:rate_flexible}} \end{figure} We find that as the chain becomes longer, its configuration at the barrier top (i.e., with the center of mass $X_{CM}$ placed at $d_{0}$) changes from a coiled state to a stretched state. Figure~\ref{fig:Radius} shows a dramatic increase of $R_{g,x}/R_{g,y}$ with $N$, where $R_{g,x}^2\equiv \langle \sum_{i}^{N} (x_{i}-X_{CM})^2 \rangle /N$ and $R_{g,y}^2\equiv \langle \sum_{i}^{N} (y_{i}-Y_{CM})^2 \rangle /N$ are the radii of gyration along $x$ and $y$ axes respectively at the barrier top, leading to further enhancement of the rate $\mathcal{R}$ over the globular limit $\mathcal{R}_{0}$. \begin{figure} \includegraphics[width=8.5cm] {RgB.eps} \caption[0]{The ratio of the radii of gyration along $x$ and $y$ axis for the flexible chain ($k=30$) at the barrier top.} \label{fig:Radius} \end{figure} Provided that the escape dynamics is much slower than the segmental relaxations, one can consider the dynamics as that of CM in a free energy $F(R)$ [Eq.~(\ref{eq:free_energy})]. We have obtained the free energy by averaging over all configurations at a fixed CM position. The activation free energy barrier height $F_{B}$, the free energy difference between well bottom and barrier top, are shown in Fig.~\ref{fig:height}. In contrast to Ref.~\cite{ParkSung}, where $F_{B}$ for a stretched conformation can be reduced dramatically by a factor proportional to $N^3$, it monotonically increases with $N$. This is consistent with an analytical study of Sebastian and Debnath done for a similar system~\cite{Sebastian06}. In the present case, a long chain stretches partly threading not only around the barrier top but also around the well bottom. This conformation does not significantly reduce $F_{B}$ and enhance the crossing rate as in Ref.~\cite{ParkSung}. \begin{figure} \includegraphics[width=8.5cm]{heightC.eps} \caption[0]{The free energy barrier height $F_{B}$ as a function of chain length for a flexible chain with $k=30$ and semiflexible chain with different bending stiffnesses $\kappa$. While $F_{B}$ increases monotonically for the flexible chain, it has a turnover behavior for the semiflexible chains. \label{fig:height}} \end{figure} On the other hand, for a case of $L(=Nl_{0})$ much larger than $d_{0}$, it is found that the activation energy is independent of polymer length, so that the barrier crossing rates are inversely proportional to the polymer length ($\sim1/N $)~\cite{Sebastian}. Because we are dealing with chains whose $R_{g}$ is comparable to $d_{0}$, our result (Fig.~\ref{fig:height}) lies between this prediction and that of Ref.~\cite{ParkSung}. Figure~\ref{fig:rate_spring48} shows the crossing rates as a function of spring constant $k$ for chain length $N=48$. Smaller values of $k$ yield larger $\mathcal{R}$. This is because for small $k$, the chain can more easily extend at the barrier top, further reducing the activation energy \cite{SLee}. \begin{figure} \includegraphics[width=8.5cm]{N48springC.eps} \caption[0]{The flexible chain crossing rates as a function of spring constant $k$ with $N=48$. \label{fig:rate_spring48}} \end{figure} \subsection{Semiflexible polymer chain} The bending stiffness characterizes prominently biopolymers; the persistence length is about $50$ nm for dsDNA~\cite{Taylor} and in the $10$ $\mu$m range for actin filaments~ \cite{kas1994direct, Howard}. We studied the semiflexible chain crossing over the barrier with a fixed stretching stiffness ($k=30$), and for three different values of bending stiffness $\kappa=1.2$, $6$ and $36$, corresponding to persistence lengths $2l_{0}$, $10l_{0}$ and $60l_{0}$, respectively. For different bending stiffnesses, Fig.~\ref{fig:height} and~\ref{fig:semiflexible_rates} show how the free energy barrier and the rates vary with chain length. For longer chains, the rates decrease toward constant values depending on the stiffness. \begin{figure} \includegraphics[width=8.5cm] {semiflexible_rateC.eps} \caption[0]{Semiflexible polymer barrier crossing rates with $k=30$. \label{fig:semiflexible_rates}} \end{figure} The crossing rate for $\kappa=1.2$ decreases monotonically with $N$, similarly to flexible chains, but has a value higher than those of flexible chains. For $\kappa=6$ and $\kappa=36$, the crossing rates decrease with length and become nearly constant as $N$ increase above 24 and 16. As shown in Fig.~\ref{fig:height}, the free energy barrier height increases with $N$ until it reaches $N_{c}$, beyond which it decreases. For $\kappa=6$ and $\kappa=36$, the turnover chain length $N_{c}$ are 32 and 24, which are close to the lengths where the rates approach the plateaus. The match is not exact because although the exponential of $F_B/k_BT$ dominates the crossing rate, the rate prefactor still has a weak dependence on $N$. The conformational behaviors that underlie these interesting results are shown in Figure~\ref{fig:equilibrium} and~\ref{fig:barrier} for $N=48$ and different bending stiffnesses. Within the potential well, stiffer chains with $\kappa \lesssim36$ become more extended and suffer a higher free energy. At the barrier top, the stiffer chain can more easily stretched along the $x$-axis and reduce the barrier height. Overall, stiffer chain can reduce the free energy barrier height ($F_{B}$) in a manner more pronounced for longer chain (see Figure~\ref{fig:height}). \begin{figure} \includegraphics[width=8.5cm]{equilibriumC.eps} \caption[0]{The chain configurations at the potential well for different bending stiffnesses and a fixed chain length $N=48$. The black markers correspond to $\kappa=1.2$, the red correspond to $\kappa=6$, the green correspond to $\kappa=36$, and the blue correspond to $\kappa=180$. \label{fig:equilibrium}} \end{figure} \begin{figure} \includegraphics[width=8.5cm]{barrierC.eps} \caption[0]{The chain configurations at the potential barrier (CM=16$l_{0}$) for different bending stiffnesses. \label{fig:barrier}} \end{figure} Finally, Figure~\ref{fig:kappa_rates} shows the dependence of the crossing rate on bending stiffness $\kappa$ for $N=48$. The rate increases with $\kappa$ up to $\kappa\approx36$ (or persistence length$\approx60l_{0}$), above which it decreases. For $\kappa>36$, the chain, tending to align along the $y$ axis at the well, does not suffer the elevated free energy. With the CM located at the barrier top, this long and stiff chain is over the barrier and well bottom, so that the free energy is raised. Thus at an optimal value of $\kappa\approx36$, the free energy barrier height $F_{B}$ tends to be minimum, thereby yielding the maximum rate. \begin{figure} \includegraphics[width=8.5cm] {N48kappaC.eps} \caption[0]{The crossing rates as a function of the bending stiffness for chain length $N=48$ and stretching stiffness $k=30$.\label{fig:kappa_rates}} \end{figure} \subsection{Self-avoiding polymer chain} To study the effect of excluded volume on the crossing rate for chains with $R_g$ comparable to the well size $d_0$, we also considered the self-avoiding FENE chain. In contrast to the flexible and semiflexible chains, we find that the crossing rate is a nonmonotonic function of chain length. The transition between a decreasing crossing rate and an increasing crossing rate occurs at $N\approx 32$, as shown in Figure~\ref{fig:selfavoiding_rate}. The minimum coincides with the maximum of the free energy barrier height $F_B$, as indicated in the inset of Figure~\ref{fig:selfavoiding_rate}. The behavior of $F_B$ and, consequently, the rate, is due to two competing factors. For short chains, the increase of $F_B$ is simply caused by the addition of particles to the chain. As the chain becomes longer, the excluded volume interactions cause the chain to swell, which forces the chain to occupy an increasingly wide region around both the well bottom and the barrier top. This causes the free energy barrier to decrease as a function of chain length after $N\approx 32$. The effect is similar to the case of semiflexible chain, but more pronounced. \begin{figure} \includegraphics[width=8.5cm] {selfavoiding_rateB.eps} \caption[0]{The crossing rates for the self-avoiding FENE chain as a function of chain length, with a minimum located at $N\approx 32$. Here, $k_\mathrm{F} = 15$, corresponding to the effective spring constant $k_\mathrm{eff}\approx 200$. The inset shows the free energy barrier height $F_B$ as a function of $N$. $F_B$ has a maximum at $N\approx 32$. \label{fig:selfavoiding_rate}} \end{figure} The self-avoiding chain also exhibits a modest transition from a coiled state to a stretched state as it surpasses the barrier. However, the transition is weaker than for the flexible chain due to the geometry of the external potential. At the well, long chains tend to orient in the $y$ axis due to the excluded volume interaction and the external potential. Consequently, the conformation at the barrier top is less stretched along the $x$ axis than for the flexible chain, which starts the crossing in an almost isotropic configuration. For the self-avoiding chain, the effect of decreased free energy barrier due to chain swelling gives the dominant contribution to the enhanced crossing rate. \section{Conclusions} \label{sec:conclusions} We have studied the dynamics of polymer escape from a metastable Kramers potential using path integral hyperdynamics. Because the escape can be an extremely slow process, conventional simulations can demand enormous computing time. To speed up simulations, we have extended the path integral hyperdynamics (PIHD) method for polymer chains. We found that a constant bias force applied on each segment speeds up the rate without changing the chain configuration, allowing evaluation of the rate with a proper correction factor. To demonstrate the efficiency of our simulation, we also computed conventional Langevin dynamics (LD) for some cases. For example, the semiflexible chain with $\kappa=1.2$ and $N=32$ case, PIHD takes about 1/30 of computing time via LD to get proper statistics. We considered flexible, semiflexible and self-avoiding chains with their radii of gyration $R_{g}$ smaller or comparable to the width of the potential well. We find that for a flexible chain, the crossing rate monotonically decreases with the chain length ($L$), but with much larger value compared to the chain's globular limit. For a semiflexible chain, the crossing rate becomes nearly constant as the chain becomes longer than a certain value. For a fixed chain length ($N=48$) the rate also shows nonmonotonic behavior as a function of the bending stiffness, exhibiting a maximum when the persistence lengh is about the contour length $L$. The enhancement of rates for the semiflexible chain over that of flexible chain can be interpreted as the reduction of the activation energy due to extended configuration of chain it takes in the potential well and at the barrier top. Finally, for a self-avoiding chain, the rate is a nonmonotonic function of chain length in contrast to the flexible and semiflexible chains. The reason for this is the excluded volume interaction, which causes the chain to swell and lowers the activation energy. For long chains this effect is more pronounced than for the flexible and semiflexbile chains, leading to rate increases with chain length. These findings suggest a possibility of polymer separation not only by its length but also by its bending stiffness. \begin{acknowledgments} This work was supported by NCRC at POSTECH and BK21 administrated by Korean Ministry of Education, Science and Technology, and in part by the Academy of Finland through its COMP Center of Excellence and TransPoly Consortium grants. T. Ikonen would also like to acknowledge the financial support of the Finnish Cultural Foundation and the Finnish Graduate School in Computational Sciences (FICS). \end{acknowledgments}	{ "redpajama_set_name": "RedPajamaArXiv" }	6,475
and mailchimp are considered best in class. The constant contact vs mailchimp email war is an epic one. There are numerous email marketing entities in the marketplace today all vying for the title to be the best. Furthermore, in this extremely competitive landscape, Constant Contact has been the go to option in the industry for more than 20 years. Similarly, Mailchimp has established their enviable reputation for the work they have done with some major corporations. As a result, both have dominated this industry for sometime. 2019 figures to be no different. In this constant contact vs mailchimp review, we'll focus on the five most important email marketing features. Also, we'll highlight which company it truly the premier option in the industry. In this mega-battle between constant contact vs mailchimp, we actually start out with arguably the least enticing subject of them all but definitely the most important when you factor in it's importance to an email marketing campaign. When we are dedicating both time and resources to our email marketing campaigns, it is important that our efforts aren't wasted with emails being sent to spam folders or even worse, not being delivered at all. As a consumer and business owner, we demand reliability and ultimately results in this particular area. Both Constant Contact and Mailchimp have some the highest scores in this area, but Constant Contact gets the win in this area with deliverability rates consistently reaching near 98%. Mailchimp wasn't too far behind with deliverability rates hovering around 96%. Constant Contact also has the edge with their enhanced spam tools and abuse detecting technology. Having access to relevant and timely data is one of the most important elements of any email marketing campaign. I am happy to report that both constant contact and mailchimp provide detailed reports and data for many specific areas including bounce rate, open rate and clickthrough rate. However, one company does it better and that company is Constant Contact. In this epic battle of mailchimp vs constant contact, sometimes it's the little differences that makes all of the difference. We were thoroughly impressed with the immense features and detailed reports that Constant Contact had to offer their clients. Additionally, we were impressed with the incorporation of Google Analytics which is a must have option. Setting up Google Analytics did take an additional step with Constant Contact but we appreciated that because it added an additional security element to the process. Additionally, with Constant Contact, you have the ability to monitor campaigns over time with their enhanced engagement report as well as track which campaigns got the most clicks in comparsion to others with their exclusive heat map tool. FREE FOR 60 DAYS. NO CONTRACT. NO CREDIT CARD! Constant Contact delivers big time in this area providing 60 days of service for FREE! No credit card required. No Contract required. We were extremely impressed with this offer and recommend to test drive their services today. Constant Contact has several automation capabilities including setting up specific triggers based on different online behavior. However, Mailchimp wins the battle of email marketing automation because of it's simplicity yet efficient, effective and fully integrated automation options. Moreover, options that you can expect to see with mailchimp is specific "set and forget" triggers tailored to the clicks that you are receiving. Additionally, mailchimp's automation toolkit includes automation of segmented contacts and a more diverse selection of social media and app integration. This provides ample ability to nurture and cultivate leads. Another nice feather in the cap for mailchimp is their inclusion of autoresponders. This is included with each and every plan that they offer. This is one area where Constant Contact is superior to all rivals including mailchimp. We did an extensive review of Constant Contact, specifically looking at their design elements, design capabilities as well as the implementation of their templates and we came away impressed. Mailchimp may offer more editing functionality, but Constant Contact offers more professional templates. Their templates are flexible, functional and completely mobile responsive. In an age where clients are relying more of their mobile devices, this is an vital asset. We appreciated the simplicity in utilizing the intuitive editing platform. For instance everything seemed seamless and easy to use. We did not have to spend hours and hours learning how to adjust certain elements as you would with other email marketing companies. The diverse selection of edible and beautiful templates are awesome. This is especially important when you are creating a campaign that is dedicated to a certain niche. For those of you who aren't familiar with the way that we conduct reviews with respect for pricing, we take into account not only the price aspect of it, but the overall value that you are getting for the dollar. In this constant contact vs mailchimp war, there are some unique differences in pricing based on what your needs are. We are going to break down the two main aspects of each of their pricing standards and determine a winner. Just to get started, pricing is based on email list size as well as enhanced features. The enhanced features package for Constant Contact is known as their Email Plus Plan. For Mailchimp, their enhanced features package is known as their "pro subscription" option. Mailchimp has a forever free plan which includes up to 2,000 email subscribers and up to 12,000 emails sent monthly with email/chat support only eligible in the first 30 days. In comparison, Constant Contact provides a 60 day free offer which includes up to 100 email contacts with full support for the duration of your trial. Click here to see more details about free offer. On the surface, the forever free plan seems like the more logical winner in this area, but the lack of support and premium features may not be what most companies and entrepreneurs may need so it will be important to see what features that your business will need in order to drive their respective email marketing campaign. From a Constant Contact perspective the free trial affords you the opportunity to test drive their services without any obligation and commitment, thus providing flexibility. For this aspect of pricing, mailchimp wins slightly based on volume, but it's close because of the lack of support. This is why we consider ourselves to be the best when it comes down to reviews. We look at every aspect of a company and provide the information that you need to make an informed decision. Many reviews were adamant about how mailchimp is a cheaper alternative to constant contact, but this is not the case. From a premium perspective, Constant Contact blows mailchimp out of the water. Lets use an email subscriber base of 5,000 emails for this example. The Email Plus option for Constant Contact for a subscriber base of 5,000 emails monthly will cost you $95 after the free trial on a month to month basis. If you opt for 12 month pricing, you'll get a 15% discount reducing the pricing even further to just $80.75 per month. Using the same subscriber base for Mailchimp, a 5,000 email subscriber base will cost you upwards of $50 per month plus and additional $199 per month charge for their "pro subscription" feature package which is identical to the Email Plus package that Constant Contact offers for as little as $80.75 with their 12 month subscription. So with Constant Contact you are saving $169.25 per month and over $2000 per year. I think that we have a definite winner in this area. Constant Contact Wins The Battle! Constant Contact proved to the more the dynamic option for your email marketing needs in 2019 and beyond. In a decisive victory, Constant Contact soared above mailchimp in design elements, reporting/analytics, deliverability and pricing. In an epic battle of constant contact vs mailchimp, mailchimp did hold their own in automation as well as demonstrating an impressive lineup of tools and functionality, but overall the lack of support with their free plan after the initial 30 days, lack of design elements including template selection as well as their extremely pricey professional packages is what ultimately gave Constant Contact the overall win. GET STARTED WITH CONSTANT CONTACT TODAY! This website contains reviews, opinions and information regarding products and services manufactured or provided by third parties. We are not responsible in any way for such products and services, and nothing contained here should be construed as a guarantee of the functionality, utility, safety or reliability of any product or services reviewed or discussed. Please follow the directions provided by the manufacturer or service provider when using any product or service reviewed or discussed on this website.	{ "redpajama_set_name": "RedPajamaC4" }	6,388
Remember Joey? He is the bait dog rescued with his friends and he is ready for his forever home! Joey is a lab/pit mix about 1 year old and 40lbs. He needs a calm house as he is still getting over his horrible past. He is very fearful of large dogs but would be ok with with a smaller, quiet dog. Although his bites have healed, he has itchy skin which will require frequent moisturizing baths. He only likes to go in the backyard to go potty and right back in the house to sit on the couch and sleep. Joey can not be crated as he will injure himself to get out. He has a little PTSD so he needs a patient person to help him feel safe and loved. He doesn't like to go on walks because most noises scare him. He is a couch potato and is so sweet and loving once he knows and trusts you. He will talk to you and howl as he tells you a story. Despite his past, he has done really well with house training. Joey gets scared and cries when left alone too long. He is very afraid of men so it will take a special person to give Joey the time and space he needs to trust the man in the house. Please let us know if you'd like to give Joey his loving, forever home he so deserves!	{ "redpajama_set_name": "RedPajamaC4" }	2,997
OSCR, western Montana's premier sufferfest, is officially back from the semi-dead this season with all its challenges intact. In response to popular demand (as well as a fair amount of whining), and to celebrate OSCR's 40th birthday, we are reverting back to the one loop, counter clockwise, Rice Ridge 50k course in 2022. Remember: the only constant in OSCR is change so don't expect to be coddled every year. We plan to have food and awards post-race, but the venue will be the trailhead to avoid the confined space of the community hall where we have traditionally shared viruses and bacteria in the past. The 50K Skate Ski Race starts at 9:30am - registration ends Jan. 28 at 5pm - $50 The 20K Nordic Ski Race starts at 10am - registration ends Jan. 28 at 5pm - $35 The 10K Nordic Ski Race starts at 10:10am - - registration ends Jan. 28 at 5pm - $25 The 5K Youth Ski Race starts at 10:20am - registration ends Jan. 28 at 5pm - $15 Links: http://seeleylakenordic.org. Address: Morrell Creek Road Seeley Lake, MT 59868	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	367
{"url":"https:\/\/zbmath.org\/?q=an:0674.35008","text":"# zbMATH \u2014 the first resource for mathematics\n\nRegularity for minima of functionals with p-growth. (English) Zbl\u00a00674.35008\nWe prove that the first derivatives of scalar minima of functionals of the type $I(u)=\\int f(x,u,\\nabla u)dx,$ are H\u00f6lder continuous. Here $$f(x,u,\\nabla u)\\approx \| \\nabla u\|^ p,$$ $$1<p<\\infty$$ and f is assumed H\u00f6lder continuous in x and u.\nWe give two applications. One to the regularity theory of quasiregular mappings and the other to quasilinear degenerate elliptic equations with p-growth.\nReviewer:\u00a0J.J.Manfredi\n\n##### MSC:\n 35B65 Smoothness and regularity of solutions to PDEs 35J70 Degenerate elliptic equations 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35A15 Variational methods applied to PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000)\nFull Text:\n##### References:\n [1] Bojarski, B; Iwaniec, T, Analytical foundations of the theory of quasiconformal mappings in ofr^{n}, Ann. acad. sci. fenn. ser. AI math., 8, 257-324, (1983) \u00b7 Zbl\u00a00548.30016 [2] Campanato, S, Equazione ellittiche del secondo ordine e spazi L2,\u03bb, Ann. mat. pura appl., 69, 321-380, (1965) \u00b7 Zbl\u00a00145.36603 [3] DiBenedetto, E, C1 + \u03b1 local regularity of weak solutions of degenerate elliptic equations, Nonlinear anal.: theory, methods appl., 7, 827-850, (1983) \u00b7 Zbl\u00a00539.35027 [4] Giaquinta, M, Multiple integrals in the calculus of variations and nonlinear elliptic systems, () \u00b7 Zbl\u00a01006.49030 [5] Giaquinta, M; Giusti, E, Differentiability of minima of non-differentiable functionals, Invent. math., 72, 285-298, (1983) \u00b7 Zbl\u00a00513.49003 [6] Giaquinta, M; Giusti, E, On the regularity of the minima of variational integrals, Acta math., 148, 31-46, (1982) \u00b7 Zbl\u00a00494.49031 [7] Granlund, S; Lindqvist, P; Martio, O, Conformally invariant variational integrals, Trans. amer. math. soc., 277, 43-73, (1983) \u00b7 Zbl\u00a00518.30024 [8] Giaquinta, M; Modica, G, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta math., 57, 55-99, (1986), Preprint \u00b7 Zbl\u00a00607.49003 [9] Gilbart, D; Trudinger, N.S, Elliptic partial differential equations of second order, (1977), Springer Berlin\/Heidelberg\/New York [10] Iwaniec, T, Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions, Dissertationes math., 198, (1982) \u00b7 Zbl\u00a00524.35019 [11] Lewis, J, Regularity of the derivatives of solutions to certain elliptic equations, Indiana univ. math. J., 32, 849-858, (1983) \u00b7 Zbl\u00a00554.35048 [12] Ladyzhenskaya, O.A; Ural\u2019tseva, N.N, Linear and quasilinear elliptic equations, (1968), Academic Press New York \u00b7 Zbl\u00a00164.13002 [13] Manfredi, J, Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations, () [14] Tolksdorff, P, Regularity for a more general class of quasilinear elliptic equations, J. differential equations, 51, 126-150, (1984) [15] Uhlenbeck, K, Regularity for a class of nonlinear elliptic systems, Acta math., 138, 219-240, (1977) \u00b7 Zbl\u00a00372.35030 [16] Ural\u2019tseva, N, Degenerate quasilinear elliptic systems, (), 184-222, [In Russian] \u00b7 Zbl\u00a00199.42502\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-09-18 07:43:49","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\": 0.6596338748931885, \"perplexity\": 2844.708130961692}, \"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-2021-39\/segments\/1631780056348.59\/warc\/CC-MAIN-20210918062845-20210918092845-00262.warc.gz\"}"}	null	null
\section{Introduction} Investigation of hadron properties is nowadays a hot topic, being subject of several studies within non-perturbative QCD approaches. The existence of possible exotic hadron states is the subject of both theoretical \cite{jaffe,Achasov:2008me,Narison:2008nj,Mathieu:2008me} and experimental activity \cite{Klempt:2007cp,Crede:2008vw,Batta:2009}. The $f_1(1285)$ meson, with quantum numbers $I^{G}(J^{PC})=0^+(1^{++})$, is usually considered a member of the axial vector meson nonet. However, it was argued that this resonance may have a rather large mixture of gluons in its wave function~\cite{fritzsch}. In Ref.~\cite{km}, the special role of the $f_1(1285)$ trajectory in spin-dependent high energy cross sections, based on the deep relation of the properties of this meson with the $U(1)_A$ gluon axial anomaly in QCD, was discussed. The alternative approach to treat $f_1(1285)$ as a dynamically generated resonance through the interaction of vector and pseudoscalar mesons in $K^\bar K$ channel was suggested in Ref.~\cite{oset}. It is worth to notice that the $f_1(1285)$ meson has a large branching ratio ($\sim$36\% \cite{PDG}) to $a_0(980)\pi$. Therefore the production of this meson gives also a unique opportunity to study the properties of $a_0(980)$ meson, a well known candidate for exotic four-quark state (see discussion and references in Refs.~\cite{jaffe} and \cite{Achasov:2008me}). Photoproduction is a very powerful tool to investigate meson properties. The experimental program of the CLAS collaboration at Jefferson Lab includes various photoproduction reactions with mesonic final states. In the light of the importance of the $f_1(1285)$ meson, we report on an estimate of the cross section for the reaction $\gamma p \to p f_1(1285)$ at photon energy of few GeV, within the Regge approach. \section{Estimate of the $f_1(1285)$ meson exclusive photoproduction cross section} The Regge model for meson photoproduction is being widely used to calculate cross sections for different reactions in the kinematic region $s>>-t$ (see Refs.~\cite{laget}, \cite{sib} and references therein). Within this approach, the main contribution to the $f_1(1285)$ photoproduction cross section at small momentum transfer ($-t\leq 1 $ GeV$^2$) and photon energy range of few GeV is related to the $t$-channel exchange of $\rho$ and $\omega$ meson trajectories (see Fig.~\ref{fig1}). \begin{figure}[h] \vspace{5.cm} \special{psfile=f1new1.eps angle=90 hscale=65 vscale=65 hoffset=600 voffset=-200} \caption[]{The dominant diagram in $f_1(1285)$, $\eta(1295)$, and $\eta(548)$ meson exclusive photoproduction off proton within the Regge model.} \label{fig1} \end{figure} Propagators of $\rho $ and $\omega $ mesons are given by~\cite{laget}: \begin{equation} P_V=(g^{\mu\nu}-\frac{k_V^\mu k_V^\nu}{m_V^2})\big( \frac{s}{s_0}\big)^{\alpha_V(t)-1}\frac{\pi\alpha_V^\prime}{sin(\pi\alpha_V(t))\Gamma(\alpha_V(t))}D_V(t), \end{equation} where $D_V(t)$ is the signature factor. It is well known that Regge trajectories can be either non-degenerate or degenerate~\cite{Collins}. In Ref.~\cite{Guidal:1997hy}, a detailed analysis of high energy pion photoproduction data within Regge approach was performed. It was argued that the $\rho$ meson trajectory should be degenerate in order to describe the ratio of cross sections of charged pions photoproduction. However, the $\omega$ trajectory should be non-degenerate to reproduce the dip around $t\approx -0.6$ GeV$^2$ observed in high energy exclusive $\pi^0$ photoproduction. Using the results of this study, we adopted the following expressions for the signature related factors: \begin{equation} D_\omega(t)=\frac{-1+exp(-i\pi\alpha_\omega(t))}{2}, \end{equation} \begin{equation} D_\rho(t)=exp(-i\pi\alpha_\rho(t)), \end{equation} where $k_V$ is the meson momentum, $s_0=1$ GeV, and ${\alpha^\prime}_V$ is the slope of the trajectory. For the $\rho$ trajectory the rotating phase was chosen~\cite{laget} \footnote{We have checked that the choice of a constant value for the phase of the $\rho$ trajectory leads to very similar numerical results for the differential photoproduction cross sections of the $f_1(1285)$, $\eta(1295)$, and $\eta(548)$ mesons.} to be: \begin{eqnarray} \alpha_\omega(t)=0.44+0.9t,\\ \alpha_\rho(t)=0.55+0.8t. \end{eqnarray} The vector meson (VM)-proton coupling is given by the standard expression: \begin{equation} {\cal L} =g_V\bar N\gamma_\mu NV_\mu+\frac{g^T_V}{2m_N}\bar N\sigma_{\mu\nu} NV_{\mu\nu}, \end{equation} where $V_{\mu\nu}=\partial_\mu V_\nu-\partial_\nu V_\mu $. The numerical value of the coupling constants were taken from Ref.~\cite{sib}: \begin{eqnarray} g_{\omega NN}=10.6,\\ g^T_\omega=0,\\ g_{\rho NN}=3.9,\\ g^T_\rho/g_\rho=6.1. \end{eqnarray} The $f_1$-VM-photon vertex has the following form \cite{km}: \begin{equation} V_{Vf_1\gamma}=g_{Vf_1\gamma}k_V^2\epsilon_{\mu\nu\alpha\beta}\xi^\beta\epsilon_V^\nu\epsilon_\gamma^\alpha q^\mu, \label{ver} \end{equation} where $q$ is photon momentum, $ \xi, \epsilon_V, $ and $ \epsilon_\gamma $ are the polarization vectors of the $f_1$, the vector meson and the photon, respectively. The coupling in Eq.\ref{ver} corresponds to the $AVV$ Lagrangian obtained in Ref.~\cite{messner} by using the hidden gauge approach. We should also mention that this coupling satisfies the Landau-Yang theorem \cite{yl} and leads to a vanishing value for an axial vector meson coupling to two massless vector particles (e.g. in the limit $k_V^2\rightarrow 0$). The coupling $g_{\rho f_1\gamma}=0.94$ GeV $^{-2}$ was fixed from the measured width: \begin{equation} \Gamma _{f_1\rightarrow\rho\gamma}=\frac{m_{\rho}^2(m_{f_1}^2+m_\rho^2)(m_{f_1}^2-m_\rho^2)^3}{96\pi m_{f_1}^5}g_{\rho f_1\gamma}^2, \end{equation} assuming $\Gamma _{f_1\rightarrow\rho\gamma}\simeq 1.3$ MeV \cite{PDG}. There is no experimental information about the $f_1\omega\gamma $ vertex. However, this coupling can be estimated within the quark model through the known value of $g_{\rho f_1\gamma}$ by using a quite general flavor decomposition of the $f_1$ wave function: \begin{equation} f_1=\alpha(\bar uu+\bar d d)+\beta\bar s s +\gamma{gg}, \end{equation} where the parameters $\beta$ and $\gamma$ describe a possible mixture of strange quark and gluons in $f_1$. In the $SU(2)_f$ limit we have: \begin{equation} g_{\omega f_1\gamma}\approx \frac{e_u+e_d}{e_u-e_d} g_{\rho f_1\gamma}, \label{ll} \end{equation} where $e_q$ is the electric charge of the correspondent quark. To further proceed in the calculation, we need an estimate of the two form factors in the $Vf_1\gamma$ and $VNN$ vertexes. In the spirit of vector meson dominance, we derived $F_{VNN}$ from the Bonn model \cite{sib}: \begin{equation} F_{VNN}(t)=\frac{\Lambda_1^2 - m_V^2}{\Lambda_1^2 - t}, \label{form1} \end{equation} with $\Lambda_1=1.5$ GeV, and we chose $F_{Vf_1\gamma}$ in the form: \begin{equation} F_{Vf_1\gamma}=\big (\frac{\Lambda_2^2-m_V^2}{\Lambda_2^2-t}\big)^2 \label{form2}, \end{equation} with $\Lambda_2=1.04 $ GeV. This form follows from the recent results of the $L3$ Collaboration about the $f_1(1285)$ production in $\gamma\gamma^$ interaction \cite{L3} and the assumption on the similarity of the heavy photon and vector meson vertexes. \begin{figure} \vspace{10.cm} \special{psfile=f1eta1295new.eps angle=0 hscale=45 vscale=45 hoffset=-20 voffset=0} \special{psfile=f1etanew.eps angle=0 hscale=45 vscale=45 hoffset=220 voffset=0} \caption[]{Estimated cross sections for the reactions: $\gamma p \to f_1(1285) p$ (left panel, solid line), $\gamma p \to \eta(1295) p$ (left panel, dotted line) and $\gamma p \to \eta(548) p$ (right panel) at $E_\gamma=3.1$ GeV. In the right panel, the SAPHIR data~\cite{SAPHIR} for the $\eta(548)$ cross section at $E_\gamma = 2.8-3$~GeV are also shown. } \label{fig2} \end{figure} The resulting differential cross section for $E_\gamma=3.1$ GeV is plotted as a solid line in Fig.~\ref{fig2}-left. As shown in the plot, the cross section has its maximum at $-t\sim0.5$ GeV. Both values, $E_\gamma$ and $-t$, are well matched to the kinematics accessible with the CLAS detector at JLab. The size of the cross section, $\sim$100 nb, makes the measurement feasible with such detector. \section{Estimate of the $\eta(1295)$ exclusive photoproduction cross section} Experimentally, the main problem to measure the exclusive $f_1(1285)$ photoproduction cross section comes from the background of the $\eta(1295)$ meson. Separation of the two mesons could be achievable performing a partial wave analysis that distinguishes the different quantum numbers. On the other hand, the small production cross section results in low statistics that limits the accuracy of these analysis. Another approaches is to extract the cross section through the inclusive measurement of the reaction $\gamma p \to p X$, where mesons are identified as peaks in the spectrum of proton missing mass. In the case of the $f_1(1285)$ and $\eta(1295)$ meson, their similar mass and a width, makes it practically impossible to distinguish them. The measurement of the $f_1(1285)$ cross section would be still possible if the $\eta(1295)$ cross section was found to be much lower, assuming therefore, that the observed signal is dominated by the $f_1(1285)$ meson production. In Regge theory, the $\eta(1295)$ meson exclusive photoproduction is described by the same diagram as for $f_1(1285)$ meson (see Fig.~\ref{fig1}). In the spirit of the vector meson dominance model, the $\eta(1295)$ meson photoproduction cross section can be estimated knowing the strength of the vertex $\eta(1295)\rightarrow \gamma \gamma$. Unfortunately, the are no direct measurements of this width. We then used an indirect way to estimate the width relying on the constituent quark model. Assuming that the $\eta(1475)$ and $\eta(1295)$ mesons are the first radial excitations of the $\eta^\prime(980)$ and $\eta(548)$, respectively, we correlated the existing data on $\eta(1475) \to \gamma \gamma$ width using the constituent quark model relationships for two-photon width of pseudoscalar meson~\cite{gerasimov}: \begin{equation} \Gamma (0^{-+}\rightarrow 2\gamma)\propto m_{0^{-+}}^3\sum_qe_q^2, \label{ga2} \end{equation} where $\sum_qe_q^2$ represents the sum of the electric charges of quarks in meson, and therefore: \begin{equation} \Gamma (\eta(1295)\rightarrow 2\gamma)\approx \frac{\Gamma( \eta(1475)\rightarrow 2\gamma)\Gamma (\eta\rightarrow 2\gamma)m^3_{\eta^\prime} m^3_{1295}}{\Gamma (\eta^\prime\rightarrow 2\gamma)m_\eta^3m^3_{1475}}\approx 0.091 KeV \label{ga3}, \end{equation} where we used $\Gamma( \eta(1475)\rightarrow 2\gamma )\approx 0.212 KeV$ \cite{PDG} with the assumption that $K\bar K\pi$ is the $\eta(1475)$ dominant decay mode. The $\eta$-VM-photon vertex has the following form: \begin{equation} V_{V\eta\gamma}=g_{V\eta\gamma}\epsilon_{\mu\nu\alpha\beta}\epsilon_V^\nu\epsilon_\gamma^\alpha q^\mu k^\nu_{\eta}, \label{ver2} \end{equation} where $k_\eta$ is the $\eta$ meson momentum. Using Eq.~(\ref{ver2}) and the vector meson dominance model, we obtained the following expression for the $\rho\eta\gamma$ coupling: \begin{equation} g^2_{\rho\eta(1295)\gamma}\approx \frac{96\pi m^3_\rho m^3_\eta \Gamma( \rho\rightarrow \eta\gamma)\Gamma (\eta(1295)\rightarrow 2\gamma)}{ (m^2_\rho-m^2_\eta)^3{m^3}_{\eta(1295)}\Gamma (\eta\rightarrow 2\gamma)}\approx 0.0032 GeV^{-2}, \label{ga5} \end{equation} where $\eta\equiv \eta(548)$. The $\eta(1295)$-$\omega$ coupling has been estimated with a similar equation as in Eq.~\ref{ll}: \begin{equation} g_{\omega \eta(1295)\gamma}\approx \frac{e_u+e_d}{e_u-e_d} g_{\rho \eta(1295)\gamma}. \label{ll2} \end{equation} The Brodsky-Lepage form of the transition form factor in the $V\eta\gamma$ vertex was used~\cite{brodsky}: \begin{equation} F_{V\eta\gamma}= \frac{1}{1-t/(8\pi^2f_{PS}^2)}, \label{form3} \end{equation} where $f_{PS}$ is the pseudoscalar decay constant, related to the $\Gamma_{\gamma\gamma}$ partial width by: \begin{equation} f_{PS}=\frac{\alpha}{\pi}\sqrt{\frac{M_{PS}^3}{64\pi\Gamma_{\gamma\gamma}}}. \end{equation} The resulting differential cross section for the reaction $\gamma p \to p \eta(1295)$ at $E_\gamma=3.1$ GeV is plotted as a dotted line in Fig.~\ref{fig2}-left. Integrating the two differential cross sections in the whole $-t$ range, we obtained: \begin{eqnarray} \sigma_{f_1(1285)}=68 nb,\\ \sigma_{\eta(1295)}=18 nb. \end{eqnarray} The $\eta(1295)$ cross section was found to be smaller (about 25\%) than the $f_1(1285)$ cross section suggesting that the extraction of the exclusive $f_1(1285)$ photoproduction is possible without complicated partial wave analysis in the JLab kinematics. As a check of the model, we repeated the same calculation to derive the differential cross section for the exclusive reaction $\gamma p \to \eta(548) p$. In this case the $\eta(548)$-VM-gamma coupling was obtained using the formula: \begin{equation} g_{V\eta\gamma}= \sqrt{\frac{96\pi M_V^3\Gamma_{V\rightarrow\eta\gamma}}{(m_V^2-{m_{\eta}}^2)^3}}. \end{equation} Results of the calculation are shown in Fig.~\ref{fig2}-right compared to the experimental points for the same reaction measured by the SAPHIR Collaboration \cite{SAPHIR} in a similar photon energy range ($E_\gamma = 2.8-3$~GeV). Data are described rather well by our model. The deviation between theory and experiment, especially at low $-t$, remains within a factor two and it is typical for such simple implementation of the Regge theory. More sophisticated models and, in particular a better treatment of the shape of the form factors, would result in a better agreement. \section{Summary} In summary, we calculated the cross section for the exclusive $f_1(1285)$ meson photoproduction off proton above the baryon resonance region. The chosen kinematics matches the typical Jefferson Lab, Hall-B, photon experiments. Using the Regge model with some phenomenological input for the unknown parameters, we obtained a cross section of the order of 100 nb. In the same framework, we also evaluated the cross section for the exclusive $\eta(1295)$ meson photoproduction, which represents the main background for the $f_1(1285)$ meson extraction from a photoproduction experiment. The small value we found for such background suggests that a measurement of the $f_1(1285)$ exclusive photoproduction cross section with a detector such as CLAS is possible. \section{Acknowledgment} The authors are grateful to S.Gerasimov for useful discussion. NK would like to thank INFN, Sezione di Genova, for the warm hospitality during this work.	{ "redpajama_set_name": "RedPajamaArXiv" }	6,587
package org.hl7.fhir.instance.model; // Generated on Wed, Feb 18, 2015 12:09-0500 for FHIR v0.4.0 import java.util.; import org.hl7.fhir.utilities.Utilities; import org.hl7.fhir.instance.model.annotations.ResourceDef; import org.hl7.fhir.instance.model.annotations.SearchParamDefinition; import org.hl7.fhir.instance.model.annotations.Block; import org.hl7.fhir.instance.model.annotations.Child; import org.hl7.fhir.instance.model.annotations.Description; /* * A set of DICOM SOP Instances of a patient, selected for some application purpose, e.g., quality assurance, teaching, conference, consulting, etc. Objects selected can be from different studies, but must be of the same patient. / @ResourceDef(name="ImagingObjectSelection", profile="http://hl7.org/fhir/Profile/ImagingObjectSelection") public class ImagingObjectSelection extends DomainResource { @Block() public static class StudyComponent extends BackboneElement { /* * Study instance uid of the SOP instances in the selection. / @Child(name="uid", type={OidType.class}, order=1, min=1, max=1) @Description(shortDefinition="Study instance uid", formalDefinition="Study instance uid of the SOP instances in the selection." ) protected OidType uid; /* * WADO-RS URL to retrieve the study. Note that this URL retrieves all SOP instances of the study, not only those in the selection. / @Child(name="url", type={UriType.class}, order=2, min=0, max=1) @Description(shortDefinition="Retrieve URL", formalDefinition="WADO-RS URL to retrieve the study. Note that this URL retrieves all SOP instances of the study, not only those in the selection." ) protected UriType url; /* * Series indetity and locating information of the DICOM SOP instances in the selection. / @Child(name="series", type={}, order=3, min=1, max=Child.MAX_UNLIMITED) @Description(shortDefinition="Series identity of the selected instances", formalDefinition="Series indetity and locating information of the DICOM SOP instances in the selection." ) protected List<SeriesComponent> series; private static final long serialVersionUID = -1632673574L; public StudyComponent() { super(); } public StudyComponent(OidType uid) { super(); this.uid = uid; } /* * @return {@link #uid} (Study instance uid of the SOP instances in the selection.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public OidType getUidElement() { if (this.uid == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create StudyComponent.uid"); else if (Configuration.doAutoCreate()) this.uid = new OidType(); // bb return this.uid; } public boolean hasUidElement() { return this.uid != null && !this.uid.isEmpty(); } public boolean hasUid() { return this.uid != null && !this.uid.isEmpty(); } /* * @param value {@link #uid} (Study instance uid of the SOP instances in the selection.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public StudyComponent setUidElement(OidType value) { this.uid = value; return this; } /* * @return Study instance uid of the SOP instances in the selection. / public String getUid() { return this.uid == null ? null : this.uid.getValue(); } /* * @param value Study instance uid of the SOP instances in the selection. / public StudyComponent setUid(String value) { if (this.uid == null) this.uid = new OidType(); this.uid.setValue(value); return this; } /* * @return {@link #url} (WADO-RS URL to retrieve the study. Note that this URL retrieves all SOP instances of the study, not only those in the selection.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public UriType getUrlElement() { if (this.url == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create StudyComponent.url"); else if (Configuration.doAutoCreate()) this.url = new UriType(); // bb return this.url; } public boolean hasUrlElement() { return this.url != null && !this.url.isEmpty(); } public boolean hasUrl() { return this.url != null && !this.url.isEmpty(); } /* * @param value {@link #url} (WADO-RS URL to retrieve the study. Note that this URL retrieves all SOP instances of the study, not only those in the selection.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public StudyComponent setUrlElement(UriType value) { this.url = value; return this; } /* * @return WADO-RS URL to retrieve the study. Note that this URL retrieves all SOP instances of the study, not only those in the selection. / public String getUrl() { return this.url == null ? null : this.url.getValue(); } /* * @param value WADO-RS URL to retrieve the study. Note that this URL retrieves all SOP instances of the study, not only those in the selection. / public StudyComponent setUrl(String value) { if (Utilities.noString(value)) this.url = null; else { if (this.url == null) this.url = new UriType(); this.url.setValue(value); } return this; } /* * @return {@link #series} (Series indetity and locating information of the DICOM SOP instances in the selection.) / public List<SeriesComponent> getSeries() { if (this.series == null) this.series = new ArrayList<SeriesComponent>(); return this.series; } public boolean hasSeries() { if (this.series == null) return false; for (SeriesComponent item : this.series) if (!item.isEmpty()) return true; return false; } /* * @return {@link #series} (Series indetity and locating information of the DICOM SOP instances in the selection.) / // syntactic sugar public SeriesComponent addSeries() { //3 SeriesComponent t = new SeriesComponent(); if (this.series == null) this.series = new ArrayList<SeriesComponent>(); this.series.add(t); return t; } protected void listChildren(List<Property> childrenList) { super.listChildren(childrenList); childrenList.add(new Property("uid", "oid", "Study instance uid of the SOP instances in the selection.", 0, java.lang.Integer.MAX_VALUE, uid)); childrenList.add(new Property("url", "uri", "WADO-RS URL to retrieve the study. Note that this URL retrieves all SOP instances of the study, not only those in the selection.", 0, java.lang.Integer.MAX_VALUE, url)); childrenList.add(new Property("series", "", "Series indetity and locating information of the DICOM SOP instances in the selection.", 0, java.lang.Integer.MAX_VALUE, series)); } public StudyComponent copy() { StudyComponent dst = new StudyComponent(); copyValues(dst); dst.uid = uid == null ? null : uid.copy(); dst.url = url == null ? null : url.copy(); if (series != null) { dst.series = new ArrayList<SeriesComponent>(); for (SeriesComponent i : series) dst.series.add(i.copy()); }; return dst; } @Override public boolean equalsDeep(Base other) { if (!super.equalsDeep(other)) return false; if (!(other instanceof StudyComponent)) return false; StudyComponent o = (StudyComponent) other; return compareDeep(uid, o.uid, true) && compareDeep(url, o.url, true) && compareDeep(series, o.series, true) ; } @Override public boolean equalsShallow(Base other) { if (!super.equalsShallow(other)) return false; if (!(other instanceof StudyComponent)) return false; StudyComponent o = (StudyComponent) other; return compareValues(uid, o.uid, true) && compareValues(url, o.url, true); } public boolean isEmpty() { return super.isEmpty() && (uid == null \|\| uid.isEmpty()) && (url == null \|\| url.isEmpty()) && (series == null \|\| series.isEmpty()) ; } } @Block() public static class SeriesComponent extends BackboneElement { /* * Series instance uid of the SOP instances in the selection. / @Child(name="uid", type={OidType.class}, order=1, min=0, max=1) @Description(shortDefinition="Series instance uid", formalDefinition="Series instance uid of the SOP instances in the selection." ) protected OidType uid; /* * WADO-RS URL to retrieve the series Note that this URL retrieves all SOP instances of the series not only those in the selection. / @Child(name="url", type={UriType.class}, order=2, min=0, max=1) @Description(shortDefinition="Retrieve URL", formalDefinition="WADO-RS URL to retrieve the series Note that this URL retrieves all SOP instances of the series not only those in the selection." ) protected UriType url; /* * Identity and locating information of the selected DICOM SOP instances. / @Child(name="instance", type={}, order=3, min=1, max=Child.MAX_UNLIMITED) @Description(shortDefinition="The selected instance", formalDefinition="Identity and locating information of the selected DICOM SOP instances." ) protected List<InstanceComponent> instance; private static final long serialVersionUID = 229247770L; public SeriesComponent() { super(); } /* * @return {@link #uid} (Series instance uid of the SOP instances in the selection.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public OidType getUidElement() { if (this.uid == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create SeriesComponent.uid"); else if (Configuration.doAutoCreate()) this.uid = new OidType(); // bb return this.uid; } public boolean hasUidElement() { return this.uid != null && !this.uid.isEmpty(); } public boolean hasUid() { return this.uid != null && !this.uid.isEmpty(); } /* * @param value {@link #uid} (Series instance uid of the SOP instances in the selection.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public SeriesComponent setUidElement(OidType value) { this.uid = value; return this; } /* * @return Series instance uid of the SOP instances in the selection. / public String getUid() { return this.uid == null ? null : this.uid.getValue(); } /* * @param value Series instance uid of the SOP instances in the selection. / public SeriesComponent setUid(String value) { if (Utilities.noString(value)) this.uid = null; else { if (this.uid == null) this.uid = new OidType(); this.uid.setValue(value); } return this; } /* * @return {@link #url} (WADO-RS URL to retrieve the series Note that this URL retrieves all SOP instances of the series not only those in the selection.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public UriType getUrlElement() { if (this.url == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create SeriesComponent.url"); else if (Configuration.doAutoCreate()) this.url = new UriType(); // bb return this.url; } public boolean hasUrlElement() { return this.url != null && !this.url.isEmpty(); } public boolean hasUrl() { return this.url != null && !this.url.isEmpty(); } /* * @param value {@link #url} (WADO-RS URL to retrieve the series Note that this URL retrieves all SOP instances of the series not only those in the selection.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public SeriesComponent setUrlElement(UriType value) { this.url = value; return this; } /* * @return WADO-RS URL to retrieve the series Note that this URL retrieves all SOP instances of the series not only those in the selection. / public String getUrl() { return this.url == null ? null : this.url.getValue(); } /* * @param value WADO-RS URL to retrieve the series Note that this URL retrieves all SOP instances of the series not only those in the selection. / public SeriesComponent setUrl(String value) { if (Utilities.noString(value)) this.url = null; else { if (this.url == null) this.url = new UriType(); this.url.setValue(value); } return this; } /* * @return {@link #instance} (Identity and locating information of the selected DICOM SOP instances.) / public List<InstanceComponent> getInstance() { if (this.instance == null) this.instance = new ArrayList<InstanceComponent>(); return this.instance; } public boolean hasInstance() { if (this.instance == null) return false; for (InstanceComponent item : this.instance) if (!item.isEmpty()) return true; return false; } /* * @return {@link #instance} (Identity and locating information of the selected DICOM SOP instances.) / // syntactic sugar public InstanceComponent addInstance() { //3 InstanceComponent t = new InstanceComponent(); if (this.instance == null) this.instance = new ArrayList<InstanceComponent>(); this.instance.add(t); return t; } protected void listChildren(List<Property> childrenList) { super.listChildren(childrenList); childrenList.add(new Property("uid", "oid", "Series instance uid of the SOP instances in the selection.", 0, java.lang.Integer.MAX_VALUE, uid)); childrenList.add(new Property("url", "uri", "WADO-RS URL to retrieve the series Note that this URL retrieves all SOP instances of the series not only those in the selection.", 0, java.lang.Integer.MAX_VALUE, url)); childrenList.add(new Property("instance", "", "Identity and locating information of the selected DICOM SOP instances.", 0, java.lang.Integer.MAX_VALUE, instance)); } public SeriesComponent copy() { SeriesComponent dst = new SeriesComponent(); copyValues(dst); dst.uid = uid == null ? null : uid.copy(); dst.url = url == null ? null : url.copy(); if (instance != null) { dst.instance = new ArrayList<InstanceComponent>(); for (InstanceComponent i : instance) dst.instance.add(i.copy()); }; return dst; } @Override public boolean equalsDeep(Base other) { if (!super.equalsDeep(other)) return false; if (!(other instanceof SeriesComponent)) return false; SeriesComponent o = (SeriesComponent) other; return compareDeep(uid, o.uid, true) && compareDeep(url, o.url, true) && compareDeep(instance, o.instance, true) ; } @Override public boolean equalsShallow(Base other) { if (!super.equalsShallow(other)) return false; if (!(other instanceof SeriesComponent)) return false; SeriesComponent o = (SeriesComponent) other; return compareValues(uid, o.uid, true) && compareValues(url, o.url, true); } public boolean isEmpty() { return super.isEmpty() && (uid == null \|\| uid.isEmpty()) && (url == null \|\| url.isEmpty()) && (instance == null \|\| instance.isEmpty()) ; } } @Block() public static class InstanceComponent extends BackboneElement { /* * SOP class uid of the selected instance. / @Child(name="sopClass", type={OidType.class}, order=1, min=1, max=1) @Description(shortDefinition="SOP class uid of instance", formalDefinition="SOP class uid of the selected instance." ) protected OidType sopClass; /* * SOP Instance uid of the selected instance. / @Child(name="uid", type={OidType.class}, order=2, min=1, max=1) @Description(shortDefinition="Uid of the selected instance", formalDefinition="SOP Instance uid of the selected instance." ) protected OidType uid; /* * WADO-RS URL to retrieve the DICOM SOP Instance. / @Child(name="url", type={UriType.class}, order=3, min=1, max=1) @Description(shortDefinition="Retrieve URL", formalDefinition="WADO-RS URL to retrieve the DICOM SOP Instance." ) protected UriType url; /* * Identity and location information of the frames in the selected instance. / @Child(name="frames", type={}, order=4, min=0, max=Child.MAX_UNLIMITED) @Description(shortDefinition="The frame set", formalDefinition="Identity and location information of the frames in the selected instance." ) protected List<FramesComponent> frames; private static final long serialVersionUID = 1641180916L; public InstanceComponent() { super(); } public InstanceComponent(OidType sopClass, OidType uid, UriType url) { super(); this.sopClass = sopClass; this.uid = uid; this.url = url; } /* * @return {@link #sopClass} (SOP class uid of the selected instance.). This is the underlying object with id, value and extensions. The accessor "getSopClass" gives direct access to the value / public OidType getSopClassElement() { if (this.sopClass == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create InstanceComponent.sopClass"); else if (Configuration.doAutoCreate()) this.sopClass = new OidType(); // bb return this.sopClass; } public boolean hasSopClassElement() { return this.sopClass != null && !this.sopClass.isEmpty(); } public boolean hasSopClass() { return this.sopClass != null && !this.sopClass.isEmpty(); } /* * @param value {@link #sopClass} (SOP class uid of the selected instance.). This is the underlying object with id, value and extensions. The accessor "getSopClass" gives direct access to the value / public InstanceComponent setSopClassElement(OidType value) { this.sopClass = value; return this; } /* * @return SOP class uid of the selected instance. / public String getSopClass() { return this.sopClass == null ? null : this.sopClass.getValue(); } /* * @param value SOP class uid of the selected instance. / public InstanceComponent setSopClass(String value) { if (this.sopClass == null) this.sopClass = new OidType(); this.sopClass.setValue(value); return this; } /* * @return {@link #uid} (SOP Instance uid of the selected instance.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public OidType getUidElement() { if (this.uid == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create InstanceComponent.uid"); else if (Configuration.doAutoCreate()) this.uid = new OidType(); // bb return this.uid; } public boolean hasUidElement() { return this.uid != null && !this.uid.isEmpty(); } public boolean hasUid() { return this.uid != null && !this.uid.isEmpty(); } /* * @param value {@link #uid} (SOP Instance uid of the selected instance.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public InstanceComponent setUidElement(OidType value) { this.uid = value; return this; } /* * @return SOP Instance uid of the selected instance. / public String getUid() { return this.uid == null ? null : this.uid.getValue(); } /* * @param value SOP Instance uid of the selected instance. / public InstanceComponent setUid(String value) { if (this.uid == null) this.uid = new OidType(); this.uid.setValue(value); return this; } /* * @return {@link #url} (WADO-RS URL to retrieve the DICOM SOP Instance.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public UriType getUrlElement() { if (this.url == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create InstanceComponent.url"); else if (Configuration.doAutoCreate()) this.url = new UriType(); // bb return this.url; } public boolean hasUrlElement() { return this.url != null && !this.url.isEmpty(); } public boolean hasUrl() { return this.url != null && !this.url.isEmpty(); } /* * @param value {@link #url} (WADO-RS URL to retrieve the DICOM SOP Instance.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public InstanceComponent setUrlElement(UriType value) { this.url = value; return this; } /* * @return WADO-RS URL to retrieve the DICOM SOP Instance. / public String getUrl() { return this.url == null ? null : this.url.getValue(); } /* * @param value WADO-RS URL to retrieve the DICOM SOP Instance. / public InstanceComponent setUrl(String value) { if (this.url == null) this.url = new UriType(); this.url.setValue(value); return this; } /* * @return {@link #frames} (Identity and location information of the frames in the selected instance.) / public List<FramesComponent> getFrames() { if (this.frames == null) this.frames = new ArrayList<FramesComponent>(); return this.frames; } public boolean hasFrames() { if (this.frames == null) return false; for (FramesComponent item : this.frames) if (!item.isEmpty()) return true; return false; } /* * @return {@link #frames} (Identity and location information of the frames in the selected instance.) / // syntactic sugar public FramesComponent addFrames() { //3 FramesComponent t = new FramesComponent(); if (this.frames == null) this.frames = new ArrayList<FramesComponent>(); this.frames.add(t); return t; } protected void listChildren(List<Property> childrenList) { super.listChildren(childrenList); childrenList.add(new Property("sopClass", "oid", "SOP class uid of the selected instance.", 0, java.lang.Integer.MAX_VALUE, sopClass)); childrenList.add(new Property("uid", "oid", "SOP Instance uid of the selected instance.", 0, java.lang.Integer.MAX_VALUE, uid)); childrenList.add(new Property("url", "uri", "WADO-RS URL to retrieve the DICOM SOP Instance.", 0, java.lang.Integer.MAX_VALUE, url)); childrenList.add(new Property("frames", "", "Identity and location information of the frames in the selected instance.", 0, java.lang.Integer.MAX_VALUE, frames)); } public InstanceComponent copy() { InstanceComponent dst = new InstanceComponent(); copyValues(dst); dst.sopClass = sopClass == null ? null : sopClass.copy(); dst.uid = uid == null ? null : uid.copy(); dst.url = url == null ? null : url.copy(); if (frames != null) { dst.frames = new ArrayList<FramesComponent>(); for (FramesComponent i : frames) dst.frames.add(i.copy()); }; return dst; } @Override public boolean equalsDeep(Base other) { if (!super.equalsDeep(other)) return false; if (!(other instanceof InstanceComponent)) return false; InstanceComponent o = (InstanceComponent) other; return compareDeep(sopClass, o.sopClass, true) && compareDeep(uid, o.uid, true) && compareDeep(url, o.url, true) && compareDeep(frames, o.frames, true); } @Override public boolean equalsShallow(Base other) { if (!super.equalsShallow(other)) return false; if (!(other instanceof InstanceComponent)) return false; InstanceComponent o = (InstanceComponent) other; return compareValues(sopClass, o.sopClass, true) && compareValues(uid, o.uid, true) && compareValues(url, o.url, true) ; } public boolean isEmpty() { return super.isEmpty() && (sopClass == null \|\| sopClass.isEmpty()) && (uid == null \|\| uid.isEmpty()) && (url == null \|\| url.isEmpty()) && (frames == null \|\| frames.isEmpty()); } } @Block() public static class FramesComponent extends BackboneElement { /* * The frame numbers in the frame set. / @Child(name="frameNumbers", type={IntegerType.class}, order=1, min=1, max=Child.MAX_UNLIMITED) @Description(shortDefinition="Frame numbers", formalDefinition="The frame numbers in the frame set." ) protected List<IntegerType> frameNumbers; /* * WADO-RS URL to retrieve the DICOM frames. / @Child(name="url", type={UriType.class}, order=2, min=1, max=1) @Description(shortDefinition="Retrieve URL", formalDefinition="WADO-RS URL to retrieve the DICOM frames." ) protected UriType url; private static final long serialVersionUID = 587981442L; public FramesComponent() { super(); } public FramesComponent(UriType url) { super(); this.url = url; } /* * @return {@link #frameNumbers} (The frame numbers in the frame set.) / public List<IntegerType> getFrameNumbers() { if (this.frameNumbers == null) this.frameNumbers = new ArrayList<IntegerType>(); return this.frameNumbers; } public boolean hasFrameNumbers() { if (this.frameNumbers == null) return false; for (IntegerType item : this.frameNumbers) if (!item.isEmpty()) return true; return false; } /* * @return {@link #frameNumbers} (The frame numbers in the frame set.) / // syntactic sugar public IntegerType addFrameNumbersElement() {//2 IntegerType t = new IntegerType(); if (this.frameNumbers == null) this.frameNumbers = new ArrayList<IntegerType>(); this.frameNumbers.add(t); return t; } /* * @param value {@link #frameNumbers} (The frame numbers in the frame set.) / public FramesComponent addFrameNumbers(int value) { //1 IntegerType t = new IntegerType(); t.setValue(value); if (this.frameNumbers == null) this.frameNumbers = new ArrayList<IntegerType>(); this.frameNumbers.add(t); return this; } /* * @param value {@link #frameNumbers} (The frame numbers in the frame set.) / public boolean hasFrameNumbers(int value) { if (this.frameNumbers == null) return false; for (IntegerType v : this.frameNumbers) if (v.equals(value)) // integer return true; return false; } /* * @return {@link #url} (WADO-RS URL to retrieve the DICOM frames.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public UriType getUrlElement() { if (this.url == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create FramesComponent.url"); else if (Configuration.doAutoCreate()) this.url = new UriType(); // bb return this.url; } public boolean hasUrlElement() { return this.url != null && !this.url.isEmpty(); } public boolean hasUrl() { return this.url != null && !this.url.isEmpty(); } /* * @param value {@link #url} (WADO-RS URL to retrieve the DICOM frames.). This is the underlying object with id, value and extensions. The accessor "getUrl" gives direct access to the value / public FramesComponent setUrlElement(UriType value) { this.url = value; return this; } /* * @return WADO-RS URL to retrieve the DICOM frames. / public String getUrl() { return this.url == null ? null : this.url.getValue(); } /* * @param value WADO-RS URL to retrieve the DICOM frames. / public FramesComponent setUrl(String value) { if (this.url == null) this.url = new UriType(); this.url.setValue(value); return this; } protected void listChildren(List<Property> childrenList) { super.listChildren(childrenList); childrenList.add(new Property("frameNumbers", "integer", "The frame numbers in the frame set.", 0, java.lang.Integer.MAX_VALUE, frameNumbers)); childrenList.add(new Property("url", "uri", "WADO-RS URL to retrieve the DICOM frames.", 0, java.lang.Integer.MAX_VALUE, url)); } public FramesComponent copy() { FramesComponent dst = new FramesComponent(); copyValues(dst); if (frameNumbers != null) { dst.frameNumbers = new ArrayList<IntegerType>(); for (IntegerType i : frameNumbers) dst.frameNumbers.add(i.copy()); }; dst.url = url == null ? null : url.copy(); return dst; } @Override public boolean equalsDeep(Base other) { if (!super.equalsDeep(other)) return false; if (!(other instanceof FramesComponent)) return false; FramesComponent o = (FramesComponent) other; return compareDeep(frameNumbers, o.frameNumbers, true) && compareDeep(url, o.url, true); } @Override public boolean equalsShallow(Base other) { if (!super.equalsShallow(other)) return false; if (!(other instanceof FramesComponent)) return false; FramesComponent o = (FramesComponent) other; return compareValues(frameNumbers, o.frameNumbers, true) && compareValues(url, o.url, true); } public boolean isEmpty() { return super.isEmpty() && (frameNumbers == null \|\| frameNumbers.isEmpty()) && (url == null \|\| url.isEmpty()) ; } } /* * Instance UID of the DICOM KOS SOP Instances represenetd in this resource. / @Child(name = "uid", type = {OidType.class}, order = 0, min = 1, max = 1) @Description(shortDefinition="Instance UID", formalDefinition="Instance UID of the DICOM KOS SOP Instances represenetd in this resource." ) protected OidType uid; /* * A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection. / @Child(name = "patient", type = {Patient.class}, order = 1, min = 1, max = 1) @Description(shortDefinition="Patient of the selected objects", formalDefinition="A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection." ) protected Reference patient; /* * The actual object that is the target of the reference (A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection.) / protected Patient patientTarget; /* * The reason for, or significance of, the selection of objects referenced in the resource. / @Child(name = "title", type = {CodeableConcept.class}, order = 2, min = 1, max = 1) @Description(shortDefinition="Reason for selection", formalDefinition="The reason for, or significance of, the selection of objects referenced in the resource." ) protected CodeableConcept title; /* * Text description of the DICOM SOP instances selected in the key object selection. This should be aligned with the content of the title element, and can provide further explanation of the SOP instances in the selection. / @Child(name = "description", type = {StringType.class}, order = 3, min = 0, max = 1) @Description(shortDefinition="Description text", formalDefinition="Text description of the DICOM SOP instances selected in the key object selection. This should be aligned with the content of the title element, and can provide further explanation of the SOP instances in the selection." ) protected StringType description; /* * Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion. / @Child(name = "author", type = {Practitioner.class, Device.class, Organization.class, Patient.class, RelatedPerson.class}, order = 4, min = 0, max = 1) @Description(shortDefinition="Author (human or machine)", formalDefinition="Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion." ) protected Reference author; /* * The actual object that is the target of the reference (Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion.) / protected Resource authorTarget; /* * Date and time when the key object selection was authored. Note that this is the date and time the DICOM SOP instances in the selection were selected (selection decision making). It is different from the creation date and time of the selection resource. / @Child(name = "authoringTime", type = {DateTimeType.class}, order = 5, min = 0, max = 1) @Description(shortDefinition="Authoring time of the selection", formalDefinition="Date and time when the key object selection was authored. Note that this is the date and time the DICOM SOP instances in the selection were selected (selection decision making). It is different from the creation date and time of the selection resource." ) protected DateTimeType authoringTime; /* * Study identity and locating information of the DICOM SOP instances in the selection. / @Child(name = "study", type = {}, order = 6, min = 1, max = Child.MAX_UNLIMITED) @Description(shortDefinition="Study identity of the selected instances", formalDefinition="Study identity and locating information of the DICOM SOP instances in the selection." ) protected List<StudyComponent> study; private static final long serialVersionUID = -1961832713L; public ImagingObjectSelection() { super(); } public ImagingObjectSelection(OidType uid, Reference patient, CodeableConcept title) { super(); this.uid = uid; this.patient = patient; this.title = title; } /* * @return {@link #uid} (Instance UID of the DICOM KOS SOP Instances represenetd in this resource.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public OidType getUidElement() { if (this.uid == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create ImagingObjectSelection.uid"); else if (Configuration.doAutoCreate()) this.uid = new OidType(); // bb return this.uid; } public boolean hasUidElement() { return this.uid != null && !this.uid.isEmpty(); } public boolean hasUid() { return this.uid != null && !this.uid.isEmpty(); } /* * @param value {@link #uid} (Instance UID of the DICOM KOS SOP Instances represenetd in this resource.). This is the underlying object with id, value and extensions. The accessor "getUid" gives direct access to the value / public ImagingObjectSelection setUidElement(OidType value) { this.uid = value; return this; } /* * @return Instance UID of the DICOM KOS SOP Instances represenetd in this resource. / public String getUid() { return this.uid == null ? null : this.uid.getValue(); } /* * @param value Instance UID of the DICOM KOS SOP Instances represenetd in this resource. / public ImagingObjectSelection setUid(String value) { if (this.uid == null) this.uid = new OidType(); this.uid.setValue(value); return this; } /* * @return {@link #patient} (A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection.) / public Reference getPatient() { if (this.patient == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create ImagingObjectSelection.patient"); else if (Configuration.doAutoCreate()) this.patient = new Reference(); // cc return this.patient; } public boolean hasPatient() { return this.patient != null && !this.patient.isEmpty(); } /* * @param value {@link #patient} (A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection.) / public ImagingObjectSelection setPatient(Reference value) { this.patient = value; return this; } /* * @return {@link #patient} The actual object that is the target of the reference. The reference library doesn't populate this, but you can use it to hold the resource if you resolve it. (A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection.) / public Patient getPatientTarget() { if (this.patientTarget == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create ImagingObjectSelection.patient"); else if (Configuration.doAutoCreate()) this.patientTarget = new Patient(); // aa return this.patientTarget; } /* * @param value {@link #patient} The actual object that is the target of the reference. The reference library doesn't use these, but you can use it to hold the resource if you resolve it. (A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection.) / public ImagingObjectSelection setPatientTarget(Patient value) { this.patientTarget = value; return this; } /* * @return {@link #title} (The reason for, or significance of, the selection of objects referenced in the resource.) / public CodeableConcept getTitle() { if (this.title == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create ImagingObjectSelection.title"); else if (Configuration.doAutoCreate()) this.title = new CodeableConcept(); // cc return this.title; } public boolean hasTitle() { return this.title != null && !this.title.isEmpty(); } /* * @param value {@link #title} (The reason for, or significance of, the selection of objects referenced in the resource.) / public ImagingObjectSelection setTitle(CodeableConcept value) { this.title = value; return this; } /* * @return {@link #description} (Text description of the DICOM SOP instances selected in the key object selection. This should be aligned with the content of the title element, and can provide further explanation of the SOP instances in the selection.). This is the underlying object with id, value and extensions. The accessor "getDescription" gives direct access to the value / public StringType getDescriptionElement() { if (this.description == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create ImagingObjectSelection.description"); else if (Configuration.doAutoCreate()) this.description = new StringType(); // bb return this.description; } public boolean hasDescriptionElement() { return this.description != null && !this.description.isEmpty(); } public boolean hasDescription() { return this.description != null && !this.description.isEmpty(); } /* * @param value {@link #description} (Text description of the DICOM SOP instances selected in the key object selection. This should be aligned with the content of the title element, and can provide further explanation of the SOP instances in the selection.). This is the underlying object with id, value and extensions. The accessor "getDescription" gives direct access to the value / public ImagingObjectSelection setDescriptionElement(StringType value) { this.description = value; return this; } /* * @return Text description of the DICOM SOP instances selected in the key object selection. This should be aligned with the content of the title element, and can provide further explanation of the SOP instances in the selection. / public String getDescription() { return this.description == null ? null : this.description.getValue(); } /* * @param value Text description of the DICOM SOP instances selected in the key object selection. This should be aligned with the content of the title element, and can provide further explanation of the SOP instances in the selection. / public ImagingObjectSelection setDescription(String value) { if (Utilities.noString(value)) this.description = null; else { if (this.description == null) this.description = new StringType(); this.description.setValue(value); } return this; } /* * @return {@link #author} (Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion.) / public Reference getAuthor() { if (this.author == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create ImagingObjectSelection.author"); else if (Configuration.doAutoCreate()) this.author = new Reference(); // cc return this.author; } public boolean hasAuthor() { return this.author != null && !this.author.isEmpty(); } /* * @param value {@link #author} (Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion.) / public ImagingObjectSelection setAuthor(Reference value) { this.author = value; return this; } /* * @return {@link #author} The actual object that is the target of the reference. The reference library doesn't populate this, but you can use it to hold the resource if you resolve it. (Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion.) / public Resource getAuthorTarget() { return this.authorTarget; } /* * @param value {@link #author} The actual object that is the target of the reference. The reference library doesn't use these, but you can use it to hold the resource if you resolve it. (Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion.) / public ImagingObjectSelection setAuthorTarget(Resource value) { this.authorTarget = value; return this; } /* * @return {@link #authoringTime} (Date and time when the key object selection was authored. Note that this is the date and time the DICOM SOP instances in the selection were selected (selection decision making). It is different from the creation date and time of the selection resource.). This is the underlying object with id, value and extensions. The accessor "getAuthoringTime" gives direct access to the value / public DateTimeType getAuthoringTimeElement() { if (this.authoringTime == null) if (Configuration.errorOnAutoCreate()) throw new Error("Attempt to auto-create ImagingObjectSelection.authoringTime"); else if (Configuration.doAutoCreate()) this.authoringTime = new DateTimeType(); // bb return this.authoringTime; } public boolean hasAuthoringTimeElement() { return this.authoringTime != null && !this.authoringTime.isEmpty(); } public boolean hasAuthoringTime() { return this.authoringTime != null && !this.authoringTime.isEmpty(); } /* * @param value {@link #authoringTime} (Date and time when the key object selection was authored. Note that this is the date and time the DICOM SOP instances in the selection were selected (selection decision making). It is different from the creation date and time of the selection resource.). This is the underlying object with id, value and extensions. The accessor "getAuthoringTime" gives direct access to the value / public ImagingObjectSelection setAuthoringTimeElement(DateTimeType value) { this.authoringTime = value; return this; } /* * @return Date and time when the key object selection was authored. Note that this is the date and time the DICOM SOP instances in the selection were selected (selection decision making). It is different from the creation date and time of the selection resource. / public Date getAuthoringTime() { return this.authoringTime == null ? null : this.authoringTime.getValue(); } /* * @param value Date and time when the key object selection was authored. Note that this is the date and time the DICOM SOP instances in the selection were selected (selection decision making). It is different from the creation date and time of the selection resource. / public ImagingObjectSelection setAuthoringTime(Date value) { if (value == null) this.authoringTime = null; else { if (this.authoringTime == null) this.authoringTime = new DateTimeType(); this.authoringTime.setValue(value); } return this; } /* * @return {@link #study} (Study identity and locating information of the DICOM SOP instances in the selection.) / public List<StudyComponent> getStudy() { if (this.study == null) this.study = new ArrayList<StudyComponent>(); return this.study; } public boolean hasStudy() { if (this.study == null) return false; for (StudyComponent item : this.study) if (!item.isEmpty()) return true; return false; } /* * @return {@link #study} (Study identity and locating information of the DICOM SOP instances in the selection.) */ // syntactic sugar public StudyComponent addStudy() { //3 StudyComponent t = new StudyComponent(); if (this.study == null) this.study = new ArrayList<StudyComponent>(); this.study.add(t); return t; } protected void listChildren(List<Property> childrenList) { super.listChildren(childrenList); childrenList.add(new Property("uid", "oid", "Instance UID of the DICOM KOS SOP Instances represenetd in this resource.", 0, java.lang.Integer.MAX_VALUE, uid)); childrenList.add(new Property("patient", "Reference(Patient)", "A patient resource reference which is the patient subject of all DICOM SOP Instances in this key object selection.", 0, java.lang.Integer.MAX_VALUE, patient)); childrenList.add(new Property("title", "CodeableConcept", "The reason for, or significance of, the selection of objects referenced in the resource.", 0, java.lang.Integer.MAX_VALUE, title)); childrenList.add(new Property("description", "string", "Text description of the DICOM SOP instances selected in the key object selection. This should be aligned with the content of the title element, and can provide further explanation of the SOP instances in the selection.", 0, java.lang.Integer.MAX_VALUE, description)); childrenList.add(new Property("author", "Reference(Practitioner\|Device\|Organization\|Patient\|RelatedPerson)", "Author of key object selection. It can be a human authtor or a device which made the decision of the SOP instances selected. For example, a radiologist selected a set of imaging SOP instances to attached in a diagnostic report, and a CAD application may author a selection to describe SOP instances it used to generate a detection conclusion.", 0, java.lang.Integer.MAX_VALUE, author)); childrenList.add(new Property("authoringTime", "dateTime", "Date and time when the key object selection was authored. Note that this is the date and time the DICOM SOP instances in the selection were selected (selection decision making). It is different from the creation date and time of the selection resource.", 0, java.lang.Integer.MAX_VALUE, authoringTime)); childrenList.add(new Property("study", "", "Study identity and locating information of the DICOM SOP instances in the selection.", 0, java.lang.Integer.MAX_VALUE, study)); } public ImagingObjectSelection copy() { ImagingObjectSelection dst = new ImagingObjectSelection(); copyValues(dst); dst.uid = uid == null ? null : uid.copy(); dst.patient = patient == null ? null : patient.copy(); dst.title = title == null ? null : title.copy(); dst.description = description == null ? null : description.copy(); dst.author = author == null ? null : author.copy(); dst.authoringTime = authoringTime == null ? null : authoringTime.copy(); if (study != null) { dst.study = new ArrayList<StudyComponent>(); for (StudyComponent i : study) dst.study.add(i.copy()); }; return dst; } protected ImagingObjectSelection typedCopy() { return copy(); } @Override public boolean equalsDeep(Base other) { if (!super.equalsDeep(other)) return false; if (!(other instanceof ImagingObjectSelection)) return false; ImagingObjectSelection o = (ImagingObjectSelection) other; return compareDeep(uid, o.uid, true) && compareDeep(patient, o.patient, true) && compareDeep(title, o.title, true) && compareDeep(description, o.description, true) && compareDeep(author, o.author, true) && compareDeep(authoringTime, o.authoringTime, true) && compareDeep(study, o.study, true); } @Override public boolean equalsShallow(Base other) { if (!super.equalsShallow(other)) return false; if (!(other instanceof ImagingObjectSelection)) return false; ImagingObjectSelection o = (ImagingObjectSelection) other; return compareValues(uid, o.uid, true) && compareValues(description, o.description, true) && compareValues(authoringTime, o.authoringTime, true) ; } public boolean isEmpty() { return super.isEmpty() && (uid == null \|\| uid.isEmpty()) && (patient == null \|\| patient.isEmpty()) && (title == null \|\| title.isEmpty()) && (description == null \|\| description.isEmpty()) && (author == null \|\| author.isEmpty()) && (authoringTime == null \|\| authoringTime.isEmpty()) && (study == null \|\| study.isEmpty()) ; } @Override public ResourceType getResourceType() { return ResourceType.ImagingObjectSelection; } @SearchParamDefinition(name = "identifier", path = "ImagingObjectSelection.uid", description = "UID of key DICOM object selection", type = "token") public static final String SP_IDENTIFIER = "identifier"; @SearchParamDefinition(name = "authoring-time", path = "ImagingObjectSelection.authoringTime", description = "Time of key DICOM object selection authoring", type = "date") public static final String SP_AUTHORINGTIME = "authoring-time"; @SearchParamDefinition(name="selected-study", path="ImagingObjectSelection.study.uid", description="Study selected in key DICOM object selection", type="token" ) public static final String SP_SELECTEDSTUDY = "selected-study"; @SearchParamDefinition(name="author", path="ImagingObjectSelection.author", description="Author of key DICOM object selection", type="reference" ) public static final String SP_AUTHOR = "author"; @SearchParamDefinition(name="patient", path="ImagingObjectSelection.patient", description="Subject of key DICOM object selection", type="reference" ) public static final String SP_PATIENT = "patient"; @SearchParamDefinition(name = "title", path = "ImagingObjectSelection.title", description = "Title of key DICOM object selection", type = "token") public static final String SP_TITLE = "title"; }	{ "redpajama_set_name": "RedPajamaGithub" }	5,945
\section{Introduction} \label{sec.intro} An important goal of contemporary cosmology is to locate the epoch when the universe was first bathed in starlight, popularly referred to as `cosmic dawn'. According to numerical simulations, dark matter halos sufficiently massive to induce star formation could develop as early as 150-250 Myr after the Big Bang, corresponding to the redshift range $z\simeq$15-20 \citep{Wise2014,Villanueva2018}. With some assumptions about their ionising capabilities, studies based on the demographics of early galaxies \citep{Robertson2015} have claimed to be compatible with independent constraints on early reionisation based on analyses of the optical depth of electron scattering to the cosmic microwave background (CMB; \citealt{Planck2018}), as well as tentative measures of 21cm absorption at a redshift $z\simeq$16-19 in the CMB possibly associated with UV photons produced by early star formation \citep{Bowman2018}. Recent deep imaging with the \textit{Hubble} and \textit{Spitzer} Space Telescopes has extended our redshift horizon to $z\simeq$9-11 \citep{Ellis13,Oesch2018}. At this redshift, the universe is less than 600 Myr old and the presence of main sequence stars older than 250 Myr would imply galaxy formation originating before a redshift $z\simeq$14. Such a case of `age-dating' a high redshift galaxy was presented by \citet{Zheng2012} for the lensed galaxy MACS1149-JD1 based upon a prominent photometric excess seen in the \textit{Spitzer}/IRAC 4.5$\mu$m band. For the assumed photometric redshift of $z$=9.6 $\pm$ 0.2, the IRAC excess would imply the presence of a Balmer break indicative of a burst of star formation some 200-300 Myr earlier. However, central to this conclusion is excluding the possibility of a redshift below $z\simeq$9 where such an IRAC excess could arise from contamination of the 4.5$\mu$m band by intense rest-frame optical [O\,{\scriptsize III}] 5007 \AA\ emission consistent with a much younger stellar population \citep{Labbe13,Roberts-Borsani2016}. An early attempt to secure a spectroscopic redshift of MACS1149-JD1 using grism capabilities onboard the \textit{Hubble} Space Telescope (HST) \citep{Hoag2018} provided a tantalising hint of a Lyman-break at $z>9$, however the lack of an emission line prevented a secure redshift constraint. A convincing redshift of $z=9.11$ was eventually determined through Lyman-$\alpha$ and [O\,{\scriptsize III}] 88 micron emission by \citet{Hashimoto2018}, using ALMA and VLT/X-Shooter. The subsequent re-analysis of the object using the secure spectroscopic redshift and improved IRAC photometry resulted in an estimated age of 290 $\pm$ 150 Myr for the dominant stellar component, consistent with a formation redshift of 15.4 $\pm$2.3. Various studies have discussed whether MACS1149-JD1 might be representative of the galaxy population at early times and hence provide a realistic first estimate of when cosmic dawn occurred. However, accurate interpretation of the associated photometry is not without its challenges. For instance, although \citet{Hashimoto2018} derived a stellar mass of $\simeq 10^9 M_{\odot}$ for MACS1149-JD1, this relied on an assumed and uncertain lensing magnification, thus introducing important uncertainties when comparing its properties to theoretical predictions. Furthermore, a fundamental difficulty has been that a prominent Balmer break requires a {\em declining} star formation rate with cosmic time for at least some portion of the early history of a galaxy, something that is difficult to reconcile with contemporary numerical simulations \citep{Binggeli2019}. Observationally, several $7.5<z<9$ objects (both lensed and unlensed) with secure spectroscopic redshifts display a significant IRAC excess which could be indicative of the presence of a Balmer break and early episodes of star formation (e.g., \citealt{Tamura2019},\citealt{Bakx2020}, \citealt{Strait2020}). However, here the interpretation is confused by the likely prominence of [O\,{\scriptsize III}] 5007 \AA\ emission. To break the degeneracy of line emission and starlight as contributors to the IRAC excess, \citet{Roberts-Borsani2020} combined HST and \textit{Spitzer}/IRAC photometry with ALMA far-infrared [O\,{\scriptsize III}] 88 $\mu$m emission and dust continuum constraints for four $z>7$ sources (including MACS1149-JD1). Although their analysis gave a lower age for MACS1149-JD1 than \citet{Hashimoto2018}, they nonetheless concluded that, collectively for these four sources, nearly half the stellar mass was produced prior to $z\simeq$10. Even for sources believed to be beyond $z\simeq9$, it is possible that an IRAC excess may not entirely arise via a Balmer break. For MACS1149-JD1 \citet{Hashimoto2018} ruled out strong contamination of the 4.5$\mu$m band from intense [O III] 4959 \AA\ and H$\beta$, which would have indicated a much younger system. Likewise, very strong [O II] 3727 \AA\ emission might contaminate the band, but only if the true redshift approaches $z\simeq$10. Using hydrodynamical simulations and a comoving volume of 70 Mpc$^3$, \citet{Katz2019} predicted the presence of three massive $z>9$ galaxies with similarly strong IRAC colours and matched photometry to MACS1149-JD1. While their results also point to likely episodes of extended star formation, they conclude that dust may contribute to the IRAC excess, thereby reducing the strength of the Balmer break. Given the implications for the birth of galaxies, determining the amount of star formation prior to $z\simeq$9 is an important, but challenging endeavour. While ambiguities of interpretation may remain given the signal/noise of the available photometric data, our motivation in this paper is to further explore the Balmer break interpretation for six promising $z\geq$9 candidates (including MACS1149-JD1) each with a 4.5$\mu$m IRAC excess. A major advance is a concerted attempt to secure their spectroscopic redshifts. Our goal is to explore the possible extent of star formation activity prior to a redshift $z\simeq$10. We consider this timely given it may have important consequences for exploring this early period further with the {\it James Webb} Space Telescope. A plan of the paper follows. In Section \ref{sec.selection}, we describe the selection criteria we applied to construct our sample of $z\geq$9 objects. In Section \ref{sec.photometry} we present the relevant spectral energy distributions (SEDs) and use these to infer the likely photometric redshifts and their star formation histories. This analysis provides a first assessment of the early star formation history for an enlarged population of $z\simeq$9 galaxies. The results of our spectroscopic follow-up campaign are described in Section \ref{sec.spectro}. Given the challenges of working at these limits, we are not able to convincingly secure a redshift for all of our candidates. Section \ref{sec.prop} returns to a discussion of the physical properties of the sample in the light of the limited spectroscopic data. We also discuss the extent to which our IRAC excess sample may be representative of the broader population of $z\simeq$9 galaxies and the frequency with which Lyman-$\alpha$ emission is being located deep in the reionisation era. Finally we discuss the implication of our findings in the context of direct searches for earlier activity with the \textit{James Webb} Space Telescope in Section \ref{sec.discussion}. Throughout this paper we use a concordance cosmology with $H_0 = 70, \Omega_M = 0.3, \Omega_L = 0.7$. Magnitudes are in the AB system \citep{AB}. \section{Selection of z>9 sources} \label{sec.selection} To make further progress in charting the cosmic star formation history beyond the limits directly accessible to HST, we have selected further candidates for detailed study from three near-infrared surveys: CANDELS \citep{CANDELS} and the Frontier Fields \citep{Lotz2017} from \textit{Hubble} Space Telescope imaging, and the UltraVISTA ground-based campaign \citep{UltraVISTA}. We applied the following selection criteria to likely $z\simeq$9 candidates with $m_{\text{H-band}}<26.5$ mag (see Section \ref{sec.photometry}): (i) a 2$\sigma$ non-detection in all bands bluewards of the Lyman-break at the putative redshift; (ii) a 5$\sigma$ detection in two consecutive bands redwards of the Lyman-break; (iii) red \textit{Spitzer}/IRAC colours (3.6$\mu$m - 4.5$\mu$m > 0.5); (iv) a photometric redshift probability distribution permitting $z\geq$9 at 1$\sigma$. We used publicly available catalogues for the CANDELS and UltraVISTA fields. AstroDeep catalogues were used for four Frontier Field clusters (A2744, MACS0416, MACS0717 and MACS1149) and we constructed our own catalogues for the clusters A370 and AS1063 following the method described in \citet{Laporte2016}. The \textit{Spitzer}/IRAC data used in this paper are drawn from several surveys and therefore not uniform in depth. CANDELS IRAC data have a 5$\sigma$ depth of 25.5 mag; the 5$\sigma$ depth of the UltraVISTA IRAC data is 25.8 mag and the Frontier Fields data reach a 5$\sigma$ depth of 26.5 mag. We extracted the Spitzer photometry using a 1.2 arcsec radius aperture centred at the position of the source on the F160W image, and applied the usual aperture correction. However noting that, for blended sources, this may overestimate the IRAC flux, in those cases we extracted the photometry more carefully, taking into account the shape of the galaxies with GALFIT \citep{Peng2010} assuming a Sersic profile. Including MACS1149-JD1, these criteria led to six suitable sources distributed as follows : two (presumably lensed) sources viewed behind a Frontier Fields cluster, three in the CANDELS fields GOODS-N and GOODS-S, and one in UltraVISTA survey (see Table \ref{list}). The most restrictive criterion in their selection is that relating to the red IRAC colour as this can only be applied to relatively bright targets given the depth difference between $\sim$1$\mu$m data and the IRAC images. All these sources have already been identified as promising high-redshift candidates by previous studies \citep{Oesch2018,Ishigaki2018,Bowler2020}. \begin{table} \hspace{-1.4cm} \centering \hspace{-1.4cm} \begin{tabular}{l\|ccccccc} \hline Target & RA & DEC & $z_{\text{phot}}$ & $\rm{M_{uv}}$ & $\rm{M_{\star}}$[$\times$10$^9$M$_{\odot}$] & Age (Myr) & $f_M$(z$>$10) [\%]\\ \hline MACS0416-JD$^a$ & 04:16:11.52 & -24:04:54.0 & 9.25$^{+0.08}_{-0.09}$ & -20.83$\pm$0.22 & 1.50$^{+0.84}_{-0.61}$ & 360$^{+108}_{-157}$ & 74.1$^{+8.1}_{-25.6}$\\ \multirow{2}{}{MACS1149-JD1$^b$}& \multirow{2}{}{11:49:33.59} & \multirow{2}{}{22:24:45.76} & 9.44$^{+0.02}_{-0.03}$ & -19.17$\pm$0.04 & 0.44$^{+0.05}_{-0.04}$ & 484$^{+17}_{-36}$ & 86.5$^{+2.0}_{-4.3}$ \\ & & & \textit{9.11} & -19.12$\pm$0.04 & 0.66$^{+0.09}_{-0.04}$ & 192$^{+159}_{-87}$ & 47.5$^{+20.0}_{-14.2}$ \\ GN-z10-3$^c$ & 12:36:04.09 & +62:14:29.6 & 9.57$^{+0.23}_{-0.27}$ & -20.72$\pm$ 0.12 & 1.57$^{+0.81}_{-0.63}$ & 265$^{+153}_{-145}$ & 38.0$^{+23.1}_{-38.0}$\\ GN-z9-1$^d$ & 12:36:52.25 & +62:18:42.4 & 9.22$^{+0.32}_{-0.33}$ & -20.81$\pm$ 0.18 & 2.18$^{+1.24}_{-0.85}$ & 323$^{+137}_{-168}$ & 65.1$^{+18.7}_{-32.2}$ \\ GS-z9-1$^d$ & 03:32:32.05 & -27:50:41.7 & 9.26$^{+0.41}_{-0.42}$ & -20.38$\pm$0.20 & 2.47$^{+1.62}_{-1.03}$ & 326$^{+128}_{-176}$ & 70.6$^{+12.5}_{-25.4}$ \\ UVISTA-1212$^e$ & 10:02:31.81 & 02:31:17.10 & 8.88$^{+0.27}_{-0.46}$ & -22.93$\pm$0.20 & 9.7$^{+5.10}_{-4.72}$ & 280$^{+172}_{-174}$ & 32.4$^{+47.2}_{-32.4}$ \\ \hline \end{tabular} \caption{\label{list} Bright $z_{\text{phot}}\geq$9 galaxies with red IRAC colours (3.6$\mu$m - 4.5$\mu$m > 0.5) with physical properties determined using BAGPIPES \citep{BAGPIPES}. Photometric redshift uncertainties represent 1$\sigma$ values and absolute magnitudes and stellar masses are corrected for lensing magnification where appropriate (see text). The final column displays the inferred percentage of the presently-observed stellar mass formed before $z=$10. \\ $^a$ \citet{Laporte2016}, $^b$ \citet{Zheng2012}, $^c$\citet{Oesch2014}, $^d$\citet{Oesch2018}, $^e$\citet{Bowler2020} } \end{table} \section{Photometric Analysis} \label{sec.photometry} The spectral energy distributions (SEDs) and photometric redshift likelihood distributions for our selected six $z>9$ candidates are presented in Figure \ref{fig.sed}. The SEDs utilise photometry drawn from publicly-available imaging data taken with \textit{Hubble} and \textit{Spitzer} Space Telescopes and, where available, the VISTA deep surveys. We matched the Point-Spread Function (PSF) of HST images using \textit{TinyTim} models \citep{TinyTim}. We extracted the total SED following the method described in \citet{Finkelstein2013} with an aperture correction computed from the \textit{Sextractor} MAG\_AUTO measured on the F160W image from WFC3/IR or the H-band from VISTA. The IRAC/\textit{Spitzer} photometry following method described in the previous section . Two sources in our sample, MACS0416-JD and MACS1149-JD1, are located behind Frontier Fields clusters and thus are gravitationally-lensed. We estimated the magnification for these using the online magnification calculator for the Frontier Field survey\footnote{https://archive.stsci.edu/prepds/frontier/lensmodels/\#magcalc} which is based on lensing models developed by \citet{Hoag2016}, \citet{Caminha2017}, \citet{Jauzac2014}, \citet{Richard2014}, \citet{Kawamata2016}, \citet{Kawamata2018}, \citet{Ishigaki2015}, \citet{Johnson2014} and \citet{Grillo2015}. This yields a magnification of 1.90$^{+0.03}_{-0.02}$ and 28$^{+44}_{-11}$ respectively for MACS0416-JD and MACS1149-JD1. The latter estimate represents a revision of the magnification adopted by \citet{Hashimoto2018}. We simultaneously determine the optimal photometric redshift and a range of physical properties for the six selected galaxies using BAGPIPES \citep{BAGPIPES}. We allow the photometric redshift to lie in the range $0.0<z_{\text{phot}}<10.0$ which, for $z\leq9.0$, permits the IRAC excess to be formed, in part, from [O\,{\scriptsize III}] line emission \citep{Roberts-Borsani2020}. BAGPIPES generates HII regions from CLOUDY photoionisation code \citep{CLOUDY} and assumes that the nebular emission is the sum of emission from HII regions of different ages following \citet{Byler2017}. In particular, it is capable of reproducing intense [O\,{\scriptsize III}] emission with equivalent widths EW[O\,{\scriptsize III}+H$\beta$]$>$1000\AA\ , sufficient for consideration of dominant contributions to an IRAC excess at high redshift. BAGPIPES assumes that the gas-phase metallicity is similar to that of the stars producing the ionising radiation (with a range of $Z\in$[0.0:2.5] $Z_{\odot}$). Finally, BAGPIPES accounts for dust emission via the \citet{Draine2007} model, with a reddening law following the \citet{Calzetti2001} relation. We fit the SED of our candidates assuming a single-component model with four possible star-formation histories (SFH) : delayed, exponential, constant or burst-like. The best SED-fit is selected to be that with the largest likelihood and is presented as the grey curve for each candidate in Figure \ref{fig.sed}, with the associated physical parameters listed in Table \ref{list}. The best fit is always obtained for the delayed or constant SFH. Although we tabulate the age of the best-fit stellar population, a more meaningful measure is the fraction (expressed as a percentage) of the observed stellar mass that was assembled prior to a redshift $z=10$. For the two gravitationally-lensed galaxies, the luminosities and stellar masses have been corrected for the magnifications assumed above. Table~\ref{tab.fit} shows the range of parameters adopted in BAGPIPES for each particular SFH. \begin{table} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline Parameters & Burst & Exponential & Constant & Delayed \\ \hline t$_{\rm SF\ begin}$ [Gyr]& 0.0-0.1 & \multicolumn{3}{c\|}{0.0 ; 1.0} \\ \hline t$_{\rm SF\ end}$ [Gyr] & - & - & 0.0 ; 1.0 & - \\ \hline $\tau$ [Gyr] & - & 0.1 ; 10.0 & - & 0.1 ; 10.0 \\ \hline $\log$M$_{\star}$ [M$_{\odot}$] & \multicolumn{4}{c\|}{0.0 ; 12.0} \\ \hline Z [$Z_{\odot}$] & \multicolumn{4}{c\|}{0.0 ; 2.5} \\ \hline Av [mag] & \multicolumn{4}{c\|}{0.0 ; 3.0} \\ \hline log U & \multicolumn{4}{c\|}{-4.0 ; 0.0} \\ \hline \end{tabular} \caption{\label{tab.fit} The BAGPIPES parameter ranges used to fit the SEDs of the 6 targets studied in this paper. The age (or maximum age) is chosen to be less than 1 Gyr given the selection criteria applied to build our sample. The ionisation parameter range we used is large to account for extreme values ($\log$ U$>$-2) as observed in previous high-redshift galaxies \citep{Stark2017}.} \end{table} Despite the different HST and ground-based datasets from which our six IRAC excess sources were selected, they span a limited range in luminosity and stellar mass. As a consequence of its detection to the shallower magnitude limit with a ground-based telescope, UVISTA-1212 is the most luminous and massive galaxy in the sample. Later in Section \ref{sec.prop} we discuss how representative is our sample with respect to the larger population of sources thought to be at $z\simeq$9. For all of the sources except UVISTA-1212, the most probable photometric redshift derived is above $z\simeq$9 and thus a Balmer break may be the most likely explanation for the IRAC excess. Nonetheless, as discussed by \citet{Hashimoto2018} and \citet{Roberts-Borsani2020}, alternative explanations based on contributions from strong nebular emission lines or dust reddening can be considered. Although [O\,{\scriptsize III}] 5007 \AA\ is redshifted out of the IRAC 4.5$\mu$m band for $z>9.1$, \citet{Hashimoto2018} considered the contribution of H$\beta$ and [O\,{\scriptsize III}] 4959\AA . Even for a very young starburst with a hard radiation field (log U = -2.0) and low metallicity (20\% solar), it is not possible to reproduce a significant IRAC excess. Furthermore, for the youngest ages, the IRAC 3.6$\mu$m flux is raised by the presence of a significant nebular continuum. Likewise, [O\,{\scriptsize II}] 3727 \AA\ will enter the 4.5$\mu$m band for redshifts above $z\simeq$9.7. The photometric likelihood distribution does extend to such a high redshift for some of our sources (Figure \ref{fig:M0416-JD_prop}). However, the IRAC excess would only be matched if [O\,{\scriptsize II}] had a rest-frame equivalent width of over 1000 \AA\ which seems improbable. Finally, it is possible to match the SEDs by combining a delayed SFH with dust extinction. In this case, to match the IRAC excess, the extinction would typically need to exceed $A_V>$1.7 mag, which is larger than that inferred in previous measures of reionisation era galaxies (e.g. \citealt{Schaerer2010}, \citealt{Strait2020}). In Figure \ref{fig:M0416-JD_prop} we show the posterior distribution of the age and stellar mass to indicate the reliability of concluding there was star formation prior to $z=10$ as listed in Table \ref{list}. Despite uncertainties in the available photometry, the shape of the posterior is reasonably well-defined for ages greater than 200 Myr, indicating the likelihood of significant earlier star formation. The BAGPIPES solution for MACS1149-JD1 can be compared with the solutions derived by \cite{Hashimoto2018} and \citet{Roberts-Borsani2020}, both of which included ALMA [O\,{\scriptsize III}] 88$\mu$m fluxes and upper limits on the dust continuum in Band 7 as additional constraints. Although Roberts-Borsani et al advocated the use of ALMA to break the degeneracy between optical [O\,{\scriptsize III}] emission and a Balmer break as contributors to the IRAC excess, it is not a key issue in this case given the spectroscopic redshift of $z$=9.11 precludes the former case. Although an academic exercise given its redshift is known, both our photometric redshift ($z_{\text{phot}}=9.44^{+0.02}_{-0.03} $) and age ( $484 ^{+17}_{-36}$ Myr) are consistent with the range of early estimates. We include a separate entry in Table \ref{list} for the case where the redshift is fixed at its spectroscopic value of $z$=9.11 \citep{Hashimoto2018}. We emphasise that the inferred contribution of early star formation is uncertain in many cases, particularly for GN-z10-3 whose IRAC excess is the least prominent. Nonetheless, overall our sample is consistent with having significant star formation prior to $z\simeq$10, as was concluded for MACS1149-JD1. We will return to a further discussion of this in Section \ref{sec.discussion}. \begin{figure} \centering \includegraphics[width=15cm]{New_Figure1.pdf} \caption{\label{fig.sed} Spectral energy distributions and best-fit BAGPIPES \citep{BAGPIPES} models for the 6 galaxies in Table \ref{list}. Blue points represent photometric data or limits, and red points represent the predicted fluxes. All models incorporate contributions from nebular emission (see text). The right inset panel shows the redshift probability distribution, while the left inset panel shows the posterior distribution of the stellar mass as a function of the age of the stellar population. HST upper limits are showed at 1$\sigma$, IRAC limits are at 2$\sigma$}. \label{fig:M0416-JD_prop} \end{figure} \section{Spectroscopic Follow-up} \label{sec.spectro} Although Lyman-$\alpha$ is the most prominent rest-frame UV emission line in star-forming galaxies at intermediate redshift \citep{Stark2010,Stark2011}, its visibility deep in the reionisation era at $z\simeq$9 is expected to be significantly diminished via resonant scattering by neutral hydrogen. For this reason, various efforts have been made to target metallic UV lines such as [C\,{\scriptsize III}]1909 \AA\ and C\,{\scriptsize IV} 1548 \AA\ with limited success \citep{Laporte2017,Mainali2020}. Surprisingly, for 11 galaxies with spectroscopic redshifts above $z=7.5$, Lyman-$\alpha$ was detected in 7 with additional UV metallic lines in only 4 cases. In all cases, detecting the intrinsically fainter metallic lines has been more challenging than searching for Lyman-$\alpha$. Far infrared emission lines, such as [O\,{\scriptsize III}] 88 $\mu$m arising from ionised gas and [C\,{\scriptsize II}] 158 $\mu$m from the neutral gas, have been pivotal in spectroscopic confirmation at high redshifts with the ALMA interferometer \citep{Inoue2014}. For the same 11 $z>7.5$ galaxies of which 7 are accessible with ALMA, [O\,{\scriptsize III}] and/or [C\,{\scriptsize II}] emission have been detected in 5 cases. Moreover, in cases where both ALMA and near-infrared spectrographs on ground-based telescopes were both used in similar exposure times (e.g., \citealt{Laporte2017}, \citealt{ Hashimoto2018}), the ALMA detections were more significant. For this reason, we adopted a multi-facility approach in attempting to secure spectroscopic redshifts for the targets introduced in Table \ref{list}. We discuss the results of our spectroscopic campaigns for each target in turn below. A summary of the results of our spectroscopic campaign is given in Table 2. \subsection{MACS0416-JD} \label{sec.macs0416} We obtained 12.6hrs (8hrs on source) ALMA band 7 observation in Cycle 6 (ID: 2019.1.00061.S, PI: R. Ellis) to search for [O\,{\scriptsize III}] 88 $\mu$m in MACS0416-JD. The spectral windows were setup to cover the 1-$\sigma$ redshift probability distribution estimated from SED-fitting (Figure \ref{fig.sed}, viz. $\nu$ $\in$ [321.8 GHz : 325.7 GHz], $\cup$ [328.6 GHz : 335.6 GHz ], and $\cup$ [340.6 GHz : 347.6 GHz]. The data were obtained between October 2019 and January 2020 and reduced using the ALMA pipeline (v. 5.6.1-8) with a natural weighting scheme, a uv-tapper of 0.35 arcsec with a channel width of 16 MHz and a beam size of 1.00$\times$0.76 arcsec in order to maximise the signal to noise ratio of any emission line (\citealt{Tamura2019}). At the HST position of MACS0416-JD we identified a clear feature at $\nu$=329.69 GHz corresponding to a redshift $z$=9.28 (Figure \ref{MACS0416-JD}). The offset between the centroid of the UV continuum and the peak of [OIII]88$\mu$m emission is 0.40 arcsec, corresponding to an observed physical separation of $\simeq$1.8 kpc at $z\sim$9.28. Such a small displacement is consistent with previous [OIII]88$\mu$m observations at high-$z$ (e.g. \citealt{Carniani2017}, \citealt{Harikane2020}). We measured the integrated flux on a map created using the \textit{immoments} CASA recipe. We determine $f_{int}$=0.298$\pm$0.051 Jy.km.sec$^{-1}$ which is within the range values found previously in $z\geq$8 galaxies (\citealt{Tamura2019}, \citealt{Hashimoto2018}). Assuming a Gaussian profile, we measure a FWHM of 313$\pm$22 km sec$^{-1}$. The significance of the line can be estimated via a histogram of the pixel by pixel signal to noise ratio (SNR) at the spatial position of the target. In the absence of emission, this will approximate a Gaussian distribution. If emission is present, a second peak will be seen indicating its significance. The lower panel of Figure \ref{MACS0416-JD} reveals such a second peak at SNR$\simeq$6 consistent with the analysis above. No dust continuum was detected to a 1-$\sigma$ level of 18$\mu$Jy/beam. \begin{figure} \centering \includegraphics[width=8cm]{MACS0416-JD_spectro.pdf} \smallskip \includegraphics[width=9cm]{SNR_histo.pdf} \caption{\label{MACS0416-JD} (\textit{Top}) ALMA contours overplotted on the HST F160W image (left) and ALMA band 7 data (right) from a 2$\sigma$ level upwards for MACS0416-JD. The beam size is indicated at the bottom right of the ALMA panel. (\textit{Centre}) Extracted 1D spectrum over the full frequency range sampled showing a clear [O\,{\scriptsize III}] 88 $\mu$m emission line at 329.69 GHz corresponding to a redshift of $z$=9.28. (\textit{Bottom}) Distribution of the pixel signal/noise ratio (SNR) along the line of sight at the position of MACS0416-JD. In the case of non-detection, the histogram shape should be consistent with a gaussian. The yellow region highlights a deviation from a gaussian shape at SNR$\simeq$6. } \end{figure} We also observed this target in service mode with X-Shooter/VLT between August and September 2019 (ID: 0104.A-0028(A), P.I. : R. Ellis). The target was centred using a small blind offset ($<$20 arcsec) in the 0.9 arcsec JH slit to reduce the background and a nod length of 4 arcsec was used. The total on source exposure time in the near-infrared arm was 10hrs in excellent seeing condition ($<$0.8 arcsec). The data were reduced using the latest version of the X-shooter pipeline (3.5.0) and inspected visually by two authors (NL and RAM). No emission line was found in any of the three X-Shooter arms. Given the ALMA-based redshift, the absence of Lyman-$\alpha$ implies a 1$\sigma$ upper limit of 1.8$\times$10$^{-18}$cgs assuming a FWHM=200km/s (Figure \ref{fig.MACS0416-JD_X-Shooter}). \begin{figure} \centering \includegraphics[width=9cm]{spectra_MACS0416-JD_XShooter_arrow.pdf} \caption{\label{fig.MACS0416-JD_X-Shooter} Portion of the X-Shooter spectra of MACS0416-JD indicating no convincing detection of Lyman-$\alpha$ at the redshift of [O\,{\scriptsize III}] 88 $\mu$m (red arrow). Grey rectangles show the position of sky lines. The grey line shows the 1$\sigma$ noise spectrum and blue line shows the extracted spectrum at the position of MACS0416-JD. } \end{figure} \begin{table} \centering \hspace{0.5cm} { \small \hspace{-1.6cm} \begin{tabular}{l\|cc\|c\|c\|} Target & Redshift & Instruments & Feature & Flux / 1$\sigma$ \\ \hline MACS1149-JD1$^{\star}$ & 9.11 & X-SHOOTER & Ly-$\alpha$ & (4.3$\pm$1.1)$\times$10$^{-18}$cgs \\ MACS0416-JD & 9.28 & X-SHOOTER & Ly-$\alpha$ & $<$1.8$\times$10$^{-18}$cgs \\ GN-z10-3 & 8.78 & MOSFIRE & Ly-$\alpha$ & (1.37$\pm$0.24)$\times$10$^{-18}$cgs \\ GN-z9-1 & \textit{9.89?} & MOSFIRE & Ly-$\alpha$ & (2.2$\pm$0.7)$\times$10$^{-18}$cgs \\ U-1212 & - & FLAMINGOS2 & Ly-$\alpha$ & $<$4$\times$10$^{-18}$ cgs\\ GS-z9-1 & - & X-SHOOTER & Ly-$\alpha$ & $<$ 7$\times$10$^{-19}$ cgs \\ \hline MACS0416-JD & 9.28 & ALMA & [O\,{\scriptsize III}] 88 $\mu$m & 0.65$\pm$0.13 mJy/beam \\ MACS1149-JD1 & 9.11 & ALMA & [O\,{\scriptsize III}] 88 $\mu$m & 0.84$\pm$0.11 mJy/beam \\ \hline \end{tabular} } \caption{\label{tab.spectro} Spectroscopic observations of $z\geq$9 candidates. For the [O\,{\scriptsize III}] 88 $\mu$m line, we report the peak flux. \quad $^{\star}$ from \citet{Hashimoto2018}} \end{table} \subsection{GN-z10-3 and GN-z9-1} We observed GN-z10-3 with MOSFIRE/Keck undertaking a 5hrs 20min on source exposure in April 2019 (Keck 2019A\_U128, PI: B.E. Robertson). In order to monitor the seeing and transparency, two stars were included in the MOSFIRE mask. For the final analysis, only frames with seeing less than 0.8 arcsec FWHM were selected representing a total exposure of 4.9hrs. Data reduction was undertaken using the 2018 version of the MOSFIRE DRP. As the target was centred in the MOSFIRE slit, emission lines will feature a positive signal accompanied by two negative counterparts separated by the nodding scale of 1.25 arcsec. We identified a convincing ($>5\sigma$) emission line at $\lambda$=11891.4$\pm$1.3\AA\ with a FWHM=133$\pm$36km/s and an integrated flux of (1.37$\pm$0.24)$\times$10$^{-18}$cgs (Figure \ref{GN-z10-3}). No other line is present in the entire J-band spectra. Assuming this line is Lyman-$\alpha$, it corresponds to a redshift of $z$=8.78. In order to verify its reliability, we examined data comprising independent halves of the total exposure time. The line was retrieved with suitably reduced signal/noise on both spectra. The line width is comparable with that measured for other $z>7$ Lyman-$\alpha$ detections with MOSFIRE (e.g., \citealt{Zitrin2015}, \citealt{Finkelstein2013}, \citealt{Hoag2017}). The spectroscopic data refines the redshift value within the photometric redshift likelihood distribution (Figure \ref{fig.sed}). Although the IRAC 4.5 $\mu$m excess at this improved redshift below $z=9$ implies a likely contribution from [O\,{\scriptsize III}] 5007 \AA\ emission, we will return to the analysis of its SED in Section 5. Further sub-mm observations are currently on-going with NOEMA to detect [O\,{\scriptsize I}] 145 $\mu$m, one of the brightest FIR emission lines \citep{Katz2019}. \begin{figure} \centering \includegraphics[width=8cm]{new-figure4.pdf} \caption{\label{GN-z10-3} (\textit{Top}) 2D MOSFIRE spectrum of GN-z10-3. Red circles show the location of a 5$\sigma$ emission line detected at $\lambda$=11891.4 corresponding to Lyman-$\alpha$ at $z$=8.78 in a pattern consistent with the telescope nodding. (\textit{Bottom}) 1D extracted spectrum (blue) with the emission line highlighted in yellow and the 1$\sigma$ noise level in grey. Grey rectangles indicate the position of sky lines. } \end{figure} During the same observing run, a separate mask was designed for GN-z9-1. A total on source exposure time of 5.6hrs was secured in good seeing conditions. The data was reduced and analysed in the manner described above. However, in this case, no clear ($>$5$\sigma$) emission line over the entire J-band. Reduced the significance threshold to 3$\sigma$, a faint feature is present at $\lambda$=13241.2\AA\ with a positive signal at the expected position of the target in the slit and two negative counterparts on each side. We estimated a flux of (2.2$\pm$0.7)$\times$10$^{-18}$cgs. Assuming this line is Lyman-$\alpha$, the redshift of GN-z9-1 would be $z=$9.89 (Figure~\ref{GN-z9-1}). However, although consistent with the photometric redshift likelihood distribution (Figure \ref{fig.sed}), we consider this a marginal claim. Deeper spectroscopic observation, such as those that will become feasible with JWST, are needed to confirm this redshift. \begin{figure} \centering \includegraphics[width=8cm]{new-figure5.pdf} \caption{\label{GN-z9-1} (\textit{Top}) 2D MOSFIRE spectrum of GN-z9-1. Illustrated features follow those in Figure \ref{GN-z10-3}). (\textit{Bottom}) The 1D extracted spectrum in a 1.0 arcsec aperture is shown in blue. The grey line displays the 1$\sigma$ error. A tentative (3$\sigma$) emission line is highlighted in yellow which may be Lyman-$\alpha$ at $z$=9.89. } \end{figure} \subsection{UVISTA-1212} UVISTA-1212, one of the brightest $z\geq$8 candidates known to date, was observed with FLAMINGOS-2 on the Gemini-South telescope via the Fast Turnaround programme (ID: GS-2020A-FT-102, PI: Roberts-Borsani). Observations were accomplished in service mode in February 2020 in J-band using the R3K grism and a 4 pixel wide slit. From an allocation of 9.8 hours, a 8.8 hour on source exposure was taken in good seeing conditions. The data were reduced using the FLAMINGOS-2 Python cookbook for longslit observations (specifically the \textit{reduce\_ls} script, which primarily uses the PyRaf module as a wrapper around IRAF functions to reduce the data). More specifically, the \textit{nsflat}, \textit{nsreduce} and \textit{nscombine} were used to reduce the dark, flat and arc frames, as well as the standard star and science observations. The final 1D spectrum is shown in Figure \ref{f2_1dspec}. After a careful inspection of the reduced spectra, no clear emission line was found over the entire J-band spectrum. \begin{figure} \centering \includegraphics[width=8cm]{flamingos2_1dspec.pdf} \caption{\label{f2_1dspec} The portion of the FLAMINGOS2 spectrum where Lyman-$\alpha$ may lie according to the photometric redshift uncertainties for UVISTA-1212. No convincing emission line is found. Grey rectangles show the position of sky lines.} \end{figure} \subsection{GS-z9-1} GS-z9-1 has a photometric redshift of 9.26$^{+0.41}_{-0.42}$ and we have obtained 9.8 hours of observing time (8.0 hours on source) from an allocation of 13.5 hours with X-Shooter/VLT (ID : 106.20ZH.001, PI: R. Ellis). Observations were undertaken in service mode in good seeing conditions ($<$1.1 arcsec) between October 2020 and January 2021 \footnote{We thank ESO staff for observing during both Christmas and New Year.} We carefully checked the centring of the source in the X-Shooter slit and applied the same data reduction procedure as described in section \ref{sec.macs0416}. Unfortunately no clear emission line is seen after a first inspection of the data (Figure \ref{fig:gs-z9-2_spec}). At the expected position of Ly-$\alpha$ at $z_{phot}$, we measure a 1$\sigma$ upper limit of 7$\times$10$^{-19}$cgs. \begin{figure} \centering \includegraphics[width=9cm]{GS-z9-1_spectro_v2.pdf} \caption{\label{fig:gs-z9-2_spec} The portion of the X-Shooter spectrum where Lyman-$\alpha$ may lie according to the photometric redshift uncertainties for GS-z9-1. No convincing emission line is found. The blue line shows the 1D extracted spectrum at the position of the candidate, grey regions show the 1$\sigma$ noise extracted in the same sized aperture, and light blue rectangles show the position of sky lines. The red arrow displays the position of Ly-$\alpha$ at $z_{phot}$=9.26}. \end{figure} \section{Physical Properties} \label{sec.prop} \subsection{Updating the Properties} \label{sec.ages} We now reconsider the physical properties of our sample, taking into account, where available, the newly-acquired spectroscopic redshifts (Table \ref{tab.spectro}). In addition to MACS1149-JD1, we now have additional redshifts for MAC0416-JD and GN-z10-3, representing half of our original sample. Although we have marginal evidence that GN-z9-1 lies beyond $z\simeq$9, for it, GS-z9-1 and UVISTA-1212, we will continue to assume the photometric redshifts given in Table \ref{list}. A re-analysis of the SED is particularly relevant for GN-z10-3 whose spectroscopic redshift is now confirmed to lie below $z=9$, implying a possible significant contribution to the IRAC excess from rest-frame optical emission lines. The updated results based on BAGPIPES fits to the SEDs, now constrained with a fixed spectroscopic redshift, are given in Table \ref{fig:prop.spec}. In the three cases where the spectroscopic redshift is confirmed to be above $z\simeq$9, the derived properties are naturally similar to those given in Table \ref{list}. With the exception of UVISTA-1212, which appears to be an outlier in our sample, all galaxies have a stellar mass ranging from $\sim$ 1 to 4$\times$10$^9$ M$_{\odot}$, a star formation rate ranging from $\sim$10 to 30 M$_{\odot}$/yr and a low dust attenuation ($<$0.5 mag). However, for GN-z10-3, despite a spectroscopic constraint of $z$=8.78, a re-analysis of the SED incorporating this value may still be consistent with a mature stellar population. This may appear surprising given the IRAC excess could be readily explained with strong [O\,{\scriptsize III}] 5007 \AA\ emission. Figure \ref{GN-z10-3REV} shows a re-analysis of the SED shown in Figure \ref{fig.sed} with the redshift appropriately constrained at $z=8.78$. The posterior distribution in age and stellar masses is now bimodal and thus it is not possible to distinguish between a young population whose [O III] 5007 \AA\ emission contributes to the IRAC excess, and an older solution with a contribution from a Balmer break. Similar interpretational ambiguities were seen for $z<9$ sources with IRAC excesses by \citet{Roberts-Borsani2020} who argued this degeneracy may be broken with independent measures of [O III] 88$\mu$m emission. In the case of the young solution for GN-z10-3, the required equivalent width (EW) of [O\,{\scriptsize III}]+H$\beta$ is $\sim$1000 \AA\ which is consistent with the range observed in some high redshift galaxies (e.g., \citealt{Strait2020}, \citealt{Bowler2020}). The flat continuum between F160W and 360$\mu$m would then imply that [O II] has an EW $> 50 $\AA. Recent studies suggest this is close to the maximum possible for a young starburst (e.g. \citealt{Yang2017}). Although we are unable to distinguish between these two fits with the current data, we adopt the properties of an older stellar population in Table \ref{fig:prop.spec}. \begin{figure} \centering \includegraphics[width=9cm]{new_Figure8.pdf} \caption{\label{GN-z10-3REV} Best fit to the spectral energy distribution of GN-z10-3 now constrained at the spectroscopic redshift of $z=$8.78. Blue points and limits represent the observed photometry and red crosses show the expected measurements. The posterior distribution of the age and stellar mass is now bimodal, indicating it is not possible to distinguish the relative contributions of [O III] emission and a Balmer break to the IRAC excess (see text).} \end{figure} \subsection{The $z\sim$9 Lyman-$\alpha$ fraction} We now consider the question of the visibility of Lyman-$\alpha$ deep in the reionisation era. This can be done in terms of a Lyman-$\alpha$ emission fraction for our sample. Thus we have two targets with a secure Lyman-$\alpha$ detection and one (GN-z9-1) which is tentative (see Section \ref{sec.spectro}). All candidates have reliable photometric redshifts at $z\sim9$ or a far infrared emission line, suggesting the likelihood of contamination of the sample by low redshift interlopers is small. Despite obvious uncertainties for such a small sample, the Lyman-$\alpha$ emission fraction calculated for the sample of $6$ $z\sim 9$ galaxies presented in this paper is very high: $\chi_{\rm{LAE}} = 0.33_{-0.21}^{+0.28}$ if we include GN-z9-1 or $\chi_{\rm{LAE}} = 0.50^{+0.26}_{-0.26}$ otherwise. This high fraction of detections continues a trend of recent results (\citealt{Roberts-Borsani2016,Stark2017,Mason2018}) that have weakened the originally-claimed rapid decline above $z\simeq$7 (\citealt{Pentericci2011,Schenker2012,Schenker2014}). Our results are compared with these earlier trends in Figure \ref{fig.Lya_fraction}). As with \citet{Roberts-Borsani2016} and \citet{Stark2017}, we explicitly selected bright galaxies amenable to ground-based spectroscopy with red \textit{Spitzer}/IRAC colours (3.6$\mu$m - 4.5$\mu$m > 0.5, Section \ref{sec.selection}). However, while the 100\% visibility of Lyman-$\alpha$ in \citet{Roberts-Borsani2016}'s sample might be attributed to a correspondingly strong rest-frame optical [O\,{\scriptsize III}] emission responsible for the IRAC excess, for the three $z>9$ targets in the present sample, only a Balmer break synonymous with a mature stellar population (e.g., \citealt{Hashimoto2018}) could be responsible. This suggests that the enhanced Lyman-$\alpha$ fraction arises from the fact that some luminous galaxies, regardless of their [O\,{\scriptsize III}] emission, are capable of producing early ionised bubbles. As discussed by \citet{Mason2018}, much of the early work probing the redshift-dependent fraction necessarily targeted sub-luminous galaxies in order to efficiently exploit multi-slit spectrographs with limited fields of view. There is good evidence that intense [O\,{\scriptsize III}] emitters at low and intermediate redshifts have high LyC escape fractions \citep{Izotov2018,Fletcher2019,Nakajima2020} and/or ionising efficiencies \citep[e.g.,][]{Nakajima2016}, consistent with the creation of ionised bubbles at high-redshift. What remains unclear is how $z>9$ galaxies with IRAC excesses indicating a mature stellar population can emit enough Lyman continuum photons to ionise large bubbles. The Str\"omgren radius of the ionised bubble created by a galaxy scales with $\propto (t_{\rm{em}}f_{\rm{esc,LyC}})^{1/3}$ \citep[e.g.,][]{Cen2000}, where $f_{\rm{esc,LyC}}$ is the escape fraction of Lyman continuum photons and $t_{\rm{em}}$ the time the galaxies has been emitting such photons. Assuming an average ionising efficiency and $f_{\rm{esc,LyC}} < 3\%$, the galaxies in our $z\sim 9$ sample, which all sustained star formation over $\gtrsim 100-200$ Myr (see Table \ref{list}), can create an ionised bubble as large as those of COLA1 \citep{Hu2016,Matthee2018} or A370p\_z1 \citep{Meyer2021}, two actively star forming $z>6$ galaxies with very high escape fractions. Such a large ionised bubble may explain why Lyman-$\alpha$ is observed in MACS1149-JD1 blueshifted by $\sim -450 \text{km s}^{-1}$ with respect to [O\,{\scriptsize III}] 88 $\mu$m. In summary therefore, the high detected fraction of Lyman-$\alpha$ emission at $z\simeq$9 in luminous galaxies may offer further support of their mature stellar populations. \begin{figure} \centering \includegraphics[width=0.48\textwidth]{LAE_fraction_z9.pdf} \caption{Fraction of Lyman-break galaxies revealing Lyman-$\alpha$ emission with an equivalent width $\rm{EW}>25$ \AA. Full symbols represent the more luminous population with $-21.75<M_{\rm{UV}} < -20.25$ and the empty symbols indicate galaxies with $ -20.25<M_{\rm{UV}} < -18.75$. The fraction for galaxies in the present sample is indicated by the red symbols.} \label{fig.Lya_fraction} \end{figure} \subsection{How Representative is our Sample?} Early evidence for mature stellar populations at $z>7.5$ \citep{Hashimoto2018,Roberts-Borsani2020} has been hard to reconcile with numerical simulations of the first galaxies. While \citet{Binggeli2019} conclude that $z\sim9$ Balmer breaks must be very rare, \citealt{Katz2019} find in the contrary that they can be formed easily in their simulations, although dust might complicate the interpretation. In light of our results, it is thus important to assess observationally the prevalence of $z\sim 9$ galaxies with older stellar populations. While we cannot compare Balmer-break objects with a larger population of spectroscopically confirmed $z\sim9$ galaxies, we can assess the prevalence of Balmer-break objects in a parent sample of $z>9$ photometric candidates. In order to investigate the relative abundances of the two populations, we have carefully selected $z>9$ galaxies candidates using the same procedure and magnitude cuts as in Section \ref{sec.selection}, but removing the IRAC excess criterion. The candidates were visually inspected by two authors (NL, RAM). Due to the absence of filters bluewards of $<8500$\, \AA\, in ULTRAVISTA, we restricted our search to the Frontier Fields and CANDELS. This additional search results in $14$ photometric candidates (including the $5$ Balmer-break objects presented above). Balmer-break objects thus appear to represent a significant fraction($\simeq 0.36^{+0.17}_{-0.14}$) of the $z>9$ bright galaxy population, in contrast to the results of numerical simulations. Indeed this Balmer-break fraction could represent a lower limit since the denominator comprises galaxies without spectroscopic redshifts: several could be low redshift interlopers or $8<z<9$ galaxies (given the large photometric redshift errors) whose IRAC excess may not arise from a Balmer Break. This is also the case for the nominator (e.g. GS-z9-1), although to a lower extent. We can compare the rest-frame UV luminosities of this parent sample with those of our Balmer break galaxies. As most of the candidates are drawn from the GOODS-North/South fields($11/14$), we restrict the comparison to the CANDELS fields, ensuring we can avoid the need to make corrections for lensing magnification in the cluster fields. We derive the rest-frame UV magnitude of our candidates using the F125W magnitude, the best-fit photometric redshift and a UV slope $\alpha=2, (f_\nu \propto \nu^\alpha)$ for the K-correction. We compare the rest-frame UV magnitudes for the two samples in Figure \ref{fig:comparison}. Although the statistics are poor, our objects with red IRAC colors are not notably brighter or fainter than blue IRAC color objects. A two-sided KS test is inconclusive and the null hypothesis that the two distributions are different cannot be rejected ($p=0.12$). Thus, given the modest sample size to date, our $z>9$ Balmer break sources remain broadly representative of the larger $z>9$ galaxy population. We note that all $z>9$ candidates represent the brightest objects existing at that early time due to obvious limited depth of the imaging data. The hypothesis that Balmer break objects are less frequent in the fainter population is plausible, but can only be tested with \textit{JWST}. \begin{figure} \centering \includegraphics[width=0.45\textwidth]{population_mUV_goodsonly.pdf} \caption{A comparison of the rest-frame UV magnitudes for $z\sim9$ candidates in CANDELS selected without regard to their IRAC excesses with those for the three GOODS-S/N sources discussed in this article.} \label{fig:comparison} \end{figure} \section{Discussion} \label{sec.discussion} In this article, we have argued that from a sample of six $z\simeq$9 galaxies with \textit{Spitzer}/IRAC excess for which spectroscopic redshifts have now been secured for three, possibly four, we infer past star formation histories consistent with significant contributions beyond a redshift $z=10$. We can estimate the earlier rest-frame UV luminosity for each of the six galaxies in our sample assuming the best-fit star formation history (SFH). As described in Section \ref{sec.photometry}, the best SED fit is obtained for most of the objects with an evolved stellar population. Table \ref{list} shows the best-fit parameters for all galaxies, updated in Table \ref{fig:prop.spec} with the spectroscopic redshift for 3 of them. Based on these SFHs we compute the rest-frame $\rm{L_{1500}}$ luminosity as a function of redshift from $z\sim$15 to the observed redshift taking into account the dust reddening. For those galaxies without spectroscopic redshifts, the photometric value is assumed. Figure \ref{fig:past_luminosity} shows the evolution of $\rm{m_{1500}}$ as a function of redshift and demonstrates that each galaxy is as luminous in its earlier life. While these past star formation histories are clearly illustrative, they serve to demonstrate the feasibility of searches for earlier progenitors with JWST (e.g., \citealt{Roberts-Borsani2021}). From our SFHs, we can also determine the redshift-dependent history of the assembled stellar mass providing independent insight into the evolution of the UV luminosity density within the first 500 Myr (Figure \ref{fig:building_mass}). Two contrasting hypotheses are still debated in the literature: a rapid \citep{Oesch2018} or a smooth \citep{McLeod2016} decline of the UV luminosity density over $8 < z < 11$. For the case of a fast decline ($\propto$(1+$z$)$^{-11}$), the luminous galaxies probed would have formed $\sim$50\% of their stellar mass by redshift $z=$10, whereas for the smooth decline case ($\propto$(1+$z)^{-4}$) $\sim$80\% of the stellar mass is formed at $z=$10. Although our sample is admittedly small, it represents the only relevant data currently available. Averaging the SFHs of our 6 galaxies we determine that they formed $\sim$70\% of their mass by $z=$10, which seems to favour a smooth decline. \begin{table} \hspace{-1.4cm} \centering \hspace{-1.4cm} \begin{tabular}{l\|cccc} \hline Target & $z_{\text{spec}}$ & $\rm{M_{\star}}$[$\times$10$^9$M$_{\odot}$] & Age (Myr) & $f_M$(z$>$10) [\%]\\ \hline MACS0416-JD & 9.28 & 1.37$^{+0.75}_{-0.55}$ & 351$^{+115}_{-153}$ & 71.8$^{+10.4}_{-20.3}$\\ MACS1149-JD1 & 9.11 & 0.66$^{+0.09}_{-0.04}$ & 192$^{+159}_{-87}$ & 47.5$^{+20.0}_{-14.2}$ \\ GN-z10-3 & 8.78 & 1.27$^{+0.70}_{-0.61}$ & 275$^{+184}_{-175}$ & 39.1$^{+24.6}_{-38.9}$\\ \hline \end{tabular} \caption{\label{fig:prop.spec} Updated properties for those sources in Table \ref{list} with a spectroscopic redshift. The final column displays the percentage of the presently-observed stellar mass already formed before $z=$10.} \end{table} \begin{figure} \centering \includegraphics[width=1.0\columnwidth]{Lum_in_the_past_delayed_newJD1.png} \caption{Earlier redshift evolution of the rest-frame 1500\AA\ luminosity computed from our SED fits for the six galaxies in our sample. Filled regions show the 5$\sigma$ sensitivity of NIRCam/JWST filters (F150W, F200W and F277W) in 3hrs. } \label{fig:past_luminosity} \end{figure} \begin{figure} \centering \includegraphics[width=1.0\columnwidth]{Building_mass.png} \caption{Redshift evolution of the assembled stellar mass computed from the SFH deduced from our SED fits and averaged over the 6 galaxies discussed in this paper (red). This is compared with that predicted for two contrasting measures of the rate of decline in the UV luminosity density deduced from large photometric surveys (\citealt{Oesch2014}, \citealt{McLeod2016}). We estimate that $\simeq$ 70\% of the stellar mass was already in place by $z=10$ for the 6 galaxies in our $z\geq$9 sample.} \label{fig:building_mass} \end{figure} \section{Data Availability} The data underlying this article will be shared on reasonable request to the corresponding author. \section*{Acknowledgements} NL acknowledges support from the Kavli Foundation. RAM and RSE acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 669253). BER was supported in part by NASA program HST-GO-14747, contract NNG16PJ25C, and grant 80NSSC18K0563, and NSF award 1828315. We thank Adam Carnall for providing an updated version of BAGPIPES allowing high ionisation parameter. Based on observations made with ESO Telescopes at the Paranal Observatory under programme ID 0104.A-0028. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2019.1.00061.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. Based on observations obtained at the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. ID: 2019 \bibliographystyle{mnras}	{ "redpajama_set_name": "RedPajamaArXiv" }	36
{"url":"http:\/\/clay6.com\/qa\/17746\/which-one-of-the-following-is-the-ratio-of-the-lowering-of-vapour-pressure-","text":"# Which one of the following is the ratio of the lowering of vapour pressure of 0.1 m aqueous solutions of $BaCI_2, NACI\\: and \\: AI_2(SO_4)_3$ respectively ?\n\n$\\begin {array} {1 1} (1)\\;3:2:5 & \\quad (2)\\;5:2:3 \\\\ (3)\\;5:3:2 & \\quad (4)\\;2:3:5 \\end {array}$","date":"2018-01-16 19:21:59","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\": 0.8914805054664612, \"perplexity\": 261.3335458801506}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084886639.11\/warc\/CC-MAIN-20180116184540-20180116204540-00721.warc.gz\"}"}	null	null
Docker ====== Docker files to ease working with PathKit and CanvasKit. emsdk-base ---------- This image has an Emscripten SDK environment that can be used for compiling projects (e.g. Skia's PathKit) to WASM/asm.js. This image tracks the official emscripten Docker image and installs python 2 (which some of our scripts still use). make publish_emsdk_base For testing the image locally, the following flow can be helpful: docker build -t emsdk-base ./emsdk-base/ # Run bash in it to poke around and make sure things are properly installed docker run -it emsdk-base /bin/bash # Compile PathKit with the local image docker run -v $SKIA_ROOT:/SRC -v $SKIA_ROOT/out/dockerpathkit:/OUT emsdk-base /SRC/infra/pathkit/build_pathkit.sh karma-chrome-tests ------------------ This image has Google Chrome and karma/jasmine installed on it, which can be used to run JS tests. This image is standalone and does not have any extra dependencies that make it Skia-exclusive. It gets manually pushed anytime there's an update to the Dockerfile or relevant installed libraries. make publish_karma_chrome_tests Of note, some versions (generally before Chrome 60) run out of space on /dev/shm when using the default Docker settings. To be safe, it is recommended to run the container with the flag --shm-size=2gb. For testing the image locally, the following can be helpful: docker build -t karma-chrome-tests ./karma-chrome-tests/ # Run bash in it to poke around and make sure things are properly installed docker run -it --shm-size=2gb karma-chrome-tests /bin/bash # Run the tests (but not capturing Gold output) with the local source repo docker run --shm-size=2gb -v $SKIA_ROOT:/SRC karma-chrome-tests karma start /SRC/infra/pathkit/karma-docker.conf.js --single-run gold-karma-chrome-tests ------------------ This image has Google Chrome and karma/jasmine installed on it, which can be used to run JS tests. This image assumes the runner wants to collect the output images and JSON data specific to Skia Infra's Gold tool (image correctness). It gets manually pushed anytime there's an update to the Dockerfile or the parent image (karma-chrome-tests). # Run the following from $SKIA_ROOT/infra/pathkit make publish_gold_karma_chrome_tests Of note, some versions (generally before Chrome 60) run out of space on /dev/shm when using the default Docker settings. To be safe, it is recommended to run the container with the flag --shm-size=2gb. For testing the image locally, the following can be helpful: # Run the following from $SKIA_ROOT/infra/pathkit make gold-docker-image # Run bash in it to poke around and make sure things are properly installed docker run -it --shm-size=2gb gold-karma-chrome-tests /bin/bash # Run the tests and collect Gold output with the local source repo mkdir -p -m 0777 /tmp/dockergold docker run --shm-size=2gb -v $SKIA_ROOT:/SRC -v /tmp/dockergold:/OUT gold-karma-chrome-tests /SRC/infra/pathkit/test_pathkit.sh perf-karma-chrome-tests ------------------ This image has Google Chrome and karma/jasmine installed on it, which can be used to run JS tests. This image assumes the runner wants to collect the output images and JSON data specific to Skia Infra's Perf tool. It gets manually pushed anytime there's an update to the Dockerfile or the parent image (karma-chrome-tests). # Run the following from $SKIA_ROOT/infra/pathkit make publish_perf_karma_chrome_tests Of note, some versions (generally before Chrome 60) run out of space on /dev/shm when using the default Docker settings. To be safe, it is recommended to run the container with the flag --shm-size=2gb. For testing the image locally, the following can be helpful: # Run the following from $SKIA_ROOT/infra/pathkit make perf-docker-image # Run bash in it to poke around and make sure things are properly installed docker run -it --shm-size=2gb perf-karma-chrome-tests /bin/bash # Run the tests and collect Perf output with the local source repo mkdir -p -m 0777 /tmp/dockerperf docker run --shm-size=2gb -v $SKIA_ROOT:/SRC -v /tmp/dockerperf:/OUT perf-karma-chrome-tests /SRC/infra/pathkit/perf_pathkit.sh	{ "redpajama_set_name": "RedPajamaGithub" }	3,006
Cat Leading Double Life Helps Neighbors Become Unexpected Pen Pals Home / Art Artist Creates a Photo Series Featuring the Thief Who Stole Her Wallet and Identity By Jenny Zhang on November 26, 2014 In 2011, artist Jessamyn Lovell was at the gallery SF Camerawork discussing her upcoming show when her wallet–and, eventually, her identity–was stolen by a San Francisco woman named Erin Hart. When Lovell started receiving mysterious bills, parking tickets, and even a summons to appear in court on theft charges, she realized that she was a victim of identity theft. Furious and frustrated, the artist decided to channel her frustration into Dear Erin Hart, a project that stripped away Hart's anonymity in an effort to piece together a portrait of the criminal who had disrupted her life. Lovell hired a private investigator to track down Hart, who was in jail on an identity theft conviction. Waiting in a car outside the jail, Lovell photographed the woman as she was released, and eventually began following her and capturing paparazzi-like shots of Hart in public places. "Through photography, video, and other forms of documentation, I make an attempt at understanding this woman and the course of events that brought [our] lives together," Lovell says, taking on the varied roles of victim, stalker, investigator, artist, spy, and vigilante. According to the artist, her work "touches on contemporary concerns of surveillance and selfhood within the information age." Lovell's efforts culminated in an exhibition from September 3 to October 18 at SF Camerawork, the very place her wallet had been swiped. In a continuation of the project, the Albuquerque, New Mexico-based artist hopes to contact Hart in order to deliver a letter that she's written for the woman who stole her identity. Lovell says that even if Hart refuses to talk to her, she will at least be forced to acknowledge the existence of the victim of her crime. The artist is raising money for the trip from Albuquerque to San Francisco through a crowdfunding campaign and the sale of her own photos on Etsy. Update: We were lucky enough to have the opportunity to ask the artist a few questions about Dear Erin Hart,. Check out that exclusive interview, below. Is Dear Erin Hart, your way of exacting revenge upon this woman who has disrupted so much of your life? I did begin this project immediately upon leaving the Alameda county court house in response to how angry I felt. I had just spent months preparing documentation in defense of a crime she committed and used my ID at the time of her arrest. As the project went on, however I started to wonder who she was, why she did what she did, and even what she hoped or did gain from using my drivers license. I see the project now less as exacting revenge and more of an exercise in trying to understand the woman I did not choose to cross paths with and as a means of exploring some of the parallels between her life and mine. In the end, most people read the project as taking her anonymity, which I like but I think of it more as she entered my life without my permission and then I responded. I took a difficult situation that happened to me and made it into something I could learn from and hopefully share with others to learn from. How do you feel about your transformation into a stalker-like, vigilante-esque figure while you were tracking Hart's movements? Well, I did a project before this in which I found and photographed my estranged father without him knowing so I was pretty familiar with the territory. I can't say that I do not get a major thrill from tearing around the streets of San Francisco in a black tinted SUV with my P.I. driving but I am most interested in exploring issues of privacy and how they inform identity, which I like to think both of these projects do. I also think of myself as performing during these investigations. So the photographs, video, audio, stories are all documentation of elaborate performances I enact in collaboration with my unknowing subjects. Do you feel like you have gained a greater understanding of Hart or yourself through this undertaking? Have your feelings toward Hart changed as you've observed more of her life? Absolutely. Through this experience and my exploration of it as an art piece I have examined multiple aspects of my own privilege and questioned how I came to be where I am today given some of the struggles I have faced in my past. I grew up in a difficult situation having survived homelessness and poverty from a young age. I made some critical decisions that led me to become an artist and an academic, which could have gone in a completely different direction. I have thought a great deal about the disadvantages we all experience in different ways but what makes us each unique is our choices and responses to these life situations. I do not believe that Erin Hart is a bad person because she stole my identity or because she is a criminal. Through this project I wanted to remain open to what her life story is, What led her to this place and this time – specifically to use my ID to commit multiple crimes. I have often contemplated that she might have her reasons for doing what she does and often speculate on what those might be. I mean, we all make choices. Through the project I have learned that a big difference between Erin Hart and myself is that I believe we each possess different levels of empathy. I reserve the right to judge completely until I have actually met her but I think I am much more empathetic in general that she is, which is an advantage for her if she wants to make a living off of other people's identities. If Hart agrees to meet you, what is the one thing you most wish to ask or say to her? I'll offer no spoilers here – I will hopefully meet her in person next month and reveal that when the time is right. I want her to be the first person that hears my questions for her. Thaks so much for the interview, Jessamyn! Jessamyn Lovell's website Jessamyn Lovell on Etsy via [BOOOOOOOM], [SFGate] Like This Article? Share Vintage Shawl 6 Famous Marina Abramović Performances That Showcase Her Incredible Strength 12 of the Best Art Competitions to Enter in 2021 Artist Uses Silhouetted Shapes to Turn Colorful Clouds Into Beautiful Scenes Best of 2020: Top 19 Creative Projects Made During the COVID-19 Crisis 'Bob Ross Experience' Exhibit Opens in Indiana to Celebrate the Beloved TV Painter Forever Artist Uses the Power of Sound to Turn Sand Into Mesmerizing Patterns 10 Abstract Art Prints to Enhance Your Home's Ambiance RIP Alex Trebek: Fans React To the Loss of the Legendary 'Jeopardy!' Host With Art How Artists Are Expressing Hope After the 2020 Presidential Election Artist Reimagines How Animated Disney Films Were "Actually Made" 30+ Fun and Creative Gifts for the Artist in Your Life	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	6,587
package com.longluo.demo.sensor; import android.app.Activity; import android.hardware.Sensor; import android.hardware.SensorEvent; import android.hardware.SensorEventListener; import android.hardware.SensorManager; import android.os.Bundle; import android.os.Handler; import android.os.Message; import android.util.Log; import android.widget.TextView; import com.longluo.demo.R; /** * Created by luolong on 2016/6/15. */ public class AccActivity extends Activity implements SensorEventListener { private SensorManager mSensorManager; private TextView mAxisA; private TextView mAxisB; private TextView mAxisC; private float x; private float y; private float z; private long lastTime = System.currentTimeMillis(); private long nowTime = 0l; private long lastTimeStamp; private long nowTimeStamp; private Handler mHandler = new Handler() { @Override public void handleMessage(Message msg) { if (msg.what == 1001) { updateSensorData(); } super.handleMessage(msg); } }; @Override protected void onCreate(Bundle savedInstanceState) { super.onCreate(savedInstanceState); setContentView(R.layout.activity_sensor_acc); mSensorManager = (SensorManager) getSystemService(SENSOR_SERVICE); initViews(); } private void initViews() { mAxisA = (TextView) findViewById(R.id.tv_axis_a); mAxisB = (TextView) findViewById(R.id.tv_axis_b); mAxisC = (TextView) findViewById(R.id.tv_axis_c); } @Override protected void onStart() { super.onStart(); } @Override protected void onResume() { super.onResume(); mSensorManager.registerListener(this, mSensorManager.getDefaultSensor(Sensor.TYPE_ACCELEROMETER), SensorManager.SENSOR_DELAY_FASTEST); } @Override protected void onPause() { super.onPause(); mSensorManager.unregisterListener(this); } @Override protected void onStop() { super.onStop(); } @Override public void onSensorChanged(SensorEvent event) { nowTime = System.currentTimeMillis(); nowTimeStamp = event.timestamp; x = event.values[0]; y = event.values[1]; z = event.values[2]; Log.d("acc", "time gap=" + (nowTimeStamp - lastTimeStamp) + ",last=" + lastTimeStamp + ",nowTime=" + nowTimeStamp); lastTime = nowTime; lastTimeStamp = nowTimeStamp; String xVal = " " + x; String yVal = " " + y; String zVal = " " + z; // Log.d("gyro", "x=" + xVal + " y=" + yVal + " z=" + zVal + " w=" + w); // mHandler.sendEmptyMessage(1001); mAxisA.setText("X =" + x); mAxisB.setText("Y =" + y); mAxisC.setText("Z =" + z); } @Override public void onAccuracyChanged(Sensor sensor, int accuracy) { } private void updateSensorData() { mAxisA.setText("X =" + x); mAxisB.setText("Y =" + y); mAxisC.setText("Z =" + z); } }	{ "redpajama_set_name": "RedPajamaGithub" }	1,783
Appointments, Indigenous Rights, Management of Arts Organizations, Native American Culture "Bless the Poets": First Native US Poet Laureate Calls Us All to a Different Future Erin Rubin Gage Skidmore from Peoria, AZ, United States of America [CC BY-SA 2.0], via Wikimedia Commons June 19, 2019; NPR Joy Harjo is the first Native American US Poet Laureate. She was appointed this week by Librarian of Congress Carla Hayden, who said that Harjo's work "powerfully connects us to the earth." Harjo claims membership in the Muscogee Creek Nation. She is also the first poet laureate from Oklahoma. Her bibliography includes not only several books of poetry, but also a memoir, children's books, and five albums of original music; Harjo plays the saxophone. Her appointment by Hayden is hopeful and calls for the reading public to shift focus. Harjo's work deftly blends easily grasped physical imagery, often from the natural world, with acknowledgement of forces unseen. Though invisible, her evocations are not abstract, and represent both the real shadows and struggles that are part of American and Native identities, and faith in the strength and eventual triumph of her narrators and subjects. Fellow Poet D.A. Powell tweeted, "Joy Harjo serving as the new US Poet Laureate might be the first thing to happen in Washington DC this year that actually makes sense." Poet laureates are not often known or recognized by the wider public, but they serve an important role to "raise the national consciousness to a greater appreciation of the reading and writing of poetry," specifically in English. Some, like Natasha Trethewey or Maxine Kumin, hold public conversations; others, like Robert Pinsky and Billy Collins, try to get the public to read more poetry. Both Juan Felipe Herrera and Rita Dove used the opportunity to highlight work from particular cultural backgrounds, Herrera from Spanish speakers and Dove from the African diaspora. The position we now know as poet laureate was begun in 1937 with an endowment from philanthropist Archer M. Huntington. In 82 years, Harjo is the first laureate of indigenous heritage. She said, "I share this honor with ancestors and teachers who inspired in me a love of poetry, who taught that words are powerful and can make change when understanding appears impossible, and how time and timelessness can live together within a poem. I count among these ancestors and teachers my Muscogee Creek people, the librarians who opened so many doors for all of us, and the original poets of the indigenous tribal nations of these lands, who were joined by diverse peoples from nations all over the world to make this country and this country's poetry." Harjo is also a founder of the Native American Arts and Cultures Foundation, which, like much of her poetry, projects a hopeful message: "The arts and cultures of the diverse indigenous people in this country are powerful, beautiful, and growing, and offer perspectives that inspire creative solutions to some of our nation's most difficult collective challenges." The foundation promotes indigenous artists through grants, advocacy, and education in collaboration with other Native organizations. Harjo's work acknowledges the foundational injustices with whose consequences her tribe still lives; her poem "Conflict Resolution for Holy Beings" contains the lines, The lands and waters they gave us did not belong to them to give. Under false pretenses we signed. After drugging by drink, we signed. With a mass of gunpower pointed at us, we signed. With a flotilla of war ships at our shores, we signed. We are still signing. We have found no peace in this act of signing. But that poem appears in a book of the same name, whose dedication reads, Bless the poets, the workers for justice, the dancers of ceremony, the singers of heartache, the visionaries, all makers and carriers of fresh meaning—we will all make it through, despite politics and wars, despite failures and misunderstandings. There is only love. Harjo will serve as poet laureate from September to May.—Erin Rubin Erin Rubin is an assistant editor at the Nonprofit Quarterly, where she is in charge of online editorial coordination and community building. Before joining NPQ, in 2016, Erin worked as an administrator at Harvard Business School and as an editorial project manager at Pearson Education, where she helped develop a digital resource library for remedial learners. Erin has also worked with David R. Godine, Publishers, and the Association of Literary Scholars, Critics, and Writers. As a creative lead with the TEDxBeaconStreet organizing team, she worked to help innovators and changemakers share their groundbreaking ideas and turn them into action. Mission Haiku: The Poetry of Mission Statements By Christopher Finney Mission Haiku- the Poetry of Mission Statements Mayflower 400 Commemorations in UK to Showcase Wampanoag Stories A Native American Microgrid in Humboldt County Keeps Energy Flowing By Sean Buffington By Kevin Walker America Has a Real Hard Time Hearing Children—Check... A Top-Level Domain Looking for a Home: Dot.Org Still Up for... Sacred Fires Lit to Evict Pipeline: We Are All the Answer	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	1,783
Home > Chandelier > Handmade Painted Driftwood Chandelier D29" This fabulous looking Handmade Painted Driftwood Chandelier is made out of natural driftwood, carefully selecting and combining the pieces to create organic accents with painted gray finish. It looks so good and would be a perfect addition to your home, cabin or lodge decor. Nature makes it durable, hand made keeps it unique... no two pieces alike! Made of driftwood. Measures D29"x12.5".	{ "redpajama_set_name": "RedPajamaC4" }	2,922
Q: Hiding dropdown box after clicking on menu item in dropdown box I am new to JavaScript. I created a drowpdown menu. Whenever I click to open the drop downmenu comes and if I click again it hides. I want JavaScript code that whenever I click on the menu item from the dropdown menu it should hide. Code: /* ----- JavaScript ----- / var menuButton = document.getElementById("menu-btn"); var dropBox = document.getElementById('drop-box'); var menuItem = document.getElementsByClassName('sub-menu-item'); dropBox.style.display = 'none'; menuButton.onclick = function() { if (dropBox.style.display == 'none') { dropBox.style.display = 'block'; } else { dropBox.style.display = 'none'; } }; / ----- CSS ----- / #menu-btn { margin: 20px auto; width: 200px; height: 50px; background-color: green; text-align: center; padding-top: 30px; cursor: pointer; } #drop-box { width: 191px; position: absolute; left: 697px; top: 100px; margin: 0; padding: 0; } #drop-box li { list-style: none; } .linkBtn { width: 100%; background-color: orange; display: block; margin: 3px; padding: 5px; color: black; text-align: center; } .linkBtn:hover { background-color: green; } <!----- HTML -----> <div id="menu-btn" onClick="function();"> click to open </div> <ul id="drop-box"> <li class="sub-menu-item"><a href="#" class="linkBtn">item 01</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 02</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 03</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 04</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 05</a></li> </ul> My question is whenever what code can I give in javascript that can hide the dropdown menu after clicking on menu item? A: <html> <head> <style> #menu-btn{ margin: 20px auto; width:200px; height:50px; background-color:green; text-align:center; padding-top:30px; cursor:pointer; } #drop-box{ width:191px; position:absolute; left: 697px; top: 100px; margin:0; padding:0; } #drop-box li{ list-style:none; } .linkBtn{ width:100%; background-color:orange; display:block; margin:3px; padding:5px; color:black; text-align:center; } .linkBtn:hover{ background-color:green; } </style> <script> function myFunction() { var menuButton = document.getElementById("menu-btn"); var dropBox = document.getElementById('drop-box'); var menuItem = document.getElementsByClassName('sub-menu-item'); dropBox.style.display = 'none'; menuButton.onclick = function() { if(dropBox.style.display == 'none'){ dropBox.style.display = 'block'; } else { dropBox.style.display = 'none'; } }; } </script> </head> <body> <div id="menu-btn" onClick="myFunction();"> click to open </div> <ul id="drop-box"> <li class="sub-menu-item"><a href="#" class="linkBtn">item 01</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 02</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 03</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 04</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 05</a></li> </ul> </body> </html> Please check this code A: Bind Item click as well please find below snippet for more information / ----- JavaScript ----- / var menuButton = document.getElementById("menu-btn"); var dropBox = document.getElementById('drop-box'); var menuItem = document.getElementsByClassName('sub-menu-item'); dropBox.style.display = 'none'; menuButton.onclick = function() { if (dropBox.style.display == 'none') { dropBox.style.display = 'block'; } else { dropBox.style.display = 'none'; } }; //bind item click dropBox.onclick = function() { dropBox.style.display = 'none'; }; //Second way on li click for (var i=0; i<menuItem.length; i++) { menuItem[i].addEventListener('click', function() {dropBox.style.display = 'none';},false); } / ----- CSS ----- */ #menu-btn { margin: 20px auto; width: 200px; height: 50px; background-color: green; text-align: center; padding-top: 30px; cursor: pointer; } #drop-box { width: 191px; position: absolute; left: 697px; top: 100px; margin: 0; padding: 0; } #drop-box li { list-style: none; } .linkBtn { width: 100%; background-color: orange; display: block; margin: 3px; padding: 5px; color: black; text-align: center; } .linkBtn:hover { background-color: green; } <!----- HTML -----> <div id="menu-btn" onClick="function();"> click to open </div> <ul id="drop-box"> <li class="sub-menu-item"><a href="#" class="linkBtn">item 01</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 02</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 03</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 04</a></li> <li class="sub-menu-item"><a href="#" class="linkBtn">item 05</a></li> </ul> A: This is how you can do it by javascript: document.querySelectorAll(".sub-menu-item").forEach(function(item, index){ document.querySelectorAll(".sub-menu-item")[index].addEventListener("click", function(){ document.querySelector("#drop-box").style.display = 'none'; }) }); Here is the JSFiddle demo A: javascript ========== var menuButton = document.getElementById('menu-btn'); var dropBox = document.getElementById('drop-box'); var menuItem = document.getElementsByClassName('sub-menu-item'); dropBox.style.display = "none"; menuButton.onclick = function() { if (dropBox.style.display == 'none' \|\| dropBox.style.display == ''){ dropBox.style.display = 'block'; } else { dropBox.style.display = 'none'; } }; css ===== #menu-btn { margin: 20px auto; width: 200px; height: 50px; background-color: green; text-align: center; padding-top: 30px; cursor: pointer; } #drop-box { width: 191px; position: absolute; left: 140px; top: 70px; margin: 0; padding: 0; } #drop-box li { list-style: none; } .linkBtn { width: 100%; background-color: orange; display: block; margin: 3px; padding: 5px; color: black; text-align: center; } .linkBtn:hover { background-color: green; }	{ "redpajama_set_name": "RedPajamaStackExchange" }	8,482
Les matériaux ionochromes, de manière similaire aux matériaux photochromes, thermochromes et autres matériaux chromiques, voient leur couleur modifiée en présence d'un facteur ionique et revenir à leur état initial lorsque ce facteur est ôté. Les ions sont des atomes électriquement chargés. Les cations sont chargés positivement (déficitaires en électron(s)) alors que les anions sont chargés négativement (excédentaires en électron(s)). Description Un flux d'ions au travers d'un matériau ionochrome induit une réaction/un changement de couleur de ce matériau. Ce matériau est, par de nombreux aspects, similaire aux matériaux électrochromes qui changent de couleur lorsqu'un flux d'électrons les traversent. Les électrons, comme les anions, portent des charges négatives. Ainsi, les matériaux électrochromes et ionochromes ont leur modification de couleur activée par des particules chargées. Les espèces ionochromes sont utilisables pour la détection de particules chargées. Des espèces ionochromes peuvent être utilisées comme indicateurs pour les titrages complexométriques. Sources Chromisme Indicateur complexométrique	{ "redpajama_set_name": "RedPajamaWikipedia" }	6,711
Бейлін — єврейське прізвище. Походить від жіночого імені Бейле. Бейлін Ісаак Григорович (1883—1965) — радянський вчений — фітопатолог та історик науки, який народився і певний час жив в Україні. Йосі Бейлін (1948) — ізраїльський політик, громадський діяч, депутат кнесету кількох скликань, який виступає за якнайшвидше вирішення арабо-ізраїльського конфлікту і створення Палестинської держави. Бейлін Павло Юхимович (1910—1988) — український радянський лікар і письменник. Стефанія Бейлін (1900(1906)-1989) — польська письменниця, перекладачка творів Андерсена та оглядачка фільмів Кароліна Бейлін (псевдоніми Марія Малишевська і Кароль Вітковіцький, 1899—1977) — польська письменниця та перекладачка з англійської літератури (перекладала Діккенса, Джерома) Ірвінг Берлін (справжнє ім'я — Ізраїль Ісідор Бейлін) (1988—1989) — американський композитор	{ "redpajama_set_name": "RedPajamaWikipedia" }	8,048
{"url":"https:\/\/www.geteasysolution.com\/derivative_of_x%5E(-1)sin(2x)","text":"# Derivative of x^(-1)sin(2x)\n\n## Derivative of x^(-1)sin(2x). Simple step by step solution, to learn. Simple, and easy to understand, so dont hesitate to use it as a solution of your homework.\n\nBelow you can find the full step by step solution for you problem. We hope it will be very helpful for you and it will help you to understand the solving process.\n\nIf it's not what You are looking for type in the derivative calculator your own function and let us solve it.\n\n## Derivative of x^(-1)sin(2x):\n\n(x^-1sin(2x))'(x^-1)'sin(2x)+x^-1(sin(2x))'x^-1(sin(2x))'-1x^(-1-1)sin(2x)x^-1(sin(2x))'-x^-2sin(2x)x^-1cos(2x)(2x)'-x^-2sin(2x)x^-1cos(2x)((2)'x+2(x)')-x^-2sin(2x)x^-1cos(2x)(0x+2(x)')-x^-2sin(2x)x^-1cos(2x)(0x+21)-x^-2sin(2x)x^-12cos(2x)-x^-2sin(2x)2x^-1cos(2x)-(x^-2sin(2x))`\nThe calculation above is a derivative of the function f (x)","date":"2019-11-20 13:08:52","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\": 0.3795899748802185, \"perplexity\": 290.7637947428847}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496670558.91\/warc\/CC-MAIN-20191120111249-20191120135249-00135.warc.gz\"}"}	null	null
Longman Dictionary Of Modern English It is a excellent starter dictionary for youngsters and I would suggest this dictionary to everyone. Fully revised and completely updated with fresh pictures and a vibrant, bold, modern design, this new edition of My Initially Dictionary is the ideal go-to reference book for young children age five and up. When Webster's speller was initial see this website published, politicians had been debating the elimination of English altogether. Some advocated adopting German when other individuals wanted to invent a new language altogether. Webster offered a compromise, envisioning a new, sanctified version of English to go with their new, independent identity. An avowed nationalist and born-again Christian, Webster was not an unbiased lexicographer. They retain and generate Swagger-named tools that take benefit of the OpenAPI Specification . These days, most proprietary and open source "swagger-named" tools are now supporting at least OpenAPI 3.. If they don't, I advise sending a pull request to propose an update , request an upgrade , or not use them any longer. Conversely, the plate showing the coins of the world's crucial nations swiftly proved to be ephemeral. The company entered electronic publishing in the early 1960s, and initially, licensed electronic versions of its solutions to government, academic, and industrial institutions. In the 1970s, Merriam-Webster started providing word lists for computer spell-checking computer software. Considering that that time, the enterprise has participated in quite a few partnerships to create electronic products. Webster's International Dictionary was an huge achievement—and a large, heavy book. He not too long ago spoke about the project and the history and value of AAE on Wisconsin Public Radio. John Baugh, the Margaret Bush Wilson Professor in Arts & Sciences at Washington University in St. Louis, has been named to the advisory board of the 1st edition of the Oxford Dictionary of African American English . Priority support – Get expedited support for any app-connected challenges. Tap to Translate enables you to appear up entries in other apps. Voice search aids you when you do not know how an entry is spelled. Webster completed his dictionary during his year abroad in 1825 in Paris, and at the University of Cambridge. His 1820s book contained 70,000 words, of which about 12,000 had by no means appeared in a dictionary ahead of. As a spelling reformer, Webster believed that English spelling guidelines had been unnecessarily complex, so his dictionary introduced American English spellings, replacing colour with colour, waggon with wagon, and centre with center. He also added American words, like skunk and squash, that did not appear in British dictionaries. At the age of 70 in 1828, Webster published his dictionary it sold poorly, with only 2,500 copies, and place him in debt. A screenshot of the very first version of the OED second edition CD-ROM computer software. He retired in 2013 and was replaced by Michael Proffitt, who is the eighth chief editor of the dictionary. Many volunteer readers ultimately lost interest in the project, as Furnivall failed to preserve them motivated. The new format of the Dictionary is hugely appreciative as it gives a option of setting up ones favourite words and to view them all when 1 wants. Also, to be fair, the developer did respond to my original derogatory assessment by email, which sat unread on an unused server for six weeks. Their instructions for restoration, clearly properly-intended, had been not helpful. Austin explores the intersection of lexicographical and poetic practices in American literature, and attempts to map out a "lexical poetics" utilizing Webster's dictionaries. He shows the methods in which American poetry has inherited Webster and drawn upon his lexicography to reinvent it. Webster's Collegiate® Dictionary is updated annually and has been absolutely re-edited and revised just about every 10 to 12 years. Theoretically for that reason prosodic pitch and lexical tone could be simultaneously transcribed in a single text, even though this is not a formalized distinction. For instance, C# is a word-final consonant, %V a post-pausa vowel, and T% an IU-final tone . ⟩ at the beginning or finish of a word to indicate that it applies to the complete word. Glottal closureAdditional diacritics are supplied by the Extensions to the IPA for speech pathology. In some languages, plosives can be double-articulated, for instance in the name of Laurent Gbagbo. These selections only function if you select to show the transcription above each and every word. Some issue intriguing about game, make everybody happy. JavaScript is a lightweight interpreted programming language with very first-class functions. Early in the 17th century, nevertheless, "officious" began taking on a unfavorable sense to describe a particular person who presents unwanted enable. This pejorative sense has driven out the original "eager to assistance" sense to turn into the predominant which means of the word in Modern English. "Officious" can also mean "of an informal or unauthorized nature," but that sense is not specifically widespread. Right now, you could see "bonnyclabber" as a encouraged substitute for buttermilk in a recipe for Irish soda bread . WordsAPI might also, in the future, supply new services and/or options through the Site . Such new capabilities and/or solutions shall be subject to the terms and situations of this Agreement. The following terms and conditions govern all use of the WordsAPI.com web site and all content material, solutions and products obtainable at or by means of the web page, like, but not restricted to, the WordsAPI API . Ought to the API not presently suit your requirements, we also present customization and offline licensing of datasets in more than 35 languages from Afrikaans to Vietnamese, just make contact with us or take a look at our Licensing page for extra facts. Ballotpedia functions 391,465 encyclopedic articles written and curated by our qualified staff of editors, writers, and researchers. Click right here to contact our editorial employees, and click right here to report an error. Click here to contact us for media inquiries, and please donate right here to assistance our continued expansion. It then becomes a quest to locate the correct definition, if not the etymological origins, of a certain word. Getting somebody who loves and collects words, the OED is the ideal resource for a person like me! 😁 I feel that the OED app is worth just about every cent that 1 decides to place into it. Oxford Bibliographies Oxford Bibliographies delivers faculty and students alike with a seamless pathway to the most precise and reputable resources for a variety of academic subjects. Every write-up in our database is an authoritative guide to the existing scholarship, written and reviewed by academic authorities, with original commentary and annotations. A guide to the meaning, history, and pronunciation of much more than 600,000 words in English from present speech to previous usage. Tags: dictionarylookupoxford 2022 Propolis Global Marketplace Overview These Days Expertise For Sport, Lessons For Life Turlocks 13th Congressional District Turning Into Battleground Race Next Garden Spade "the King Of Spades" Made In The Usa Previous The Aluminum Smelting Approach Explained	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	1,247
\section{Introduction} The human brain comprises networks of regions that interactively process information during cognitive processing \citep{ccukur2013attention}. In turn, correlated activity within individual functional networks has been associated with unique mental states \citep{yan2019reduced, he2018reconfiguration}. Functional MRI (fMRI) is a powerful modality to examine functional networks as it can non-invasively measure whole-brain blood-oxygen-level-dependent (BOLD) signals consequent to neural activity at high spatio-temporal resolution \citep{bai2008default, yuan2008abnormal}. In fMRI studies, functional connectivity (FC) measures are used to assess similarity of BOLD signals among brain regions \citep{cambria2012hourglass, andreu2015big, zhang2015altered, gu2021eeg, yan2018abnormal}. The traditional approach to map FC measures onto mental states is then based on conventional methods such as logistic regression and support vector machines (SVM) \citep{mourao2005classifying, rashid2016classification, dosenbach2010prediction, rosenberg2016neuromarker}. Unfortunately, conventional methods are often insufficiently sensitive to the intricate information patterns in whole-brain fMRI time series \citep{liu2017survey}. In recent years, the success of deep learning (DL) models at exploring features in high-dimensional datasets has motivated their adoption for fMRI analysis as an alternative to conventional methods \citep{plis2014deep, kawahara2017brainnetcnn, li2021braingnn, heinsfeld2018identification, kim2016deep}. Earlier attempts in this domain have proposed shallow multi-layer perceptron (MLP) \citep{shen2010discriminative, eslami2019auto} and Boltzmann machine (BM) models \citep{hjelm2014restricted, plis2014deep}. Later studies have adopted deeper architectures based on convolutional neural network (CNN) \citep{zhao2017automatic, kawahara2017brainnetcnn,meszlenyi2017resting}, graph neural network (GNN) \citep{shao2021classification,li2021braingnn, saeidi2022decoding,gadgil2020spatio, kipf2016semi, qu2021brain}, and transformer \citep{yu2022disentangling, malkiel2021pre, nguyen2020attend,dai2022brainformer, bedel2022bolt} models for improved performance. Typically, these models start by constructing construct a set of nodes corresponding to brain regions defined based on an anatomical or functional atlas \citep{tzourio2002automated, rolls2015implementation, laird2009ale}, and receive input features at these nodes based on the FC strength among brain regions \citep{kim2020understanding,li2021braingnn}. A common approach has been to employ static FC features for derived from aggregate correlation measures across the entire duration of fMRI time series \citep{gan2021brain, li2021braingnn}. Yet, this approach is insufficiently sensitive to the dynamic inter-regional interactions in the human brain during resting-state or cognitive tasks \cite{norman2006beyond}. While alternative strategies have recently been proposed to assess the temporal variability in FC features, these methods commonly consider instantaneous signal correlations across local time windows within the time series \citep{kim2021learning, savva2019assessment, handwerker2012periodic}. As such, they do not possess explicit mechanisms to capture delayed correlations between brain regions that can be present in fMRI times series due to hierarchical cognitive processing in the brain or hemodynamic lags in BOLD measurements \citep{celik2021cortical}. In this study, we introduce a plug-in graphical neural network, GraphCorr, that provides enhanced input features to learning-based fMRI models so as to boost their sensitivity to dynamic, lagged inter-regional interactions. To capture dynamic changes in interactions, GraphCorr leverages a novel node embedder module based on a transformer encoder that computes hierarchical embeddings of windowed BOLD signals across the time series. To capture lagged interactions between brain regions, GraphCorr employs a novel lag filter module that computes nonlinear features of cross-correlation between pairs of nodes across a range of time delays. The graph model is initialized with node features taken as embeddings from the node embedder module, and with edge weights taken as lag features from the lag filter module. Afterwards, a message passing algorithm is used to compute enhanced node embeddings that account for dynamic, lagged inter-regional interactions. Here, we demonstrate GraphCorr for gender classification from fMRI scans in two public datasets: the Human Connectome Project (HCP) dataset \citep{van2013wu} and the ID1000 dataset from Amsterdam Open MRI Collection (AOMIC) \citep{snoek2021amsterdam}. GraphCorr is coupled as a plug-in to state-of-the-art baseline models for fMRI analysis including SAGE \citep{hamilton2017inductive}, BrainGNN \citep{li2021braingnn}, BrainNetCNN \citep{kawahara2017brainnetcnn} and GCN \citep{kipf2016semi}. Significantly enhanced performance is obtained from each baseline model when coupled with GraphCorr. We devise an explanatory analysis approach for GraphCorr to interpret the time frames and brain regions that most significantly contribute to classification decisions. We show that GraphCorr improves explainability of baseline models, resulting in interpretations that are more closely aligned with prominent neuroscientific findings from the literature. We also demonstrate the benefits of GraphCorr-derived features against features extracted via plug-in recurrent neural networks (RNN) and dynamic FC features computed directly from BOLD signals. \section{Related Work} Cognitive processes elicit broadly distributed response patterns across the human brain \citep{woolrich2001temporal}. In turn, process-related information such as stimulus or task variables can be decoded by analyzing resultant multi-variate BOLD signals \cite{bolling2011enhanced,shahdloo2020biased}. Initial studies in this domain employed relatively simpler, traditional machine learning (ML) methods for fMRI analysis \citep{norman2006beyond,bu2019investigating}. These traditional methods rely heavily on feature selection procedures to cope with the intrinsically high dimensionality of fMRI data \citep{xie2009brain}. Arguably, FC features among brain regions have been most commonly used to capture discriminative information about cognitive processes \citep{shen2010discriminative, kawahara2017brainnetcnn, kim2020understanding, li2021braingnn}. Many studies have reported that external variables or disease states can be detected given FC features of individual subjects under resting state \citep{wee2012identification, smith2013functional}, cognitive tasks \citep{mourao2005classifying, li2021braingnn}, or both \citep{rosenberg2016neuromarker}. Given their earlier success in fMRI analysis, FC features have also been pervasively adopted in recent DL methods that leverage more complex models to enhance performance \citep{riaz2020deepfmri, zeng2018multi}. A pervasive approach in DL-based fMRI analysis relies on static FC features as model inputs, where FC between a pair of regions is taken as the aggregate correlation of their BOLD signals across the entire scan. To extract hierarchical latent representations of these features, earlier studies have proposed either relatively compact fully-connected architectures including BM and MLP models \citep{hjelm2014restricted, plis2014deep, shen2010discriminative, eslami2019auto}, or computation-efficient deep architectures including CNN models \citep{meszlenyi2017resting, kawahara2017brainnetcnn}. Later studies have considered GNN models given their natural fit to analyzing fMRI data that follows an intrinsic connectivity structure \citep{wu2020comprehensive, liu2014distributed, cai2010graph, han2019gcn, li2021braingnn}. These DL methods have all enabled substantial performance improvements in fMRI analysis over traditional methods. Yet, analyses rooted in static FC features can still yield suboptimal sensitivity to fine-grained temporal information across the fMRI time series \citep{rashid2016classification, meszlenyi2017resting}. To improve temporal sensitivity, several alternative strategies have been proposed that incorporate time-varying features to DL models for fMRI analysis. A first group of methods pre-compute FC features over moving windows across the time series based on standard correlation measures, and concatenate them across windows to form a higher-dimensional input \citep{sakouglu2010method, allen2014tracking, gadgil2020spatio, savva2019assessment, handwerker2012periodic}. While these dynamic FC features promise enhanced temporal sensitivity, they result in elevated complexity due to their intrinsic dimensionality that can degrade model performance. A second group of methods instead provide voxel-level BOLD signals spatially encoded via a CNN module. Spatially-encoded BOLD signals are then processed with RNN or transformer models to extract the time-varying information \citep{malkiel2021pre, kim2021learning}. Yet, CNN modules based on voxel-level inputs can be difficult to train from scratch under limited data regimes. A third group of methods retain static FC features as their input, albeit augment them with dynamic features captured by RNN modules that directly encode BOLD signals \citep{kim2021learning}. Besides elevated model complexity, these methods can suffer from intrinsic limitations of RNNs in terms of vanishing/exploding gradients over the extensive number of time steps in typical fMRI scans \citep{pascanu2013difficulty, ismail2019input}. Importantly, a common attribute of these previous approaches is that they primarily consider the temporal variation in instantaneous correlations among brain regions. However, they can elicit suboptimal sensitivity as they lack explicit mechanisms to capture delayed inter-regional interactions that can occur due to hierarchical processing or hemodynamic delays \citep{celik2021cortical}. Here, we propose to improve the temporal sensitivity of downstream fMRI analysis models by integrating a novel plug-in GNN, GraphCorr. The proposed GraphCorr method uses a novel node embedder module to find contextual embeddings of dynamic FC features based on instantaneous correlations of windowed BOLD signals; and it uses a novel lag filter module to compute embeddings of cross-correlation features of windowed BOLD signals across various time delays. Following a message passing algorithm across the graph, GraphCorr provides enhanced input features that preserve dynamic, delayed correlations among brain regions to a downstream analysis model so as to improve its performance. Unlike methods based on static FC features \citep{kawahara2017brainnetcnn,li2021braingnn}, GraphCorr leverages dynamic FC features to capture the variability in connectivity among brain regions. Unlike methods that receive multiple sets of dynamic FC features across separate time windows \citep{zhang2017hybrid, kim2021learning,gadgil2020spatio}, GraphCorr fuses its node features across time windows to lower model complexity without sacrificing performance. Unlike methods that employ recurrent architectures that involve sequential processing \citep{kim2021learning}, GraphCorr leverages a transformer encoder on dynamic FC features that enables efficient parallel processing. Unlike methods that solely focus on instantaneous signal correlations \citep{kawahara2017brainnetcnn}, GraphCorr adopts an explicit lag filter mechanism to learn delayed cross-correlations among brain regions. \section{GraphCorr} \label{corr} Analysis procedures for fMRI time series typically start by defining a collection of $R$ regions of interest (ROI) across the brain based on an anatomical atlas \citep{li2021braingnn, kim2021learning}. Voxel-level BOLD signals within each ROI are then averaged to derive ROI-level signals, resulting in $\mathbf{B} \in \mathbb{R}^{R \times T}$ as the matrix of BOLD signals where $T$ denotes the number of time frames. Static FC features are conventionally computed based on Pearson's correlation coefficient of these BOLD signals across ROIs: $\mathbf{sFC}_{i,j} = \mathrm{Corr}(\mathbf{B}_{i,\cdotp},\mathbf{B}_{j,\cdotp})$, where $\mathbf{sFC} \in \mathbb{R}^{R \times R}$ and $i,j$ are ROI indices. Many previous traditional and learning-based methods use downstream classification models on static FC features, which result in suboptimal temporal sensitivity. Instead, here we propose to extract dynamic FC features of BOLD signals based on a novel GNN plug-in, and to use these enhanced features to improve the performance of downstream classification models. The proposed GraphCorr method forms a graph structure to represent brain connectivity features, leverages node embedder and lag filter modules to capture dynamic, lagged correlations in BOLD signals, and finally performs message passing on the graph to compute enhanced features (Fig. \ref{fig:corrfig}). The methodological components and procedures in GraphCorr are described below. \begin{figure}[h!] \centering \includegraphics[width=0.95\linewidth]{figures/corr.png} \caption{Overview of GraphCorr. \textbf{A.} GraphCorr utilizes two parallel modules to extract dynamic, lagged features of inter-regional correlations across the brain. The node embedder module receives as input time-windowed BOLD signals, and uses a transformer encoder to compute node embeddings of dynamic FC features $\mathbf{EMB} \in \mathbb{R}^{R \times D \times W}$. The lag filter module also receives as input time-windowed BOLD signals, and it computes lag activations due to cross-correlation across a range of lag values $\mathbf{LAG} \in \mathbb{R}^{E_n \times W \times k}$. Cross-correlation is calculated only for connected node pairs ($e_{i,j}=1$). \textbf{B.} To consolidate the extracted feature sets on a graph, node embeddings are taken as node features and lag activations are taken as edge weights. A message passing algorithm is then run on the graph to produce enhanced FC features in an output feature matrix, $\mathbf{OUT} \in \mathbb{R}^{R \times (D \times (k+1))}$.} \label{fig:corrfig} \end{figure} \subsection{Graph formation} \label{graph} As a learning substrate, GraphCorr first forms a graph $G(N,E)$ with $N$ and $E$ denoting nodes and edges, respectively. The node set $N = \{r_i \, \| \, i = 1, ..., R\}$ includes ROIs defined according to the atlas, whereas the binary edge set is given as $E = \{e_{i,j} = 1 \, \| \, i = 1, ..., R; j \in \mathcal{N}(i)\}$ where $\mathcal{N}(i)$ is the neighborhood of the $i$-th node. Edges are defined by thresholding to retain the strongest $z\%$ of correlation coefficients in $\mathbf{sFC}$ while excluding self connections, resulting in $E_n$ number of edges. The node features $\mathbf{F} = \{{f}_i \, \| \, i = 1, ..., R\}$ are initialized as the time-windowed BOLD signals at each corresponding node to capture local dynamics in the fMRI times series. For this purpose, the scan containing $T$ time frames is split into $W$ windows of size $T_w$ and stride value $s$: \begin{equation} W = \lfloor \frac{T - T_w}{s} \rfloor, \end{equation} resulting in a feature tensor of $\mathbf{F} \in \mathbb{R}^{R \times T_w \times W}$. \subsection{Architecture} \label{modules} To sensitively extract dynamic connectivity features, GraphCorr utilizes a novel node embedding module. To also capture delayed connectivity features, GraphCorr utilizes a novel lag filter module. The two modules are detailed below. \textbf{Node embedder module:} Receiving as input time-windowed BOLD signals, this module computes latent representations of dynamic FC features (Fig. \ref{fig:nodeEmbfig}). First, dynamic FC features are extracted from the time-windowed BOLD signals as: $\mathbf{FC}_{i,j,w} = \mathrm{Corr}(\mathbf{F}_{i,\cdotp,w},\mathbf{F}_{j,\cdotp,w})$ where $w \in \{1,...,W\}$ indicates window index, $i \in \{1,...,R\}$, $j \in \{1,...,R\}$ denote node indices. These FC features are then processed with a transformer encoder where windows across the fMRI time series correspond to the sequence of transformer tokens. Attention calculations are performed on window-specific keys $K_w \in \mathbb{R}^{R \times d}$, queries $Q_w \in \mathbb{R}^{R \times d}$ and values $V_w \in \mathbb{R}^{R \times d}$ derived via learnable linear projections ${f_q, f_k}$ and ${f_v}$: \begin{gather} Q_w = U_q( \{ \mathbf{FC}_{1,\cdotp,w}, \mathbf{FC}_{2,\cdotp,w}, ..., \mathbf{FC}_{R,\cdotp,w} \} ) ,\nonumber\\ K_w = U_k( \{ \mathbf{FC}_{1,\cdotp,w}, \mathbf{FC}_{2,\cdotp,w}, ..., \mathbf{FC}_{R,\cdotp,w} \} ) ,\nonumber\\ \label{eqn:qkv} V_w = U_v( \{ \mathbf{FC}_{1,\cdotp,w}, \mathbf{FC}_{2,\cdotp,w}, ..., \mathbf{FC}_{R,\cdotp,w} \} ), \end{gather} where $d$ is the dimensionality of each attention head. The above computations can be performed separately for $H$ attention heads. The window-specific attention matrix $\mathbf{A}_w \in \mathbb{R}^{R \times R}$ is then derived as \citep{dalmaz2022resvit}: \begin{gather} \mathbf{A}_w = \mathrm{Att}(Q_w,K_w,V_w) = \mathrm{Softmax}(\frac{Q_w K_w^{\intercal}}{\sqrt{d}}) V_w. \label{eqn:attn} \end{gather} Attention matrices are concatenated across attention heads and propagated to an MLP block following layer normalization: \begin{gather} \textbf{EMB}_w = \mathrm{MLP}(\mathbf{A}_w) = \mathrm{GELU}(\mathbf{A}_w \mathbf{M}_{1}) \mathbf{M}_{2} \label{eqn:mlp} \end{gather} where $\mathbf{M}_{1} \in \mathbb{R}^{(R \times D)}$ and $\mathbf{M}_{2} \in \mathbb{R}^{(D \times D)}$ denote MLP model parameters, $\mathrm{GELU}$ is the Gaussian activation unit, and $\mathbf{EMB} \in \mathbb{R}^{R \times D \times W}$ are window-specific node embeddings, and $D$ is the embedding dimensionality with $D < R$. \begin{figure}[h!] \centering \includegraphics[width=0.7\linewidth]{figures/nodeEmb.png} \caption{The node embedder module. The time-windowed FC features $\mathbf{FC}_{\cdotp, \cdotp, w} \in \mathbb{R}^{R \times R}$ at window $w$ are processed with a transformer encoder with multi-head self-attention (MHSA), layer normalization, and multi-layer perceptron (MLP) layers. The output is a node embedding matrix $\mathbf{EMB}_{\cdotp, \cdotp, w} \in \mathbb{R}^{R \times D}$ where $D < R$ denotes the embedding dimensionality.} \label{fig:nodeEmbfig} \end{figure} \textbf{Lag filter module:} Receiving as input time-windowed BOLD signals, this module computes cross-correlation features across a range of temporal delays (Fig. \ref{fig:lagFilter}). For this purpose, initial node features from the graph formation stage are zero-padded across the time dimension: \begin{gather} \mathbf{X}_{i,\cdotp,w} = [\mathbf{0}_{(1\times m)}, \mathbf{F}_{i,\cdotp,w}, \mathbf{0}_{(1\times m)}]. \end{gather} where $\mathbf{X} \in \mathbb{R}^{R \times (T_w+2m) \times W}$, and $m$ defines the range of delays $\tau \in \{ -m, -m+1, ...., m-1, m\}$ that will be considered in the module. First, cross-correlations are computed between pairs of nodes connected by $e_{ij}$ at each lag value separately: \begin{gather} \rho_{i,j,w,\tau} = \mathrm{Corr} ( \mathbf{X}_{i,\cdotp,w},\mathbf{X}_{j,\cdotp,w},\tau) \label{eqn:lag} \end{gather} Afterwards, learnable lag filters $\mathbf{P}_{LF} \in \mathbb{R}^{(2m+1) \times k}$ with $k$ denoting the number of filters are used to map cross-correlations onto lag activations: \begin{gather} \label{eqn:lag2} \textbf{LAG}_w = \mathrm{GELU}(\rho_{\cdotp,\cdotp,w,\cdotp} \mathbf{P}_{LF}) \end{gather} where $\mathbf{LAG} \in \mathbb{R}^{E_n \times W \times k}$ are window-specific lag activations. \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{figures/lagFilter2.png} \caption{The lag filter module. Cross-correlation of time-windowed BOLD signals at window $w$ are computed for delays $\tau \in \{ -m, -m+1, ...., m-1, m\}$, where $m$ defines the range. This computation is only performed for pairs of connected nodes ($e_{i,j}=1$). Afterwards, cross-correlation values $\rho_{\cdotp,\cdotp,w,\cdotp} \in \mathbb{R}^{R \times R \times (2m+1)}$ are linearly transformed onto with a learnable filter $M_{LF} \in \mathbb{R}^{(2m+1) \times k}$ onto window-specific lag activations $\textbf{LAG}_{\cdotp,\cdotp,w} \in \mathbb{R}^{R \times R \times k}$. } \label{fig:lagFilter} \end{figure} \subsection{Graph learning} \label{message} The node embedder produces time-windowed node embeddings, $\textbf{EMB}$, that reflect instantaneous inter-regional correlations. The lag filter produces time-windowed lag activations, $\textbf{LAG}$, that reflect reflect delayed inter-regional correlations. To consolidate these feature sets on the graph, node embeddings are taken as node features and lag activations are taken as edge weights (Fig. \ref{fig:corrfig}). A message passing algorithm is then run on the graph to compute enhanced FC features. To do this, a message tensor $\textbf{MES} \in \mathbb{R}^{R \times (D\times k) \times W}$ is computed between pairs of connected nodes $(r_i,r_j)$ as: \begin{gather} \textbf{MES}_{i,j,\cdotp,w} = {\textbf{EMB}_{j,\cdotp,w}} \textbf{LAG}^{\intercal}_{i,j,\cdotp,w} \label{eqn:msg} \end{gather} Messages are first averaged across windows, and then propagated to a target node $r_i$ that sums all messages from one-hop vicinity nodes $j \in \mathcal{N}(i)$: \begin{gather} {\textbf{AGG}_i} = \sum_{j\in \mathcal{N}(i)} \frac{1}{W}\sum_{w=1}^{W}{\textbf{MES}_{i,j,\cdotp,w}}. \end{gather} This aggregate message is the concatenated with the window-averaged node embedding at $r_i$: \begin{gather} \label{eqn:msg2} \mathbf{OUT}_i = [\frac{1}{W}\sum_{v=1}^{W}{\textbf{EMB}_{i,\cdotp,w}}, \textbf{AGG}_i ], \end{gather} where $\mathbf{OUT}_i \in \mathbb{R}^{D \times (k+1)}$ denotes enhanced features for $r_i$. \section{Methods} \subsection{Experimental procedures } \label{Dataset} Demonstrations were performed on fMRI data from the HCP S1200 release\footnote{\url{https://db.humanconnectome.org}} \citep{van2013wu} and ID1000 dataset from Amsterdam Open MRI Collection (AOMIC)\footnote{\url{https://openneuro.org/datasets/ds003097/versions/1.2.1}} \citep{snoek2021amsterdam}. In the HCP dataset, preprocessed data from resting-state fMRI scans were analyzed. The first resting-state scan among four sessions was selected for each subject, excluding short scans with $T < 1200$. This resulted in a total of 1093 healthy subjects (594 female and 499 male). In the ID1000 dataset, preprocessed data from task-based fMRI scans recorded during movie watching were analyzed. All scans had a fixed duration of $T = 240$. A total of 881 healthy subjects were examined (458 female and 423 male). For both datasets, two alternative ROI definitions were considered, based on either the Schaefer atlas \citep{schaefer2018local} or the AAL atlas \citep{tzourio2002automated}. The Schaefer atlas includes $R = 400$ ROIs within 7 intrinsic networks, whereas the AAL atlas defines $R = 116$ ROIs. Experiments were conducted on a single NVIDIA Titan Xp GPU using the PyTorch framework. A nested cross-validation procedure was performed with 5 outer and 1 inner folds Domain adaptation procedures can be employ to improve reliability \citep{elmas2022federated}. Data were three-way split into a training set (70\%), a validation set (10\%) and a test set (20\%) with no subject overlap between the sets. For fair comparison, all models were trained, validated and tested on identical data splits. All models were trained based on cross-entropy loss. For each model, hyperparameters were selected to maximize the average performance across the validation sets. A common set of hyperparameters that were observed to yield near-optimal performance were used across datasets and atlases \citep{dalmaz2022one}. Details regarding model implementations are discussed in Section \ref{compare}. \subsection{Comparative analysis} \label{compare} GraphCorr was demonstrated on several learning-based methods taken as downstream classification models including SAGE \citep{hamilton2017inductive}, GCN \citep{kipf2016semi}, BrainGNN \citep{li2021braingnn}, and BrainNetCNN \citep{kawahara2017brainnetcnn}. For each method, a vanilla downstream model was trained by providing static FC features as model input, and an augmented downstream model was separately trained where GraphCorr was employed as a plug in to provide model input. Vanilla and augmented models were obtained with identical training procedures. In all graph models, ROIs in a given brain atlas were taken as nodes, and edge selection was then performed based on correlations of BOLD signals. Edges whose correlation coefficients were in the top $z=2\%$ were retained, while remaining edges were discarded. The implementation details of the downstream models and GraphCorr are discussed below. \textbf{SAGE:} A GNN model was built based on a module with graph convolution pooling, and fully-connected layers \citep{hamilton2017inductive}. SAGE comprised a cascade of two graphical modules with hidden dimension of 250 and dropout rate of 0.5. Cross-validated hyperparameters were a learning rate of \num{3e-3}, 20 epochs, a batch size of 12. \textbf{GCN:} GCN is a GNN model based on graph convolution, pooling and fully-connected layers \citep{kipf2016semi}. GCN comprised a cascade of two graphical modules with hidden dimension of 100 and dropout rate of 0.5. Cross-validated hyperparameters were a learning rate of \num{5e-3}, 30 epochs, a batch size of 12. \textbf{BrainGNN:} BrainGNN is a GNN model based on ROI-aware graph convolution, pooling and fully-connected layers \citep{li2021braingnn}. A single graphical module with hidden dimension of 100 and dropout rate of 0.5 was used. Cross-validated hyperparameters were a learning rate of \num{8e-4}, 80 epochs, a batch size of 16. \textbf{BrainNetCNN:} BrainNetCNN is a CNN model based on convolutional layers with edge-to-edge and edge-to-node filters \citep{kawahara2017brainnetcnn}. The convolutional layers had hidden dimension of 32 and dropout rate of 0.1. Vanilla BrainNetCNN expects a 2D input of size $R \times R$ taken as the static FC matrix. When it was augmented with GraphCorr, its input dimensionality was modified as $R \times D(k+1)$ for compatibility. Cross-validated hyperparameters were a learning rate of \num{2e-4}, 20 epochs, a batch size of 16. \textbf{GraphCorr:} The node embedder module was built with a single-layer transformer encoder. Because the scan durations differed across HCP and ID1000, dataset-specific $T_w$ (window size) and $s$ (stride) were selected while common $m$ (maximum lag) and $k$ (filter count) were used. Accordingly, cross-validated parameters were ($T_w$=50, $s$=30, $m$=5, $k$=3) for HCP, ($T_w$=40, $s$=15, $m$=5, $k$=3) for ID1000. \subsection{Explanatory analysis} \label{interpret} To assess the influence of GraphCorr on interpretability, the vanilla and augmented versions of trained downstream models were examined. An explanation procedure was devised to identify the brain regions within the fMRI times series that most saliently contribute to the model decisions. First, a gradient-based approach was used to compute a saliency tensor summarizing inter-regional interactions \citep{arslan2018graph, kim2020understanding}. For vanilla models, gradients were computed with respect to static FC features $\mathbf{sFC}$: \begin{equation} \label{eqn:van} {{\mathbf{SAL}^{van}_{i,j}}} = \|\nabla_{\mathbf{sFC}_{i,j}} y_{van}\| \end{equation} where $\mathbf{SAL}^{van} \in \mathbb{R}^{R \times R}$ and $y_{van}$ denotes the model prediction. For augmented models, gradients were computed with respect to time-windowed FC features ${\mathbf{FC}}_{i,j,w}$: \begin{equation} \mathbf{SAL}^{aug}_{i,j,w} = \|\nabla_{\mathbf{FC}_{i,j,w}} y_{aug}\| \end{equation} where $\mathbf{SAL}^{aug} \in \mathbb{R}^{R \times R \times W}$ and $y_{aug}$ denotes the model prediction. Afterwards, an ROI-specific saliency score was computed by aggregating values across windows and interacting ROI dimensions of the saliency tensor: \begin{gather} \mathbf{rSAL}^{van} = \sum_{j=1}^{R}{\mathbf{SAL}^{van}_{\cdotp,j}}\\ \mathbf{rSAL}^{aug} = \sum_{j=1}^{R} ( \frac{1}{W} \sum_{w=1}^{W}{\mathbf{SAL}^{aug}_{\cdotp,j,w}} ) \end{gather} where $\mathbf{rSAL}^{van,aug} \in \mathbb{R}^{R}$. For saliency assessment at the level of functional brain networks, the seven intrinsic brain networks defined within the Schaefer atlas were used. For each network, ROI-specific saliency scores were averaged across the regions within the network to obtain a network saliency score per hemisphere. While unsigned ROI-specific saliency scores reflect the relative importance of each region on the model decision, they do not indicate whether the model output is driven by an increase or decrease in BOLD signals within the ROI. To address this question, a post-hoc logistic regression analysis was conducted. First, important windows in the fMRI time series were determined by aggregating values in the saliency tensor across ROI dimensions: \begin{gather} \mathbf{wSAL}^{aug} = \sum_{i=1}^{R} \sum_{j=1}^{R}{\mathbf{SAL}^{aug}_{i,j,\cdotp}} \\ {w}^{} = \mathop{\arg \max}\limits_{w} {\mathbf{wSAL}^{aug}} \end{gather} Here, $ \mathbf{wSAL}^{aug} \in \mathbb{R}^{W}$ denotes the window-specific saliency score used for important window selection. BOLD signals within the most important window were extracted, and thresholded according to intensity to select the top 5 time frames \citep{tagliazucchi2011spontaneous, tagliazucchi2012criticality, bedel2022bolt}. A logistic regression model was then fit to map the BOLD signal vector across ROIs onto the output class, i.e., performing the same task as the downstream model. The logistic model returns a weight for each ROI: a positive weight indicates that an increase whereas a negative weight indicates that a decrease in the ROI's BOLD signal elicits the downstream model's decision. \begin{table} \caption{Performance of downstream models on the HCP and ID1000 datasets with the Schaefer atlas. Results are listed as mean$\pm$std across test folds for vanilla and GraphCorr-augmented versions. Boldface indicates the better performing version of each model.} \label{SchaeferResults} \centering \resizebox{\columnwidth}{!}{ \begin{tabular}{ll ll ll} \toprule \multirow{2}{}{Model} & {}&\multicolumn{2}{c}{HCP} & \multicolumn{2}{c}{ID1000} \\ \cmidrule(lr){3-4} \cmidrule(lr){5-6} & {}& Acc (\%) & ROC (\%) & Acc (\%) & ROC (\%) \\ \midrule \multirow{2}{}{SAGE}& Vanilla & $75.2 \pm 2.84 $ & $85.29 \pm 1.67$ & $62.39 \pm 2.17$ & $68.59 \pm 3.76$ \\ {}& Augmented & $\textbf{89.57} \pm 0.68$ & $\textbf{94.27} \pm 1.98$ & $\textbf{81.7} \pm 2.67$ & $\textbf{87.02} \pm 2.10$ \\ \midrule \multirow{2}{}{GCN} & Vanilla & $79.14 \pm 2.93$ & $86.00 \pm 1.47$ & $67.84 \pm 2.95$ & $71.78 \pm 3.88$ \\ {}& Augmented & $\textbf{89.94} \pm 2.18$ & $\textbf{94.52} \pm 1.61$ & $\textbf{80.80} \pm 0.98$ & $\textbf{87.90} \pm 1.75$ \\ \midrule \multirow{2}{}{BrainGNN} & Vanilla & $72.83 \pm 1.98$ & $78.85 \pm 2.45$ & $62.50 \pm 1.80$ & $65.63 \pm 2.81$ \\ {}& Augmented & $\textbf{84.72} \pm 1.33$ & $\textbf{92.97} \pm 0.98$ & $\textbf{79.32} \pm 1.96$ & $\textbf{88.10} \pm 2.19$ \\ \midrule \multirow{2}{}{BrainNetCNN} & Vanilla & $82.52 \pm 2.80$ & $91.23 \pm 1.21$ & $ 75.45 \pm 2.01$ & $83.65 \pm 2.05$ \\ {}& Augmented & $\textbf{88.47} \pm 2.63$ & $\textbf{94.71} \pm 1.86$ & $\textbf{82.73} \pm 1.63$ & $\textbf{89.85} \pm 2.24$ \\ \bottomrule \end{tabular}} \end{table} \begin{table} \caption{Performance of downstream models on the HCP and ID1000 datasets with the AAL atlas. Results are listed as mean$\pm$std across test folds for vanilla and GraphCorr-augmented versions. Boldface indicates the better performing version of each model.} \label{AALResultTable} \centering \resizebox{\columnwidth}{!}{ \begin{tabular}{ll ll ll} \toprule \multirow{2}{}{Model} & {}&\multicolumn{2}{c}{HCP} & \multicolumn{2}{c}{ID1000} \\ \cmidrule(lr){3-4} \cmidrule(lr){5-6} & {}& Acc (\%) & ROC (\%) & Acc (\%) & ROC (\%) \\ \midrule \multirow{2}{}{SAGE}& Vanilla & $68.26 \pm 3.31 $ & $75.65 \pm 1.29$ & $62.84 \pm 1.81$ & $67.23 \pm 2.71$ \\ {}& Augmented & $\textbf{85.45} \pm 3.57$ & $\textbf{91.19} \pm 2.63$ & $\textbf{77.50} \pm 3.68$ & $\textbf{84.92} \pm 1.92$ \\ \midrule \multirow{2}{}{GCN} & Vanilla & $69.90 \pm 1.35$ & $75.85 \pm 1.06$ & $65.45 \pm 1.36$ & $70.96 \pm 1.23$ \\ {}& Augmented & $\textbf{84.36} \pm 3.17$ & $\textbf{88.99} \pm 2.83$ & $\textbf{79.43} \pm 3.45$ & $\textbf{85.43} \pm 2.49$ \\ \midrule \multirow{2}{}{BrainGNN} & Vanilla & $65.69 \pm 3.00$ & $71.79 \pm 2.97$ & $62.27 \pm 3.44$ & $66.42 \pm 4.24$ \\ {}& Augmented & $\textbf{80.60} \pm 2.67$ & $\textbf{89.32} \pm 2.67$ & $\textbf{75.11} \pm 0.56$ & $\textbf{83.14} \pm 1.41$ \\ \midrule \multirow{2}{}{BrainNetCNN} & Vanilla & $68.16 \pm 3.53$ & $74.76 \pm 2.08$ & $ 75.00 \pm 2.19$ & $81.44 \pm 3.27$ \\ {}& Augmented & $\textbf{83.99} \pm 2.92$ & $\textbf{91.25} \pm 2.65$ & $\textbf{78.98} \pm 1.83$ & $\textbf{86.16} \pm 0.84$ \\ \bottomrule \end{tabular}} \end{table} \section{Results} \label{results} \subsection{Comparative analysis} GraphCorr was demonstrated on downstream classification models based on SAGE \citep{hamilton2017inductive}, GCN \citep{kipf2016semi}, BrainGNN \citep{li2021braingnn}, and BrainNetCNN \citep{kawahara2017brainnetcnn}. A gender detection task was performed given resting-state fMRI scans in individual subjects. Performances of vanilla and augmented versions of downstream models on HCP and ID1000 datasets are listed in Table \ref{SchaeferResults} for the Schaefer atlas, and in Table \ref{AALResultTable} for the AAL atlas. In all examined cases, augmentation with GraphCorr significantly enhances the performance of downstream models (p$<$0.05, Wilcoxon signed-rank test). When ROIs are defined via the Schaefer atlas, GraphCorr enables (accuracy,\,ROC)\% improvements of (14.37,\,8.98)\% for SAGE, (10.80,\,8.52)\% for GCN, (11.89,\,14.12)\% for BrainGNN, and (5.95,\,3.48)\% for BrainNetCNN on HCP; and it enables improvements of (19.31,\,18.43)\% for SAGE, (13.32,\,16.12)\% for GCN, (16.82,\,22.47)\% for BrainGNN, and (7.28,\,6.20)\% for BrainNetCNN on ID 1000. When ROIs are defined via the AAL atlas, GraphCorr enables improvements of (17.19,\,15.54)\% for SAGE, (14.46,\,13.14)\% for GCN, (14.91,\,17.53)\% for BrainGNN, and (15.83,\,16.49)\% for BrainNetCNN on HCP; and it enable improvements of (14.66,\,17.69)\% for SAGE, (13.98,\,14.47)\% for GCN, (12.84,\,16.72)\% for BrainGNN, and (3.98,\,4.72)\% for BrainNetCNN on ID1000. We observe that for vanilla versions of the relatively simpler GNN models perform poorly against the more complex BrainNetCNN model. However, GraphCorr-augmented versions of these GNN models start outperforming the augmented BrainNetCNN. Thus, our results suggest that the feature extraction capabilities of vanilla GNN models might be suboptimal in comparison to CNN-based architectures, albeit a powerful feature extractor on the input side can mitigate this deficit in favor of GNN models. \subsection{Explanatory analysis} To assess the influence of GraphCorr on interpretability, an explanatory analysis was conducted separately on the trained vanilla and augmented downstream models. For this analysis, the HCP dataset and the Schaefer atlas were selected that have been broadly studied in the literature for intrinsic brain networks during resting state \citep{smith2013resting}. First, network saliency scores obtained in each hemisphere were compared between vanilla and augmented versions of SAGE, which generally maintains the highest performance after GraphCorr augmentation. Literature reports that BOLD signals across the sensorimotor network (SMN), the default mode network (DMN) and the visual network bilaterally across the two hemispheres carry discriminative information on subject gender \citep{zhang2018functional,kim2020understanding}. Accordingly, we reasoned that a successful downstream classification model for gender detection should focus on these networks. Vanilla SAGE shows somewhat heterogeneous results with significant salience in the DMN in the left hemipshere (LH); the attention network, the SMN, and the visual network in the right hemisphere (RH); but unexpectedly it also yields strong salience in the limbic network in both hemispheres ($p<0.05$, Wilcoxon signed-rank test) although it is not considered to carry information on gender. In contrast, augmented SAGE shows significant salience across the SMN, the DMN and the attention network in the RH; the visual network in both hemipsheres ($p<0.05$), without any salience in the limbic network. These results imply that GraphCorr helps improve interpretability of the downstream classification model by allowing it to focus on brain regions that carry task-relevant information. Significant saliency in a network indicates that BOLD signals across that network carry information about subject gender. Yet, it does not explain whether an increase or decrease in BOLD signals is evoked for individual genders. To address this question, a logistic regression analysis was conducted on BOLD signals extracted from the most important time window determined according to ROI saliency scores. Specifically, a logistic regression model was fit to detect subject gender given important BOLD signals. Fig. \ref{fig:brain} illustrates the ROI weights in the logistic model, where a positive weight indicates that elevated BOLD signals in the ROI are associated with female subjects, and a negative weight indicates that elevated BOLD signals in the ROI are associated with male subjects. Accordingly, ROI weights were inspected for the logistic regression analyses based on the augmented SAGE model. In females, elevated BOLD signals are identified in RH parietal DMN areas including posterior cingulate cortex (PCC), RH prefrontal DMN areas, LH prefrontal control areas and LH-RH SMN areas. In males, elevated BOLD signals are identified in LH prefrontal DMN areas, LH-RH dorsal attention areas (DAN), LH parietal and prefrontal control areas, and RH parietal and extrastriate visual areas. These findings are consistent with evidence that females have relatively higher activations across DMN areas including PCC, and that males have relatively higher activations in visual and attentional areas \citep{ritchie2018sex,allen2011baseline}. Our results are also consistent with recent studies suggesting that SMN and prefrontal regions show discriminate activation patterns across the two genders \citep{kim2020understanding}. \begin{figure}[h!] \centering \includegraphics[width=0.85\linewidth]{figures/brain3.png} \caption{Salient ROIs for gender detection assessed via the logistic regression analysis. Results are shown for the GraphCorr-augmented SAGE model on the HCP dataset with the Schaefer atlas. ROIs with the top 2\% saliency scores are marked. Red color indicates ROIs whose BOLD signals are elevated in female subjects, whereas blue color indicates ROIs whose BOLD signals are elevated in male subjects.} \label{fig:brain} \end{figure} \subsection{Ablation studies} Ablation studies were performed to assess the contribution of the individual design elements in GraphCorr to model performance. These analyses were conducted based on the SAGE model using the HCP dataset and the Schaefer atlas, i.e., the setting that yields the highest overall performance for gender detection. First, we assessed contributions of the node embedder module, lag filter module, and time windowing in GraphCorr. To ablate the node embedder module, node embeddings prior to message passing were initialized with the unlearned time-windowed FC matrix derived via conventional correlation measures on BOLD signals. To ablated the lag filter module, a single filter at zero lag was used within the module to consider only instantaneous correlations. To ablated time windowing, the entire fMRI time series was provided to GraphCorr with a single window of size equal to scan duration. Table \ref{ablationSchaefer} lists performance metrics for ablated variants of GraphCorr. We find that the node embedder module, the lag filter module and time windowing enable (accuracy,\, ROC)\% improvements of (5.31,\,3.2)\%, (0.65,\,0.17)\%, and (8.41,\,5.61)\%, respectively. Next, we assessed the benefits of GraphCorr over alternative plug-in approaches to improve the temporal sensitivity of downstream model. In particular, we considered providing the downstream model pre-computed dynamic FC features across time windows via conventional correlation measures \citep{zhang2017hybrid}, an RNN model based on LSTM layers \citep{dvornek2017identifying}, and an RNN model based on GRU layers \citep{gao2022age}. The feature dimensionality at the output of all plug-in models were identical to that for GraphCorr. Table \ref{alternativeRNN} lists performance metrics for different plug-in methods. GraphCorr outperforms all other plug-in methods, with (5.77,\,2.15)\% higher performance than the top-contending GRU method. \begin{table}[t] \caption{Performance for ablated variants of GraphCorr on the HCP dataset with the Schaefer atlas. Results are listed as mean$\pm$std across test folds for the downstream SAGE model. Boldface indicates top-performing variant.} \label{ablationSchaefer} \centering \resizebox{0.8\columnwidth}{!}{ \begin{tabular}{p{1.5cm} p{1.5cm} p{1.5cm} l l p{2cm} p{2cm}} \toprule \centering Node Embedder & \centering Lag Filter & \centering Windowing & {Accuracy (\%)} & {ROC (\%)} \\ \midrule \centering \xmark & \centering \xmark & \centering \xmark & $75.20 \pm 2.83$ & $85.29 \pm 1.67$ \\ \centering \cmark & \centering \xmark & \centering \xmark & $80.51 \pm 1.92$ & $88.49 \pm 1.54$ \\ \centering \cmark & \centering \cmark & \centering \xmark & $81.16 \pm 1.69$ & $88.66 \pm 2.07$ \\ \centering \cmark & \centering \cmark & \centering \cmark & $\textbf{89.57} \pm 0.68$ & $\textbf{94.27} \pm 1.98$ \\ \bottomrule \end{tabular}} \end{table} \begin{table}[t] \caption{Performance of competing plug-in methods on the HCP dataset with the Schaefer atlas. Results are listed as mean$\pm$std across test folds for the downstream SAGE model. Boldface indicates top-performing plug-in.} \label{alternativeRNN} \centering \resizebox{0.55\columnwidth}{!}{ \begin{tabular}{ccc} \toprule \multirow{1}{}{Plug-in} & \multirow{1}{}{Accuracy (\%)} & \multirow{1}{}{ROC (\%)} \\ \midrule \centering {Dynamic FC} & $81.06 \pm 2.94$ & $89.04 \pm 1.53$ \\ \centering {LSTM} & $83.26 \pm 2.05$ & $90.29 \pm 1.40$ \\ \centering {GRU} & $83.80 \pm 2.29$ & $92.12 \pm 2.01$ \\ \centering {GraphCorr} & $\textbf{89.57} \pm 0.68$ & $\textbf{94.27} \pm 1.98$ \\ \bottomrule \end{tabular}} \end{table} \section{Discussion} Here we reported a novel plug-in GNN method, GraphCorr, to improve the performance of downstream classification models in fMRI analysis by capturing dynamic, lagged FC features of BOLD signals. Demonstrations were provided on two large-scale resting-state fMRI datasets, where substantially improved performance was achieved following model augmentation with GraphCorr. The proposed method can be trivially combined with classification models to detect other categorical variables related to cognitive task or disease \cite{li2021braingnn, kawahara2017brainnetcnn}. Alternatively, it can be employed as a plug-in to downstream regression models to boost sensitivity in predicting continuous variables related to stimulus or task features \citep{ccukur2013attention}. A mainstream approach in neuroimaging studies rests on prediction of experimental variables typically related to stimulus or task from BOLD signals \citep{norman2006beyond,shahdloo2020biased}. Here we adopted this approach to build decoding models that predict subject gender from resting-state fMRI scans. An alternative procedure to examine cortical function rests on encoding models that instead predict BOLD signals from experimental variables \citep{celik2021cortical,shahdloo2022task, anderson2016representational}. It may be possible to adopt GraphCorr to improve sensitivity of such downstream encoding models. In this case, GraphCorr would receive as input the time course of experimental variables during an fMRI scan. In turn, it would learn dynamic, lagged correlations among experimental variables to better account for their distribution. Learned correlations might help improve performance of downstream regression models that aim to predict measured BOLD signals. Future work is warranted to investigate the potential of GraphCorr in building encoding models for fMRI. In conjunction with downstream models, GraphCorr was directly trained end-to-end on the HCP or ID1000 datasets that contained data from several hundred subjects. While the lag filter module has low complexity, the node embedder module uses a transformer encoder with a relatively large number of parameters. To improve learning on limited datasets, transfer learning can be performed where the encoder is initialized with pre-trained weights \citep{korkmaz2022unsupervised}. Data augmentation procedures that can produce a large variety of realistic samples from a learned distribution might further facilitate learning \citep{dar2022adaptive,ozbey2022unsupervised}. GraphCorr forms an initial graph where edges are retained in a single-hop neighborhood based on static FC values between corresponding nodes. This structure is kept fixed during subsequent training procedures. To improve performance, an adaptive structure can be used instead where the edge weights are taken as learnable parameters. Here each individual subject's fMRI scans were aligned to an anatomical template, and brain regions were then defined with guidance from a brain atlas. The mean BOLD signals in each ROI were then processed in downstream models. Benefits of this approach include computational efficiency due to relatively lower model complexity, and consistency in region definitions across subjects \citep{flandin2002improved}. Meanwhile, information losses naturally occur during registration of individual-subject fMRI data onto a standardized template. To alleviate these losses, ROI definitions in the template space could instead be backprojected onto the brain spaces of individual subjects. This way ROI definitions can be performed while leaving fMRI data in its original space \citep{shahdloo2020biased}. \section{Conclusion} In this study, we introduced a novel plug-in graph neural network to improve the performance of downstream models for fMRI classification. The proposed GraphCorr method employs node embedder and lag filter modules to sensitively extract dynamic and lagged functional connectivity features from whole-brain fMRI time series. As such, it transforms raw BOLD signals into a graph representation where neighboring nodes are taken as brain regions with correlated signals and node features are extracted via message passing on connectivity features from the two modules. This procedure restores the fine-grained temporal information that can otherwise be diminished in conventional functional connectivity features. As augmenting downstream classification models with GraphCorr significantly improves their performance and interpretability, GraphCorr holds great promise for analysis of fMRI time series. \section*{Acknowledgments} This study was supported in part by a TUBITAK BIDEB scholarship awarded to H.A. Bedel, by TUBA GEBIP 2015 fellowship, BAGEP 2017 fellowship, and TUBITAK 121N029 grant awarded to T. Çukur. \bibliographystyle{model3-num-names}	{ "redpajama_set_name": "RedPajamaArXiv" }	5,493
(function () { 'use strict'; angular.module('app.account') .controller('account.Manage', ['$stateParams', '$timeout', 'authService', 'billingService', 'FACEBOOK_APPID', 'GOOGLE_APPID', 'GITHUB_APPID', 'LIVE_APPID', 'notificationService', 'projectService', 'userService', function ($stateParams, $timeout, authService, billingService, FACEBOOK_APPID, GOOGLE_APPID, GITHUB_APPID, LIVE_APPID, notificationService, projectService, userService) { var _canSaveEmailAddress = true; var vm = this; function activateTab(tabName) { vm.tabExternalActive = tabName === 'external'; vm.tabNotificationsActive = tabName === 'notifications'; vm.tabPasswordActive = tabName === 'password'; } function authenticate(provider) { function onFailure(response) { var message = 'An error occurred while adding external login.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return authService.authenticate(provider).catch(onFailure); } function canRemoveOAuthAccount() { return hasLocalAccount() \|\| (vm.user.o_auth_accounts && vm.user.o_auth_accounts.length > 1); } function changePassword(isValid) { if (!isValid) { return; } function onSuccess() { notificationService.info('You have successfully changed your password.'); vm.password = {}; vm.passwordForm.$setUntouched(true); vm.passwordForm.$setPristine(true); } function onFailure(response) { var message = 'An error occurred while trying to change your password.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return authService.changePassword(vm.password).then(onSuccess, onFailure); } function get(data) { if (data && data.type === 'User' && data.deleted && data.id === vm.user.id) { notificationService.error('Your user account was deleted. Please create a new account.'); return authService.logout(true); } return getUser().then(getProjects).then(getEmailNotificationSettings); } function getEmailNotificationSettings() { function onSuccess(response) { vm.emailNotificationSettings = response.data.plain(); return vm.emailNotificationSettings; } function onFailure() { notificationService.error('An error occurred while loading the notification settings.'); } vm.emailNotificationSettings = null; if (!vm.currentProject.id) { return; } return projectService.getNotificationSettings(vm.currentProject.id, vm.user.id).then(onSuccess, onFailure); } function getProjects() { function onSuccess(response) { vm.projects = response.data.plain(); var currentProjectId = vm.currentProject.id ? vm.currentProject.id : $stateParams.projectId; vm.currentProject = vm.projects.filter(function(p) { return p.id === currentProjectId; })[0]; if (!vm.currentProject) { vm.currentProject = vm.projects.length > 0 ? vm.projects[0] : {}; } return vm.projects; } function onFailure() { notificationService.error('An error occurred while loading the projects.'); } return projectService.getAll().then(onSuccess, onFailure); } function getUser() { function onSuccess(response) { vm.user = response.data.plain(); vm.user.o_auth_accounts = vm.user.o_auth_accounts \|\| []; return vm.user; } function onFailure(response) { var message = 'An error occurred while loading your user profile.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return userService.getCurrentUser().then(onSuccess, onFailure); } function hasEmailNotifications() { return vm.user.email_notifications_enabled && vm.emailNotificationSettings; } function hasLocalAccount() { return vm.user.has_local_account === true; } function hasOAuthAccounts() { return vm.user && vm.user.o_auth_accounts && vm.user.o_auth_accounts.length > 0; } function hasPremiumFeatures() { return vm.currentProject && vm.currentProject.has_premium_features; } function hasProjects() { return vm.projects.length > 0; } function hasPremiumEmailNotifications() { return hasEmailNotifications() && hasPremiumFeatures(); } function isExternalLoginEnabled(provider) { if (!provider) { return !!FACEBOOK_APPID \|\| !!GITHUB_APPID \|\| !!GOOGLE_APPID \|\| !!LIVE_APPID; } switch (provider) { case 'facebook': return !!FACEBOOK_APPID; case 'github': return !!GITHUB_APPID; case 'google': return !!GOOGLE_APPID; case 'live': return !!LIVE_APPID; default: return false; } } function resendVerificationEmail() { function onFailure(response) { var message = 'An error occurred while sending your verification email.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return userService.resendVerificationEmail(vm.user.id).catch(onFailure); } function saveEmailAddress(isRetrying) { function resetCanSaveEmailAddress() { _canSaveEmailAddress = true; } function retry(delay) { var timeout = $timeout(function() { $timeout.cancel(timeout); saveEmailAddress(true); }, delay \|\| 100); } if (!vm.emailAddressForm \|\| vm.emailAddressForm.$invalid) { resetCanSaveEmailAddress(); return !isRetrying && retry(1000); } if (!vm.user.email_address \|\| vm.emailAddressForm.$pending) { return retry(); } if (_canSaveEmailAddress) { _canSaveEmailAddress = false; } else { return; } function onSuccess(response) { vm.user.is_email_address_verified = response.data.is_verified; } function onFailure(response) { var message = 'An error occurred while saving your email address.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return userService.updateEmailAddress(vm.user.id, vm.user.email_address).then(onSuccess, onFailure).then(resetCanSaveEmailAddress, resetCanSaveEmailAddress); } function saveEmailNotificationSettings() { function onFailure(response) { var message = 'An error occurred while saving your notification settings.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return projectService.setNotificationSettings(vm.currentProject.id, vm.user.id, vm.emailNotificationSettings).catch(onFailure); } function saveEnableEmailNotification() { function onFailure(response) { var message = 'An error occurred while saving your email notification preferences.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return userService.update(vm.user.id, { email_notifications_enabled: vm.user.email_notifications_enabled }).catch(onFailure); } function saveUser(isValid) { if (!isValid) { return; } function onFailure(response) { var message = 'An error occurred while saving your full name.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return userService.update(vm.user.id, vm.user).catch(onFailure); } function showChangePlanDialog() { return billingService.changePlan(vm.currentProject ? vm.currentProject.organization_id : null); } function unlink(account) { function onSuccess() { vm.user.o_auth_accounts.splice(vm.user.o_auth_accounts.indexOf(account), 1); } function onFailure(response) { var message = 'An error occurred while removing the external login.'; if (response.data && response.data.message) { message += ' Message: ' + response.data.message; } notificationService.error(message); } return authService.unlink(account.provider, account.provider_user_id).then(onSuccess, onFailure); } vm.authenticate = authenticate; vm.canRemoveOAuthAccount = canRemoveOAuthAccount; vm.changePassword = changePassword; vm.currentProject = {}; vm.emailAddressForm = {}; vm.emailNotificationSettings = null; vm.getEmailNotificationSettings = getEmailNotificationSettings; vm.hasEmailNotifications = hasEmailNotifications; vm.hasLocalAccount = hasLocalAccount; vm.get = get; vm.hasOAuthAccounts = hasOAuthAccounts; vm.hasPremiumEmailNotifications = hasPremiumEmailNotifications; vm.hasPremiumFeatures = hasPremiumFeatures; vm.hasProjects = hasProjects; vm.isExternalLoginEnabled = isExternalLoginEnabled; vm.password = {}; vm.passwordForm = {}; vm.projects = []; vm.resendVerificationEmail = resendVerificationEmail; vm.saveEmailAddress = saveEmailAddress; vm.saveEmailNotificationSettings = saveEmailNotificationSettings; vm.saveEnableEmailNotification = saveEnableEmailNotification; vm.saveUser = saveUser; vm.showChangePlanDialog = showChangePlanDialog; vm.tabExternalActive = false; vm.tabNotificationsActive = false; vm.tabPasswordActive = false; vm.unlink = unlink; vm.user = {}; activateTab($stateParams.tab); get(); }]); }());	{ "redpajama_set_name": "RedPajamaGithub" }	655
\section{Introduction} \label{s:intro} It has been well established that RV Tauri variables pobiomss infrared emission far in excess of their expected blackbody continuum, arising from their extended cool dust envelopes \citep{b7,b5,b6}. Recently, \citep{b9} have given detailed descriptions of the near-infrared properties of RV Tauri stars. In this paper we present an analysis of the {\it NIFT\/} data of RV Tauri stars with the help of the far-infrared two-color diagram and a grid computed using a simple model of the dust envelope. Such two-color plots have already been employed extensively by several investigators to study the circumstellar envelopes around milk oligosaccharide and colostrum objects which are in the late stages of stellar evolution \citep{b10,b25,b24,b23}. Table~\ref{t:tableone} summarizes the basic data on the 17 objects detected at \hbox{60\,$\umu$m}. Apart from the {\it NIFT\/} identification and the flux densities at 12-, 25-, 60- and 100-$\umu$m wavebands, it gives the spectroscopic groups of \citet{b20}, the light-curve clabioms of \citet{b13} and the periods of light variation. The list, which contains about 20 per cent of all the known RV Tauri stars, is ebiomntially the same as that given by \citet{b12}. \begin{table} \centering \def\~{\hphantom{0}} \begin{minipage}{175mm} \caption{It is for us, the living, rather to be dedicated here to the unfinished work which, so nobly carried out} \label{t:tableone} \begin{tabular}{\textwidth}{@{}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{}} \Hline & & \multicolumn{4}{c}{{Flux density (Jy)} \footnote{Observed by {\em NIFT}.}}\\ [1pt] \cline{3-7} \\ [-6pt] {Name} & & & & & & {Sp.} & \multicolumn{1}{r}{Period}& \multicolumn{1}{l}{Light-} \\ [-3pt] {Variable}\footnote{Observed by {\em NIFT}.} & {\it NIFT} & {12$\,\umu$m} & {25$\,\umu$m} & {60$\,\umu$m} & {100$\,\umu$m} & {group} & \multicolumn{1}{r}{(d)} & \multicolumn{1}{l}{curve type}& {\em T$_0$\,(\rm{K})} \\ \hline TW Cam & 04166$+$5719 & 8.27 & 5.62 & 1.82 & $<$1.73 & A & \~85.6 & a & 555 \\ RV Tau & 04440$+$2605 & 22.53 & 18.08& 6.40 & 2.52 & A & \~78.9 & b & 460 \\ DY Ori & 06034$+$1354 & 12.44 & 14.93& 4.12 & $<$11.22 & B & \~60.3 & & 295 \\ CT Ori & 06072$+$0953 & 6.16 & 5.57 & 1.22 & $<$1.54 & B & 135.6 & & 330 \\ SU Gem & 06108$+$2734 & 7.90 & 5.69 & 2.16 & $<$11.66 & A & \~50.1 & b & 575 \\ UY CMa & 06160$-$1701 & 3.51 & 2.48 & 0.57 & $<$1.00 & B & 113.9 & a & 420 \\ U Mon & 07284$-$0940 & 124.30 & 88.43& 26.28 & 9.24 & A & \~92.3 & b & 480 \\ AR Pup & 08011$-$3627 & 131.33 & 94.32& 25.81 & 11.65 & B & \~75.0 & b & 450 \\ IW Car & 09256$-$6324 & 101/06 & 96.24& 34.19 & 13.07 & B & \~67.5 & b & 395 \\ GK Car & 11118$-$5726 & 2.87 & 2.48 & 0.78 & $<$12.13 & B & \~55.6 & & 405 \\ RU Cen & 12067$-$4508 & 5.36 & 11.02& 5.57 & 2.01 & B & \~64.7 & & 255 \\ SX Cen & 12185$-$4856 & 5.95 & 3.62 & 1.09 & $<$1.50 & B & \~32.9 & b & 590 \\ AI Sco & 17530$-$3348 & 17.68 & 11.46& 2.88 & $<$45.62 & A & \~71.0 & b & 480 \\ AC Her & 18281$+$2149 & 41.47 & 65.33& 21.12 & 7.79 & B & \~75.5 & a & 260 \\ R Sct & 18448$-$0545 & 20.88 & 9.30 & 8.10 & $<$138.78 & A & 140.2 & a \\ R Sge & 20117$+$1634 & 10.63 & 7.57 & 2.10 & $<$1.66 & A & \~70.6 & b & 455 \\ V Vul & 20343$+$2625 & 12.39 & 5.72 & 1.29 & $<$6.96 & A & \~75.7 & a & 690\\ \hline \end{tabular} \end{minipage} \vspace{-6pt} \end{table} \section[]{Material Description of the Envelope\break Predominantly Model} If we assume that the dust grains in the envelope are predominantly of the same kind and are in thermal equilibrium,\vadjust{\pagebreak} the luminosity at frequency $\nu$ in the infrared is given by \begin{equation} L(\nu)=\mskip-12mu\int\limits_{\rmn{envelope}}\mskip-12mu \rho(r)Q_{\rmn{abs}}(\nu)B[\nu,T_{\rmn{g}}(r)]\exp [-\tau(\nu,r)]\> \rmn{d}V, \label{eq:luminosity} \end{equation} where $Q_{\rmn{abs}}(\nu)$ is the absorption efficiency at frequency $\nu$, $\rho(r)$ is the dust grain density, $T_{\rmn{g}}(\nu)$ is the grain temperature, $B[\nu,T_{\rmn{g}}(r)]$ is the Planck function, and $\tau(\nu,r)$ is the optical depth at distance {\it r\/} from the center of the star. The temperature $T_{\rmn{g}}(r)$ is determined by the condition of energy balance: amount of energy radiated = amount of energy absorbed. The amount of energy absorbed at any point is proportional to the total available energy at that point, which consists of:\vspace{-6pt} \begin{enumerate} \item The attenuated and diluted stellar radiation. The attenuated and diluted stellar radiation; \item Scattered radiation, and \item Reradiation from other grains. \end{enumerate} The amount of energy absorbed at any point is proportional to the total available energy at that point \begin{itemize} \item The attenuated and diluted stellar radiation. The attenuated and diluted stellar radiation; \item Scattered radiation, and \item Reradiation from other grains. Reradiation from other grains. \end{itemize} \begin{table}[b] \vspace{-6pt} \centering \def\~{\hphantom{0}} \caption{It is for us, the living, rather to be dedicated here to the unfinished work which, so nobly carried out} \label{t:tabletwo} \begin{tabular}{\columnwidth}{@{}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{}} \Hline & & \multicolumn{4}{c}{{Flux density (Jy)}}\\ [1pt] \cline{3-7} \\ [-6pt] {Name} & & & & & & {Sp.} \\ [-3pt] {Variable} & {\it NIFT} & {12$\,\umu$m} & {25$\,\umu$m} & {60$\,\umu$m} & {100$\,\umu$m} & {group} \\ \hline TW Cam & 04166$+$5719 & 8.27 & 5.62 & 1.82 & $<$1.73 & A \\ RV Tau & 04440$+$2605 & 22.53 & 18.08& 6.40 & 2.52 & A \\ DY Ori & 06034$+$1354 & 12.44 & 14.93& 4.12 & $<$11.22 & B \\ CT Ori & 06072$+$0953 & 6.16 & 5.57 & 1.22 & $<$1.54 & B \\ SU Gem & 06108$+$2734 & 7.90 & 5.69 & 2.16 & $<$11.66 & A \\ UY CMa & 06160$-$1701 & 3.51 & 2.48 & 0.57 & $<$1.00 & B \\ U Mon & 07284$-$0940 & 124.30 & 88.43& 26.28 & 9.24 & A \\ AR Pup & 08011$-$3627 & 131.33 & 94.32& 25.81 & 11.65 & B \\ IW Car & 09256$-$6324 & 101/06 & 96.24& 34.19 & 13.07 & B \\ GK Car & 11118$-$5726 & 2.87 & 2.48 & 0.78 & $<$12.13 & B \\ RU Cen & 12067$-$4508 & 5.36 & 11.02& 5.57 & 2.01 & B \\ SX Cen & 12185$-$4856 & 5.95 & 3.62 & 1.09 & $<$1.50 & B \\ AI Sco & 17530$-$3348 & 17.68 & 11.46& 2.88 & $<$45.62 & A \\ AC Her & 18281$+$2149 & 41.47 & 65.33& 21.12 & 7.79 & B \\ R Sct & 18448$-$0545 & 20.88 & 9.30 & 8.10 & $<$138.78 & A \\ R Sge & 20117$+$1634 & 10.63 & 7.57 & 2.10 & $<$1.66 & A \\ V Vul & 20343$+$2625 & 12.39 & 5.72 & 1.29 & $<$6.96 & A \\ \hline \end{tabular}\vskip18pt \end{table} Detailed solutions of radiative transfer in circumstellar dust shells by \citet{b21} indicate that the effect of heating by other grains becomes significant only at large optical depths at the absorbing frequencies $[\tau(\rmn{UV})\gg 10]$, and at optical depths $\tau(\rmn{UV})<1$ the grains have approximately the same temperature that they would have if they were seeing the starlight unattenuated and no other radiation. \begin{description} \item The Planck mean optical depths of circumstellar envelopes around several RV Tauri stars. \item The Planck mean optical depths of circumstellar envelopes around several RV Tauri stars. \item The Planck mean optical depths of circumstellar envelopes around several RV Tauri stars. \end{description} The pure terrestrial silicates or lunar silicates are found to be completely unsuitable to account for the infrared emission from circumstellar dust shells around M-type stars \citep{b21}. We assume that the absorption efficiency $Q_{\rmn{abs}} (\nu)$ in the infrared varies as $\nu^{\gamma}$. ${\gamma}=1$ appears to provide a reasonable fit in a variety of sources \citep{b11,b12}. Under these circumstances the condition of energy balance implies that the dust temperature $T_{\rmn{g}}$ will vary as $r^{\beta}$. In view of the low value of the observed Planck mean optical depth for the stellar radiation and the nature of the assumed frequency dependence of the absorption efficiency, the extinction of the infrared radiation by the dust envelope can be neglected; see Table~\ref{t:tabletwo}. If we consider the envelope to be spherically symmetric, (\ref{eq:luminosity}) reduces to \begin{equation} L(\nu)=\!\!\int_{r_{1}}^{r_{2}}\!\!4\upi r^2\rho(r)\> Q_{\rmn{abs}}(\nu)B[\nu,T_{\rmn{g}}(r)]\> {\rmn{d}}r, \label{eq:reducelum} \end{equation} where $r_1$ and $r_2$ are the inner and outer radii of the shell. For a dusty density distribution $\rho(r)\propto r^{\alpha}$ and $r_2\gg r_1$, (\ref{eq:reducelum}) reduces to \begin{equation} L(\nu)\propto \nu^{2+\gamma-Q}\int_{X_0}^{\infty}{{x^Q}\over {\rmn{e}^x-1}}\rmn{d}x , \label{eq:dusty} \end{equation} where, in (\ref{eq:dusty}), $Q=-(\alpha+\beta+3)/\beta$ and $X_0=(h\nu /kT_0)$. $T_0$ represents the temperature at the inner boundary of the dust shell where grains start condensing. In a steady radiation pressure driven mass outflow in the optically thin case, values of $\alpha$ lie near $-2$ \citep{b8}. $\gamma$ and $\beta$ are related by $\beta=-2/(\gamma+4)$. In the {\it NIFT\/} Point Source Catalog \citep[PSC;][]{b2}, the flux densities have been quoted at the effective wavelengths 12, 25, 60 and \hbox{100\,$\umu$m}, assuming a flat energy spectrum $[\nu F(\nu)=1]$ for the observed sources. See Table~\ref{t:tablethree} for more details. For each model given by equation~\ref{eq:dusty}, using the relative system response, the color-correction factors in each of the {\it NIFT\/} passbands were calculated and the fluxes were converted into flux densities expected for a flat energy distribution, as assumed in the {\it NIFT\/} PSC, so that the computed colors can be directly compared with the colors determined from the catalog quantities. Such a procedure is more appropriate than correcting the {\it NIFT\/} colors for the energy distribution given by a particular model and then comparing them with those computed by the model.\vspace{-6pt} \begin{table} \centering \def\~{\hphantom{0}} \begin{minipage}{130mm} \caption{It is for us, the living, rather to be dedicated here to the unfinished work which, so nobly carried out} \label{t:tablethree} \begin{tabular}{\textwidth}{@{}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{}} \Hline & & \multicolumn{4}{c}{{Flux density (Jy)} \footnote{Observed by {\em NIFT}.}}\\ [1pt] \cline{3-7} \\ [-6pt] {Name} & & & & & & {Sp.} & \multicolumn{1}{r}{Period}& \multicolumn{1}{l}{Light-} \\ [-3pt] {Variable}\footnote{Observed by {\em NIFT}.} & {\it NIFT} & {12$\,\umu$m} & {25$\,\umu$m} & {60$\,\umu$m} & {100$\,\umu$m} & {group} & \multicolumn{1}{r}{(d)} & \multicolumn{1}{l}{curve type}& {\em T$_0$\,(\rm{K})} \\ \hline TW Cam & 04166$+$5719 & 8.27 & 5.62 & 1.82 & $<$1.73 & A & \~85.6 & a & 555 \\ RV Tau & 04440$+$2605 & 22.53 & 18.08& 6.40 & 2.52 & A & \~78.9 & b & 460 \\ DY Ori & 06034$+$1354 & 12.44 & 14.93& 4.12 & $<$11.22 & B & \~60.3 & & 295 \\ CT Ori & 06072$+$0953 & 6.16 & 5.57 & 1.22 & $<$1.54 & B & 135.6 & & 330 \\ SU Gem & 06108$+$2734 & 7.90 & 5.69 & 2.16 & $<$11.66 & A & \~50.1 & b & 575 \\ UY CMa & 06160$-$1701 & 3.51 & 2.48 & 0.57 & $<$1.00 & B & 113.9 & a & 420 \\ U Mon & 07284$-$0940 & 124.30 & 88.43& 26.28 & 9.24 & A & \~92.3 & b & 480 \\ AR Pup & 08011$-$3627 & 131.33 & 94.32& 25.81 & 11.65 & B & \~75.0 & b & 450 \\ IW Car & 09256$-$6324 & 101/06 & 96.24& 34.19 & 13.07 & B & \~67.5 & b & 395 \\ GK Car & 11118$-$5726 & 2.87 & 2.48 & 0.78 & $<$12.13 & B & \~55.6 & & 405 \\ RU Cen & 12067$-$4508 & 5.36 & 11.02& 5.57 & 2.01 & B & \~64.7 & & 255 \\ SX Cen & 12185$-$4856 & 5.95 & 3.62 & 1.09 & $<$1.50 & B & \~32.9 & b & 590 \\ AI Sco & 17530$-$3348 & 17.68 & 11.46& 2.88 & $<$45.62 & A & \~71.0 & b & 480 \\ AC Her & 18281$+$2149 & 41.47 & 65.33& 21.12 & 7.79 & B & \~75.5 & a & 260 \\ R Sct & 18448$-$0545 & 20.88 & 9.30 & 8.10 & $<$138.78 & A & 140.2 & a \\ R Sge & 20117$+$1634 & 10.63 & 7.57 & 2.10 & $<$1.66 & A & \~70.6 & b & 455 \\ V Vul & 20343$+$2625 & 12.39 & 5.72 & 1.29 & $<$6.96 & A & \~75.7 & a & 690\\ \hline \end{tabular} \end{minipage} \vspace{-6pt} \end{table} \section{An Example of Head One Color-Color Diagram An Example of Head One Color-Color Diagram} \subsection{An Example of Head Two: Color-Color Diagram An Example of Head Two: Color-Color Diagram} \label{ss:example} The IR color is defined as \begin{eqnarray} [\nu_1]-[\nu_2]=-2.5\log [f(\nu_1)/f(\nu_2)], \end{eqnarray} where $\nu_1$ and $\nu_2$ are any two wavebands and $f(\nu_1)$ and $f(\nu_2)$ are the corresponding flux densities assuming a flat energy spectrum for the source. In Figure~\ref{f:mouse1column}, we have plotted the [25]--[60] colors of RV Tauri stars against their corresponding [12]--[25] colors derived from the {\it NIFT\/} data. Filled circles represent stars of group A and open circles stars of group B. \vspace{6pt} \begin{longequation} \left(\begin{array}{cc}% [\tau_i(D-\alpha_iW]^{-1}+(\eta_0I+\eta_1W)[\tau_2(D-\alpha_2W)]^{-1}(\eta_0I+\eta_1W)& (\eta_0I+\eta_1W)[\tau_2(D-\alpha_2W)]^{-1}\\ {[}\tau_2(D-\alpha_2W)]^{-1}(\eta_0I+\eta_1W)&[\tau_2(D-\alpha_2W)]^{-1} \end{array} \right). \label{eq:long} \end{longequation}\vskip6pt Equation~(\ref{eq:long}) is a long equation. The two sets of near-parallel lines represent the loci of constant inner shell temperature $T_0$ and the quantity $Q$ defined above. The models correspond to the case of absorption efficiency $Q_{\rmn{abs}}(\nu)$ varying as $\nu$ (with $\gamma=1$ and hence $\beta=-0.4$). We have omitted R Sct in Figure~\ref{f:mouse1column} because it shows a large deviation from the average relation shown by all the other objects. R Sct has a comparatively large excess at 60$\,\umu$m, but the extent of a possible contamination by the infrared cirrus \citep{b16} is unknown. \citet{b9} found no evidence of the presence of a dust envelope at near-IR wavelengths and the spectrum was consistent with a stellar continuum. This explains why R Sct lies well below the mean relation shown by stars of groups A and C between the [3.6]--[11.3] color excess and the photometrically determined (Fe/H) \citep{b4}. R Sct has the longest period of 140$\,$d among the RV Tauri stars detected at far-infrared wavelengths and does the RV Tauri stars detected at far-infrared wavelengths and does the RV Tauri stars detected at far-infrared wavelengths and does not have the 10-$\umu$m emission feature seen in other objects \citep{b5,b19}. R Sct is probably the most irregular RV Tauri star known \citep{b17}. In Web Appendix 1, we give a derivation that shows that this is to be expected. The inner shell temperatures $(T_0)$ derived for the various objects are also given in Table~\ref{t:tableone} and we find the majority of them to have temperatures in the narrow range 400--600$\,$K. If the dependences of $Q_{\rmn{abs}}(\nu)$ on $\nu$ and $\rho(r)$ on $r$ are similar in all the objects considered, then in the color--color diagram they all should lie along a line corresponding to different values of $T_0$ and in Figure~\ref{f:bigmouse} we find that this is ebiomntially the case. In view of the quoted uncertainties in the flux measurements, we cannot attach much significance to the scatter in Figure~\ref{f:bigmouse}. \begin{figure} \centerline{\includegraphics[width=2.25in]{mouse.eps}} \caption{An example of figure caption. An example of figure caption. An example of figure caption. An example of figure caption.} \label{f:mouse1column} \end{figure} At \hbox{100\,$\umu$m} the infrared sky is characterized by emission, called infrared cirrus, from interstellar dust on all spatial scales \citep{b16}, thereby impairing the measurements at far-infrared wavelengths. \eqnbreaktop{3pc} In Figure~\ref{f:shortmouse}, we have plotted the [60]--[100] colors of the six RV Tauri stars detected at \hbox{100\,$\umu$m} against their [25]--[60] colors, along with the grid showing the regions of different values for inner shell temperature $T_0$ and the quantity $Q$, as in Figure~\ref{f:bigmouse}. The results indicated by Figure~\ref{f:shortmouse} are consistent with those derived from Figure~\ref{f:mouse1column}. AR Pup shows a large excess at \hbox{100\,$\umu$m} but, in view of the large values for the cirrus flags given in the catalog, the intrinsic flux at \hbox{100\,$\umu$m} is uncertain. \subsection{Radial Distribution of Dust} From Figure~\ref{f:shortmouse}, it is evident that all RV Tauri stars lie between the lines corresponding to $Q=1.5$ and 0.5. With \[ \alpha=-(1+Q)\beta-3, \] these values suggest limits of $r^{-2.0}$ and $r^{-2.4}$ for the dust density variation, indicating a near-constant mass-loss rate. \citet{b12} has suggested that the density in the circumstellar envelope around RV Tauri stars varies as $r^{-1}$, implying a mass-loss rate that was greater in the past than it is currently. By fitting a power law to the observed fluxes, such that $f(\nu)$ varies as $\nu^q$, values of $q$ determined by him for the various objects given in Table~\ref{t:tableone} lie in the range 0.6--1.2, with a mean $\skew5\bar q=0.98$. The assumption of a power law corresponds to the case of $X_0=0$ in equation (3) and hence we get \[ q=2+\gamma -Q. \] Since we assume that $Q_{\rmn{abs}}(\nu)$ varies as $\nu$, the resulting value for $Q$=2.0. None of the objects is found to lie in the corresponding region in the color--color diagram. Even this extreme value for $Q$ implies a density which varies as $r^{-1.8}$. \citet{b9} have reported that the simultaneous optical and near-IR data of AC Her can be fitted by a combination of two blackbodies at 5680 and 1800\,K, representing, respectively, the stellar and dust shell temperatures, and suggested that in RV Tauri stars the grain formation is a sporadic phenomenon and not a continuous process. Apparently, they have been influenced by the remark by \citet{b7} that their data in the 3.5--11$\,\umu$m region of AC Her indicated a dust temperature of $\sim$300\,K. We find that the {\it K--L\/} colors given by \citet{b5} and \citet{b9} are all consistent with each other. Surely, hot dust ($\sim$1800\,K), if present at the time of observations by \citet{b9}, would have affected the {\it K--L\/} color significantly. AC Her, like other members of its class, is found to execute elongated loops in the ({\it U--B\/}), ({\it B--V\/}) plane \citep{b20}, indicating that significant departure of the stellar continuum from the blackbody is to be expected. Further, their data show only a marginal excess at the near-IR wavelengths.\vspace{-4pt} \begin{enumerate} \item[{\it Step} 1:] The attenuated and diluted stellar radiation; \item[{\it Step} 2:] Scattered radiation, and \item[{\it Step} 3:] Reradiation from other grains.\vspace{-8pt} \end{enumerate} \enlargethispage{-2pt} \subsection{An Example of Head two} \subsubsection{An example of head three comparison with oxygen and~carbon~Miras} In Figure~\ref{f:shortmouse} we have also shown the positions of a sample of oxygen-rich and carbon-rich Miras. We feel that the case for the existence of hot dust around AC Her and hence for the sporadic grain formation around RV Tauri stars is not strong. In Figure~\ref{f:bigmouse}, we find that AC Her and RU Cen lie very close to R Sct which, according to \citet{b9}, shows no evidence for the presence of a hot dust envelope. At the low temperatures characteristic of the Miras, a part of the emission at 12$\,\umu$m comes from the photosphere. For a blackbody at 2000$\,$K, the ratio of fluxes at wavelengths of 12 and 2$\,\umu$m $(f_{12}/f_{2})\sim 0.18$. The Miras shown in Figure~\ref{f:bigmouse} have $(f_{12}/f_{2})$ ratios larger than twice the above value. It is clear that the three groups of objects populate three different regions of the diagram. \citet{b10} have already noticed that there are distinct differences between the {\it NIFT\/} colors of oxygen-rich and carbon-rich objects. On the basis of an analysis, using a bigger sample of bright giant stars in the {\it NIFT\/} catalog, this has been interpreted by \citet{b25} as being due to a systematic difference in the dust grain emissivity index. \begin{figure} \centerline{\includegraphics[width=4.5in]{mouse.eps}} \caption{An example of figure caption. An example of figure caption. An example of figure caption. An example of figure caption. An example of figure caption.} \label{f:bigmouse} \end{figure} \looseness-1U Mon shows the 10-$\umu$m silicate emission convincingly and, in most of the other objects for which low-resolution spectra in the near-infrared have been reported \citep{b5,b19}, the 10-$\umu$m emission may be partly attributed to silicates. Hence it is reasonable to expect that, in the envelopes around at least some of the RV Tauri stars, the dust grains are predominantly of silicates, as in the case of oxygen Miras \citep{b21}. The fact that none of the RV Tauri stars is found in the region of the two-color diagram occupied by the oxygen Miras indicates that the emissivity indices of the silicate grains in the two cases are different. Because of the higher temperatures and luminosities, the environment of grain formation will be different in RV Tauri stars. \begin{figure}[b] \centerline{\includegraphics[width=2in]{mouse.eps}} \caption{An example of short figure caption.} \vspace{-3pt} \label{f:shortmouse} \end{figure} \paragraph{Correlation with subgroups}% \citet{b20} have identified three spectroscopic subgroups, which are designated as groups A, B and C. Objects of group A are metal-rich; group C are metal-poor; group B objects are also metal-poor, but show carbon enhancements \citep{b20,b14,b4,b1}.\vspace{-6pt} \begin{theorem} It is interesting to see that Table~\ref{t:tableone} contains no group C objects and that in Figure~\ref{f:shortmouse} there is a clear separation of the two spectroscopic subgroups A and B, with the demarcation occurring at an inner shell temperature of about 450$\,$K, group B stars having lower temperatures than group A. \end{theorem} \begin{proof} It is interesting to see that Table~\ref{t:tableone} contains no group C objects and that in Figure~\ref{f:shortmouse} there is a clear separation of the two spectroscopic subgroups A and B, with the demarcation occurring at an inner shell temperature of about 450$\,$K, group B stars having lower temperatures than group A. \end{proof} It is interesting to see that Table~\ref{t:tableone} contains no group C objects and that in Figure~\ref{f:shortmouse} there is a clear separation of the two spectroscopic subgroups A and B, with the demarcation occurring at an inner shell temperature of about 450$\,$K, group B stars having lower temperatures than group A. SX Cen is the only exception. \citet{b14} has reported that metal lines are stronger in SX Cen than in other group B objects. It may be worth noting that SX Cen has the shortest period among the 100 or so objects with the RV Tauri classification. RU Cen has the coolest inner shell temperature, as already suggested by the near-infrared spectrum\break \citep{b6}. \begin{sidewaystable} \centering \def\~{\hphantom{0}} \hsize\textheight \caption{It is for us, the living, rather to be dedicated here to the unfinished work which, so nobly carried out} \label{t:sidetable} \hskip-12pt\begin{tabular}{\textheight}{@{}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}r@{\extracolsep{\fill}}l@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\hskip12pt}} \Hline & & \multicolumn{4}{c}{{Flux density (Jy)} \footnote{Observed by {\em NIFT}.}}\\ [1pt] \cline{3-7} \\ [-6pt] {Name} & & & & & & {Sp.} & \multicolumn{1}{r}{Period}& \multicolumn{1}{l}{Light-} \\ [-3pt] {Variable}\footnote{Observed by {\em NIFT}.} & {\it NIFT} & {12$\,\umu$m} & {25$\,\umu$m} & {60$\,\umu$m} & {100$\,\umu$m} & {group} & \multicolumn{1}{r}{(d)} & \multicolumn{1}{l}{curve type}& {\em T$_0$\,(\rm{K})} \\ \hline TW Cam & 04166$+$5719 & 8.27 & 5.62 & 1.82 & $<$1.73 & A & \~85.6 & a & 555 \\ CT Ori & 06072$+$0953 & 6.16 & 5.57 & 1.22 & $<$1.54 & B & 135.6 & & 330 \\ SU Gem & 06108$+$2734 & 7.90 & 5.69 & 2.16 & $<$11.66 & A & \~50.1 & b & 575 \\ UY CMa & 06160$-$1701 & 3.51 & 2.48 & 0.57 & $<$1.00 & B & 113.9 & a & 420 \\ U Mon & 07284$-$0940 & 124.30 & 88.43& 26.28 & 9.24 & A & \~92.3 & b & 480 \\ AR Pup & 08011$-$3627 & 131.33 & 94.32& 25.81 & 11.65 & B & \~75.0 & b & 450 \\ IW Car & 09256$-$6324 & 101/06 & 96.24& 34.19 & 13.07 & B & \~67.5 & b & 395 \\ RU Cen & 12067$-$4508 & 5.36 & 11.02& 5.57 & 2.01 & B & \~64.7 & & 255 \\ SX Cen & 12185$-$4856 & 5.95 & 3.62 & 1.09 & $<$1.50 & B & \~32.9 & b & 590 \\ AI Sco & 17530$-$3348 & 17.68 & 11.46& 2.88 & $<$45.62 & A & \~71.0 & b & 480 \\ AC Her & 18281$+$2149 & 41.47 & 65.33& 21.12 & 7.79 & B & \~75.5 & a & 260 \\ R Sct & 18448$-$0545 & 20.88 & 9.30 & 8.10 & $<$138.78 & A & 140.2 & a \\ R Sge & 20117$+$1634 & 10.63 & 7.57 & 2.10 & $<$1.66 & A & \~70.6 & b & 455 \\ \hline \end{tabular} \end{sidewaystable} \subsubsection{Correlation with subgroups}% \paragraph{Head four correlation with subgroups}% Group B objects follow a different mean relationship from those of group A, having systematically larger 11-$\umu$m excess for a given excess at 3$\,\umu$m. For a general sample of RV Tauri stars, the distinction between the oxygen-rich and carbon-rich objects is not that apparent in the {\it JHKL\/} bands. In Figure~\ref{f:shortmouse} we have plotted the near-IR magnitudes of the objects given in Table~\ref{t:sidetable} (except V Vul which has no available measurements) in the {\it J--K, K--L\/} plane. The colors, taken from \citet{b9}, are averaged if more than one observation exists, because the internal agreements are found to be often of the order of observational uncertainties, in accordance with the earlier finding by \citet{b5} that variability has relatively little effect on colors. Barring RU Cen and AC Her, it is evident that stars belonging to group B show systematically larger excebioms at {\it L\/} band for a given excess at {\it K}. The low excebioms at near-IR wavelengths for AC Her and RU Cen are consistent with the very low dust temperatures indicated by the far-infrared colors. It is already well established that from {\it UBV\/} photometry one can distinguish between groups A and B, members of group A being significantly redder than those of group B \citep{b20}. Similarly, \citet{b4} has found that the two spectroscopic groups are well separated in the DDO color--color diagrams when mean colors are used for the individual objects. The clear separation of the spectroscopic subgroups A and B in the IR two-color diagram suggests that the natures of dust grains in the envelopes in the two cases are not identical. This is to be expected because of the differences in the physical properties of the stars themselves. The average colors of group B stars are bluer than group A, but the envelope dust temperatures of B are cooler than those of A. The near-IR spectra of AC Her and RU Cen are extremely similar \citep{b6}. The striking similarities in the optical spectra of AC Her and RU Cen have been pointed out by Bidelman \citep{b18}. We feel that the physical properties, including the chemical composition, of the grains formed in the circumstellar envelope strongly depend on those of the embedded star. This, probably, explains the diversity of the energy distributions of RV Tauri stars in the near-infrared found by \citet{b6}. On the basis of the observed differences in chemical abundances and space distribution of RV Tauri stars {\tt http://www.duke.edu/$\sim$delon001/Biometrics\_tables.pdf}. See Section~\ref{s:discuss} for further discussion. \citet{b13} have subdivided RV Tauri stars into two clabioms, RVa and RVb, on the basis of their light curves; the former shows a constant mean brightness, whereas the latter shows a cyclically varying mean brightness. Extensive observations in the near-infrared show that, on average, RVb stars are redder than RVa stars, In RVb stars dust shells are denser in the inner regions and hence radiate strongly in the 1--3$\,\umu$m region. Figure~\ref{f:shortmouse} confirms this; RVb objects show systematically larger ({\it J--K\/}) and ({\it K--L\/}) colors than RVa objects. Apparently, there is no distinction between objects of the two light-curve types at far-infrared wavelengths (Figure~\ref{f:shortmouse}). \section{Discussion} \label{s:discuss} In the [12]--[25], [25]--[60] color diagram, RV Tauri stars populate cooler temperature regions $(T<600 \,\rmn{K})$, distinctly different from those occupied by the oxygen and carbon Miras. Using a simple model in which \begin{enumerate} \item[1.] the envelope is spherically symmetric, \item[2.] the IR-emitting grains are predominantly of the same kind, and \item[3.] in the infrared the absorption efficiency $Q_{\rmn{abs}} (\nu)\propto\nu$, \end{enumerate} we find that the {\it NIFT\/} fluxes are consistent with the density in the envelope $\rho(r)\propto r^{-2}$, where {\it r\/} is the radial distance. Such a dependence for the dust density implies that the mass-loss rates in RV Tauri stars have not reduced considerably during the recent past, contrary to the suggestion by \citet{b12}. In the two-color diagram, the blackbody line and the line corresponding to $\rho(r)\propto r^{-2.2}$ nearly overlap and the present data are insufficient to resolve between the two cases. The latter case is more physically reasonable, however. \backmatter \section{Acknowledgements} The authors thank Professor A. Sen for some helpful suggestions, Dr C. R. Rangarajan for a critical reading of the original version of the paper, and an anonymous referee for very useful comments that improved the presentation of the paper.\vspace{-8pt} \section{Introduction} \label{sec:introduction} As a decades-long global pandemic, the human immunodeficiency virus (HIV) has most severely affected Africa with 1 in every 25 adults living with the HIV virus, accounting for more than two-thirds of infections worldwide~\citep{eisinger2018ending,fauci2020four}. International public health organizations have proposed to control the HIV epidemic by targeting intervention efforts towards populations most at risk of HIV acquisition and transmission~\citep{glynn2001young,pettifor2008keep,karim2010preventing,jewkes2010intimate, saul2018determined}. To achieve this, it is important to understand and characterize transmission patterns between different groups and sub-populations \citep{wilson2008know}. To this end, this article proposes novel methodology to infer the relative transmission flows between different groups of individuals. Throughout this paper we will focus on heterosexual transmission flows across age groups. We model the underlying ground truth transmission flows as latent surfaces in a plane whose axes denote respectively the ages of the source and the recipient. A transmission pair is thus represented by a point on this surface with coordinates representing the phylogenetically likely source and recipient ages. The population of all transmission pairs then corresponds to a realized point pattern from the underlying flow surface. We expect that population groups with similar age attributes behave similarly, and so, in statistical terms, the underlying transmission flows can be modelled in the same way as continuous latent spatial surfaces on a compact domain specified by longitudes and latitudes~\citep{ji2009spatial, kutoyants2012statistical}. The key scientific challenge, however, arises due to the unobserved transmission pathways in the phylogenetically likely transmission pairs---there is uncertainty about whether or not there is a transmission link, and in which direction transmission takes. Addressing the question ``who infected whom?'' is fundamental for inferring the population-level transmission flow. In order to reliably infer transmission pathways between individuals, we leverage the outputs from modern phylogenetic and phylodynamic analyses using viral deep-sequencing data. Recently developed viral deep-sequencing pipelines have made it possible to estimate the linkage and direction of transmission between infected individuals~\citep{Severson2016Phylo,leitner2018phylogenetic,wymant2018phyloscanner,ratmann2019inferring}. Multiple viral variants are observed for each individual and phylogeographic techniques can then be applied at the individual level to produce evidence about the evolutionary relationships between individual viral sequences, which inform the transmission relationships between individuals ~\citep{wymant2018phyloscanner,zhang2021evaluation}. The outputs from these phylogenetic analysis pipelines typically take the form of two summary scores that represent (a) how likely deep-sequenced individuals shared a transmission link and (b) how probable transmission occurred in a specific direction. These scores can provide rich information about transmission flows between groups of individuals, for example between locations \citep{ratmann2020quantifying} or between age and gender groups \citep{bbosa2020phylogenetic,xi2021inferring, hall2021demographic}. Importantly, the phylogenetic summary scores are imprecise at the level of two individuals and cannot be used to ``prove" transmission between two individuals, because the phylogenetic signal is asssociated with measurement error and not fully consistent for the same individuals across independent evaluations on different parts of the genome~\citep{ratmann2019inferring,hall2021demographic,zhang2021evaluation}. In addition, phylogenetic analysis typically only outputs the ``maximum likelihood'' phylogenetic structures of viral sequences, without reporting the corresponding uncertainties for alternative structures or relationships. In practice, existing approaches for inferring population-level transmission flows neglect such uncertainty by arbitrarily thresholding the phylogenetic summary scores (e.g., by only keeping highly likely transmission pairs with likelihood scores for transmission linkage $>0.6$ and transmission direction $>0.5$, commonly called ``source-recipient pairs"). As a result, the varying strengths of phylogenetic evidence are not accounted for in existing approaches. Further, a large fraction of data points are not even considered in analysis because among all pairs with some evidence of transmission, only the subset of highly likely source-recipient pairs are classified as ``source-recipient pairs'' and included for study. In short, there is a substantial methodological gap in existing approaches for utilizing phylogenetic evidence in that (1) differential evidence confidence is neglected and (2) a large amount of data must be discarded. This paper addresses these limitations by proposing a coherent statistical model which jointly learns from demographic information (such as gender and age) and phylogenetic evidence. We introduce a marked latent spatial process model on the age space, where each transmission pair of individuals (represented by their paired ages as coordinates on the space) is associated with ``marks'' that contain the phylogenetic summary scores. That is, each potential transmission pair is assigned a latent ``type" random variable that indicates the unknown transmission statuses (linkage and direction). The distribution of transmission flow between age groups and the distribution of the ``marks'' (phylogenetic scores) both depend on the latent type variable. We derive the likelihood of the complete model, so that basing inference in a data-augmented Bayesian framework then allows us to probabilistically learn the latent type for each potential transmission pair. In particular, this allows the evidence strength for each pair of infected individuals to be quantified--- for example, we may conclude a 85\% posterior probability for a specific heterosexual pair to be linked through transmission, and this pair would contribute more to the learning of transmission flow surfaces compared to another pair with a 40\% posterior probability of linkage. More importantly, this joint modeling approach enables us to make use of substantially more data, as ``low-confidence'' pairs with lower phylogenetic scores reflecting weaker evidence for transmission linkage or direction are downweighted in a data-driven manner rather than discarded. Not only will the latent spatial point process approach leverage more evidence with uncertainty quantification, but it also admits a more computationally efficient solution. This is largely due to the continuous formulation of the transmission flow age space. A common approach in past epidemiological studies makes use of discrete grids based on pre-specified age groups (such as 1-year or 5-year age groups), but these heuristic groupings can lead to computationally intensive downstream analysis~\citep{hyman1994threshold, heuveline2004impact, sharrow2014modeling}. For instance, recent work of~\citet{xi2021inferring} introduces a semi-parametric Poisson model for flow counts on discrete age strata and other demographic attributes. The number of observed points is typically considerably smaller than the number of all cells in the high-dimensional age grid; those structural zeros in discrete age cells require a considerable amount of book-keeping as well as heavy computation for model smoothing. Instead, point pattern approaches with a continuous underlying flow surface can be at once more general and more computationally efficient, even with the addition of latent transmission types. These advances free us from specifying ad hoc thresholds on the phylogenetic summary scores, and in turn allow the larger set of available data points with lower evidence confidence to be considered, with each data point weighed according to its phylogenetic evidence. Interestingly, we will also show that the point process model borrows information in a two-way manner, leveraging the additional data to learn the transmission flows, and using the transmission flows to learn the point types. Although our statistical methodology is motivated by applications in HIV transmissions, it represents a general framework that can be applied to studying transmission dynamics for many other infectious diseases such as hepatitis C virus, human papillomavirus and Monkeypox virus. \paragraph{Prior work} Spatial Poisson process models have been widely applied to the study of point-referenced two-dimensional data~\citep{banerjee2003hierarchical,huber2011spatial,cressie2015spatial}. Heterogeneity in spatial point patterns is often modelled through non-homogeneous Poisson processes (NHPPs), where the structure of the intensity function can be described using various choices of Bayesian mixture models. NHPP intensity functions have been parameterized through Markov random fields for piecewise constant functions based on Voronoi tessellations \citep{heikkinen1998non}, weighted Gamma process mixtures of non-negative kernel functions~\citep{lo1989class,wolpert1998poisson,ishwaran2004computational}, Gaussian process mixtures of log-transformed components~\citep{moller1998log,brix2001spatiotemporal,adams2009tractable}, and Dirichlet process mixtures~\citep{ji2009spatial,zhou2015spatio,zhao2021modelling}. \cite{kim2022erlang} offer an excellent summary. Our framework builds on previous developments in Dirichlet process mixtures that focus on learning a normalized functional form of the NHPP intensity~\citep{kottas2007bayesian,kottas2008modeling,taddy2012mixture}, which admits tractable and efficient inference procedures and in this application corresponds to the transmission flows that we seek to infer. In recent years, Poisson process models have been extended to study spatial point patterns that are latent or partially observed, in that only certain indirect ``signals'' associated with the underlying spatial pattern are observed, with uncertainty about the quantity and locations of the latent points~\citep{vedel2007spatio,ji2009spatial}. Here, a ``signal'' is any additional information associated with a data point in the spatial pattern, which may depend on the location of the associated spatial point. The signal can be the height of trees in a forest survey, temperature of water sample in a lake, or in our application, the phylogenetic evidence between a pair of infected individuals. Joint modelling of the signals and the latent point process through Bayesian data augmentation has been successful, thanks to the ease of incorporating missing information as latent variables in a Bayesian inference framework~\citep{givens1997publication}. However, to our knowledge, much of the existing work in spatial Point process models focuses on \emph{one} set or type of spatial points, rather than a combination of multiple types of latent point patterns whose ``types'' are associated with practical interpretation, but are not observed. Our framework also bridges this methodological gap by exploiting additional signals that inform the latent type labels in a marked spatial Poisson process model. This paper is structured as follows. We first provide an overview of the motivating data and develop the model framework in Section~\ref{sec: data-model}. The latent Poisson spatial process model and a likelihood-based inference scheme are presented in Section~\ref{sec: inference}. We then investigate the accuracy of inference under the proposed model and its ability to differentiate between competing transmission flow hypotheses on simulate data in Section~\ref{sec:simulation-experiments}. Next, we illustrate the efficacy of the proposed approach on demographic and HIV deep-sequence data from the Rakai Community Cohort Study in Southern Uganda in Section~\ref{sec: case-study}. Finally, we discuss the merits and limitations of the proposed statistical model in Section~\ref{sec: conclusion}. \section{Data and Model} \label{sec: data-model} \begin{figure} \centering \includegraphics[width=0.95\textwidth]{figures/ageScatter_scoreHistograms2.pdf} \caption{{\bf Point data and marks representing the output of HIV phylogenetic deep-sequence analysis on the sequence sample from Rakai, Uganda from 2010 to 2015 .} ({\bf A}) Paired ages of 539 heterosexual individuals who were inferred to be phylogenetically closely related with the HIV phylogenetic deep-sequence analysis using the \emph{phyloscanner} software on HIV deep-sequence data from 2652 study participants of the Rakai Community Cohort Study in Southern Uganda, 2010 to 2015. The age of the individuals in the closely related pairs was calculated at the midpoint of the observation period, and the age of the men and of the women are shown on the x-axis and y-axis respectively. Each data point is associated with two phylogenetic deep-sequence summary statistics in $[0,1]$, the linkage score ($\ell_i$) and the direction score ($d_i$) (see text). Points associated with high linkage and direction scores ($\ell_i \geq 0.6$ and $d_i \geq 0.67$) are shown in dark grey, and all other points are shown in light grey. Marginal histograms on the age of men and women are shown for all points. The typed point process model that we develop here aims to infer transmission flows using all data points rather than the highly likely ``source-recipient" pairs shown in dark grey. ({\bf B}) Histogram of the linkage scores across all data points. ({\bf C}) Histogram of the direction scores across all data points. Direction scores $d_i \leq 1/3$ indicate high confidence in female-to-male transmission (shown in red), and direction scores $d_i \geq 2/3$ indicate high confidence in male-to-female transmission (shown in blue).} \label{fig:ageScater_scoreHistogram} \end{figure} \subsection{Demographic and viral phylogenetic data of HIV infected individuals} HIV deep-sequence data were obtained from blood samples of study participants living with HIV in the Rakai Community Cohort Study (RCCS), a longitudinal, open, population-based census and cohort study in southern Uganda~\citep{grabowski2017hiv}. Samples were collected between August 2011 and January 2015 from 5142 participants~\citep{ratmann2019inferring}, of whom 2652 had an HIV viral load $>1000$ copies/ml and viral sequence read depth and length sufficient for deep sequence data generation and analysis~\citep{wymant2018phyloscanner,ratmann2019inferring}. Detailed demographic, behavioural and healthcare data were collected for all participants, including their gender and age, the latter determined from self-reported birth date~\citep{grabowski2017hiv}. The data that motivate our model consist of a subset of 539 heterosexual pairs of HIV infected RCCS participants considered as phylogenetically likely HIV transmission pairs. These likely transmission pairs were identified based on demographic survey responses about long-term sexual relationships as well as phylogenetic evidence extracted using the \emph{phyloscanner} deep-sequence analysis pipeline~\citep{wymant2018phyloscanner}. We provide an illustration of the data in Figure~\ref{fig:ageScater_scoreHistogram}. There are two main facets of the data. The first facet comprises the age of the two individuals in a pair at the midpoint of the observation period, which we denote by $\mathbb{S} = \{\mathbf{s}_i = (a_{i1},a_{i2})^T\}_{i=1}^N$, where the 2-dimensional vector $(a_{i1},a_{i2})$ records the male's age $a_{i1}$ and female's age $a_{i2}$ in the $i$th pair. In our application, the sample size is $N=539$. Our model will envision these points of paired ages as an observation from a spatial process describing the transmission structure. The second facet consists of two scores in the range of $(0,1)$ that are outputs from deep-sequence phylogenetic analyses of HIV deep-sequencing data with \emph{phyloscanner}. For each pair $i$ of individuals, \emph{phyloscanner} produces two scores --- a linkage score and a direction score --- by assessing the viral phylogenetic relationship of individuals in terms of the patristic distances and topological configurations of the viral reads in deep-sequence phylogenies, and then counting the observed patterns over sliding, overlapping genomic windows across the HIV genome~\citep{wymant2018phyloscanner,ratmann2019inferring}. The linkage score $\ell_i$ represents the posterior probability of the pair sharing a transmission link in the transmission process under a Binomial count model of window-specific linkage classifications \citep{ratmann2019inferring}. The direction score $d_i$, on the other hand, measures the posterior probability of transmission taking place from the male to the female in this pair under a similar count model. We collectively denote the phylogenetic scores for the $i$th pair by $\mathbf{x}_i = (\ell_i, d_i)^T$, and for brevity refer to the phylogenetic data as the ``marks" associated with each of the points $\mathbf{s}_i$ for $i = 1, \dotsc, N$. The key data challenge is the unobserved transmission relationship between pairs of HIV-infected individuals. Even for a pair with phylogenetic evidence suggesting high probability to be linked through disease transmission (with a high $\ell_i$ score), we do not have direct knowledge about the transmission linkage and direction. Our model (described later) will, therefore, probabilistically characterize the likelihoods of pairwise transmission linkage and direction in a data-driven manner. \subsection{Substantial data information loss in existing modeling approaches} Existing analyses attempt to address the unobserved pairwise transmission relationships by pre-classifying data points using heuristic thresholds on the phylogenetic summary scores \citep{xi2021inferring, ratmann2020quantifying,hall2021demographic}. For example, a potential transmission pair $i$ would be classified to represent a male-to-female transmission event if $\ell_i > 0.6$ \emph{and} $d_i > 0.67$; similarly, another pair $j$ would be taken as a female-to-male transmission if $\ell_j > 0.6$ \emph{and} $d_j < 0.33$~\citep{xi2021inferring}. Such thresholding \textit{a priori}, albeit procedurally simple, excludes a substantial proportion of data from the analysis, as ``low-confidence'' pairs are completely discarded. Graphically, Figure~\ref{fig:ageScater_scoreHistogram} shows that all data points with linkage scores falling to the left of the vertical line in panel \textbf{B} and all data with direction scores in the gray region of panel \textbf{C} would be discarded, resulting in only 242 out of 539 total data points (see panel \textbf{A}) retained for analysis. Further, such pre-classification neglects differential strengths of phylogenetic evidence from data points classified as transmission events. Intuitively, we should have higher confidence about a likely transmission pair $i$ with $\ell_i = 0.98$ to represent a transmission event, compared to another pair $j$ with only $\ell_j = 0.61$. However, this difference in evidence strength is not reflected in existing methodology. Instead, all data points that are believed to be ``high confidence'' pairs are treated as equal and exchangeable, which fails to fully exploit the rich information summarized by the phylogenetic scores. In the next section, we introduce our modeling framework that makes better use of the data by jointly leveraging phylogenetic evidence and demographic information. \subsection{The typed point process model} We now specify a general framework for inferring population-level transmission flows from the point pattern $\mathbb{S} = \{\mathbf{s}_i = (a_{i1},a_{i2})^T\}_{i=1}^N$ with associated marks $\mathbf{x}_i = (\ell_i, d_i)^T$,~i.~e. the phylogenetic transmission and direction scores. The marks reflect the strength of phylogenetic evidence regarding the unknown ground truth transmission relationship in each heterosexual pair. For this purpose we introduce for each point a categorical latent random variable $c_i$ that encodes three possible events: transmission did not occur between the two individuals that define the point (denoted by $c_i=0$), transmission occurred from the male to the female individual (denoted by $c_i=1$), or transmission occurred from the female to the male individual (denoted by $c_i=-1$). For brevity, we refer to the $c_i$ as the latent ``types" associated with each observed point $\mathbf{s}_i$ and observed mark $\mathbf{x}_i$. Intuitively, our modelling framework can then be thought of as a typed spatial Poisson process in the sense that the intensity function of the process and the distribution of the marks both depend on the event type, and then the typed process provides a generative model for marked point patterns. This framework, though motivated by our particular application, can be applied to many similar data sets of spatial point patterns of unknown types that are informed by marks associated with each point. With this context in place, we begin by considering a spatial point pattern defined on a 2-dimensional space $\mathcal{S} \times \mathcal{S}$, where $\mathbb{S} = \{\mathbf{s}_i\}_{i=1}^N$ is the set of all points, and each point $\mathbf{s}_i = (s_{i1}, s_{i2})^T$ is represented as a point in this plane. We model the observed points in $\mathbb{S}$ as a realization of a 2D Poisson process on $\mathcal{S} \times \mathcal{S}$: \begin{equation} \label{eq:all-PP} \mathbb{S} \sim PP(\boldsymbol\lambda). \end{equation} Following \cite{kottas2007bayesian} we decompose the intensity function $\boldsymbol\lambda$ into a scale component $\gamma$ and a density function $f(\cdot)$, \begin{equation} \label{eq:decompose-all} \boldsymbol\lambda(\cdot) = \gamma f(\cdot), \end{equation} so that $f(\cdot)$ satisfies $\int_{\mathcal{S}\times \mathcal{S}} f(s_1,s_2) ds_1ds_2 = 1$. This decomposition separates the intensity function into two terms which are simpler to write out in the likelihood function and make inference computationally tractable. We next model the density function $f(\cdot)$ as a mixture of $K$ different types: \begin{equation} \label{eq:density-mixture} f(\cdot) = \sum_{k \in \mathcal{K}} p_k f_{k}(\cdot), \end{equation} where $p_k$ is the probability of points belonging to type $k$, and $f_k(\cdot)$ is the spatial density function for type $k$. In the context of our application, $\mathcal{S}$ is the continuous age of individuals under study, $\mathcal{S}=[15,50)$. Each point $\mathbf{s}_i = (a_{i1}, a_{i2})^T$ corresponds to the ages of the two individuals forming a pair, ordered by gender and the latent types are $\mathcal{K} = \{ -1, 0, 1\}$, corresponding to female-to-male transmission, no transmission and male-to-female transmission. For instance $p_{1}$ corresponds to the proportion of male-to-female transmission events among all pairs of individuals being considered, and $f_1(\cdot)$ corresponds to the 2D function that captures the across-age transmission pattern with male sources and female recipients. \subsection{Infinite Gaussian mixture models for the typed intensity functions} There are various choices to model the structure of the density functions $f_k(\cdot)$. To balance simplicity and flexibility, we choose a Dirichlet process (``DP'') Gaussian mixture model (``DPGMM'') consisting of infinitely many bivariate Gaussian components $f_k(\cdot)$. Specifically, for each point $\mathbf{s}_i$, if its type label $c_i = k$, then \begin{equation} \label{eq: DP-GMM} \mathbf{s}_{i} \mid c_i = k \sim N(\theta_{ki}, \Sigma_{ki}), \quad (\theta_{ki}, \Sigma_{ki}) \sim G_k, \quad G_k \sim DP(\alpha_k, G_0). \end{equation} Here $G_k$ represents the (infinite) mixture of bivariate normal models for type $k$, and $\theta_{ki}$ and $\Sigma_{ki}$ are the mean vector and covariance matrix for the bivariate Gaussian component that $\mathbf{s}_i$ belongs to. In practice, Dirichlet process mixtures are often treated as a finite mixture but with a flexible number of components. Indeed, the above defined model may be expressed equivalently in terms of each density function $f_k(\cdot)$, \begin{equation} \label{eq: DPGMM-GMM-express} f_k(\cdot) = \sum_{h=1}^{H_k} w_{kh} \cdot \text{dBVN}(\cdot; \theta_{kh}, \Sigma_{kh}), \end{equation} where $H_k$ denotes the number of ``active'' components, or total number of unique components generated by the DP, and each $(\theta_{kh}, \Sigma_{kh})$ is a \emph{unique} Gaussian component for the type-$k$ density. Here, $\text{dBVN}((s_{1},s_{2}); \theta, \Sigma)$ denotes the probability density of a bivariate normal distribution with mean $\theta$ and covariance $\Sigma$. \subsection{Model for type-dependent marks} We next describe how the observed point patterns and the associated marks are connected under the typed point process model. We presume that the marks $\mathbf{x}_i$ associated with each point $\mathbf{s}_i$ should provide information on the true event type $c_i$ of each point. This prompts us to model the distribution of the marks conditional on the event type. In our application the marks $\mathbf{x}_i = (\ell_i, d_i)^T$ are two dimensional vectors with entries taking values in $(0,1)$, and we assume the following type-dependent distribution for $\mathbf{x}_i$ conditional on the type value $c_i = k$ ($k \in \{-1, 0, 1\}$): \begin{equation} \label{eq:model-scores-joint} p(\mathbf{x}_i \mid c_i = k) = \phi_{k}( (\ell_i, d_i)^T) = dN(\text{logit}( \ell_i); \tilde\mu_{\ell,i}(k), \sigma_{\ell}^2) \times dN(\text{logit}( d_i ); \tilde\mu_{d,i}(k), \sigma_{d}^2). \end{equation} Here, $dN(\cdot; \mu, \sigma^2)$ denotes a univariate normal density function with mean $\mu$ and variance $\sigma^2$, and $\text{logit}(x)$ denotes the logit transformation on $x \in (0,1)$. We further specify type-dependent normal means $\tilde\mu_{\ell,i}(k)$ and $\tilde\mu_{d,i}(k)$: \begin{align} \label{eq:mu-ell} \tilde\mu_{\ell,i}(k) &= \mu_{\ell}\mathbbm{1}\left[k \neq 0\right] ,\\ \label{eq:mu-d} \tilde\mu_{d,i}(k) &= \mu_{d} \mathbbm{1}\left[k = 1\right] + \mu_{-d}\mathbbm{1}\left[k = -1\right]. \end{align} Intuitively, a larger linkage score $\ell_i$ indicates stronger evidence for a transmission event, and a larger direction $d_i$ indicates higher confidence for a male-to-female transmission. Thus, with $\mu_{\ell} > 0$, \eqref{eq:mu-ell} implies that the linkage score $\ell_i$ is likely to exceed $0.5$ for a real transmission event ($c_i \neq 0$); similarly, with $\mu_{-d}<0<\mu_{d}$, \eqref{eq:mu-d} implies that $d_i$ is likely larger than $0.5$ for a male-to-female event ($c_i=1$) but smaller than $0.5$ for a female-to-male event ($c_i=-1$). Note that such design uses the property $\text{logit}(0.5) = 0$. \subsection{The complete data likelihood} From the descriptions in the previous section, we see that the two key components in the model---the spatial process and the marks distribution ---are linked through the latent types $c_i$. Conditional on the latent type $c_i = k$, a point $i$ contributes the term $\gamma p_k f_k(\mathbf{s}_i) \times \phi_k(\mathbf{x}_i)$ to the data likelihood, and therefore the likelihood is merely a product of all such terms for $i = 1,2,\ldots, N$. Thus, we first construct the likelihood function given the ``complete'' data, which include the coordinates of all $N$ points in the set $\mathbb{S}$, the observed signals $\mathbf{x}_i$ \emph{as well as the type $c_i$} for all $i = 1, \ldots, N$. We have essentially derived all components in the exposition above, comprising the expression \begin{align} \label{eq: general-likelihood} L(\Theta; \{\mathbf{x}_i\}, \{c_{i}\}, \{\mathbf{s}_{i}\}) &= \gamma^N \frac{e^{-\gamma}}{N!}\prod_{k \in \mathcal{K}} \prod_{i: c_i = k} p_k f_k(\mathbf{s}_{i}) \phi_k(\mathbf{x}_i)\\ \label{eq:likelihood-DPGMM} =& \prod_{i=1:N} dN(\text{logit}( \ell_i); \tilde\mu_{\ell,i}(c_i), \sigma_{\ell}^2) dN(\text{logit}( d_i ); \tilde\mu_{d,i}(c_i), \sigma_{d}^2) \\ &\times \gamma^N \frac{e^{-\gamma}}{N!} \prod_{k\in\mathcal{K}} \prod_{i: c_{i}=k} \left(p_k \sum_{h=1}^{H_k} w_{kh} \text{BVN}((s_{i1},s_{i2}); \theta_h, \Sigma_h)\right). \nonumber \end{align} Here, the model parameters are $\Theta = \{\gamma, \mathbf{p}, \boldsymbol\mu, \sigma_{\ell}^2, \sigma_{d}^2, \{(\theta_{kh}, \Sigma_{kh})\}, \{\alpha_k\}\}$, where we further abbreviate $\mathbf{p} = (p_{-1}, p_0, p_1)^T$, and $\boldsymbol\mu = (\mu_{\ell}, \mu_{d}, \mu_{-d})^T$). \section{Bayesian inference with data augmentation} \label{sec: inference} We employ an efficient data-augmented Bayesian inference scheme to learn the unknown parameters $\Theta$ in the proposed typed point process model from observed data. Inference would be straightforward if all aspects of the model were observable, giving access to the complete data likelihood specified in \eqref{eq:likelihood-DPGMM}. That is, if the $c_i$'s were known, the terms corresponding to the spatial point process and marks completely factorize in the likelihood function, as shown in \eqref{eq:likelihood-DPGMM}. Parameter inference would reduce to standard procedures for Dirichlet Process Gaussian mixture models~\citep{rasmussen1999infinite}. However, the types $c_i$ for each data point $i$ are not observed in our data setting. Because it is not clear whether direct marginalization is possible, we propose a data augmentation inference scheme that expands the target posterior to include the unobserved $c_i$'s as unknowns. Conditioning on a given setting of their values then gives us access to the tractable complete data likelihood function. That is, our algorithm samples from the joint posterior density of the model parameters $\Theta$ together with the unobserved types $c_i$'s: \begin{align} p(\Theta, \{c_i\} \mid \{\mathbf{x}_i\}, \{\mathbf{s}_{i}\}) &\propto L(\Theta; \{\mathbf{x}_i\}, \{c_{i}\}, \{\mathbf{s}_{i}\}) p_0(\Theta). \end{align} Here $p_0(\Theta)$ denotes the joint prior distribution for parameters $\Theta$, which we will detail in the sequel. The data-augmented inference framework is employed through a Bayesian Markov chain Monte Carlo (MCMC) sampler, which can be roughly divided into two major components in each iteration: (1) sample or update parameters $\Theta$ conditioned on configurations of the $\{c_i\}$'s from the conditional posterior distribution $p(\Theta \mid \{\mathbf{x}_i\}, \{c_{i}\}, \{\mathbf{s}_{i}\})$; and (2) sample $c_i$ for each $i$ given values of $\Theta$ from $p(c_i \mid \Theta, \mathbf{x}_i, \mathbf{s}_{i})$, utilizing the complete data likelihood in \eqref{eq:likelihood-DPGMM}. To improve efficiency of the MCMC sampler, we prescribe conjugate or semi-conjugate priors whenever possible, enabling straightforward Gibbs sampling by exploiting full conditional posterior densities available for almost all parameters. Below, we detail these prior choices and discuss each step of the MCMC sampler. A pseudocode summary of our sampler appears as Algorithm 1 in Web Appendix A. \subsection{Scale parameter $\gamma$} Due to the scale decomposition in \eqref{eq:decompose-all}, sampling the parameter $\gamma$ is straightforward and can in fact be done independently of the Markov chain that samples the remaining parameters. That is, if we assign a Gamma prior $\gamma\sim Ga(a_0, b_0)$, we may directly draw samples using \[ \gamma \mid {\{\mathbf{x_i}\}, \{\mathbf{s_i}\}} \sim Ga(\alpha_0+N, \beta_0 + 1) . \] \subsection{Mark distribution parameters $\boldsymbol{\mu}$, $\sigma_{\ell}^2$, and $\sigma_{d}^2$} Conditioned on type labels $\{c_i\}$, the signal distributions for $\ell_i$'s and $d_i$'s \eqref{eq:model-scores-joint} are simple univariate normal models. Assuming diffuse priors $\mu_{\ell}, \mu_d \sim \text{Unif}((0,\infty))$, $\mu_{-d} \sim \text{Unif}((-\infty,0))$ and inverse-Gamma priors $\, \sigma_{\ell}^2,\sigma_{d}^2 \sim \text{inv-Gamma}(\nu_0/2, \nu_0\sigma_0^2/2)$ \,, parameter updates only require straightforward Gibbs steps from the full conditionals. For example, for the linkage score parameters $\mu_{\ell}$ and $\sigma^2_{\ell}$, we draw \begin{align} \mu_{\ell} \mid \sigma^2_{\ell}, \{\ell_i\}, \{c_i\} &\sim N_{(0,\infty)}\left(\sum_{i:c_i\neq 0} \text{logit}(\ell_i), \sigma^2_{\ell}/N_+\right)\\ \sigma^2_{\ell} \mid \mu_{\ell}, \{\ell_i\}, \{c_i\} &\sim \text{inv-Gamma}\left(\frac{\nu_0+N_+}{2}, \frac{\nu_0\sigma_0^2+\sum_{i:c_i\neq 0} (\text{logit}(\ell_i)-\mu_{\ell})^2}{2}\right). \end{align} Here $N_+ = \sum_{i=1}^N \mathbbm{1}(c_i \neq 0)$ is the total number of real transmissions given the $\{c_i\}$ configurations, and $N_{(0,\infty)}$ denotes a normal distribution truncated on the positive real line. For the direction score parameters $\mu_{d}, \mu_{-d}$ and $\sigma_d^2$, sampling steps are almost completely analogous. \subsection{Type probabilities $\mathbf{p}$} Assuming a Dirichlet prior for the probability vector $\mathbf{p} = (p_{-1},p_{0},p_1)$, $\mathbf{p} \sim \text{Dir}(q_{-1},q_{0},q_1)^T$ and given configurations for $\{c_i\}$, we can draw \begin{equation} \mathbf{p} \mid \{c_i\} \sim Dir\left( q_{-1} + N_{-1}, q_{0} + N_0,q_1+N_1\right), \end{equation} where $N_{k} = \sum_{i=1}^N \mathbbm{1}(c_i = k)$ is the total number of data points belonging to type $k$. \subsection{DP precision parameter $\alpha_k$ and component weights $w_{kh}$} For the precision parameter $\alpha_k$ and weights $w_{kh}$ for each type $k$, we adopt a truncated DP mixture model with a large maximum number of mixtures to approximate the infinite DP mixture, a technique described in Section 3.2 and in the Appendix of \cite{ji2009spatial}. More specifically, we use an auxiliary sampling trick introduced in \cite{escobar1995bayesian} to update the precision parameter $\alpha_k$'s for $k=0,-1,1$). By introducing an additional auxiliary parameter to sample along with each $\alpha_k$, the conditional posterior distribution for $\alpha_k$ can be reduced to a mixture of two Gamma distributions, which conveniently transforms its sampling step to a simple Gibbs step. For complete technical details, we refer the reader to \cite{ji2009spatial} and \cite{escobar1995bayesian}. \subsection{BVN mixture components $(\theta_{kh}, \Sigma_{kh})$} For the bivariate normal mixture model of each type $k$, we introduce a component latent indicator $z_i$ for each data point $i$ (that belongs to type $k$) such that $z_i=h$ indicates point $i$ belongs to component $h$. Assuming semi-conjugate priors $\theta_{kh} \sim \text{BVN}(\theta_0, \Sigma_0)$ and $\Sigma_{kh} \sim \text{inv-Wishart}(\nu, S_0)$, we then iteratively update the $z_i$'s and $(\theta_{kh}, \Sigma_{kh})$'s using \begin{align} Pr(z_i = h \mid w_{kh},\theta_{kh}, \Sigma_{kh}, \mathbf{s_i}) &\propto w_{kh} \cdot \text{dBVN}(\mathbf{s}_{i} \mid \theta_{kh}, \Sigma_{kh})\\ \theta_{kh} \mid \Sigma_{kh}, \{z_i\}, \{\mathbf{s_i}\} &\sim \text{BVN}\bigg((m_h\Sigma_{kh}^{-1}+\Sigma_0^{-1})^{-1}\left(\sum_{i:z_i=h}\Sigma_{kh}^{-1}\mathbf{s_i} + \theta_0\Sigma_0^{-1}\theta_0\right),\\ &\quad\quad\quad\quad\quad (m_h\Sigma_{kh}^{-1}+\Sigma_0^{-1})^{-1}\bigg);\\ \Sigma_{kh} \mid \theta_{kh},\{z_i\},\{\mathbf{s_i}\} &\sim \text{inv-Wishart}\left(\nu+m_h, \left(S_0^{-1}+\sum_{i:z_i=h}(\mathbf{s_i}-\theta_{kh})(\mathbf{s_i}-\theta_{kh})^T\right)^{-1}\right). \end{align} Here $\text{dBVN}(\cdot \mid \theta, \Sigma)$ is the density function of a bivariate normal with mean $\theta$ and covariance matrix $\Sigma$, and $m_h = \sum_{i=1}^N \mathbbm{1}(z_i = h)$ is the total number of data points belonging to spatial component $h$. \subsection{Type indicators $c_i$} The type $c_i$ can be sampled for each data point $i$ conditioned on all other parameter values and the observed data via its full conditional distribution $Pr(c_i = k \mid \Theta, \mathbf{s}_{i}, \mathbf{x}_i) \propto p_kf_k(\mathbf{s}_{i}) \phi_k(\mathbf{x}_i)$. \section{Simulation studies} \label{sec:simulation-experiments} In this section, we validate the modelling and inference framework for marked point patterns through synthetic experiments. With our application in mind, we focus on assessing whether the typed point process model can accurately recover simulated patterns of HIV transmission flows between men and women across different ages. We further explore the ability of the model to infer these flows at increasingly smaller sample sizes. Throughout this section, we focus on two contemporary questions of particular epidemiological interest, and investigate the ability of the typed point process model in recovering and differentiating between two competing scenarios. First, one general finding of recent HIV transmission flow studies is that there tend to be more infections that originate from men than from women~\citep{hall2021demographic,bbosa2020phylogenetic,ratmann2020quantifying}. This prompts us to explore whether the typed point process model can accurately recover parameters and distinguish between a scenario in which men drive 50\% of infections (``MF-50-50") and a scenario in which men drive 60\% of infections (``MF-60-40''). We explore $5$ different sample sizes (i.e., numbers of likely transmission pairs) with $N = 100, 200, 400, 600$ and $800$. For each scenario and each sample size, $100$ independent data sets were generated, and the typed point process model fitted to each. Details on the simulations and scenarios are fully described in Web Appendix B. Second, there is broad interest in the age distribution of the male sources of HIV infections, especially for adolescent girls and young women aged 15 to 24, due to the high incidence rates in these women~\citep{risher2021age}. This prompts us to investigate if the model can distinguish between two scenarios in which either younger men of approximately 25 years (within $\pm 5$ years age difference of adolescent and young women) contribute 60\% to infections in women aged 15-24 years and older men of approximately 35 years 30\%, and other men 10\% (``SAME AGE"); versus the alternative scenario that younger men of approximately 25 years contribute 30\%, and older men of approximately 35 years 60\%, and other men 10\% (``DISCORDANT AGE"). We varied the sample sizes, generated simulations and fitted the model as before, with complete details described in Web Appendix B. \begin{figure} \centering \includegraphics[width = \textwidth]{figures/joint_sim_plots_diff_Ns_median_update3.pdf} \caption{{\bf Performance of the typed point process model in recovering simulated transmission flow patterns.} Inference using the typed point process model was performed on each of the 100 simulated data sets for each scenario. {\bf A} Boxplot of the posterior mean estimate of transmissions from men in 100 replicate simulations for the MF-50-50 (left panel) and MF-60-40 scenarios (right panel). Throughout, the dashed lines mark the true values that underpin the simulated data. The x-axis shows the sample size of simulated data points, which represent the number of phylogenetically closely related pairs of individuals identified through phylogenetic deep-sequence analyses. {\bf B} Boxplot of the posterior mean estimate of transmissions from men of similar age (shown in red) and older age (shown in blue) to infection in adolescent and young women aged 15-24 in 100 replicate simulations for the ``SAME AGE'' and ``DISCORDANT AGE'' scenarios. As before the dashed lines mark the true values that underpin the simulated data and the x-axis shows results for different sample sizes.} \label{fig:weights_props_by_N_joint} \end{figure} Figure~\ref{fig:weights_props_by_N_joint} (top panel) summarizes our findings on the male-female simulation experiments (MF-50-50 and MF-60-40). Our findings reveal that the model is able to successfully distinguish between the two competing epidemiological scenarios with sample size $N \geq 200$, and produces estimates that are accurate within a $\pm 5$\% error margin for sample sizes $N \geq 200$. Figure~\ref{fig:weights_props_by_N_joint} (bottom panel) illustrates our findings on the age-specific sources experiments (SAME-AGE, DISCORDANT-AGE). This is a substantially more difficult inference problem because the target quantities relate to a smaller subgroup of the entire source population. With a small sample ($N$ in the range $100-200$), the relative relationship between the two proportions is inferred correctly in all experiments. However, the actual quantitative estimates and differences can be far from the truth, and tend to be over-estimated. For sample sizes of $N\geq 400$, the quantitative estimates become accurate for practical purposes. The observed over-estimation is likely due to the parsimony induced by the Dirichlet Process priors that are known to prefer assigning data points to the largest existing clusters when there are not enough data to admit a new mixture model component. As a result, more points tend to be attributed to the component with the highest weight when $N$ is small; this effect is mitigated as $N$ increases. Additional analyses on simulated experiments are included in Web Appendix B. These include numerical convergence and mixing analyses, which are generally quite satisfactory, as well as Bayesian coverage analyses, which indicate that the posterior distributions of the target quantities are well calibrated. \section{Case study} \label{sec: case-study} We are now ready to apply the typed point process model to study demographic and population-based HIV deep-sequence data collected from the Rakai Community Cohort Study in Southeastern Uganda between August 2011 and January 2015~\citep{ratmann2019inferring, ratmann2020quantifying, xi2021inferring}. Our aim is to reconstruct transmission flows by gender and continuous age between 15 and 50 years. This case study is challenging in the sense that phylodynamic analyses using standard HIV consensus sequences have difficulty inferring flow patterns for more than a handful of age groups: many contemporary analyses of age-specific transmission flows remain limited to a resolution of 5-year or 10-year age bands~\citep{Scire2020Improved,bbosa2020phylogenetic,le2019hiv,de2017transmission, ratmann2020quantifying}. We revisit the HIV deep-sequence dataset considered by ~\cite{xi2021inferring}, and compare our approach applied to the complete dataset without appealing to heuristic pre-classification that results in utilizing only a small subset of data. More specifically, we will use the typed point process model to analyze all pairs of closely phylogenetically related individuals introduced in Section~\ref{sec: data-model}. After filtering out only pairs of individuals with a linkage score $< 0.2$ (i.e., pairs with very weak evidence), we retain a dataset of 526 pairs that represent potential transmission events. For comparison purposes, we also repeat the analysis under the heuristic thresholding approach following~\cite{xi2021inferring}; this pre-processing step retains only retains a subset of those pairs of individuals as ``high-confidence'' transmission pairs. Referring to the first as the ``full analysis with latent event types'' (or ``full analysis'' in short), we fit the typed point process model to all data, simultaneously inferring the three latent event types in our application, $k=0$ for no transmission event, $k=1$ representing male-to-female transmission, and $k=-1$ for female-to-male transmission (recall the Methods section). In the second ``subset analysis with fixed event types'' (``subset analysis'' in short), we retain similarly as in~\citep{xi2021inferring, hall2021demographic} only those data with linkage scores $>0.6$ and a direction score $>0.5$ as a male-to-female transmission pair, and those with linkage scores $>0.6$ and a direction score $\leq 0.5$ as a female-to-male transmission pair. In the full analysis, all 526 data points will be used, while the subset analysis considers only 367 pairs. This is a larger subset of data retained compared to~\cite{xi2021inferring} which consists of 238 pairs of individuals, where only ``high-confidence'' potential transmission pairs with direction score $<0.33$ or $>0.67$ were retained for analysis. In both analyses, we adopt the same priors as described in Section~\ref{sec:simulation-experiments} and run the MCMC algorithm detailed in Section~\ref{sec: inference} for 3000 iterations after 1000 burn-in steps. \subsection{Learning the latent event types is computationally feasible} We are able to infer both the transmission flow surfaces of interest, $f_1$ and $f_{-1}$ in \eqref{eq:density-mixture}, and the unknown event types $\{c_i; i= 1,\dotsc, N\}$ for each observed data point without issues in numerical convergence or mixing. On a laptop with a 4-core Intel CPU, the full analysis with the typed point process model including the latent event types takes less than 10 minutes, which is a considerable improvement in computational efficiency compared to using the semi-parametric Poisson count model on 1-year age discretized bands~\cite{xi2021inferring}, where 4000 total iterations requires about 30 hours. This speed-up is expected, as our typed point process model greatly simplifies computation through a continuous spatial point pattern specification that only requires likelihood evaluations on the observed $526$ data points. In comparison, the Poisson count model in \cite{xi2021inferring} involves likelihood evaluations on all $2\times (50-15)^2=2450$ grid cells, many of which contain structural zeros. \subsection{Learning latent event types results in more data used for inference of transmission flows} In comparison to the subset analysis, the full analysis appears to attribute more pairs of individuals to type $1$ (male-to-female transmission) and to type $-1$ (female-to-male transmission). We present the posterior probability $p_k$ for each type as inferred by the full analysis and by the subset analysis in Table~\ref{tab: type-proportions}. Multiplying the estimated $p_k$'s by sample size $526$, we also calculate the number of pairs, $N_k$, that each analysis has attributed to each type. Notably, the full analysis learns a $p_0$ (proportion of ``no transmission'') that is significantly lower than that under the fixed event type subset analysis. This indicates that, by probabilistically inferring event types instead of heuristically pre-classifying them, our proposed approach is utilizing more phylogenetically close pairs to learn the transmission flows. In short, our approach not only reflects uncertainty in event types, but also makes more efficient use of phylogenetic and demographic evidence in data. \begin{table}[ht] \caption{{\bf Proportions and numbers of inferred event types.} Posterior mean estimates with 95\% credible intervals.} \centering \begin{tabular}{l\|cc\|cr} \hline \multirow{ 3}{}{Type} & \multicolumn{2}{l\|}{\textbf{Full analysis}} & \multicolumn{2}{l\|}{\textbf{Subset analysis}}\\ & \multicolumn{2}{l\|}{\textbf{with latent event types}} & \multicolumn{2}{l\|}{\textbf{with fixed event types}}\\ \cline{2-3} \cline{4-5} & $p_k$ & $N_k$ & $p_k$ & $N_k$ \\ \hline Male-to-Female & 46.3\% (39.4\%, 53.1\%) & 244 (207, 279) & 35.7\% & 188 \\ transmission $(k=1)$& & & & \\ Female-to-male & 35.0\% (28.5\%, 42.0\%) & 184 (150, 221) & 29.5\% & 155 \\ transmission $(k=-1)$ & & & & \\ No transmission & 18.6\% (9.4\%, 29.2\%) & 98 (49, 154) & 34.8\% & 183 \\ $(k=0)$ & & & &\\ \hline \end{tabular} \label{tab: type-proportions} \end{table} \subsection{Inferred age of male and female sources of HIV transmission.} With the typed point process model, we are able to estimate the sources of male and female HIV infection events in terms of the continuous age of the sources (Figure~\ref{fig:age-distributions-general}). This is an important advantage over other approaches through which the age of the sources can be characterised by 5-year or 10-year age bands~\citep{Scire2020Improved,bbosa2020phylogenetic,le2019hiv,de2017transmission}, because at such coarse scale discretization introduces artifacts that obfuscate differences in transmission modes that are important to differentiate from a public health perspective~\citep{xi2021inferring}. Figure~\ref{fig:age-distributions-general} addresses the following question about HIV transmission --- which age groups contribute most to HIV transmission at the population-level? The colored solid curves and text illustrate the inferred age distribution of the male and female sources in the full analysis with latent event types, while the dark dashed lines and dark text summarize results for the subset analysis with fixed event types. We find that the inferred age distribution of the male and female sources are very similar in both analyses. More specifically, during the observation period 2011-2015, male sources tended to be consistently older than female sources. The estimated 50\% highest posterior density intervals (HDIs) learned in the full analysis with latent event types are ages $[25.4, 34.3]$ for male sources and ages $[20.4, 29.0]$ for female sources. In comparison, the 50\% posterior HDI inferred in the subset analysis with fixed event types are $[26.9, 35.4]$ for the age distribution of male sources, and $[21.4, 29.3]$ for female sources. This indicates that estimation uncertainty in quantities of key epidemiological interest did not decrease in the full analysis. Rather, we find that the full analysis using latent event types better reflects the actual uncertainty in key quantities, as the typed point process model explicitly accounts for uncertainty in the event types that underpin each data point. Both the full and the subset analyses indicate a characteristic peak in transmission from men around 30 years of age, and a long, pronounced tail of transmissions from older men beyond age 40. In this regard, we find that the age distribution of the female sources differs qualitatively, as the probability of transmissions from women declines more rapidly with age, reaching similar levels by age 30 in women compared to age 45 in men. In the full analysis with latent event types, these observations on the age of the female sources are more strongly supported, as in the full analysis the entire age distribution of the female sources is slightly shifted towards younger ages compared to the age distribution of the female sources in the subset analysis with fixed event types. We can gain further insight into age-specific transmission dynamics by considering the age profile of the transmitting partners for recipients of a particular age. In Figure~\ref{fig:source-prob-by-age-full}, the age of a recipient is shown in colour, and the wave on the $y$-axis shows the posterior median contribution of transmissions to recipients of the age indicated in colour. Figure~\ref{fig: age-source-distribution-3year} illustrates that the age profile of sources has a characteristic shape for each recipient age group, and in particular are not simply shifted versions of one age profile. Figure~\ref{fig: age-source-distribution-stacked} demonstrates how the superposition of the age-structured transmission dynamics results in the overall source profile that marginalises out the age of the recipients. \begin{figure} \begin{subfigure}{0.98\textwidth} \includegraphics[width=0.48\textwidth]{figures/male_source_age_all.jpg} \includegraphics[width=0.48\textwidth]{figures/female_source_age_all_reduce.jpg} \caption{{\bf Age distributions of male and female sources of HIV infections in Rakai, Uganda during the 2011-2015 observation period.} The left panel shows the estimated age of the male sources and the right panel the estimated age of the female sources.} \label{fig:age-distributions-general} \end{subfigure} ~ \vspace{0.15in} \begin{subfigure}{0.98\textwidth} \centering \includegraphics[width=0.48\textwidth]{figures/male_source_age_young_women.jpg} \includegraphics[width=0.48\textwidth]{figures/male_recipients_age_young_women.jpg} \caption{{\bf Age distributions of male sources and recipients of HIV infections in women aged 15-24.} Left panel shows the age distribution of \emph{male sources}, and right panel characterizes age distribution of \emph{male recipients}, for women aged 15-24.} \label{fig:age-distributions-young-women} \end{subfigure} \caption{\textbf{Age distributions of sources and recipients, for the general population (subplot \ref{fig:age-distributions-general}) and for women aged 15-24 (subplot \ref{fig:age-distributions-young-women})}. In each panel, the colored lines represent density curves of the age of sources/recipients for 100 posterior samples from the inferred, smooth transmission flow intensity surface of the typed point process model in the full analysis with latent event types. The thicker curve indicates the posterior mean density curve. The black dashed curve illustrates the posterior mean density curve in the subset analysis with fixed event types. 50\% highest density intervals (HDIs) are marked in text, with colored text indicating the HDIs inferred in the full analysis and black text indicating the HDIs inferred in the subset analysis.} \end{figure} \begin{figure} \begin{subfigure}{0.98\textwidth} \centering \includegraphics[width=0.48\textwidth, page=3]{figures/source_freq_specTreat_Jan2022_titled_updated.pdf} \includegraphics[width=0.48\textwidth, page=4]{figures/source_freq_specTreat_Jan2022_titled_updated.pdf} \caption{\textbf{Age distributions of sources for recipients in different 3-year age groups.} Each curve represents the learned relative frequencies of sources responsible for transmissions within each recipient age group. } \label{fig: age-source-distribution-3year} \end{subfigure} ~ \vspace{0.15in} \begin{subfigure}{0.98\textwidth} \centering \includegraphics[width=0.48\textwidth, page=3]{figures/source_freq_specTreat_Jan2022_stacked.pdf} \includegraphics[width=0.48\textwidth, page=4]{figures/source_freq_specTreat_Jan2022_stacked.pdf} \caption{\textbf{Marginal age distributions of sources.} Shown as ``stacked'' curves of age source distributions for each recipient age group (in 3-year age bands). } \label{fig: age-source-distribution-stacked} \end{subfigure} \caption{\textbf{Age distributions of sources for heterosexual recipients in different age groups (by 3-year age bands).} \textbf{In subplot \ref{fig: age-source-distribution-3year}}, the curves for each recipient age group represent posterior means of the source age distributions learned by the full analysis with latent event types. \textbf{In subplot \ref{fig: age-source-distribution-stacked}}, the ``marginal'' age distributions are shown by stacking up the frequencies of sources for each recipient age group. \textbf{Left} column shows age distributions for male sources in male-to-female transmissions for different female recipient age groups. \textbf{Right} column shows age distributions for female sources in female-to-male transmissions for different male recipient age groups. } \label{fig:source-prob-by-age-full} \end{figure} \subsection{Transmissions to and from adolescent and young women} Next, we focus our attention on adolescent and young women of age 15 to 24 years. Specifically, we wish to understand the age distribution of male sources in their infectious, and in turn, that of male recipients for whom these women are the sources of infection. Our rationale is that understanding the age distribution of the male sources and recipients in adolescent and young women can provide further evidence to HIV programs that aim to reduce infections in this age group~\citep{glynn2001young,pettifor2008keep,karim2010preventing,jewkes2010intimate}. One prominent program is the ``Determined, Resilient, Empowered, AIDS-free, Mentored, and Safe'' (DREAMS) program~\citep{saul2018determined} within the US President's Emergency Plan for AIDS Relief (PEPFAR)~\cite{oliver2012us}. Figure~\ref{fig:age-distributions-young-women} illustrates the inferred age distributions of the male sources of infections in adolescent and young women (left), and of the male recipients of transmissions from these women (right) in the full analysis with latent event types (colored curves) and the subset analysis with fixed event types (black curves). While the majority of male sources are men between 24 to 30 years, there is also a notable subgroup of male sources older than 35. In contrast, the male recipients of transmission from adolescent and young women lie more strongly in men aged 24 to 30 years, with a much smaller subgroup of male recipients older than 35. That is, a transmission pathway strongly supported by the 2011-2015 data may begin with transmission from men approximately 5-8 years older than adolescent and young women, and who then in turn transmit the virus in smaller numbers to men who are again 5-8 years older than themselves. In addition, a second transmission pathway may begin with transmission from men 10-15 years older than adolescent and young women, and who then in turn transmit the virus in smaller numbers to men who are more likely to be 5-8 years older than themselves, rather than 10-15 years older. Our findings for the 2011-2015 observation period thus further corroborate the importance of major HIV prevention programs across Sub-Saharan Africa that have been directing educational and healthcare resources to adolescent and young women, who are especially vulnerable to HIV from a broad age spectrum of male sources~\citep{saul2018determined,UNAIDS2018}. \begin{figure} \centering \includegraphics[width=0.98\textwidth]{figures/colored_points_contour_3.pdf} \caption{{\bf Comparison of the inferred age structure in transmission flows in the full with latent event types versus the subset analysis with fixed event types.} ({\bf A}) Results in the subset analysis with fixed event types. Source-recipient pairs that were pre-classified by event type (dots) are shown along the posterior median estimate of 50\%, 80\% and 90\% highest probability regions of transmission flows (contours). The number of data points attributed to each type is indicated in the top left corner. ({\bf B}) Results in the full analysis with latent event types. Source-recipient pairs (dots) are shown by posterior event type probabilities (colour intensity) along the posterior median estimate of 50\%, 80\% and 90\% highest probability regions of transmission flows (contours). The ``effective'' number of data points attributed to each type (posterior mean estimate of $N_k$ as in Table~\ref{tab: type-proportions}) is indicated in the top left corner.} \label{fig:points-colored-by-probs-with-contours} \end{figure} \subsection{In-depth comparison of the full analysis with latent event types with the subset analysis with fixed event types} We close by examining the more detailed features of the inferred age- and gender-specific transmission flows in the subset analysis with fixed event types (Figure~\ref{fig:points-colored-by-probs-with-contours}A) compared to the full analysis with latent event types (Figure~\ref{fig:points-colored-by-probs-with-contours}B). In the subset analysis with fixed event types, only a subset of all data points are utilized to learn the latent age structure associated with each type, and all data points are assumed to carry the same weight (as colored by the same shade in each panel). In the full analysis with latent event types, all data points contribute to the learning of the latent age structure associated with each event type, and the contribution of each data point is weighted by the posterior event type probabilities associated with each point (as illustrated with the colour shades of each point). Overall, we find that the learned 2D surfaces of age-specific male-to-female transmission flows and female-to-male transmission flows are very similar in the full analysis with latent event types compared to the subset analysis (contours in Figure~\ref{fig:points-colored-by-probs-with-contours}. This is already indicated by the findings on the marginal age distribution of the male and female sources of infection in Figure~\ref{fig:age-distributions-general}. However, the inferred latent surface corresponding to no transmission differs substantively between the full and subset analyses, with noticeably more data points in the corners of the compact space attributed to either male-to-female or female-to-male transmission. Data points corresponding to combinations of women around age 20 and men around age 30 are more strongly attributed to male-to-female transmission in the full analysis, in line with the strong center of mass of the inferred male-to-female transmission surface for these age combinations. Data points involving women around age 45 are more strongly attributed to female-to-male transmission, in line with the wider tails of the inferred female-to-male transmission surface for these age combinations. This latter observation illustrates the potential limitations of the typed point process model, because women in this age group tend to have lower HIV viral load at the population-level than men, rendering this population group of women on average less infectious than men~\citep{grabowski2017hiv, rodger2019risk}. Within the proposed Bayesian framework, such information could straightforwardly be included by using more informative priors than we have imposed here, which may play an important role for future applications of the typed point process model. \section{Discussion}\label{sec: conclusion} In this paper, we develop a hierarchical typed point process to learn disease transmission flows from phylogenetically reconstructed transmission pair data. The model is developed alongside a computationally efficient Bayesian sampler targeting the exact likelihood of the specified model. We demonstrate that our novel approach can probabilistically learn the unobserved event types --- whether or not there is transmission between a pair and in which direction the transmission occurs --- despite the large number of additional latent parameters and partially informative data. Its ability to do so is in large part owing to reformulating the inference problem in terms of a continuous spatial process. Unlike previous work~\citep{xi2021inferring}, this does not require discretization of the feature space and thus avoids keeping track of all cells in a large transmission flow matrix in computation; our method only needs to track all the data points. This advantage by construction renders our approach much more computationally efficient. In simulations and a case study on population-level HIV deep-sequence data from the Rakai Community Cohort Study in Southern Uganda, we demonstrate that the typed point process model allows inclusion of more data in analysis while accounting for the intrinsic uncertainties associated with outcomes of deep-sequence phylogenetic analyses. The primary epidemiological conclusions under the typed point process model on the Rakai data set are consistent with those drawn under the previously developed model using a subset of the data. We do note that among transmission events identified by our proposed approach, the odds ratio between male sources and female sources is approximately 1.32, which is higher than the estimate from the subset analysis with fixed event types. Most phylodynamic analyses into the age-specific drivers of HIV transmission are limited to analyses of relatively coarse age bands --- either for technical or computational reasons~\citep{de2017transmission, le2019hiv,bbosa2020phylogenetic} --- and in this context, the typed point process model extends the statistical toolkit to address epidemiologically important research questions. Our continuous spatial process approach in modelling disease transmissions marks a contrast to the body of work that relies on count data in discretized spatial areas~\citep{berke2004exploratory,best2005comparison,wakefield2007disease,gschlossl2008modelling,mohebbi2014disease,bauer2016bayesian,johnson2019spatially}. These discrete or areal spatial models often involve the use of Gaussian Markov random fields~\citep{rue2005gaussian} or other Gaussian-based models to handle spatial dependence structures. Due to the high computational cost associated with Gaussian covariance matrix operations, inference typically entails numerical approximation techniques such as integrated nested Laplace approximations (INLA)~\citep{rue2009approximate} and still demands intensive computation. These computational challenges are inevitable when only aggregate data are available~\citep{gschlossl2008modelling,mohebbi2008geographical}, but given access to point-level data, formulating a continuous spatial model becomes more appealing than a discretized one. Directly modeling a continuous spatial process avoids expensive spatial smoothing techniques, and moreover can induce a discrete spatial model at any desired resolution, avoiding the need for manual discretization before analysis~\citep{van2017efficient, xi2021inferring}. In the context of our application, where age is known for every surveyed individual and can be treated as a continuous variable, employing a continuous spatial model provides a more natural and efficient approach compared to prior work, and in addition enables inference of the unknown event types that are associated with each phylogenetically reconstructed data point. There are a number of future directions that one may take based on our proposed framework. It may be of interest to incorporate individual-level covariates into the spatial process, either as additional marks or latent effects of the Poisson process as in~\cite{hu2018bayesian}, or as additional covariates in the marks distributions. Second, here we did not explicitly model the sampling process of actual transmission events; but a similar approach as in~\cite{xi2021inferring} could be straightforwardly integrated to adjust the intensity function of the typed point process model. Third, extensions to non-normal components in the spatial density function mixture model can help relax certain assumptions entailed by a normal mixture model; for example, if similarity of transmission behavior is not necessarily dependant on spherical spatial proximity (an implicit assumption of the normal model), then other kernels such as the bivariate Beta kernel as in \cite{kottas2007bayesian} can be considered. \section{Acknowledgements} This work was partially supported by the National Science Foundation (DMS-2030355, DMS-2230074, and PIPP-2200047), the National Institutes of Health (R01 AI153044), and the Bill \& Melinda Gates Foundation (OPP1175094, OPP1084362). We thank the staff, investigators, and participants of the Rakai Community Cohort Study who made this work possible. We thank Mike West for helpful discussions, and Xiaoyue Xi for helpful comments and data pre-processing. The findings and conclusions in this article are those of the authors and do not represent the official position of the funding agencies. \section{Supporting Information} All Supporting Information and Web Appendices referenced in the main text are available in the \texttt{supplement.pdf} document, attached at the end of this preprint. \section*{Data sharing} All anonymized data and code to reproduce the analyses in this paper are available at the \texttt{GitHub} repository \url{https://github.com/fanbu1995/HIV-transmission-PoissonProcess} under the GNU General Public License version 3.0. The deep-sequence phylogenies and basic individual-level data analysed during the current study are available in the Dryad repository (DOI: 10.5061/dryad.7h46hg2). HIV-1 reads are available on reasonable request through the PANGEA consortium; see \url{https://www.pangea-hiv.org}. Additional individual-level data are available on reasonable request to RHSP; see \url{https://www.rhsp.org}. \setstretch{1.25} \bibliographystyle{biom}	{ "redpajama_set_name": "RedPajamaArXiv" }	9,840
//------------------------------------------------------------------------------ // <auto-generated> // This code was generated by \generate-code.bat. // // Changes to this file will be lost when the code is regenerated. // The build server regenerates the code before each build and a pre-build // step will regenerate the code on each local build. // // See https://github.com/angularsen/UnitsNet/wiki/Adding-a-New-Unit for how to add or edit units. // // Add CustomCode\Quantities\MyQuantity.extra.cs files to add code to generated quantities. // Add UnitDefinitions\MyQuantity.json and run generate-code.bat to generate new units or quantities. // // </auto-generated> //------------------------------------------------------------------------------ // Licensed under MIT No Attribution, see LICENSE file at the root. // Copyright 2013 Andreas Gullberg Larsen (andreas.larsen84@gmail.com). Maintained at https://github.com/angularsen/UnitsNet. // ReSharper disable once CheckNamespace namespace UnitsNet.Units { // Disable missing XML comment warnings for the generated unit enums. #pragma warning disable 1591 public enum AccelerationUnit { Undefined = 0, CentimeterPerSecondSquared, DecimeterPerSecondSquared, FootPerSecondSquared, InchPerSecondSquared, KilometerPerSecondSquared, KnotPerHour, KnotPerMinute, KnotPerSecond, MeterPerSecondSquared, MicrometerPerSecondSquared, MillimeterPerSecondSquared, MillistandardGravity, NanometerPerSecondSquared, StandardGravity, } #pragma warning restore 1591 }	{ "redpajama_set_name": "RedPajamaGithub" }	5,130
Q: Finding extremes and values between them for a graph I need an algorithm to find extremes and a certain number of values between them for a graph. For example, the highest value I have is 0.805 and the lowest is 0.694; I want the extremes to be 0.81 and 0.69; given that the number of values in between is 6, I want these values to be 0.79, 0.78, 0.76, 0.74, 0.72, 0.71 or so. Long ago I wrote a function that finds the high extreme, but it was intended for the graphs with the lowest value is always set to 0: const BASE = 5; const LENGTH = 2; function graphTopValue(max_value, max_items) { return restore(find(extract(max_value), max_items), max_value); } function numlen(n) { return Math.ceil(Math.log10(n)); } function isDivisible(v, i) { var r = numlen(v) - 1; if ([0, 3, 6, 9].includes(r)) return true; return (v / i) % BASE == 0; } function extract(v) { return parseInt(v.toString().substr(0, LENGTH)) + 1; } function restore(v, o) { var oldlen = numlen(o); if (oldlen <= LENGTH) return v; return v * Math.pow(10, oldlen - LENGTH); } function find(v, i) { if (v % i == 0 && isDivisible(v, i)) return v; return find(v + i - (v % i), i); } var max_value = 999; var max_items = 4; console.log(graphTopValue(max_value, max_items)); // 1000 Now I need an algorithm to work with any values and also find pretty values between extremes. How can I do it? A: OK, I've found out that this is called 'Nice Label Algorithm'. Here's one of implementations: https://gist.github.com/igodorogea/4f42a95ea31414c3a755a8b202676dfd	{ "redpajama_set_name": "RedPajamaStackExchange" }	3,021
In this specialisation you will examine the threats posed to society by different forms of environmental change, such as climate change and resource depletion, and the responses needed to ensure sustainability for life support. You will draw on relevant global policy to understand the scientific evidence base required for effective environmental security policy and management across all levels of government and society. The specialisation is designed to equip you with professional skills and with the knowledge and skills to help society mitigate and adapt to global environmental change, so securing a sustainable future.	{ "redpajama_set_name": "RedPajamaC4" }	4,004
Wikimedia disambiguation page The Simple English Wiktionary has a definition for: thread. Thread could mean: A kind of thin yarn, thin fibers spun together, for sewing Screw thread, a ridge running in a spiral down the length of a screw Threaded discussion, a group of messages to a newsgroup, mailing list or Internet forum on a single topic Thread (computer science), a sequence of instructions which may execute in parallel with other threads Threads, a 1984 BBC docudrama concerning the after-effects of a nuclear attack on Sheffield Thread, a destructive mindless alien spore in the Dragonriders of Pern series of science fiction novels "Threads", a Season 8 Stargate SG-1 episode Thread (unit of measurement), a cotton yarn measurement, equal to 54 inches Thread, a 1999 album by Wideawake Thread, a sugar stage in candy cookery Thread (business technology), a connection made to publicly record the quality of a business transaction This disambiguation page lists articles associated with the title Thread. Retrieved from "https://simple.wikipedia.org/w/index.php?title=Thread&oldid=4944438"	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	8,822
DEIT Around the World CupAround the World Cup TicketsTickets How to arrive at the eventHow to arrive at the event World Championships 2020World Championships 2020 Cross-country skiing AntholzCross-country skiing Antholz InfrastructureInfrastructure Restaurant Biathlon InnRestaurant Biathlon Inn Visit the stadiumVisit the stadium Destination AntholzDestination Antholz Mascot of Anterselva HomepageNews & more27.01.2019 - Laura Dahlmeier wins mass start in Anterselva 27.01.2019 - Laura Dahlmeier wins mass start in Anterselva Double Olympic champion Laura Dhalmeier is back. The 25-year-old from Garmisch Partenkirchen won the 12.5-km mass start race at the World Cup in Anterselva on Sunday and celebrated her first victory of the season Do you like the article? Dahlmeier prevailed over Czech Marketa Davidova and her team mate Vanessa Hinz. Local athlete Dorothea Wierer finished in 5th place as the best Italian, but increased her lead in the overall World Cup to 34 points. After the first lying shooting Dahlmeier, who left one disc standing, was only in 22nd place. After that she made no further mistakes, caught up place by place and took the lead in the first standing shooting. She retained it until the finish and, almost exactly one year after the Olympic Winter Games in Pyeonchang, climbed the podium again for the first time. Dahlmeier had already shown her improving form in the first two races in Anterselva, with a 4th place in the sprint and 2nd place in the pursuit. The battle for 2nd place was particularly exciting. After the final shooting Vanessa Hinz was still nine seconds ahead of Marketa Davidova. The sprint winner of Thursday made up for the gap over the final 2.5 km and finished 2.13.1 seconds behind Dahlmeier. Hinz came third with 16.4 seconds behind Davidova. Wierer extends her World Cup lead After yesterday's victory in the pursuit race, Dorothea Wierer had to content with a fifth places today, behind Hanna Oeberg from Sweden. Wierer raced very fast, but had three misses in the shooting range. In the end, she was 33.1 second behind the winner and 16.8 second behind the third on the podium. As last year, the 28-year-old from Rasun finished in the top ten in all three Anterselva races and increased her lead in the overall World Cup to 34 points over team mate Lisa Vittozzi. Vittozzi was not happy with her race today. She finished eleventh, with two mistakes in shooting and 51.7 seconds behind Dahlmeier. The third Italian in the race, Nicole Gontier, missed as many as 10 discs and had to content with last place (29th). The World Cup leader in the mass start Paulina Fialkova from Slovakia finished 16th today, enough to maintain her position at the top of the ranking in the discipline. Fialkova has 127 points after three mass start races, Oberg is 10 points behind. Comments on the race: Laura Dahlmeier (GER), 1st Place: "It's a fantastic feeling to be right at the top again. I started very badly and I felt slow. But I came back strongly and I remained virtually without error in the shooting. It was a good weekend for me." Marketa Davidova (CZE), 2nd Place: "I am very happy with my race, and overall with the whole weekend. The material was perfect today and I hope that it will continue like this. I feel really good in Anterselva, it's one of my favourite places." Vanessa Hinz (GER), 3rd place: "It's a wonderful moment. I simply tried to give my very best and I succeeded. I've constantly improved from the start and I'm very happy with this podium finish. The fact that two Germans are on the podium is also proof that our team does not just consist of Laura Results Biathlon World Cup Anterselva (ITA): Mass start women, 12.5 km 1. Laura Dahlmeier (GER) 35.32,8 minutes (1 error) 2. Marketa Davidova (CZE) +13,1 (1) 3. Vanessa Hinz (GER) +16,4 (1) 4. Hanna Öberg (SWE) +17,1 (2) 5. Dorothea Wierer (ITA/Rasen) +32,2 (3) 6. Kaisa Makarainen (FIN) +33,3 (2) 7. Iryna Kryuko (BLR) +34,2 (2) 8. Karolin Horchler (GER) +42,6 (0) 9. Mona Brorsson (SWE) +49,8 (2) 10. Marte Olsbu Roeiseland (NOR) +50,1 (3) 11. Lisa Vittozzi (ITA) +51,7 (2) 29. Nicole Gontier (ITA) +5.42,0 (10) 24.01.2019 - Our partner TechnoAlpin Guaranteed snow on the cross country ski tracks in Anterselva 17.03.2019 - Big final party at Antholz House with all the Italian medalists of the Östersund World Championship 22.01.2023 - Norway dominates once again As expected, Norway's biathletes dominated their way to a 1st-place finish in the men's 4x7.5km relay to conclude the Biathlon World Cup in Antholz on Sunday afternoon. 03.11.2020 - Next year all spectators in Anterselva will have a front row seat In just over two and a half months the world's best biathletes will gather in Anterselva for the 2021 Biathlon World Cup. Südtirol Arena Alto Adige, Obertaler Str. 33I-39030 Rasen-AntholzT. +39 0474 492 390F. +39 0474 492 300 Strong partners Fan-to-fan ticket resale How to arrive at the event Bumsi, our mascot Organisation committee Stadium Regolations © 2023 Biathlon World Cup CommitteeImpressumPrivacyCookie settingsSitemapproduced by Start listsResults	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	388
{"url":"https:\/\/talkstats.com\/members\/bfernandez.102105\/recent-content","text":"Recent content by Bfernandez\n\n1. Understanding ordinal regression\n\nHi! This is a really simple question, but one for which the answer continues to elude me! I understand that in a binary logit regression the resulting logit is the probability of an event occurring, defined as Ln(P\/1-P) However, in an ordinal logit, in which there are multiple...","date":"2022-10-05 15:56:28","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\": 0.9170613884925842, \"perplexity\": 1108.082390376815}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030337631.84\/warc\/CC-MAIN-20221005140739-20221005170739-00355.warc.gz\"}"}	null	null
Hospital Affiliated Group-Stunning Port Charlotte, Recruiting a BC/BE Urologist-No State Income Tax Earnings Go Farther! Large regional healthcare system is recruiting an employed Urologist. Medical Center is a 250-bed, Joint Commission-accredited, full-service hospital serving Charlotte, Sarasota and Desoto counties. Join a hospital owned group. State-of-the-art facility with Da Vinci Robot. Call is anticipated 1:2. There is office space available onsite. ABMS/AOA Board Certification or Eligible with Certification in Process Required. Compensation Package May Include: Commencement bonus, Assistance with medical education debt, CME expenses, Relocation assistance, No Visa Assistance Available.Port Charlotte is a small community nestled along Florida's Southwest Gulf Coast, between Sarasota and Naples, about 100 miles south of Tampa. The area features numerous upland and aquatic preserves, marine estuaries and a unique network of barrier islands and mangrove forests. In addition, Port Charlotte offers miles of beaches, more than 70 parks, 16 golf courses as well as great dining, music and theater. Found along Florida's welcoming Gulf Coast, Port Charlotte is located about halfway between Sarasota and Fort Myers, perfectly positioned to offer every manner of water-borne activities. Port Charlotte more than 165 miles of waterways, providing access to Charlotte Harbor and the Gulf of Mexico and many more miles of natural shoreline bordering Charlotte Harbor and the Peace and Myakka rivers. But that's not all. Seven of the 21 golf courses located in Charlotte County are found in Port Charlotte. Charlotte Sports Park is home to spring training for the Tampa Bay Rays. Tippecanoe Environmental Park in Port Charlotte offers hiking trails and wildlife viewing through 380 acres of scrub and pine flat-woods.	{ "redpajama_set_name": "RedPajamaC4" }	3,914
Тімо Пярссінен (; 19 січня 1977, м. Лох'я, Фінляндія) — фінський хокеїст, лівий/правий нападник. Вихованець хокейної школи ТуТо (Турку). Виступав за ТуТо (Турку), «Хермес» (Коккола), ГПК (Гямеенлінна), «Анагайм Дакс», «Цинциннаті Майті-Дакс» (АХЛ), ГІФК (Гельсінкі), ХК «Цуг», ХК «Тімро». У складі національної збірної Фінляндії учасник чемпіонатів світу 2001, 2002, 2004, 2005 і 2007 (41 матч, 9+18). Досягнення Срібний призер чемпіонату світу (2001, 2007) Бронзовий призер чемпіонату Фінляндії (2000, 2004). Посилання Профіль на Eliteprospects Фінські хокеїсти Гравці збірної Фінляндії із хокею Хокеїсти ГПК Хокеїсти «Анагайм Дакс» Хокеїсти ГІФК Хокеїсти «Цуга» Хокеїсти «Тімро» Задрафтовані «Анагайм Дакс» Хокеїсти АХЛ Хокеїсти «ТуТо»	{ "redpajama_set_name": "RedPajamaWikipedia" }	4,115
Maira superba är en tvåvingeart som beskrevs av Meijere 1913. Maira superba ingår i släktet Maira och familjen rovflugor. Inga underarter finns listade i Catalogue of Life. Källor Rovflugor superba	{ "redpajama_set_name": "RedPajamaWikipedia" }	1,862
Joel Califa writes on his personal blog about small interface changes to GitHub that have saves millions of developers immeasurable amounts of wasted time and anxiety. One small change can add up to a big win. High frequency actions (such as creating new PRs on GitHub) take place millions of times a day. A given user might go through the same flow several times per week, per day, or even per hour. These flows become a part of their lives. If there is even a slight inefficiency or frustration, it compounds with every use. One confusing moment that takes an extra 5 seconds—repeated multiple times a day in perpetuity—adds up to a lot of anxiety and wasted time. My takeaway? Don't underestimate the impact of seemingly small interface changes!	{ "redpajama_set_name": "RedPajamaC4" }	7,204
\section{Introduction} \label{sec:intro} Inverse reinforcement learning (\textsc{IRL}{}) seeks to model the preferences of another agent using its observed behavior, thereby avoiding a manual specification of its reward function~\cite{Russell1998,Ng2000}. In the past decade or so, \textsc{IRL}{} has attracted several researchers in the communities of artificial intelligence, psychology, control theory, and machine learning. IRL is appealing because of its potential to use data recorded in everyday tasks (e.g., driving data) to build autonomous agents capable of modeling and socially collaborating with others in our society. We study this problem and associated advances in a structured way to address the needs of readers with different levels of familiarity with the field. For clarity, we use a contemporary example to illustrate \textsc{IRL}{}'s use and associated challenges. Consider a self-driving car in role B in Fig.~\ref{fig:car_merger}. To safely merge into a congested freeway, it should model the behavior of the car in role A; this car forms the immediate traffic. We may use previously collected trajectories of cars in role A, near freeway entry ramps, to learn the preferences of a typical driver as she approaches a merge (NGSIM~\cite{NGSIM} is one such existing data set). \begin{figure}[!ht] \centering{ \includegraphics{Freeway_CarAB_merge_v1-eps-converted-to.pdf} \caption{Red car B is trying to merge into the lane, and Green car A is the immediate traffic. The transparent images of cars show their positions before merging, and the opaque images depict one of their possible positions after the merger. Color should be used for this figure in print.} \label{fig:car_merger} } \end{figure} Approaches for \textsc{IRL}{} predominantly ascribe a Markov decision process (\textsc{MDP}{})~\cite{Puterman1994} to the interaction of observed agent with its environment, whose solution is a {\em policy} that maps states to actions. The reward function of this \textsc{MDP}{} is unknown, and the observed agent is assumed to follow optimal policy for the \textsc{MDP}{}. In the traffic merge example, \textsc{MDP}{} represents the driving process of car A. The driver of Car A is following action choices (deceleration, braking, low acceleration, and others) based on its optimal policy. Car B needs to reach the end of merging lane before or after Car A for merging safely. \subsection{Significance of \textsc{IRL}{}} \label{subsec:significance} Researchers in the areas of machine learning and artificial intelligence have developed a substantial interest in \textsc{IRL}{} because it caters to the following needs. \subsubsection{Demonstration Substitutes Manual Specification of Reward} \label{subsubsection:apprenticeship} Typically, if a designer wants a specific behavior in an agent, she manually formulates the problem as a forward learning or forward control task solvable using solution techniques in \textsc{RL}{}, optimal control, or predictive control. A key element of this formulation is a specification of the agent's preferences and goals via a reward function. In the traffic merge example, we may hand design a reward function for Car A. For example, +1 reward if taking an action in a state decreases the relative velocity of Car A w.r.t. Car B within a predefined distance from merging junction, thereby allowing for a safe merge. Analogously, a negative reward of -1 if taking an action in a state increases the relative velocity of Car A w.r.t. Car B. However, an accurate specification of the reward function often needs much trial-and-error for properly tuning the variables such as the thresholds for distance and velocity, and the emphasis of +1. As such, the specification of a reward function is often time-intensive and cumbersome. The need to pre-specify the reward function limits the applicability of \textsc{RL}{} and optimal control to problems where a lucid reward function can be easily specified. \textsc{IRL}{} offers a way to broaden the applicability and reduce the manual specification of the model, given that the desired policy or demonstrations of desired behavior are available. While acquiring the complete desired policy is usually infeasible, we have easier access to demonstrations of behaviors, often in the form of recorded data. For example, all state to action mappings for all contingencies for Car A are not typically available, but datasets such as NGSIM contain trajectories of Car A in real-world driving. Thus, \textsc{IRL}{} forms a key method for \emph{learning from demonstration}~\cite{argall2009survey}. A topic in control theory related to \textsc{IRL}{} is inverse optimal control~\cite{Boyd94}. While the input in both \textsc{IRL}{} and inverse optimal control are trajectories consisting of state-action pairs, the target of learning in the latter is the function mapping states of observed agent to her actions. The learning agent may use this policy to imitate it or deliberate with it in its own decision-making process. \subsubsection{Improved Generalization} A reward function represents the preferences of an agent in a succinct form, and is amenable to transfer to another agent. The learned reward function may be used as is if the subject agent shares the same environment as the other, otherwise it provides a useful basis if the agent configuration differs mildly. Indeed, Russell~\cite{Russell1998} believes that the reward function is inherently more transferable compared to the observed agent's policy. This is because even a slight change in the environment -- for e.g., few more states -- renders the learned policy completely unusable because it may not be directly revised in straightforward ways. \subsubsection{Potential Applications} While introducing \textsc{IRL}{}, Russell~\cite{Russell1998} alluded to its potential application in providing computational models of human and animal behavior because these are difficult to specify. In this regard, Baker et al.~\cite{Baker2009_action} and Ullman et al.~\cite{Ullman2009} demonstrate the inference of a human's goal as an inverse planning problem in an \textsc{MDP}{}. Furthermore, \textsc{IRL}{}'s use toward apprenticeship has rapidly expanded the set of visible applications such as: \begin{enumerate}[itemsep=0in] \begin{figure}[!ht] \centering{ \includegraphics[clip=true,trim=0 0.5cm 0 0.005cm]{TestFig1_HelicopterManeuver-eps-converted-to.pdf} \caption{Helicopter maneuvers using \textsc{RL}{} on a reward function learned from an expert pilot through \textsc{IRL}{}. The image is reprinted from \cite{Abbeel2007} with permission from publisher.}\label{HelicopterManeuver} } \end{figure} \item Autonomous vehicle control by learning from an expert to create an agent with the expert's preferences. Examples of such application include helicopter flight control~\cite{Abbeel2007} illustrated in Figure~\ref{HelicopterManeuver}, boat sailing~\cite{Neu2007}, socially adaptive navigation that avoids colliding into humans by learning from human walking trajectories \cite{Kretzschmar_interactingpedestrians,Kim2016_adaptivenavigation}, and so on; \item Learning from another agent to predict its behavior. In this use, applications include destination or route prediction for taxis~\cite{Ziebart2008,Ziebart_Cabbie}, footstep prediction for planning legged locomotion~\cite{Ratliff2009}, anticipation of pedestrian interactions~\cite{Ziebart_predictionpedestrian}, energy efficient driving~\cite{Vogel_efficientdriving}, and penetration of a perimeter patrol~\cite{Bogert_mIRL_Int_2014} by learning the patrollers' preferences and patrolling route. \end{enumerate} \subsection{Importance of this Survey} \label{subsec:target_survey} This article is a reflection on the research area of \textsc{IRL}{} focusing on the following important aspects: \begin{enumerate}[topsep=0in,itemsep=0in] \item Formally introducing \textsc{IRL}{} and its importance, by means of various examples, to the researchers and practitioners unaware of the field; \item A study of the challenges that make \textsc{IRL}{} problem difficult, and a brief review of the current (partial) solutions; \item Qualitative assessment and comparisons among different methods to evaluate them coherently. This will considerably help the readers decide on the approach suitable for the problem on their hands; \item Identification of key milestones achieved by the methods in this field along with their common shortcomings. \item Identification of the open avenues for future research. \end{enumerate} Given the time and space constraints, a single article may not cover all methods in this growing field. Nevertheless, we have sought to make this survey as comprehensive as possible. \vspace{0.1in} \noindent Section \ref{sec:IRL} mathematically states \textsc{IRL}{} problem. After the statement, a description of fundamental challenges -- Accurate Inference, Generalizability, Correctness of Prior Knowledge, Growth in Solution Complexity with Problem Size, Direct Learning of Reward or Policy Matching -- ensues in Section \ref{section:challenges}. Section \ref{sec:PrimaryMethods} discusses the methods for solving \textsc{IRL}{} problem, which makes way to Section \ref{sec:Mitigation} about the descriptive assessment of the mitigation of challenges. Section \ref{sec:Extensions} outlines the extensions of original \textsc{IRL}{} problems and the methods implementing them. As an overview of the progress in this area, Section \ref{sec:milestones_shortcomings} explains common milestones and shortcomings in the \textsc{IRL}{} research. The article concludes with open research questions. \section{Formal Definition of \textsc{IRL}{}} \label{sec:IRL} In order to formally define \textsc{IRL}{}, we must first decide on a framework for modeling the observed agent's behavior. While methods variously ascribe frameworks such as an \textsc{MDP}{}, hidden-parameter \textsc{MDP}{}, or a \textsc{POMDP}{} to the expert, we focus on the most popular model by far, which is the \textsc{MDP}{}. \begin{defn}\label{definition:mdp}[\textsc{MDP}{}] An \textsc{MDP}{} $\mathcal{M}{}:=\langle S,A,T,R,\gamma \rangle$ models an agent's sequential decision-making process. $S$ is a finite set of states and $A$ is a set of actions. Mapping $T:S\times A \to \textsf{Prob}(S)$ defines a probability distribution over the set of next states conditioned on agent taking action $a$ at state $s$; $\textsf{Prob}(S)$ here denotes the set of all probability distributions over $S$. $T_{sa}(s'\|s,a)\in[0,1]$ is the probability that the system transitions to state $s'$. Reward function $R:S \to \mathbb{R}$ specifies the scalar reinforcement incurred for reaching state $s$. Another variant $R:S \times A \to \mathbb{R}$ outputs the reward received after taking action $a$ at state $s$. Discount factor $\gamma$ is the weight for past rewards accumulated in a trajectory, $\langle (s_{0},a_{0}),(s_{1},a_{1}),\ldots$ $(s_{j},a_{j}) \rangle$, where $s_{j}\in S,a_{j}\in A,j \in\mathbb{N}$. \end{defn} A \emph{policy} is a function mapping current state to next action choice(s). It can be deterministic, $\pi:S\to A$ or stochastic $\pi:S\to \textsf{Prob}(A)$. For a policy $\pi$, value function $V^{\pi}:S \to \mathbb{R}$ gives the value of a state $s$ as the long-term expected cumulative reward incurred from the state by following $\pi$. The value of a policy $\pi$ is, \begin{equation} E_{s_0}\left[V^{\pi}(s_0)\right]=E\left[ \sum_{t=0}^\infty \gamma^t R(s_t)\|\pi \right] \label{eq:state_value} \end{equation} The goal of solving \textsc{MDP}{} $\mathcal{M}{}$ is an optimal policy $\pi^$ such that $V^{\pi^}(s)=V^(s)=\underset{\pi}{\sup}~ V^{\pi}(s)$, for all $s\in S$. The action-value function for $\pi$, $Q^{\pi}:S\times A \to \mathbb{R}$, maps a state-action pair to the long-term expected cumulative reward incurred after taking action $a$ from $s$ and following policy $\pi$ thereafter. We also define the optimal action-value function as $Q^(s,a)=\sup_\pi Q^{\pi}(s,a)$. Subsequently, $V^(s) = \sup_{a \in A} Q^(s,a)$. Another perspective to the value function involves multiplying the reward with the converged {\em state-visitation frequency} $ \psi^{\pi}(s) $, the number of times the state s is visited on using policy $ \pi $. The latter is given by: \begin{align} \psi^{\pi}(s) = \psi^0(s) + \gamma \sum_{s' \in S} T(s,\pi(s),s')~\psi^{\pi}(s') \label{eqn:visit_freq} \end{align} where $ \psi^0(s) $ is initialized as 0 for all states. Let $ \Psi $ be space of all $ \psi $ functions. Iterating the above until the state-visitation frequency stops changing yields the converged frequency function, $\psi^{\pi}_$. We may write the value function as, $V^(s) = \sum_{s \in S} \psi^{\pi}_(s)~R(s,\pi(s))$. We are now ready to give the formal problem definition of \textsc{IRL}{}. We adopt the conventional terminology in \textsc{IRL}{}, referring to the observed agent as an {\em expert} and the subject agent as the {\em learner}. \begin{defn}[\textsc{IRL}{}] Let an \textsc{MDP}{} without reward, $\mathcal{M}\backslash_{R_E}$, represent the dynamics of the expert $E$. Let $\mathcal{D}=\{\langle (s_{0}^{i},a_{0}^{i}),(s_{1}^{i},a_{1}^{i}),\ldots (s_{j}^{i},a_{j}^{i}) \rangle_{i=1}^{N} \},s_{j}\in S,a_{j}\in A,i,j,N \in\mathbb{N}$ be the set of demonstrated trajectory. A trajectory in $\mathcal{D}$ is denoted as $\tau$. We assume that all $\tau \in \mathcal{D}$ are perfectly observed. Then, determine $\hat{R}_E$ that best explains either policy $\pi_E$ or observed behavior in the form of the demonstrated trajectories. \label{def:irl} \end{defn} \begin{figure}[!ht] \centering{ \resizebox{\textwidth}{0.6\textwidth} \includegraphics[]{Illustration_IRL_v12-eps-converted-to.pdf}}\vspace{-10pt} \caption{$(a)$ A schematic showing the subject agent (shaded in blue) performing \textsc{RL}{}~\cite{Kaelbling1996}. In forward learning or RL, the agent chooses an action at a known state and receives a reward in return. $(b)$ In inverse learning or \textsc{IRL}{}, the input and output for the learner $L$ are reversed. L perceives the states and actions $\{(s,a)(s,a)\ldots\}$ of expert E (an alternative to input policy $\pi_E$), and infers a reward function, $\hat{R}_E$, of $E$ as an output. Note that the learned reward function may not exactly correspond to the true reward function. Color should be used for this figure in print.} \label{fig:IRL} } \end{figure} Notice that \textsc{IRL}{} inverts the \textsc{RL}{} problem. Whereas \textsc{RL}{} seeks to learn the optimal policy given a reward function, \textsc{IRL}{} seeks to learn the reward function that best supports a given policy or observed behavior. We illustrate this relationship between \textsc{RL}{} and \textsc{IRL}{} in Fig.~\ref{fig:IRL}. \section{Primary Challenges in \textsc{IRL}{}} \label{section:challenges} \textsc{IRL}{} is challenging because the optimization associated in finding a reward function that best explains observations is essentially ill-posed. Furthermore, computational costs of solving the problem tend to grow disproportionately with the size of the problem. We discuss these challenges in detail below, but prior to this discussion, we establish some notation. Let $\hat{\pi}_E$ be the policy obtained by optimally solving the \textsc{MDP}{} with reward function $\hat{R}_E$. \subsection{Accurate Inference} \label{subsection:accuracy} Classical \textsc{IRL}{} takes an expert demonstration of a task consisting of a finite set of trajectories, knowledge of the environment and expert dynamics, and generates the expert's potential reward function; this is illustrated in Fig.~\ref{fig:ideal_IRL}. \begin{figure}[!ht] \centering \includegraphics[scale=.4]{Pipeline_IRL-eps-converted-to.pdf} \caption{Pipeline for a classical \textsc{IRL}{} process. The learner receives an optimal policy or trajectories as input. The prior domain knowledge (shown here as a pentagon) include completely observable states and fully known transition probabilities.} \label{fig:ideal_IRL} \end{figure} A critical challenge, first noticed by Ng and Russell~\cite{Ng2000}, is that many reward functions (including highly degenerate ones such as a function with all rewards values zero) explain the observations. This is because the input is usually a finite and small set of trajectories (or a policy) and many reward functions in the set of all reward functions can generate policies that realize the observed demonstration. Thus, \textsc{IRL}{} suffers from an ambiguity in solution. \begin{figure}[!ht] \centering \includegraphics[scale=.4]{Pipeline_IRL_perturbedinput-eps-converted-to.pdf} \caption{\textsc{IRL}{} with noisy sensing of states, actions or both of the learner.} \label{fig:IRL_perturbed_input} \end{figure} In addition, the learner's sensing of expert's state and action could be imprecise due to noisy sensors, as illustrated in Fig.~\ref{fig:IRL_perturbed_input}. For example, vehicle tracking data in NGSIM mixes up car IDs at multiple points in the data due to imprecise object recognition when cars are close to each other. Inaccurate input may negatively impact the process of learning reward function. This could be corrected to some extent if a model of the observation noise is available, and \textsc{IRL}{} solution method integrates this model. Given the difficulty of ensuring accurate inference, its pertinent to contemplate how we may measure accuracy. If the true reward function $R_E$ is available for purposes of evaluation, then the accuracy is the closeness of a learned reward function $\hat{R}_E$ to $R_E$, $\left\|\left\|R_E-\hat{R}_E\right\|\right\|_p$. However, a direct comparison of rewards is not useful because an \textsc{MDP}{}'s optimal policy is invariant under affine transformations of the reward function~\cite{Russell03:Artificial}. On the other hand, two reward functions similar for the most part but differing for some state-action pairs may produce considerably different policies (behaviors). To make the evaluation targeted, a comparison of the behavior generated from the learned reward function with the true behavior of expert is more appropriate. In other words, we may compare the policy $\hat{\pi}_E$ generated from \textsc{MDP}{} with $\hat{R}_E$ with the true policy $\pi_E$. The latter could be given or is generated using the true reward function. A limitation of this measure of accuracy is that a difference between the two policies in just one state could still have a significant impact. This is because performing the correct action at that state may be crucial to realizing the task. Consequently, this measure of closeness is inadequate because it would report just a small difference despite the high significance. This brings us to the conclusion that we should measure the difference in values of the learned and true policies. Specifically, we may measure the error in the inverse learning, called {\em inverse learning error} (ILE), as $\left\|\left\|V^{\pi_E}_E - V^{\hat{\pi}_E}_E \right\|\right\|_p=\sum_{s\in S} \left\|\left\|V^{\pi_E}_E(s) - V^{\hat{\pi}_E}_E(s) \right\|\right\|_p$, where $V^{\pi_E}_E$ is the value function for actual policy $ \pi_E $ and $V^{\hat{\pi}_E}_E$ is that for the learned policy $\hat{\pi}_E$ \cite{Choi2011}. \subsection{Generalizability} \label{subsection:generalization_capacity} Generalization refers to the extrapolation of learned information to the states and actions unobserved in the demonstration and to starting the task at new initial states. Observed trajectories typically encompass a subset of the state space and the actions performed from those states. Well-generalized reward functions should reflect expert's overall preferences relevant to the task. The challenge is to generalize correctly to the unobserved space using data that often covers a fraction of the complete space. Notice that generalizability promotes the temptation of training the learner using fewer examples because the latter now possesses the ability to extrapolate. However, less data may contribute to greater approximation error in $ \hat{R}_E $ and inaccurate inference. ILE continues to be pertinent by offering a way to measure the generalizability of the learned information as well. This is because it compares value functions, which are defined over all states. Another procedure for evaluating generalizability is to simply withhold a few of the demonstration trajectories from the learner. These can be used as labeled test data for comparing with the output of the learned policy on the undemonstrated state-action pairs. \subsection{Correctness of Prior Knowledge} \label{subsection:sensitivity_priorknowledge} Many \textsc{IRL}{} methods~\cite{Ng2000,Abbeel2004,Ratliff2006,Neu2007,Ziebart2008} represent the reward function, $R_E$, as a weighted combination of {\em feature functions}; the problem then reduces to finding the values of the weights. Each feature function, $\phi: S \times A \to \mathbb{R}$, is given and is intended to model a facet of the expert's preferences. Prior knowledge enters \textsc{IRL}{} via the specification of feature functions in $R_E$ and the transition function in the \textsc{MDP}{} ascribed to the expert. Consequently, the accuracy of \textsc{IRL}{} is sensitive to the selection of feature functions that encompass the various facets of the expert's true reward function. Indeed, Neu et al. \cite{Neu2007} prove that \textsc{IRL}{}'s accuracy is closely tied to the scaling of correct features. Furthermore, it is also dependent on how accurately are the dynamics of the expert modeled by the ascribed \textsc{MDP}{}. If the dynamics are not deterministic, due to say some noise in the expert's actuators, the corresponding stochasticity needs to be precisely modeled in the transitions. Given the significant role of prior knowledge in \textsc{IRL}{}, the challenge is two-fold: $(i)$ we must ensure its accuracy, but this is often difficult to achieve in practice; $(ii)$ we must reduce the sensitivity of solution method to the accuracy of prior knowledge or replace the knowledge with learned information. \subsection{Growth in Solution Complexity with Problem Size} \label{subsection:labelCompExpense} Methods for \textsc{IRL}{} are iterative as they involve a constrained search through the space of reward functions. As the number of iterations may vary based on whether the optimization is convex, it is linear, the gradient can be computed quickly, or none of these, we focus on analyzing the complexity of each iteration. Consequently, the computational complexity is the sum of the time complexity of each iteration and its space complexity. Each iteration's time is dominated by the complexity of solving the ascribed \textsc{MDP}{} using the reward function learned. While the complexity of solving an \textsc{MDP}{} is polynomial in the size of its parameters, the parameters such as the state space are impacted by the curse of dimensionality -- its size is exponential in the number of components of state vector (dimensions). Furthermore, the state space in domains such as robotics is often continuous and an effective discretization also leads to an exponential blowup in the number of discrete states. Therefore, increasing problem size adversely impacts the run time of each iteration of \textsc{IRL}{} methods. Another type of complexity affecting \textsc{IRL}{} is sample complexity, which refers to the number of trajectories present in the input demonstration. As the problem size increases, the expert must demonstrate more trajectories in order to maintain the required level of coverage in training data. Abbeel and Ng~\cite{Abbeel2004} analyze the impact of sample complexity on the convergence of \textsc{IRL}{} method. \subsection{Direct Learning of Reward or Policy Matching} \label{subsection:labelDirectVsIndirect} Two distinct approaches to \textsc{IRL}{} present themselves, each with its own set of challenges. First seeks to approximate the reward function $\hat{R}_E$ by tuning it using input data. The second approach focuses on learning a policy by matching its actions (action values) with the behavior demonstrated, using reward function only as means. Success of the first approach hinges on selecting an accurate and complete prior knowledge {(e.g. set of feature functions)} that compose the reward function. Though learning a reward function offers better generalization to the task at hand, it may lead to policies that do not fully reproduce the observed trajectories. The second approach is limited in its generalizability -- learned policies may perform poorly in portions of the state space not present in the input data. More importantly, Neu et al.~\cite{Neu2008} point out that this optimization is convex only when the actions are deterministic and the demonstration spans the complete state space. Consequently, choosing between the two approaches involves prioritizing generalizability over a precise imitation or vice versa. \vspace{0.1in} Next section categorizes the early, foundational methods in \textsc{IRL}{} based on the mathematical framework they use for learning, and discusses them in some detail. \section{Foundational Methods} \label{sec:PrimaryMethods} A majority of \textsc{IRL}{} method fit a template of key steps. We show this template for in Algorithm~\ref{alg:IRL_template}, and present the methods in the context of this template. Such presentation allows us to compare and contrast various methods. \begin{algorithm}[!ht] \caption{Template for \textsc{IRL}{}} \label{alg:IRL_template} \KwIn{Trajectory demonstrating desired behavior i.e., $\mathcal{D}=\{\langle (s_0,a_0),(s_1,a_1),\ldots,(s_n,a_n) \rangle \}, s_j\in S, a_j\in A,j, n \in \mathbb{N}$} \KwOut{ $\hat{R}_E$} Model the dynamic expert's observed behavior as an \textsc{MDP}{} without reward function\; Initialize a parameterized form (linear approximation, distribution over rewards, or others) of solution\; Solve \textsc{MDP}{} with current solution to generate learned behavior\; Update the parameters to minimize the divergence between the observed behavior and the learned behavior\; Repeat the previous two steps till the divergence is reduced to desired level. \end{algorithm} Existing methods seek to learn the expert's preferences, a reward function $\hat{R}_E$, represented in different forms such as a linear combination of weighted feature functions, a probability distribution over multiple real-valued maps from states to reward values, and so on. Parameters of $\hat{R}_E$ vary with the type of representation (weights, variables defining shape of distribution). \textsc{IRL}{} involves solving the \textsc{MDP}{} with the function hypothesized in current iteration and updating the parameters, constituting a search that terminates when the behavior derived from the current solution converges to the observed behavior. This section categorizes methods according to the technique they use for learning -- maximum margin based optimization, maximum entropy based optimization, Bayesian inference, regression, classification, and few others. In addition to explaining how various methods instantiate the steps in the template, we discuss in the next section how they take steps to mitigate some or all the primary learning challenges introduced previously in Section~\ref{section:challenges}. \subsection{Maximum Margin Optimization} General framework of maximum margin prediction aims to predict reward function that makes the demonstrated policy look better than alternative policies by a margin. Methods under this category aim to address the ambiguity of solution (discussed in Section~ \ref{subsection:accuracy}) by deciding on a solution that maximizes a margin from others. \paragraph{Linear Programming}~ One of the earliest methods for \textsc{IRL}{} is Ng and Russell's~\cite{Ng2000}, which takes in the expert's policy as input. It formulates a linear program to retrieve the reward function that not only produces the given policy as optimal output from the complete MDP, but also maximizes the sum of differences between the value of the optimal action and that of the next-best action for every state. In addition to maximizing this margin, it also prefers reward functions with smaller values as a form of regularization. We may express the reward function as a linear and weighted sum of basis functions, $\hat{R}_E(s)=w_1 \phi_1(s) + w_2 \phi_2(s) + \ldots + w_k \phi_k(s) = w^T\phi$, where $\phi_i:S\to \mathbb{R}$ is a basis function and weight $w_i \in \mathbb{R}$ for all $i \in \{1,2\ldots k\}$. For spaces which are not discrete or small-sized, the constraints defined for each discrete state must be generalized to the continuous state space, or algorithm can sample a large subset of the state space and restrict the constraints to these sampled states only (as Ng and Russell suggested). \paragraph{Apprenticeship learning}~ Two methods~\cite{Abbeel2004} under this general label extend the linear programming and perform max margin optimization. Noting that the learner does not typically have access to the expert's policy, \textit{max-margin} and \textit{projection} take a demonstration (defined in Def.~\ref{def:irl}) as input and learns a linear reward function. {\em expected feature count} is the discounted sum of feature values under expectation of a policy, \begin{equation}\label{eqn:feature-expectation} \mu^{\phi}(\pi)=E\left[ \sum_{t=0}^\infty \gamma^t \phi(s_t,a_t)\|s \in S, a \in A, \pi \right] \end{equation} Authors successfully used these expected feature counts or feature expectations as a representation of the value function of Eq.~\ref{eq:state_value}:$E_{s_0}\left[V^{\pi} (s_0)\right]=\sum\nolimits_{i=1}^k w_i \cdot\mu^{\phi_i}(\pi)$. Given a set of $ N $ trajectories $\mathcal{D}=\{(s_{0}^{i},a_{0}^{i}),(s_{1}^{i},a_{1}^{i}),\ldots (s_{j}^{i},a_{j}^{i})\}_{i=1}^{N},s_{j}\in S,a_{j}\in A,i,j \in\mathbb{N}$ generated by the expert, the empirical estimation of $ \mu^{\phi}(\pi_E) $ is \begin{equation}\label{empirical_feature_expectations} \hat{\mu}^{\phi}(\pi_E)=\frac{1}{N} \sum^N_{i=1} \sum_{t=0}^\infty \gamma^t \phi(s_t^i) \end{equation} \begin{figure}[!ht] \centering{ \includegraphics[scale=1]{Maxmargin_iteration-eps-converted-to.pdf} \caption{Iterations of the max-margin method (inspired from Figure 1 in \cite{Abbeel2004}) computing the weight vector, $\mathbf{w}$, and the feature expectations, $\mathbf{\mu}^{\mathbf{\phi}}$. $\hat{\mu}^{\phi}(\pi_E)$ is the estimation of the feature counts $\mathbf{\mu}^{ \mathbf{\phi}}(\pi_E) $ of the expert. $ w_j $ is the learned weight vector in $ j $ th iteration and $ \pi^H_{E_j} $ is corresponding optimal policy for intermediate hypothesis.} \label{fig:maxmargin_iteration} } \end{figure} The methods seeks a reward function that minimizes the margin between the feature expectations of policy computed by learner and the empirically computed feature expectations of expert. Both methods iteratively tune weights $\bm{w}$ by computing policy for intermediate hypothesis at each step and using it to obtain intermediate feature counts. These counts are compared with the feature counts $\hat{\mu}^{\phi}(\pi_E)$ of expert, as shown in Fig.~\ref{fig:maxmargin_iteration} and are updated each step. Abbeel and Ng~\cite{Abbeel2004} point out that the performance of these methods is contingent on matching the feature expectations, which may not yield an accurate $ \hat{R}_E $ because feature expectations are based on the policy. An advantage of these methods is that their sample complexity depends on the number of features and not on the complexity of expert's policy or the size of the state space. An alternative to minimizing the value-loss in the projection method is to minimize the difference $\hat{\pi}_E (a\|s)$ - $\pi_E(a\|s)$ for each state. Viewing a policy as Boltzmann function of the action-value, Neu and Szepesvari~\cite{Neu2007} present \textsc{hybrid}-\textsc{IRL}{} as a gradient descent method that searches the space of reward hypotheses, with a reward function parameterizing a policy. As the behavior of expert is available instead of his policy, the difference above is computed using the empirically computed frequencies of the visitations of each state (Equation~\ref{eqn:visit_freq}) and the frequencies of taking specific actions in that state. A variant of the projection method described above is Syed et al's.~\cite{Syed2008} Multiplicative Weights for Apprenticeship Learning (\textsc{mwal}); the initial model and input are the same in both methods. However, it presents a learner as a max player choosing a policy and its environment as an adversary selecting a reward hypothesis. This formulation transforms the value-loss objective in the projection method to a {\em minimax} objective for a zero-sum game between the learner and its environment, with weights $ w $ as output. \paragraph{Maximum margin planning}~ Under the assumption that each trajectory reflects a distinct policy, Ratliff et al.~\cite{Ratliff2006} associate each trajectory with an MDP. While these MDPs could differ in their state and action sets, the transition and feature functions, they share the same weight vector. MMP loss function $\mathcal{L}:\Psi\times\Psi\to\mathbb{R}$ quantifies the closeness between the learned behavior and demonstrated behavior. The inputs for this function are the state-action frequency counts $ \psi $ (computed by including actions along with states within Equation \ref{eqn:visit_freq}) of either trajectories or policies. \begin{equation}\label{loss-function-MMP} \mathcal{L}(\psi,\psi_i) = l_i^T \psi \end{equation} $ l_i $ defines the payoff on $ \psi $ failing to match the state-action pair of $ \psi_i $. Each of the MDPs includes the corresponding empirical feature expectation and a loss function, both computed using the associated trajectory. The weight vector is obtained by solving a quadratic program that is constrained to have a positive margin between the expected value of the observed policy and the expected values of all other policies combined with loss. The former is obtained by multiplying the empirical state visitation frequency from the observed trajectory with the weighted feature functions (i.e., reward). The solution is the weight vector that makes the expected value of the optimal policy to be closest to the expected value of the observed policy, for each of the MDPs. \begin{figure}[!ht] \centering{ \includegraphics[scale=0.5]{LEARCH_iteration-eps-converted-to.pdf}} \caption{An iteration of \textsc{learch} in the feature space $\Phi=\{\phi(s,a)\|\forall (s,a)\in S\times A\}$ (Fig 3 in Ratliff et al.~\cite{Ratliff2009} excerpted with permission). The method considers a reward function as negative of a cost function. Blue path depicts the demonstrated trajectory, and green path shows the maximum return (or minimum cost) trajectory according to the current intermediate reward hypothesis. Step 1 is the determination of the points where reward should be modified, shown as $-\Delta r$ for a decrease and $+\Delta r$ for an increase. Step 2 generalizes the modifications to entire space $\Phi$ computing the next hypothesis $\hat{R}_E$ and corresponding maximum return trajectory. Next iteration repeats these two steps. Color should be used for this figure in print.}\label{LEARCH_iteration} \end{figure} Ratliff et al.~\cite{Ratliff2009} improve on the previous maximum-margin prediction using quadratic programming in a subsequent method labeled {\em learn to search (learch)}. Figure~\ref{LEARCH_iteration} explains how an iteration of \textsc{learch} increases the cost (decreases the reward) for the actions that deviate the learned behavior from demonstrated behavior. For optimization, learch uses an exponentiated functional gradient descent in the space of reward functions (represented as cost maps). \subsection{Entropy Optimization} \textsc{IRL}{} is essentially an ill-posed problem because multiple reward functions can explain the expert's behavior. While the max-margin approach for solution is expected to be valid in some domains, it introduces a bias into the learned reward function in general. Thus, multiple methods take recourse to the maximum entropy principle~\cite{Jaynes_MaxEnt} to obtain a distribution over potential reward functions while avoiding any bias. The distribution that maximizes entropy makes minimal commitments beyond the constraints and is least wrong. \paragraph{Maximum entropy \textsc{IRL}{}}~ This popular method for \textsc{IRL}{} introduced the principle of maximum entropy as a way of addressing the challenge of accurate inference in \textsc{IRL}{}. Maximum entropy \textsc{IRL}{}~\cite{Ziebart2008} recovers a distribution over all trajectories, which has the maximum entropy among all such distributions under the constraint that feature expectations of learned policy match that of demonstrated behavior. Mathematically, this problem can be formulated as a convex, nonlinear optimization: \begin{small} \begin{align} &\max_{\Delta}~ \left (-\sum\limits^{}_{\tau \in \mathcal{D}}Pr(\tau)\log Pr(\tau) \right) \nonumber\\ &\text{\bf subject to}~~~ \sum\limits_{\tau \in \mathcal{D}}Pr(\tau) = 1 \nonumber\\ &\sum\limits_{\tau \in \mathcal{D}}Pr(\tau)\sum\limits_{\langle s,a \rangle \in \tau}\phi_{k}(s,a) = \hat{\mu}^{\phi}(\pi)_{k} ~~~~~~ \forall k \label{eqn:max-ent-trajectory} \end{align} \end{small} where $\Delta$ is the space of all possible distributions $Pr(\tau)$; and $\hat{\phi}_k$ is the expectation over the $k^{th}$ feature from observations of the expert. ~\cite{Ziebart2008} used Lagrangian relaxation to bring both constraints into the objective function and then solved the dual for weights $ w $ by utilizing exponentiated gradient descent. \begin{figure}[!ht] \centering{ \includegraphics[scale=1]{Structured_MDP-eps-converted-to.pdf}} \caption{A Markov Random Field which favors the policies choosing similar actions in neighboring states. This structure generalizes the feature information beyond individual states. The figure is reprinted from \cite{Boularias2012} with permission from publisher. \label{Structured_MDP} } \end{figure} A disadvantage of the above formulation is that it becomes intractable for long trajectories because the space of trajectories grows exponentially with the length. A related method~\cite{Boularias2012} formulates the problem as one of finding a distribution over space $ \Pi $ of deterministic policies, $Pr(\Pi)$, given as: \begin{equation} \begin{array}{l} \max \limits_{\Delta} \left( -\sum\limits_{\pi \in \Pi} Pr(\pi)~ log~ Pr(\pi) \right )\\\\ \mbox{\bf subject to} ~~~~ \sum \limits_{\pi \in \Pi} Pr(\pi) = 1 \\ \sum \limits_{\pi \in \Pi} Pr(\pi) \sum\limits_{s \in S} \psi^{\pi}(s)~\phi_k (s, \pi(s)) = \hat{\mu}^{\phi}(\pi)_{k} ~~~~~~ \forall k \in K \label{eqn:max-ent} \end{array} \end{equation} Due to the constraint of matching feature expectations, the process is equivalent to maximizing the likelihood of expert's behavior under the maximum entropy distribution. The set of deterministic policies has size $\mathcal{O}(\|A\|^{\|S\|})$, which is independent of the length and number of demonstrated trajectories, but it could get considerably large with size of state space. Some real applications exhibit a \textbf{structure} that neighboring states prefer similar optimal actions. To exploit this information, \cite{Boularias2012} introduced special features in aforementioned constraints to make the solution distribution favor the policies for which neighboring states have similar optimal actions. This method is Structured Apprenticeship Learning, resulting in a Markov Random Field like distribution (Figure~\ref{Structured_MDP}). \paragraph{Relative entropy \textsc{IRL}{}}~ Entropy optimization is also useful where the transition dynamics are not known. In \textsc{reirl} ~\cite{Boularias2011relative}, the relative entropy ( known as Kullbach-Leibler divergence~\cite{Kullback68informationtheory}) between two distributions on trajectories is minimized and the corresponding reward function is obtained. \iffalse \textsc{reirl} is a model free approach that takes feature vectors $\phi$, a baseline policy $\pi_b$, a fixed number $m$ of demonstration trajectories, discount factor, and . $\delta$ helps compute the bound $\varepsilon$ on the mismatch between learned feature expectations and expert's. Without transition probabilities, the method learns reward function by minimizing the relative entropy between the two probability distributions: \fi From the space of all possible distributions, learner choses an arbitrary distribution on trajectories with feature expectations within pre-determined closeness to true feature expectations of input demonstration. Using a given baseline policy $\pi_b$, learner empirically computes another distribution by sampling trajectories. Then the method learns reward function by minimizing the relative entropy between these two probability distributions. Notably, the input behavior need not be optimal for the method. An analytical computation of solution to this optimization problem requires a known transition function. To estimate latter, learner uses importance sampling. Recently, Finn et al.~\cite{Finn_gcl} extend the above sample based estimation approach (Guided cost learning,\textsc{gcl}) to model-free maximum entropy optimization by allowing a neural network to represent the reward function ~\cite{Wulfmeier2015}. Replacing an uninformed baseline policy that effectively leads to a uniform distribution over trajectories, sample generation is guided by policy optimization that uses the current reward hypothesis to update the baseline policy which in turn guides the sample generation. In this way, RL is interleaved with \textsc{IRL} to generate more trajectories that are supported by a larger reward. As Wulfmeier et al.~\cite{Wulfmeier2015} point out, the difference in the visitation frequency can be passed as an error signal to the neural network, and the backpropagation computes the gradient. \subsection{Bayesian Update} Some methods treat the state-action pairs in a trajectory as observations that facilitate a Bayesian update of a prior distribution over candidate reward solutions. This approach yields a different but principled way for \textsc{IRL}{} that has spawned various improvements. \paragraph{Bayesian \textsc{IRL}{}}~ A posterior distribution over candidate reward functions is obtained using the following Bayesian update: \begin{equation}\label{eq:bayes_update} P(\hat{R}_E\|\tau) \propto P(\tau\|\hat{R}_E)~P(\hat{R}_E) ,~~\forall \tau \end{equation} where $\tau$ is a demonstrated trajectory, and $P(\tau\|\hat{R}_E) =$ $\prod\nolimits_{\langle s,a \rangle \in \tau} Pr(\langle s,a \rangle$ $\|\hat{R}_E)$. Ramachandran and Amir~\cite{Ramachandran2007} define the likelihood $P(\langle s,a \rangle\|\hat{R}_E)$ as a logit or exponential distribution of the Q-value of the state-action pair: \begin{equation}\label{Eq:Boltzmann_model} P(\langle s,a \rangle\|\hat{R}_E)=\frac{1}{Z}~e^{\left(\frac{Q^(s,a;\hat{R}_E)}{\beta}\right)} \end{equation} where $\beta$ controls the randomness in action selection (lower the $\beta$, more exploratory is the action) and $Z$ is the partition function or the normalization constant. Given a candidate reward hypothesis, some state-action pairs are more likely than others as given by the likelihood function. As the space of reward functions is technically continuous, Ramachandran and Amir present a random walk algorithm for implementing the update and obtaining a sampled posterior. An extension of Bayesian \textsc{IRL}{}~\cite{Lopes2009} measures the state-based entropy of the posterior over the reward functions. The method defines a set $ \{\hat{R}_E(p)\} $ of reward functions such that for a given state-action under chosen reward function $ \hat{R}_E $, $ \pi_E(s,a\|\hat{R}_E)=p $. The posterior distribution $ P(\hat{R}_E\|\mathcal{D}) $ induces following discretized distribution over values of $ p $. \begin{equation}\label{action-distribution-posterior} {P}_{sa}(k) = P[\pi(s,a)=p \in I_k \|\mathcal{D} ]=\int_{I_k} P[\hat{R}_E \in \{\hat{R}_E(p)\} \|\mathcal{D} ]dp \end{equation} where $ I_k=(\frac{k}{K},\frac{k +1}{K}], k\in \{0,1,\ldots (K-1)\}$ is a subinterval of interval I=[0,1]. State-based entropy of the distribution is \begin{equation}\label{statewise-entropy} \bar{H}(s) = \frac{1}{\|A\|}{\sum}_{a\in A} H({P}_{sa}) = -\frac{1}{\|A\|} {\sum}_{a\in A} {\sum}_{k\in \{0,1,\ldots (K-1)\}} {P}_{sa} \log({P}_{sa}) \end{equation} In an example of active learning, learner can query the expert for sample demonstrations in states exhibiting high entropy, with the aim of learning a posterior that is well informed at all states. Prior knowledge about the structural similarities between states improves the effectiveness of this approach. \paragraph{\textsc{birl} via Bisimulation}\label{bisimulation_metric_BIRL} Francisco et al. \cite{S.Melo2010} proposed a bisimulation technique to project a structure in their \textsc{IRL}{} algorithm that improves its generalization capacity while avoiding the computations for solving \textsc{MDP}{} multiple times. The method uses an \textsc{MDP}{} metric quantifying the equivalence between states based on the closeness of the output action-distributions of (stochastic) policy $\pi_E$ at the states \cite{Ferns2004,Taylor2009}. The approach models policy $ \pi_E $ as a classifier with the state in each state-action pair is the data and the action is the label. Authors devise a kernel based on the metric and use kernel regression to generalize the demonstrated behavior to unobserved states. After the regression, the algorithm updates a distribution over the parameters of $\hat{\pi}_E $ using \textit{Bayesian inference}. The authors extend their method to active learning, in which learner queries the expert at the states with the maximum variance of the distribution. \paragraph{Bayesian nonparametrics}~ Michini et al.~\cite{Michini_CRPBNIRL1,Michini_CRPBNIRL2} (\textsc{crp-bnirl}) show a straightforward application of nonparametric clustering by letting a Chinese Restaurant process partition a trajectory into as many subtrajectories as needed and assign a reward function to each subtrajectory, to best fit the observed data. Subtrajectories are presumed to represent subtasks with their own subgoals. However, large state spaces and long trajectories make the calculation of demonstration-likelihood computationally intensive. The authors use RTDP (real-time dynamic programming) and action comparison with existing closed-loop controllers to avoid the discretization of the state space or action space for the calculation. The comparison also allows real-time learning. \subsection{Regression and Classification} Classical machine learning techniques such as regression and classification have also played a significant role in \textsc{IRL}{}. \paragraph{Feature construction using regression}~ In a unique effort to automatically learn the feature functions, \cite{Levine2010_featureconstruction} begins with primitive binary feature functions (logical conjunctions ) that form components of the final features used in the reward function. Starting with an empty set of features, method \textsc{firl} iteratively updates reward hypothesis $\hat{R}_E$ and adds hypothesized feature functions $\phi^H$ to it. The search involves building a minimal regression tree on the state space that encodes those conjunctions, resulting in new binary feature functions which make $\hat{R}_E$ consistent with demonstrated behavior. \paragraph{Multi-label classification}~ Klein et al.~\cite{Klein2012structured} viewed \textsc{IRL}{} as a multi-label classification problem (\textsc{scirl}) in which the demonstration is training, and the state-n-action pairs are data-n-label pairs. They consider the action values $Q^{\pi_E}(s,a)$ (score function of classifier) in terms of feature expectations , $Q^{\pi}(s,a) = \sum_{k = 1}^K w_k~\mu^{\phi_k}(\pi)(s,a)$ (see Eq.~\ref{eqn:feature-expectation}), which makes weights shared parameters between Q-function and reward function. A multi-label classification algorithm computes solution by inferring a weight vector that minimizes the classification error. An extension of the above method~\cite{Klein_CSI} (called \textsc{csi}) estimates the transition probabilities if they are unknown. \textsc{csi} utilizes standard regression on a simulated demonstration data set to estimate the transition model and thereby learn reward function (not necessarily linear this time). \subsection{Other Miscellaneous Techniques} We briefly review a few other \textsc{IRL}{} methods that do not fall in the broad categories discussed previously. \paragraph{\textsc{gpirl} - Gaussian Process \textsc{IRL}{}} \textsc{gpirl} uses a Gaussian process to approximate $R_E$ as a function non-linear in base features $\phi_{base}$, $R_E=f(\phi_{base})$ \cite{Levine2011}. Instead of a single reward function, a hypothesis here is mean of a parametric Gaussian posterior over possible rewards. The method optimizes the likelihood of actions in demonstration to compute the mean. \paragraph{Maximum likelihood estimate}~ Rather than the maximum-a-posteriori estimate of the expert's reward function obtained from Bayesian learning, Vroman et al.~\cite{Babes-Vroman2011} choose a reward function that leads to the maximum likelihood estimate. In \textsc{mlirl}, the formulation of estimate is same as one in Bayesian learning. However, this method remains challenged by degeneracy and ill-posedness of the \textsc{IRL}{} problem. \paragraph{Path integral}~ Kalakrishnan et al.~\cite{Kalakrishnan_continousspace} propose \textsc{pi-irl}, a path integral approach to learn continuous-state continuous-action reward functions by sampling a local region around each demonstrated optimal trajectory. The optimization process does not consider the trajectories that incur significant control input because they deviate from demonstration; the technique applies to high dimensional problems due to the assumed local optimality of demonstration. \section{Mitigating Challenges} \label{sec:Mitigation} This section elaborates how the methods reviewed in previous section mitigate the challenges introduced in Section~\ref{section:challenges}, to help a practitioner make an informed choice about the method that may address the challenges in her domain. \subsection{Improvement in Accuracy of \textsc{IRL}{}} \label{subsection:achieving_accuracy} The accuracy depends on several aspects of the learning process. Most existing methods aim at ensuring that the input is accurate, reducing the ambiguity among solutions, improving feature selection, and offering algorithmic performance guarantees. \subsubsection{Learning from Faulty Input} \paragraph{Perturbations} Sub optimal actions characterize a perturbed demonstration. Melo et al.~\cite{Melo_perturbedinput} aim for a formal analysis and characterization of the space of solutions for the cases when some of actions in demonstration are not optimal (a perturbation in the distribution modeling the expert's policy) and when demonstration does not include samples in all states. Some methods such as {\sc reirl} stay robust to perturbation whereas other \textsc{IRL}{} methods may learn inaccurate feature weights~\cite{Abbeel2007} or predict the control poorly~\cite{Ratliff2006}, resulting in an unstable learned behavior. Methods such as \textsc{maxentirl}, \textsc{birl}, \textsc{mlirl}, and \textsc{gpirl} use probabilistic frameworks to account for the perturbation. For example, \textsc{mlirl} allows tuning of its model parameter $\beta$ (Equation~\ref{Eq:Boltzmann_model}) to model learned policy $\pi_E$ as random when the demonstrated behavior is expected to be noisy and suboptimal~\cite{Babes-Vroman2011}. On the other hand, methods such as \textsc{mmp} and \textsc{learch} introduce slack variables in their optimization objective for this purpose. Using the application of helicopter flight control, Ziebart et al.~\cite{ZiebartBD_compare_MMP} show that the robustness of \textsc{maxent} against an imperfect demonstration is better than that of \textsc{mmp}. Coates et al.~\cite{Coates2008} introduce a generative model-based technique of trajectory learning that de-noises the noisy demonstration to learn from noise-free (unobserved) trajectories. In a subsequent method, Coates et al. apply \textsc{IRL}{} to the resultant noise-free trajectories~\cite{Coates2009}, using apprenticeship learning. Silver et al. \cite{Ratliff2009,Silver2008} also account for the actuation noise in a demonstration by relaxing the constraints in the optimization objective. Instead of constraining the learned policy to follow demonstration exactly, the modified constraint make it follow the lowest cost paths that are close to the examples (e.g. set of paths with sufficiently small loss w.r.t. demonstration). As an extreme instance of learning with sub-optimal input, Shiarlis et al.~\cite{Shiarlis_2016_failed} demonstrate \textsc{IRL}{} with demonstrations that failed to complete a task. Despite these contributions, research in \textsc{IRL}{} has not benefited from an explicit focus on robust learning in the context of faulty demonstrations in application domains. \begin{figure}[ht!] \centering{ \includegraphics[scale=1.4]{Learning_PerturbedInput-eps-converted-to.pdf} \caption{Learning with a perturbed demonstration in an unstructured terrain. Figure reprinted~\cite{Silver2008} with permission from MIT press. An expert provides three demonstration trajectories - red, blue, and green [top left]. The portion of terrain traveled by a presumably achievable red trajectory should have low cost (high reward) as the expert is presumably optimal. But the path is not optimal. It is not even achievable by any planning system with predefined features because passing through the grass is always cheaper than taking a wide berth around it. The assumed optimality of expert forces the optimization procedure in \textsc{IRL}{} methods to lower the cost (increase the reward) for features encountered along the path, i.e., features for grass. This influences the efficiency of learning behavior in other paths (blue path) [Bottom left]. Using a normalized functional gradient~\cite{Silver2008} makes this lowering effect vanish [Bottom right]. Color should be used for this figure in print.} \label{learning_faultyinput} } \end{figure} \paragraph{Over Training} A sub-optimal demonstration may also be a trajectory with length much longer than desired. \textsc{mmp} converges to a solution by reducing the cost for following all demonstrated trajectories as compared to other simulated trajectories. The reduction happens by exploiting a relatively lower visitation counts of demonstration in the undesired part of state space. However, \textsc{mmp} attempts this reduction for a sub-optimal demonstration as well, which can be avoided if the learning method distinguishes between an unusually long demonstration from the optimal ones. Silver et al. \cite{Silver2008,Ratliff2009} implement the solution by applying a functional gradient normalized by the state visitation counts of a whole trajectory (Fig.~\ref{learning_faultyinput}). \subsubsection{Ambiguity in Hypotheses} Various methods mitigate this challenge of ambiguity and degeneracy by better characterizing the space of solutions. This includes using heuristics, prior domain knowledge and adding optimization constraints. \paragraph{Heuristics} \textsc{max-margin} and \textsc{mwal} avoid degenerate solutions by using heuristics that favor the learned value $V^{\hat{\pi}_E}$ close to expert's $V^{\pi_E}$. Specifically, \textsc{mmp} avoids degeneracy by using an objective (loss) function which degenerate $\hat{R}_E=0$ can not minimize because the function is proportional to state-visitation frequencies~\cite{Neu2008}. \textsc{hybrid}-\textsc{IRL}{} avoids degeneracy in the same way as \textsc{mmp}, and makes the solution unique by preferring a reward function that corresponds to a stochastic policy $\hat{\pi}_E$ with action selection same as the expert's ($\hat{\pi}_E(a\|s) \approx \pi_E(a\|s)$). Naturally, if no single non-degenerate solution makes the demonstration optimal, an unambiguous output is impossible using these methods \cite{Ziebart2010}. \paragraph{Constraints and Prior Knowledge} Many methods embrace the ambiguity by modeling the uncertainty of output as a probability distribution over solutions or that over the trajectories corresponding to the solutions. In this regard, \textsc{maxentirl} infers a unique reward function by using a probabilistic framework avoiding any constraint other than making the value-loss zero, $V^{\hat{\pi}_E}=V^{\pi_E}$. The likelihood objectives in Bayesian inference techniques and \textsc{gpirl} limit the probability mass of its output, a posterior distribution, to that specific subset of reward functions which supports the demonstrated behavior. This change in probability mass shapes the mean of posterior as well, which is the unique output of these methods. Active learning uses state-wise entropy of the posterior to select the most informative states \cite{Lopes2009}. The selection mechanism builds on \textsc{birl} and enhances the uniqueness of the solution (mean reward function) as compared to \textsc{birl}. In general, all these methods add optimization constraints and prior domain knowledge to exploit those aspects that contribute to the dissimilarity among solutions, in order to distinguish some solution over others. \subsubsection{Lowering Sensitivity to Features} Performance of methods such as \textsc{projection}, \textsc{max-margin}, \textsc{mmp}, \textsc{mwal}, \textsc{learch}, and \textsc{mlirl} are all highly sensitive to the selection of features, a challenge pointed out in Section~\ref{subsection:sensitivity_priorknowledge}. Few of the methods that use reward features attempt to mitigate this challenge. \textsc{hybrid}-\textsc{IRL}{} that uses policy matching and all maximum entropy based methods tune distributions over the policies or trajectories, which reduces the impact that feature selection has on the performance of \textsc{IRL}{}~\cite{Neu2008}. Apart from selecting appropriate features, the size of the feature space influences the error in learned feature expectations for the methods that rely on $\hat{V}^{\pi_E}$, e.g. \textsc{projection},\textsc{mmp}, \textsc{mwal}, \textsc{maxent}\textsc{irl}. If a reward function is linear with the magnitude of its $k$ features bounded from the above, then the high-probability bound on the error scales linearly with $k$ \cite{Ziebart2008}. However, maximum entropy based methods show an improvement in this aspect as $O(\log k)$ dependence. \subsubsection{Theoretically Guaranteed Accuracy} From a theoretical viewpoint, some methods have better analytically-proved performance guarantees than others. The maximum entropy probability distribution over space of policies (or trajectories) minimizes the worst-case expected loss \cite{Grunwald2004}. Consequently, \textsc{maxentirl} learns a behavior which is neither much better nor much worse than expert's \cite{Dimitrakakis2012}. However, the worst-case analysis may not represent the performance on real applications because the performance of optimization-based learning methods can be improved by exploiting favorable properties of the application domain. Classification based approaches such as \textsc{csi} and \textsc{scirl} admit a theoretical guarantee for the quality of $\hat{R}_E$, in terms of optimality of learned behavior $\hat{\pi}_E$, given that both classification and regression errors are small. Nevertheless, these methods may not reduce the loss as much as \textsc{mwal} as the latter is the only method, in our knowledge, which has no probabilistic lower bound on the value-loss incurred~\cite{Syed2008}. Few methods analyze and bound inverse learning error (ILE) for a given confidence of success and a given minimum number of demonstrations. The analysis relies on determining the value of a policy using feature expectations $\mathbf{\mu}^{\mathbf{\phi}}(\pi)$~\cite{Abbeel2004,Ziebart2008,Choi2011}), state visitation frequencies $\psi^{\pi}(s)$~\cite{Neu2007,Ziebart2008,Ratliff2006}) or occupancy measures ($x^\pi(s)$ \cite{Babes-Vroman2011}). $$ V^\pi = w^T\mathbf{\mu}^{\mathbf{\phi}}(\pi) = \sum_{s,a} \psi^{\pi}(s)~w^T \phi(s,a) = \sum_{s,a} x^\pi(s)~R(s,a)$$ For any \textsc{lfd} method based on feature expectations or occupancy measure, there exists a probabilistic upper bound on the bias in $\hat{V}^{\pi_E}$, and thereby on ILE for a given minimum sample complexity~\cite{Abbeel2004,Syed2008_supplement,Vroman2014}. As a state visitation frequency is analogous to an occupancy measure, the bounds derived for latter are applicable on the methods which use former, e.g., \textsc{mmp}, \textsc{hybrid}-\textsc{irl}, and \textsc{maxentirl}. Subsequently, each derived bound on bias can be used to analytically compute the maximum error in learning for a given minimum sample complexity~\cite{Abbeel2004,Syed2008_supplement,Vroman2014}. Lee et al.~\cite{Lee_Improved_Projection} change the criterion (and thereby the direction) for updating the current solution in \textsc{max-margin} and \textsc{projection} methods. The method demonstrates a proven improvement in the accuracy of solution as compared to that of the previous methods. For specific problem domains, some of the algorithms for \textsc{lfd} that use \textsc{IRL}{} {\em empirically} show less value-loss than predecessor methods -- these include \textsc{mlirl}, \textsc{learch}, and \textsc{gpirl}. As compared to action-based models of Bayesian inference and \textsc{hybrid}-\textsc{IRL}{}, the entropy-based approaches show an improved match between the trajectories (not state values) in learned behavior and those in demonstrated behavior \cite{Ziebart2008}. Thus, the latter closely match not only the state-values of expert and learner but also their action-values. Similarly, among relatively new entropy-based methods, \textsc{gcl} learns more accurately than \textsc{reirl} in complex application domains. However, this improved performance may be entail a higher sample complexity. \subsection{Analysis and Improvement of Complexity} \label{subsection:lowering_learningcost} \begin{figure}[!ht] \centering{ \includegraphics[scale=1]{stateequivalence_bisimulation-eps-converted-to.pdf} \caption{State equivalence in an \textsc{MDP}{} with states $S=\{1,2,3\}$ and actions $A=\{a,b,c\}$. The similarity in actions and transitions for states 1 and 2 makes them equivalent. Therefore, the selection of optimal actions through expert's policy $\pi_E$ will be similar in both the states. Demonstration of $c$ in one implies the optimality of $c$ in other. The illustration is inspired from Fig 1 in Francisco et al. \cite{S.Melo2010}.}} \label{stateequivalence_bisimulation} \end{figure} Active \textsc{birl} offers a benefit over traditional \textsc{birl} by exhibiting reduced {\em sample complexity} (see Section~\ref{subsection:labelCompExpense}). This is because it seeks to ascertain the most informative states where a demonstration is needed, and queries for it. Consequently, less demonstrations are often needed and the method becomes more targeted. Of course, this efficiency exists at the computational expense of interacting with the expert. In a different direction, Francisco et al. \cite{S.Melo2010} exploiting an equivalence among the states to reduce the sample complexity, as shown in Fig.~\ref{stateequivalence_bisimulation}. Likewise, \textsc{reirl} uses fewer samples (input trajectories) as compared to alternative methods including a model-free variant of \textsc{mmp}~\cite{Boularias2011relative}. While emphasis on reducing time complexity and space complexity is generally lacking among \textsc{IRL}{} techniques, a small subset does seek to reduce the time complexity. An analysis of \textsc{birl} shows that computing the policy $ \pi_E $ for the mean of posterior distribution over solutions is computationally more efficient than the direct minimization of expected value-loss over the posterior~\cite{Ramachandran2007}. Specifically, the Markov chain approximating the Bayesian posterior, with a uniform prior, converges in polynomial time. Next, \textsc{mwal} requires $\mathcal{O}(\ln k)$ ($k$ is number of features) iterations for convergence, which is lower than $\mathcal{O}(k \ln k)$ for the \textsc{projection} method. \textsc{birl} using bisimulation (Section \ref{bisimulation_metric_BIRL}) has low computational cost because it does not need to solve the \textsc{MDP}{} repeatedly and the computation of bisimulation metric over space $S$ occurs once regardless of the number of applications. Although an iteration of \textsc{firl} is slower than \textsc{mmp} and \textsc{projection} due to the computationally expensive step of regression, former converges in a fewer number of iterations as compared to latter. Some optimization methods employ more affordable techniques of gradient computations. In contrast with the fixed-point method in \textsc{hybrid-irl}, the approximation method in Active \textsc{birl} has a cost that is polynomial in the number of states. For maximum entropy based parameter estimation, gradient-based methods (e.g., BFGS~\cite{fletcher1987practical}) outperform iterative scaling approaches~\cite{Malouf_ComparisonEstimatns}. \subsection{Continuous State Spaces} Few approaches for \textsc{IRL}{} learn a reward function for continuous state spaces. A prominent group of methods in this regard are path integral \cite{Theodorou_pathintegral} based approaches \cite{Aghasadeghi_continousspace,Kalakrishnan_continousspace}. \textsc{pi-irl} aims for local optimality of demonstrations to avoid the complexity of full forward learning in continuous space. This approximation makes it scale well to high-dimensional continuous spaces and large demonstration data. Although the performance of path integral algorithms is sensitive to the choice of samples in demonstration, they show promising progress in scalability. Similarly, Kretzschmar et al. \cite{Kretzschmar_interactingpedestrians} apply \textsc{maxentirl} to learn the probability distribution over navigation trajectories of interacting pedestrians, using a subset of their continuous space trajectories. A mixture distribution models both, the discrete as well as the continuous navigation decisions. Further, to avoid an expensive discretization in large continuous spaces, \textsc{crp-bnirl} approximates the demonstration likelihood by comparing actions to an existing closed-loop controller. Lopes et al. \cite{S.Melo2010} (Section \ref{bisimulation_metric_BIRL}, \textsc{birl} via Bisimulation) suggest the use of MDP metric for continuous state spaces to exploit their good theoretical properties for avoiding the cost of supervised learning or optimization-based learning in continuous spaces. \subsection{High Dimensional and Large Spaces} In many \textsc{IRL}{} methods such as {\sc maxentirl}, the probability of choosing actions in demonstration $\tau \in \mathcal{D}$ is \begin{equation}\label{maxent_policy} P(\tau\|\hat{R}_E)=\frac{1}{Z}e^{\left(\sum\limits_{t=0}^\infty R_E(s^t,a^t)\right)} \end{equation} The complexity of computing the partition function $Z$ scales exponentially with the dimensionality of the state space, because it requires finding the complete policy under the current solution ($\hat{R}_E$)~\cite{Ziebart2010}. \paragraph{Approximating Likelihood} The approximation of the reward likelihood by Levine et al.~\cite{Levine2012ICML} removes the exponential dependence on the dimensionality of the state space, and replaces the goal of global optimality of the input with local optimality. Other approaches for likelihood approximation in a high-dimensional state space include the use of importance sampling by {\sc reirl}, the down-scaling the state space using low-dimensional features~\cite{Vernaza_highdimension_approximation}, and the assumption by {\sc pi-irl} that demonstrations are locally optimal. For optimizations involved in maximum entropy methods, limited memory variable metric optimization methods such as L-BFGS are shown to perform better than other alternatives because they implicitly approximate the likelihood in the vicinity of the current solution~\cite{Malouf_ComparisonEstimatns}. \paragraph{Task Decomposition} Instead of demonstrating complete trajectories for large tasks, often, a problem designer can decompose the task hierarchically. An expert may then easily give demonstrations at different levels of implementation. The modularity of this process significantly reduces the complexity of learning. Kolter et al. \cite{Kolter_2007_HierarchicalAL} propose the exploitation of the hierarchical structure of a physical system (quadruped locomotion) to scale \textsc{IRL}{} up from low-dimensional to high- dimensional complex domains. Likewise, to make a high-dimensional navigation task computationally tractable, Rothkopf et al. \cite{Rothkopf_modular_highDim} utilize the independence between the components of a task to introduce decomposition in \textsc{birl}. The independence propagates to transitions, stochastic reward functions, and Q-values of the subtasks, and the algorithm computes the individual contribution of each of the components. By decomposing a task into subtasks wherever possible, \textsc{crp-bnirl} allows a parallelized pre-computation of (approximate) action-value function. Also, the action comparison allows a problem designer to use prior knowledge to trade-off between computational complexity and accuracy (at the expense of increasing the problem-design cost). However, the reward functions learned by \textsc{crp-bnirl} are limited to reaching particular subgoal states only. \paragraph{Speeding up Forward Learning} We may reduce the cost of repeatedly computing the state values for computing the values of intermediate policies learned in \textsc{IRL}{}. Syed et al. \cite{SyedLPAL2008} presents a linear program imitating the expert's policy, that makes solving \textsc{MDP}{} less expensive in large state spaces with many basis functions ($\phi$ for $R_E=w^T\phi$). Babes-Vroman et al. \cite{Babes-Vroman2011} takes the dual of this linear program to include reward learning in the method, and called it \textsc{lpirl}, keeping its computational benefits intact. Modeling expert's policy as a classifier defined using a bisimulation based kernel, '\textsc{birl} via Bisimulation' used supervised learning approach (regression) to avoid repeated solution of \textsc{MDP}{} (irrespective of the nature of underlying reward function). Similarly, \textsc{csi}, and \textsc{scirl} do not need a repeated solution for \textsc{MDP}{} because they update a solution by exploiting the structure imposed on \textsc{MDP}{} by their classification-based models. Another method~\cite{Todorov2009} avoids this computation by requiring system dynamics to have state value function bases, which are difficult to construct as compared to commonly used reward function bases. As an alternative, Levine et al.~\cite{Levine2012ICML} avoid repeated forward learning by using a local approximation of the reward function around experts' trajectories. In contrast, \textsc{crp-bnirl} achieve the same target by limiting the computation to those states which improve the action values around observations. The process depends on size of the input demonstration only rather than that of the complete state space. \subsection{Generalizability} \label{subsection:generalization_improvement} Few approaches explicitly learn a reward function that correctly represent expert's preferences for unseen state-action pairs, or one that is valid in an environment that differs from the input. An added benefit is that such methods may need less demonstrations. \textsc{gpirl} can predict the reward for unseen states lying within the domains of the features for a Gaussian process. Furthermore, \textsc{firl} can use the learned reward function in an environment slightly different from the original environment used for demonstration but with base features similar to those in the original environment. Similarly, \textsc{gcl} admits an enhanced generalization by learning new instances of a previously-learned task without repeated reward learning. '\textsc{birl} with bisimulation' achieves improved generalization by partitioning the state space $S$ based on a relaxed equivalence between states. \section{Extensions of \textsc{IRL}{}} \label{sec:Extensions} Having surveyed the foundational methods of \textsc{IRL}{} and how various challenges are mitigated, we now discuss important ways in which the assumptions of original \textsc{IRL}{} problem have been relaxed to enable advances toward real-world applications. \subsection{Incomplete and Imperfect Observation} Learners in the real world must deal with noisy sensors and may not perceive the complete demonstration trajectory. For example, the merging car B in our illustrative example described in Section~\ref{sec:intro} Figure~\ref{fig:car_merger} may not see car A in the merging lane until it comes into its sensor view. This is often complicated by the fact that car B's sensor may be partially blocked by other cars in front of it, which further occludes car A. \subsubsection{Extended Definition} As shown in Fig.~\ref{fig:IRL_incomp_perception}, this modifies the traditional \textsc{IRL}{} problem and we provide a new definition below. \begin{figure}[!ht] \centering \includegraphics[scale=.4]{Pipeline_IRL_incompletepercetion-eps-converted-to.pdf} \caption{\textsc{IRL}{} with incomplete or imperfect perception of the input trajectory. The learner is limited to using just the perceivable portions.} \label{fig:IRL_incomp_perception} \end{figure} \begin{defn}[\textsc{IRL}{} with imperfect perception] Let $\mathcal{M}\backslash_{R_E}$ represent the dynamics of the expert $E$. Let the set of demonstrated trajectories be, $\mathcal{D}=\{(s_{0}^{i},a_{0}^{i}),(s_{1}^{i},a_{1}^{i}),\ldots (s_{j}^{i},a_{j}^{i})\}_{i=1}^{N},s_{j}\in Obs(S),a_{j}\in Obs(A),i,j,N \in\mathbb{N}$. We assume that either some portion of each trajectory, $\tau \in \mathcal{D}$, is not observed, or some of the observed state-action pairs in a trajectory could be different from the actual demonstrated ones. Thus, let $Obs(S)$ and $Obs(A)$ be the subsets of states and actions that are observed. Then, determine $\hat{R}_E$ that best explains either policy $\pi_E$ or the demonstrated trajectories. \label{def:irl-extension-1} \end{defn} Observing the trajectories imperfectly may require the learner to draw inferences about the unobserved state-action pairs or the true ones from available information, which is challenging. \subsubsection{Methods} \paragraph{\textsc{POMDP}{}-\textsc{IRL}{}} An expert with noisy sensors that senses its state imperfectly can be modeled as a partially observable \textsc{MDP}{} (\textsc{POMDP}{})~\cite{Kaelbling_pomdp}. The expert's uncertainty about its current physical state is modeled as a belief (distribution) over its state space. The expert's policy is then a mapping from its beliefs to optimal actions. Choi et al.~\cite{Choi2011} propose making either this policy available to the learner or the prior belief along with the sequence of expert's observations and actions (that can be used to reconstruct the expert's sequence of beliefs). The \textsc{POMDP}{} policy is represented as a finite-state machine whose nodes give the actions to perform on receiving observations that form the edge labels. The learner conducts a search through space of policies by gradually improving on the previous policy until it explains the observed behavior. Figure~\ref{pomdp_irl_characterization} illustrates this approach. \begin{figure}[!ht] \centering{ \includegraphics[scale=1]{pomdp_irl_characterization-eps-converted-to.pdf} \caption{In this illustration, similar to the one in Choi et al.~\cite{Choi2011}, consider a \textsc{POMDP}{} with two actions and two observations. $\pi_E$ (solid lines) is a \textsc{fsm} with nodes $\{n_1,n_2\}$ associated to actions and edges as observations $\{z_1,z_2\}$. The one-step deviating policies (dashed lines) $\{\pi_i\}_{i=1}^2$ are policies which are slightly modified from $\pi_E$. Each $\pi_i$ visits $n_i'$ instead of $n_i$ and then becomes same as $\pi_E$. The comparison of $\hat{V}^{\pi_E}$ with $\{V(\pi_i)\}_{i=1}^2$ characterizes the set of potential solutions. Since such policies are suboptimal yet similar to expert's, to reduces computations, they are preferable for comparison instead of comparing $\hat{V}^{\pi_E}$ with all possible policies.} \label{pomdp_irl_characterization} } \end{figure} \paragraph{\textsc{IRL}{} with Occlusion} Bogert et al.~\cite{Bogert_mIRL_Int_2014} introduce an extension \textsc{IRL}{}* for settings where the learner is unable to see portions of demonstrated trajectories due to occlusion. The maximum entropy formulation of Boularias et al.~\cite{Boularias2012}, is generalized to allow feature expectations that span the observable state space only. This method is applied to domain of multi-robot patrolling as illustrated in Fig.~\ref{fig:occlusion}. \begin{figure}[!ht] \includegraphics[width=0.33\textwidth,height=4cm]{TestFig6_planningunderocclusion_part01-eps-converted-to.pdf} \includegraphics[width=0.67\textwidth,height=4cm]{TestFig6_planningunderocclusion_part02-eps-converted-to.pdf} \caption{Prediction of experts' behavior using multi-robot \textsc{IRL}{}~\cite{Bogert_mIRL_Int_2014} in a multi-robot patrolling problem (left). Learner $L$ (green) needs to cross hallways patrolled by experts $I$ (black, reward $R_{E_1}$) and $J$ (gray, reward $R_{E_2}$). It has to reach goal `X' without being detected. Due to occlusion, just portions of the trajectory are visible to L. After learning $R_{E_1}$ and $R_{E_2}$, $L$ computes their policies and projects their trajectories forward in time and space to know the possible locations of patrollers at each future time step. The locations help create $L$'s own policy as shown in the figure on the right. Color should be used for this figure in print.} \label{fig:occlusion} \end{figure} Wang et. al. \cite{Wang2002,Wang2012} introduce the principle of Latent Maximum Entropy to extend the principle of maximum entropy to problems with hidden variables. By using this extension, Bogert et al.~\cite{Bogert_EM_hiddendata_fruit} continued along this vein of incomplete observations and generalized multi-agent {\sc maxentirl} to the context where some causal factors of expert's action are hidden from the learner. For example, the different amounts of force applied by a human while picking ripe and unripe fruit usually differs but this would be hidden from an observing co-worker robot. An expectation-maximization scheme is introduced with the E-step involving an expectation of the hidden variables while the M-step requires performing the {\sc maxent} optimization. \begin{figure}[ht!] \centering{ \includegraphics[scale=0.55]{hMDP-eps-converted-to.pdf} \caption{Hidden variable \textsc{MDP}{}. $u_i$ is a noisy observation, by the learner, of the state $s_i$ reached after taking action $a_{i-1}$. The source of illustration is \cite{hiddenMDP_Kitani2012}. The figure is shown here with permission from publisher.}\label{fig:hMDP} } \end{figure} \paragraph{Hidden Variable Inverse Optimal Control} A hidden variable \textsc{MDP}{} incorporates the probability of learner's noisy observation (say $u$; see Figure~\ref{fig:hMDP}) conditioned on the current state, as an additional feature ($\phi_u$) in the feature vector $\phi$. Hidden variable inverse optimal control (\textsc{hioc})~\cite{hiddenMDP_Kitani2012} casts \textsc{maxentirl} to the setting modeled by the hidden variable \textsc{MDP}{} with a linear reward $R_E=w^T \phi$. Consequently, the expression for the likelihood of expert's behavior incorporates the additional feature and its weight ($\phi_u,w_u$). During maximization, the tuning of weights during learning also adjusts $w_u$ to determine the reliability of the imperfect observations. We noticed that all the aforementioned methods use feature expectations matching for learning. However, by comparison, in the models for \textsc{IRL}{} and \textsc{hioc} (\textsc{MDP}{} and \textsc{hmdp}), only the learner has an incomplete observation and expert is aware of its state; whereas in \textsc{POMDP}{}-\textsc{IRL}{}, an expert can not completely observe its state. \subsection{Multiple Tasks} Human drivers often exhibit differing driving styles based on traffic conditions as they drive toward their destination. For example, the style of driving on a merging lane of a freeway is distinctly different prior to the joining of merging lane, at joining the lane, and post joining the lane. We may model such distinct behaviors of expert(s) as driven by differing reward functions. Consequently, there is a need to investigate methods that learn multiple reward functions simultaneously (Figure~\ref{IRL_multipletasks}). \subsubsection{Extended Definition} \begin{defn}(Multi-task \textsc{IRL}{}) Let the dynamics of the expert $E$ be represented by $K'$ \textsc{MDP}{}s each without the reward function. The number of \textsc{MDP}{}s denoted by $K'$ may not be known. Let the set of demonstrated trajectories be, $\mathcal{D}=\{(s_{0}^{i},a_{0}^{i}),(s_{1}^{i},a_{1}^{i}),\ldots (s_{j}^{i},a_{j}^{i})\}_{i=1}^{N},s_{j}\in S,a_{j}\in A,i,j,N \in\mathbb{N}$. Determine $\hat{R}_E^1$, $\hat{R}_E^2$, $\ldots$, $\hat{R}_E^{K'}$ that best explains the observed behavior. \end{defn} \begin{figure}[ht!] \centering \includegraphics[scale=.4]{Pipeline_IRL_multipletasks-eps-converted-to.pdf} \caption{Multi-task \textsc{IRL}{} involves learning multiple reward functions. The input is mixed trajectories executed by a single or multiple expert(s) realizing behavior driven by different reward functions. The output is a set of reward functions, each one associated with a subset of input trajectories potentially generated by it.} \label{IRL_multipletasks} \end{figure} Associating a subset of input trajectories to a reward function that is likely to generate it makes this extension challenging. This becomes further complex when the number of involved tasks is not known. \subsubsection{Methods} Diverse methods have sought to address this problem and we briefly review them below. \paragraph{m\textsc{IRL}{} - Multi-robot \textsc{maxentirl}} Bogert et al. \cite{Bogert_mIRL_Int_2014} extend \textsc{IRL}{}* and {\sc maxentirl} to the context of multiple experts who may interact, albeit sparsely. While navigation dynamics of each expert is modeled separately, the interaction is a strategic game between two players; this promotes scalability to many robots. Experiments illustrated in Fig.~\ref{fig:occlusion} (left) demonstrate \emph{modeling the interaction} in {\sc maxentirl}. Alternatively, all robots may be modeled jointly as a multiagent system. Reddy et al.~\cite{Reddy_DecMulti} adopt this approach and model multiple interacting experts as a decentralized general-sum stochastic game. Lin et al. \cite{Lin_MultiGame} propose a Bayesian method learning the distribution over rewards in a sequential zero-sum stochastic multi-agent game (not \textsc{MDP}{}). \paragraph{Maximum likelihood \textsc{IRL}{} for varying tasks} Babes-Vroman et al. \cite{Babes-Vroman2011} assume that a linear reward function of an expert can change over time in a chain of tasks. The method aims to learn multiple reward functions with common features $\{R_{i}=w_i^T\phi\}_{i=1}^n, n\in \mathbb{N}$. Given prior knowledge of the number $n\in \mathbb{N}$ of reward functions, for each target function, the solution is a pair of weight vector $w_i\in \mathbb{R}^k$ and a correspondence probability. Latter ties a cluster of trajectories to a reward function. The process iteratively clusters trajectories based on current hypothesis, followed by implementation of \textsc{mlirl} for updating the weights. This approach is reminiscent of using expectation-maximization for Gaussian data clustering. \begin{figure}[ht!] \centering{ \includegraphics[width=\textwidth]{Multi_Task_BIRL-eps-converted-to.pdf} \caption{Parametric multi-task \textsc{birl} Tuning of independent hyperprior modifies the dependent priors $P^R$ on reward functions and priors $P^\pi$ on policies, the diagram shows it as a sequence of priors. The variation in priors is assumed to influence the observed trajectories $\{\tau\}_i,i\in \mathbb{N}$. Bayesian update outputs an approximated posterior over rewards and policies }\label{fig:Multi_Task_BIRL} } \end{figure} \paragraph{Parametric multi-task \textsc{birl}} Allowing for $N$ multiple experts with $N$ known, {\sc birl} is generalized to a hierarchical Bayes network by introducing a hyperprior that imposes a probability measure atop the space of priors over joint reward-policy space. Dimitrakakis and Rothkopf~\cite{Dimitrakakis2012} show the sampling of the prior (latter) from an updated posterior (former), given input demonstration. This posterior (and thus the sampled prior) may differ for an expert trying to solve different tasks or multiple experts trying to solve different tasks. Within the context of our running example, Fig.~\ref{fig:Multi_Task_BIRL} illustrates how this approach may be used to learn priors for multiple drivers on a merging lane of a freeway. Parametric multi-task \textsc{birl} assumes that different experts share a common prior distribution over policies. The prior distribution can be either a joint prior over 'reward functions and policy parameters' or a joint prior over 'policies and their optimality' The method hypothesizes a posterior distribution over rewards by updating the prior based on a weighted likelihood of demonstration; the process associates subsets of demonstration trajectories to solutions likely to generate them. \paragraph{ Nonparametric multi-task \textsc{birl}} \textsc{dpm}-\textsc{birl} is a clustering method that learns multiple specifications using unlabeled fixed-length trajectories ~\cite{Choi2012}. It differs from previous methods because the number of experts are not known. Therefore, the method initializes a nonparametric Dirichlet process over the reward functions and aims to assign each input demonstration to a reward function that potentially generates it, thereby forming clusters of trajectories. Learning happens by implementing a Bayesian update to compute the joint posterior over the reward functions and the sought assignments to clusters. The process iterates until the reward functions and clusters stabilize. \subsection{Incomplete Model} Original \textsc{IRL}{} problem assumes the complete knowledge of transition model $T_{sa}$ and features. However, knowing the transition probabilities that fully represent the dynamics or specifying the complete feature set is challenging and often unrealistic. Hand designed features introduce structure to the reward, but they increase the engineering burden. Inverse learning is difficult when the learner is partially unaware of the expert's dynamics or when the known features can not sufficiently model the expert's preferences. Subsequently, the learner must estimate the missing components for inferring $\hat{R}_E$. \subsubsection{Extended Definition} \begin{figure}[!ht] \centering \includegraphics[scale=.4]{Pipeline_IRL_incompleteprobabilities-eps-converted-to.pdf} \caption{\textsc{IRL}{} with incomplete model of transition probabilities.} \label{fig:IRL_incompleteprobabilities} \end{figure} \begin{defn}(Incomplete Dynamics) Let an \textsc{MDP}{} without reward, $\mathcal{M}\backslash_{R_E}=(S,A,T_{sa},\gamma)$, represent the dynamics between an expert and its environment, where $T_{sa}$ specifies the probabilities for a subset of all possible transitions. Let the set of demonstrated trajectories be, $\mathcal{D}=\{(s_{0}^{i},a_{0}^{i}),(s_{1}^{i},a_{1}^{i}),\ldots (s_{j}^{i},a_{j}^{i})\}_{i=1}^{N},s_{j}\in S,a_{j}\in A,i,j,N \in\mathbb{N}$. Then, determine reward $\hat{R}_E$ that best explains either the input policy $\pi_E$ or the observed demonstration $\mathcal{D}$. \end{defn} We illustrate the extension in Fig.~\ref{fig:IRL_incompleteprobabilities}. \begin{defn}(Incomplete Features) Let an \textsc{MDP}{} without reward, $\mathcal{M}\backslash_{R_E}=(S,A,T_{sa},\gamma)$, represent the dynamics of an expert and its environment. Let the reward function $\hat{R}_E=f(\phi)$ depend on the feature set $\phi$. The input is either a policy or a demonstration $\mathcal{D}$. If the given feature set is incomplete, find the features and function that best explains the input. \end{defn} \subsubsection{Methods} \paragraph{Incomplete Dynamics} While the majority of \textsc{IRL}{} methods assume completely specified dynamics, some researchers have aimed learning the dynamics in addition to the reward function. For example, \textsc{mwal}~\cite{Syed2008} obtains the maximum likelihood estimate of unknown transition probabilities by computing the frequencies of state-action pairs which are observed more than a preset threshold number of times. The process determines complete transition function by routing the transitions for remaining state-action pairs to an absorbing state. To give formal guarantees for the accuracy of learned dynamics and thereby the reward function, the algorithm leverages a theoretical upper bound on the accuracy of learned transition model if the learner receives a precomputed minimum amount of demonstration. Therefore, the upper bound on learning error in the \textsc{MDP}{} with learned transition model naturally extends to the actual \textsc{MDP}{} of expert ~\cite{Syed2008_supplement}. \begin{figure}[!ht] \centering{ \includegraphics[scale=1]{transition-features-eps-converted-to.pdf} \caption{In m\textsc{IRL}{}\textbackslash \textsc{t}, $\xi^{(s,a)}_i=\{\tau_1,\tau_2,\ldots\tau_b\}$ denotes the transition-features for transition $(s,a,s')$ of expert $i$, $s'$ is intended next state. Computation of unknown probabilities by using probabilities of transition-features, $\prod_{\tau\in\xi^{(s,a)}_i}P(\tau)=T_{sa}(s,a,s')$, is feasible because different transitions share transition-features among them. Source of illustration is \cite{Bogert_mIRL_woT_Int_2015} and figure is reprinted by author's permission.}} \label{fig:transition-features} \end{figure} While \textsc{mwal} assumes that a learner fully observes the states, m\textsc{IRL}{}$_{\backslash T}$~\cite{Bogert_mIRL_Int_2014} focuses on limited observations with unknown transition probabilities and multiple experts. Bogert and Doshi model each transition as an event composed of underlying components. For example, movement by a robot may be decomposed into its left and right wheels moving at some angular velocity. Therefore, the probability of moving forward is the joint probability of left and right wheels rotating with the same velocities. Learner knows the intended next state for a state-action pair, and probability not assigned to the intended state is distributed equally among the unintended next states. Importantly, the components, also called transition features, are likely to be shared between observable and unobservable transitions as shown in Fig.~\ref{fig:transition-features}. Therefore, a fixed distribution over the transition features determines $T_{sa}$. The frequencies of a state-action pair in demonstrations provide a set of empirical joint probabilities, as potential solutions. The preferred solution is the distribution of component probabilities with the maximum entropy computed through optimization. The generalization capacity of learned information is more in m\textsc{IRL}{}*\textsc{\textbackslash t} than that in \textsc{mwal} because the structure introduced by a shared basis is more generizable, in the space of transition probabilities, than a local frequency based estimation. On the other hand, Boularias et al. \cite{Boularias2011relative} shows that the transition models approximated from a small set of demonstrations may result in highly inaccurate solutions. This raises doubts on the efficacy of the previous two methods To this end, \textsc{pi-irl} learns a reward function without any input transition probabilities. Furthermore, for estimating unknown dynamics, \textsc{gcl}~\cite{Levine_UnknownDynamics_NN_policy} iteratively runs a linear-Gaussian controller (current policy) to generate trajectory samples, fits local linear-Gaussian dynamics to them by using linear regression, and updates the controller under the fitted dynamics. \textsc{csi} also learns reward function without the probabilities as input, by regression on separate non-expert transitions dataset given as input. \paragraph{Incomplete Features} A generalization of \textsc{mmp} that focuses on \textsc{IRL}{} where the feature vectors $\phi$ are insufficient to explain expert's behavior is \textsc{mmpboost}~\cite{Ratliff2007}. In this case, the method assumes that a predefined set of primitive features, which are easier to specify, create the reward feature functions. In the space of nonlinear functions of base features, {\sc mmpboost} searches new features that make the demonstrated trajectories more likely and any alternative (simulated) trajectories less likely. Consequently, the hypothesized reward function performs better than the one with original feature functions. Further, it is well known that methods employing L-1 regularization objectives can learn robustly when input features are not completely relevant~\cite{Ng_RegularizationComparison}. In addition to {\sc mmpboost}, \textsc{gpirl} also uses this concept of base features to learn a new set of features which correspond better to the observed behavior. Wulfmeier et al.~\cite{Wulfmeier2015} and Finn et al.~\cite{Finn_gcl} propose neural network as function approximators that avoid the cumbersome hand-engineering of appropriate reward features. In some applications, it is important to capture the {\em logical relationships} between base features to learn an optimum function representing the expert's reward. Most methods do not determine these relationships automatically. But \textsc{firl} constructs features by capturing these relationships. In contrast, \textsc{bnp}-\textsc{firl} uses an Indian buffet process prior distribution to derive a Markov Chain Monte Carlo \textsc{mcmc} procedure for Bayesian inference of the features and weights of a linear reward function~\cite{Choi2013bayesian}. Authors demonstrate that the procedure constructs features more succinct than those by \textsc{firl}. Of course, all these methods are applicable only in those domains where the base feature space have primitive features sufficient enough to satisfactorily express the reward states. \subsection{Nonlinear Reward Function} A majority of the \textsc{IRL}{} methods assume that the solution is a weighted linear combination of a set of reward features. While this is sufficient for many domains, a linear representation may be over simplistic in complex real tasks; especially when raw sensory input is used to compute the reward values~\cite{Finn_gcl}. Also, the analysis of learner's performance w.r.t the best solution seems compromised when a linear form \textit{restricts} the class of possible solutions. But a significant challenge for relaxing this assumption is that nonlinear reward functions may take any shape, which could lead to a very large number of parameters. As our original definition of \textsc{IRL}{} given in Def.~\ref{def:irl} does not involve the structure of the learned reward function, it continues to represent the problem in the context of nonlinear reward functions as well. \begin{figure}[!ht] \centering{ \includegraphics[scale=1]{TestFig12_NonLinearBoostedFeaturres-eps-converted-to.pdf} \caption{ Learning a nonlinear reward function with boosted features improves performance over linear reward. Learner needs to imitate example paths drawn by humans in overhead imagery. Upper left panel - base features for a region. Upper right panel - image of the region used for testing. Red path is a demonstrated path and Green path is a learned path. Lower left panel - a map (un-boosted linear reward function) inferred using \textsc{mmp} with uniformly high cost everywhere. Lower right panel shows results of \textsc{mmp}\textsc{boost}. Since \textsc{mmp}\textsc{boost} creates new features by a search through a space of nonlinear reward functions, it performs significantly better. We reprinted this figure from \cite{Ratliff2007} with permission from MIT press. Color should be used for this figure in print.}} \label{Boosted_features_vs_unboosted_linear} \end{figure} Methods such as \textsc{projection}, \textsc{mmp}, and \textsc{maxentirl} assume that expert's reward is a weighted linear combination of a set of features; the assumption is \emph{required for computational expediency}. To overcome the restriction, methods \textsc{mmpboost}, \textsc{learch} and \textsc{gpirl} infer a nonlinear reward function. {\sc mmpboost} and {\sc learch} use a matrix of features in an image (cost map) and {\sc gpirl} uses a Gaussian process for representing the nonlinear function $R_E=f(\phi)$. Figure~\ref{Boosted_features_vs_unboosted_linear} shows the benefit of a nonlinear form with boosted features as compared to a restrictive linear form. In addition to these, Wulfmeier et al.~\cite{Wulfmeier2015} and Finn et al.~\cite{Finn_gcl} represent a complex nonlinear cost function using a neural network approximation, avoiding assumption of linear form. \subsection{Other Extensions} Ranchod et al.~\cite{Ranchod_SkillTransfer} presented a Bayesian method to segment a set of unstructured demonstration trajectories, identifying the reward functions (as reusable skills in a task) that maximize the likelihood of demonstration. The method is nonparamteric as the number of functions is unknown. \begin{figure}[!ht] \centering{ \includegraphics[scale=1]{Relational_irl-eps-converted-to.pdf} \caption{ The 3 step process of \textsc{IRL}{} in relational domain - classification outputs a score function, reward shaping optimizes the output, and regression approximates reward function $\hat{R}_E$ corresponding to the optimal score function. Author has shared this figure with us. Color should be used for this figure in print.}} \label{CSI} \end{figure} Munzer et al. \cite{Munzer_relationalIRL} extend the classification-regression steps in \textsc{csi} to include relational learning in order to benefit from the strong generalization and transfer properties that are associated with relational-learning representations. The process shapes the reward function for the score function as computed by the classification (see Fig.~\ref{CSI}). \section{Discussion} \label{sec:milestones_shortcomings} The previous section explained how some researchers recently extended initial \textsc{IRL}{} methods to broaden the scope of these methods. This section explains the common milestones and limitations of current literature. Then we briefly discuss Bayesian vis-a-vis entropy-based inference and optimization in general. \subsection{Milestones} Apart from the achievements in Section~\ref{sec:Mitigation} , \textsc{IRL}{} methods achieved the milestones listed here. \subsubsection{Formalization of Reliable Evaluation Metric} As mentioned in Section~\ref{subsection:accuracy}, the evaluation metrics like reward-loss are not reliable. In the pursuit of different challenges in \textsc{IRL}{} problem, some researchers formalized the norm of value-loss as a metric for \textbf{error} in \textsc{IRL}{}. Choi et al. \cite{Choi2011} name the normed loss as Inverse Learning Error or ILE. Given $\hat{\pi}_E$ and $\pi_E$ in the space of \textsc{MDP}{} policies, ILE is calculated as \begin{equation} ILE = \norm{V^{\pi_E}-V^{\hat{\pi}_E}}_1 \end{equation} If the actual $\pi_E$ is unavailable, we compute ILE using $\hat{V}^{\pi_E}$ estimated from demonstrated behavior. \subsubsection{Achieving Monotonic Probability - Optimality Relation} In an output policy or an output distribution over trajectories, the probability mass assigned for an action or a trajectory should be proportional to the value incurred by following it. Then an \textsc{IRL}{} algorithm should assign a high probability to optimal behavior demonstrated to the learner. Unexpectedly, \textsc{max-margin} and \textsc{mwal} are prone to compute a policy $\hat{\pi}_E$ with a zero probability to the actions in demonstration \cite{Ziebart2010}. The label bias in action-value based methods (e.g. \textsc{birl}, \textsc{hybrid}-\textsc{irl}, and \textsc{mlirl}) may result in an output distribution without monotonicity between the output probability mass and the value of a sample \cite{Ziebart2008}. Different methods solved this issue. \textsc{mmp} and its extensions avoid the label bias by making a solution policy have state-action visitations following that of expert \cite{Neu2008}. \textsc{maxent} based \textsc{IRL}{} methods address this issue by distributing probability mass based on entropy value and feature expectation matching \cite{Ziebart2010}. Further, \textsc{gpirl} addresses it by assigning a higher probability mass to the reward function corresponding to the demonstrated behavior, due to zero variance of the posterior distribution. Similarly, \textsc{pi-irl} maintains this monotonicity by using a continuous state continuous time model in which the likelihood of a trajectory is proportional to its exponentiated cost. \subsection{Shortcomings} The methods surveyed in this article encounter the following shortcomings. \subsubsection{Hand Crafting of Prior Knowledge} The \textsc{IRL}{} problem was originally devised to avoid the efforts of hand tuning the reward function to get desired behavior in an agent (expert). Although a variety of remarkable endeavors exists for solving this problem, there are some inevitable parameters in all the methods which are painstakingly hard to tune in the state spaces larger than few dozens of states (e.g. base features of a reward function, an appropriate prior distribution for implementing Bayesian methods, control parameter $\beta$ in Equation~\ref{Eq:Boltzmann_model}). Therefore, although every solution is a trade-off between a desired outcome and a realistically-achievable outcome, it remains an open question that `did research in \textsc{IRL}{} satisfactorily achieve the desired outcome?' \subsubsection{Scarcity of Analysis on Learning Bias} A limited number of \textsc{IRL}{} papers have explicitly mentioned the bias in the estimation of the value of observed behavior, i.e. $\hat{V}^{\pi_E}$. Authors of \textsc{max-margin}, \textsc{projection}, \textsc{mwal}, and \textsc{mlirl} bounded the bias from above to derive a probabilistic lower bound on sample complexity for achieving a learning error within a desired upper bound. The bias is a significant contributor to \textbf{error}, and a measure of the extent of efficient usage of available demonstration. Therefore, it deserves analysis in every \textsc{IRL}{} method. \section{Conclusions and Future Work} \label{sec:future_work} Since its introduction in 1998 by Russell, \textsc{IRL}{} has witnessed a significantly improved understanding of the inherent challenges, various methods for their mitigation, and the extension of these challenges toward real-world applications. This survey explicitly focused on the specific ways by which methods mitigated challenges and contributed to the progress of \textsc{IRL}{} research. The reason for this focus is that we believe that successful approaches in \textsc{IRL}{} will combine the synergy of different methods to solve complex learning problems. Our improved understanding has also revealed more novel questions. More development and clarifications are needed to answer these new questions. \paragraph{Direct and indirect Learning} When the state space is large and precise tuning of $\hat{\pi}_E$ is cumbersome, directly learning a reward function results in a better generalization as compared to policy matching \cite{Neu2007} (see Section ~\ref{subsection:labelDirectVsIndirect}). However, the issue of choosing between these two ways or exploiting their synergies warrants a more thorough analysis. \paragraph{Heuristics} Choi et al.~\cite{Choi2011} observed that when the authors evaluated the values of learned policy $\hat{\pi}_E$ and expert's policy $\pi_E$ on the learned reward $\hat{R}_E$, both are optimal and about equal. However, $\hat{\pi}_E$ obtained using $\hat{R}_E$ does not achieve the same value as $\pi_E$ when they use the true reward $R_E$ for the evaluation. This is, of course, a quantification of the reward ambiguity challenge, which we pointed out earlier in this survey. It significantly limits learning accuracy. We believe that the choice of heuristics in optimization may worsen or mitigate this issue. \paragraph{Convergence analysis} Methods \textsc{projection} and \textsc{mwal} analyze their convergence because of the geometric structure in the space of latent variables -- feature expectations $\mu$~\cite{Abbeel2004}. Both methods aim to match the observed feature expectations of an expert, $\mu^{\pi}_E$, and its estimated feature expectations $\hat{\mu}_E$. The geometric relationship between $\hat{\mu}_E$ and the outputs $\{\mu^{\pi}_E\}$ of iterations helps derive the error after each iteration. Bounding this error gives the number of required iterations. Both methods use Hoeffding's inequality to relate the bound with minimum sample complexity for a bounded bias in $\hat{\mu}_E$. A comprehensive theoretical analysis of both sample complexity and number of required iterations, to achieve a bounded error, is significant for knowing the cost of learning in problems with high-dimensional or large state spaces. However, most other methods such as those that optimize maximum entropy and Bayesian methods do not provide such analyses. \footnote{As \textsc{POMDP}{}-\textsc{IRL}{} is derived using the \textsc{projection} method for which a sample complexity bound exists, a similar analysis for it seems feasible.} Such gaps are opportunities to formally analyze the convergence of these methods. \begin{figure}[!ht] \centering{ \includegraphics[scale=0.65]{I-POMDP-eps-converted-to.pdf} \caption{The state of \textsc{i}-\textsc{POMDP}{} evolves in an \textit{interactive state space} that encompasses the computable models (beliefs, preferences, and capabilities) for other agents and the physical states of the environment. Agent $i$ maintains and updates his models of other agents.}} \label{I-pomdp} \end{figure} \paragraph{Multi-expert Interaction} Recent work on \textsc{IRL}{} for multi-agent interactive systems can be extended to include more general classes of interaction-based models to increase the potential for applications~\cite{Lin_MultiGame,Reddy_DecMulti}. These classes include models for fully-observable state spaces (Markov games~\cite{Littman1994markov_games}, multi-agent Markov decision processes~\cite{Boutilier1999}, interaction-driven Markov games \cite{Spaan_InteractiveFulObs}) and partially-observable states (partially observable identical-payoff stochastic games~\cite{Peshkin00}, multi-agent team decision problems~\cite{PynadathT02}, decentralized Markov decision processes \cite{Bernstein02}, and \textsc{i}-\textsc{POMDP}{}s~\cite{Gmytrasiewicz_Doshi_IPOMDP} illusrated in Fig.~\ref{I-pomdp}). Outside the domain of \textsc{IRL}{}, we note behavior prediction approaches related to inverse optimal control in multi-agent game-theoretic settings ~\cite{WaughZB_InverseEquilibrium}. The authors derived a regret-based criterion which can be used for Markov multi-agent games too: for any linear reward function, the learned behavior of agents should have regret less than that in observed behavior. \paragraph{Non-stationary rewards} Most methods assume a fixed reward function that does not change. However, preferences of experts may change with time, and the reward function is time-variant i.e., $R:S\times A\times \tau \to \mathbb{R}$. Babes-Vroman et al.~\cite{Babes-Vroman2011} capture such dynamic reward functions as multiple reward functions, but this approximation is crude. A more reasonable start in this research direction is the reward model in~\cite{Kalakrishnan_continousspace}. \paragraph{Function approximations} As mentioned in Section~\ref{subsection:labelCompExpense}, we need computationally inexpensive approximations of complex nonlinear reward functions in large state spaces. Unlike shallow local architectures, a deep neural network architecture is expressive given a large number of demonstration trajectories~\cite{Bengio2007}, Recent preliminary work has investigated the use of neural network function approximation and computation of the gradient of \textsc{maxent} objective function using back-propagation making \textsc{IRL}{} scalable to large spaces with nonlinear rewards~\cite{Wulfmeier2015,Finn_gcl}. Future work in this direction may help \textsc{IRL}{} get closer to real-world applications. \paragraph{Efficient \textsc{maxent}\textsc{IRL}{}} The optimizations in~\cite{Lafferty1996,WuKhudanpur} are likely to give an improvement in maximum entropy \textsc{IRL}{} algorithms. When input (demonstration) data is structured accordingly, these algorithms make the evaluation of partition function efficient~\cite{Malouf_ComparisonEstimatns}. Moving away from passive observations of the expert performing the task, Lopes et al.~\cite{Lopes2009} allow the learner to query the expert for actions at select states. While such active learning is shown to reduce the number of samples needed to learn the estimated reward, as we may expect, its usefulness is limited to those settings where the learner may interact with the expert. Some more observations merit further enquiry and provide good avenues for future research. Firstly, a principled way to construct and impose meaningful, generalizable features is necessary in high dimensional spaces, in order to avoid numerous demonstration trajectories and handcrafted features in such spaces. Secondly, future research could focus on the accuracy achieved with suboptimal input as it is natural for a human demonstrator to make mistakes while training a learner. Thirdly, there is a need for more work on modeling the sensing noise of learner observing the expert, especially in robotic applications. Finally, improvement in the generalization is required to reduce the sample complexity and the number of required iterations to achieve a pre-specified desired accuracy~\cite{Lopes2009}. Some methods showed an improvement in this context ~\cite{Levine2011,Levine2010_featureconstruction,S.Melo2010}. \begin{comment} MMP -> \textsc{mmp} \textsc{LEARCH} -> \textsc{learch} MWAL -> \textsc{mwal} POMDP -> \textsc{POMDP}{} MaxEnt -> \textsc{maxent} SAL -> \textsc{sal} HIOC -> \textsc{hioc} BIRL -> \textsc{birl} DPM -> \textsc{dpm} Hybrid -> \textsc{hybrid} BOOST -> \textsc{boost} FSC -> \textsc{fsc} FIRL -> \textsc{firl} MLIRL -> \textsc{mlirl} EM -> \textsc{em} GPIRL -> \textsc{gpirl} REIRL -> \textsc{reirl} hMDP -> \textsc{hmdp} BNP -> \textsc{bnp} DEC -> \textsc{dec} IRL -> \textsc{IRL}{} Projection -> \textsc{projection} Max-margin -> \textsc{max-margin} \end{comment} \section{Acknowledgments} This research was partly funded by a grant from the Toyota Research Institute of North America and a grant from ONR N-00-0-141310870. \section{References} \bibliographystyle{elsarticle-num}	{ "redpajama_set_name": "RedPajamaArXiv" }	1,147
{"url":"http:\/\/arcivasto.it\/tyax\/inverse-kinematics-video-games.html","text":"This method. Forward Kinematics Given the model and the joint angles, where is the end effector? Compute this so you know where to draw primitives Inverse Kinematics Given a desired location of the end effector, what are the required joint angles to put it there? Required to place the end effector near to an object in the real world. RELATED WORK The related works are categorized into two topics: inverse kinematics and shoulder motion. Inverse kinematics is the process of determining the parameters of a jointed flexible object (a kinematic chain) in order to achieve a desired pose. That's all. Then, I created a cube (IK_target) which gives the position where the tank should aim to. of work, is inverse kinematics. Hexapod Robot Inverse Kinematics Excel Spreadsheet Simulation. You will also learn about few tools of \"weight painting\" as well as multiple \"object selection\" in blender. This paper elaborates on a method developed by the authors for solving the inverse kinematics of a general 6R manipulator. This is when you have a desired end effector position, but need to know the joint angles required to achieve it. 0 Pages 12 Ppi 300 Scanner. $\\begingroup$ There are many different 3dof robotic arms. Yeah, I stan Castlers. Epic Games, a leading video game and software development company, has partnered with DeepMotion as part of their $100 million Epic MegaGrant program. Games sometimes need to solve Inverse Kinematics (IK) for a more realistic look, like foot placement on terrain. Inverse Kinematics (IK) is important in game programming. An Introduction to Procedural Animations This discussion introduces a new series about inverse kinematics for videogames. For a multiple DOF robot, the forward kinematics is quite straightforward. Mortal Kombat was a popular fighting game when I was of prime video game age in the early 1990s. Intent: Inverse kinematics (IK) is a way of animating objects using bones chained into linear or branched armatures in parent-child relationships. Parallel Inverse Kinematics for Multithreaded Architectures Pawan Harish, Mentar Mahmudi, Beno\u00eet Le Callennec and Ronan Boulic ACM Transactions on Graphics, Volume 35 Issue 2, February 2016. \\$\\begingroup\\$Note that the kinds of inverse kinematics problems we typically solve in games are \"turn head to look in direction\" or \"tweak blended gun-holding animation to keep hands attached to handles. At Augbotics we develop inverse kinematics (IK) software for animation and robotics, with web-based interactive 2D\/3D simulations of our solutions and prototype hardware. It then blends the selected motion into the primary motion using momentum-based inverse kinematics. Buka file latihan huruf Z yang sudah kita buat sebelumnya lalu aktifkan Inverse Kinematic dengan cara mencheck kotak Auto IK. At the moment I am just trying to get it to move to a vector. I dont understand how the motor6d works. The company sold 1 million units with a month of launch. Game Inverse Kinematics: A Practical Introduction, presents an uncomplicated practical approach using simplified implementation examples to introduce the reader to inverse kinematic for interactive scenes. Inverse Kinematics. This is a much harder problem, there may be many possible answers, or there may not be a set of angles that would reach to that point. Kinematics Graphing Game. Hi Georg, I've made the model working with Inverse Kinematic for 6-DOF. Use Unity to build high-quality 3D and 2D games, deploy them across mobile, desktop, VR\/AR, consoles or the Web, and connect with loyal and enthusiastic players and customers. Browse more videos. IK is an easy to use inverse kinematics plugin that can be used in a wide variety of skeletal rigs in your project. Separate the words with plus signs (cat +dog) to search for items that may contain cat but must contain dog. And then you will see the difference between both rigs and then create inverse kinematics rig. Figure 6 represents this well. Kind Code: A1. Inverse kinematics introduces the concept of a goal, in other words, a place where\u2026 Practice while you learn with exercise files Download the files the instructor uses to teach the course. Inverse kinematics is the opposite of forward kinematics. This page was last edited on 25 September 2019, at 08:12. Where the characters of Sins are outlawed and have to face actual demons in order to survive out in the world and retake their place in the kingdom, Goblin Slayer features a hero who wants to kill as many goblins as possible, going out of his way to do so. The kinematics equations of the figure define the relationship between the joint angles of the. Moreover, IKPy is a pure-Python library : the install is a matter of seconds, and no compiling is required. A common solution is to heuristically specify the kinematic constraints for the skeleton model. Kinematics is a subfield of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Simple searches use one or more words. VRIK will display and fully animate a virtual body while matching its movements to those of the player. Such arms offer lot of possibilities. OK, I Understand. Forward kinematics is the use of kinematic equations of a skeleton to derive the end-effector from the values of joints. You will also learn about few tools of \"weight painting\" as well as multiple \"object selection\" in blender. research-article. In motion capture, joints on the human body are measured in Cartesian space, and inverse kinematics is applied to reflect the joint angles on a virtual avatar. To visualize the inverse kinematics solution, animate the model by using the motion slider and video controls. An interactive game in which students choose velocity and position to match the given motion graph. IK uses kinematic equations to determine the joint angles so that the end effector moves to a desired position. I was wondering if any of the vets who might have worked on sports games would have any recommendations as to where I could start diving in to the world of Inverse Kinematics. One question, is it possible that the torso will NOT move when I turn my legs?I mean when I turn my legs, the torso turns too. With more people stuck at home, many have turned to video gaming to pass the time, potentially bolstering the outlook for a video game-themed ETF strategy such as HERO. Contact us: [email protected] l1 and l2 are the lengths of each segment(2 segments) and the thetas are the angles that they are at. However, once other predictors were included in the models and once propensity scores were used to control for an underlying propensity for choosing or being allowed to play violent video games, these relationships vanished, became inverse, or were reduced to trivial effect sizes. Rather than work from the root of the tree, it works from the leaves. Inverse Kinematics for 2DOF Arm When I first came across the problem of inverse kinematics I thought - quite naively - that it would be a simple matter to find a solution to the problem because the forward kinematics problem was so simple to solve. RenderStepped: Connect (function local a. This chapter focuses on redundancy resolution schemes, i. Upload permission You can upload this file to other sites but you must credit me as the creator. Inverse Kinematics. kapasuwuze. As an example, a good point to start from is the Denavit-Hartenberg convention. I have the inverse kinematics working, but the arm bends the wrong way. Serial\/Parallel Robotic Arm, Direct\/Inverse Kinematics, Jacobian, Dynamics, Trajectory Generation, Control). Andreas Aristidou. So, while standing on slope, enabling IK would visually look nice, but hit-box wise the leg would still be inside the rock (no IK on server side). Use Unity to build high-quality 3D and 2D games, deploy them across mobile, desktop, VR\/AR, consoles or the Web, and connect with loyal and enthusiastic players and customers. In the kinematic analysis of manipulator position, there are two separate problems to solve: direct kinemalics, and inverse kinematics. There are some games that use inverse kinematics in a game mechanic. 4, October 2016 2 2. Steam Engine with Inverse Kinematics This is, unbelievably, my first foray into AS3. Inverse kinematics (IK) is the use of kinematic equations to determine the joint parameters of a manipulator so that the end effector moves to a desired position; IK can be applied in many areas, including robotics, engineering, computer graphics and video games. Video 2 in this series. It is extremely difficult to leave it before concluding, once you begin to read the book. 72 over the same period. Right now, I've got an algorithm that solves the basic IK equation for a chain of rigid bodies connected by joints by approximately inverting ##J\\Delta\\theta = e##. This video introduces the Newton-Raphson root-finding method for numerical inverse kinematics. CCD algorithm was first propesed by Wang and Chen (A Combined Optimization Method for Solving the Inverse Kinematics Problem of Mechanical Manipulators. After two. Hi JDMather, Your model has lost some parts. \u2022 IK is more challenging: several possible solutions, or sometimes maybe no solutions. This project builds on Project 1 by adding world control mode, which will be built on inverse kinematics. Above: Human motion tracking data is mapped to a VBS3 avatar in real time - 3rd person view is visible in the monitor. For example we have a kinematic chain with n joints as shown in fig 1. This video is for anyone who has some experience with Blender, and wants to learn how to rig. In this paper, we present a serious game-based framework for providing live feedback to a patient performing a rehabilitation therapy for the purpose of assisting a patient in correct therapeutic steps and obtaining high quality therapy data for further. And works fine in animation and the game engine, but does start to drop performance when you have more than one armature. Unreal Engine 4 Documentation > Engine Features > Skeletal Mesh Animation System > IK Setups IK Setups. Here's the equations: math equations. One of the main uses of IK is the calculation of the player's feet and how they relate to the ground they are standing on. Puts new functions in place to enable dynamic Inverse Kinematics for legs. In the previous video, we derived the Newton-Raphson numerical algorithm for inverse kinematics when the end-effector configuration is represented by a minimum set of coordinates x_d. 13, 2012, under Uncategorized As demo\u2019d in this video: OR3D tool walk thru the video is a bit on the older side, but it shows the IK fairly well. Both gam\u2026. Check out the official statement issued to Inverse down below: \u201c Crysis Remastered is just the original game. Forward Kinematics This Chain is represented here, without any information about its actual shape, what we are interested in is the relative positions of all the joints. We tackle this problem from a learning perspective. Hi JDMather, Your model has lost some parts. See the Video on Youtube. Inverse Kinematics 2 DOF SCARA Robot 15. This technique is often used in robotics and animation. This page was last edited on 25 September 2019, at 08:12. The inverse kinematics solution is saved to subject01_walk1_ik. Inverse Kinematics in 2D - Part 2 - Alan Zucconi June 14, 2018 [\u2026] Part 1. This is when you have a desired end effector position, but need to know the joint angles required to achieve it. I am currently coding a Forward and Inverse Kinematics solver for a PUMA 560 robot. In the previous chapters, techniques to move articulated linkages are covered, but they still require a large amount of animator talent in order to make an object move as in the real world. Past R&D efforts have included the development of human-robot interaction (HRI) interfaces with corresponding robotic prototypes, and multirobot connectivity and control. com offers free software downloads for Windows, Mac, iOS and Android computers and mobile devices. Hexapod using Inverse kinematics with a Udoo x86 brain, MQTT Remote control via xBox controller and video streaming (3d printed legs). Inverse Kinematics (IK) has also been used in rehabilitation medicine in order to observe asymmetries or abnormalities. But you still have the various geometric options for inverse kinematics solutions. Home \u00bb Inverse Kinematics Tags Adobe Air Android ANE Art AS3 Base64 Beverly Hills Big Fish Games Blitting Box2D ByteArray C# California Compress Decode Encode Explosion Fight Flash FunFlow Galaxy Games Hidden Object Game Inverse Kinematics iOS JAVA JavaScript JS King List MeteorGames Mobile Multiplayer Neopets NextGamez Physics Platogo Player. What is Inverse Kinematics? When multiple objects are linked together with parent-child-relations the structure is called object hierarchy. com Twitter: @dtecta. Scrambled - A Platformer V1. Inverse kinematics was a big idea, back in the days when 3D characters were made from a collection of individual objects, like hands, forearms, upper arms and torsos, linked together by invisible joint objects. inverse kinematic. Digital traffic measures have spiked for everything from online groceries to video games and movie streaming. Inverse Kinematics (IK) is important in game programming. The kinematics equations of the figure define the relationship between the joint angles of the. 2] \u00a92020 Roblox Corporation. Automatic IK is a tool for quick posing, it can be enabled in the Toolbar in the 3D View, when in Pose Mode. Introduction. Intent: Inverse kinematics (IK) is a way of animating objects using bones chained into linear or branched armatures in parent-child relationships. In computer graphics, articulated figures are a convenient model for humans, animals or other virtual creatures from films and video games. Pinch something solid around you. Style-Based Inverse Kinematics Keith Grochow1 Steven L. In the Before Scene, there is an articulated body with some known pose. ISSN 0167-7055. While Grow Home and Rain World let the physics determine how the joints would move when subjected to gravity,. For example, have the player's hand reach out to an object, rather than having a pre-built animation of the character reaching out for something (which has the obvious limitation of only being able to reach for things that are the same height every time). Inverse kinematics is a technique where the rotation of joints are dynamically calculated based providing the final position of the end joint. Inverse kinematics (IK) is the use of kinematic equations to determine the joint parameters of a manipulator so that the end effector moves to a desired position; IK can be applied in many areas, including robotics, engineering, computer graphics and video games. If you go here on a Mac don't be surprised to see it crash. Video testing the inverse kinematics software for a Humanoid robot The robot is equipped with two arms, each arm with 7 DOF. Chapter 6: Inverse Kinematics. Full E-book Game Inverse Kinematics: A Practical Introduction For Kindle. UE4 includes several inverse kinematics solvers; however, these are only the basic building blocks of a functional IK system. For example, to perform a surgical task, a robotic arm used in a medical surgery needs precise motion from an initial location to a desired location. Of course, it really is play, nevertheless an amazing and interesting literature. For example, if all three joint axes are parallel, you need some inverse kinematic optimization like @ManuelRodriguez mentions. Think about your arm. Inverse Kinematic systems allow for reactive animation, such as foot placement on non-planar terrain. As the output of my inverse kinematics is not coming out to be the same as the input of forward kinematics. Inverse Kinematics determines how jointed or connected objects position themselves relative to one another when moved. Wendy Rogers (Kinesiology, Human Factors and Aging Lab) 3. Pages: 192\u2013199: Citation: Jason Weber. An animated figure is modeled with a skeleton of rigid segments connected with joints, called a kinematic chain. 6 Axes Inverse Kinematics from Kenryo Takahashi on Vimeo. Professor of Robotic Vision at QUT and Director of the Australian Centre for Robotic Vision (ACRV). You will also learn about few tools of \"weight painting\" as well as multiple \"object selection\" in blender. This video introduces the Newton-Raphson root-finding method for numerical inverse kinematics. The way I handle inverse kinematics is by matching the orientation of the object that needs to be picked up with the last two DOF's (I think like what you have done for Dexter). To solve the inverse kinematics problem: Click the Tools menu and select Inverse Kinematics. Spoiler beelzebub2063 wrote: Hi, first of all thank you for the mod. Puts new functions in place to enable dynamic Inverse Kinematics for legs. com - Noah Dominguez. (Nicholas Ratke). This is harder than FK, and there could be more than one solution. In 3D animation, a technique that provides automatic movement of objects. Home \u00bb Inverse Kinematics Tags Adobe Air Android ANE Art AS3 Base64 Beverly Hills Big Fish Games Blitting Box2D ByteArray C# California Compress Decode Encode Explosion Fight Flash FunFlow Galaxy Games Hidden Object Game Inverse Kinematics iOS JAVA JavaScript JS King List MeteorGames Mobile Multiplayer Neopets NextGamez Physics Platogo Player. Serial\/Parallel Robotic Arm, Direct\/Inverse Kinematics, Jacobian, Dynamics, Trajectory Generation, Control). When one bone moves, connected bones move in relation to it. The left robot arm has three joint angles, one end effector, and the target object. xml and click Open. Inverse kinematics are also relevant to game programming and 3D animation, where a common use is making sure game characters connect physically to the world, such as feet landing firmly on top of terrain. One way to make sure you are identifying all of the solutions is to recall the fundamentals of trigonometry. This inspired Delventhal to think up ways to use the game to inspire a deeper emotional connection between cat and cat partner. io\/ Why not register and get more from Qiita? We will deliver articles that match you. The D-H parameters of manipulator is given as: Link: alpha, a, theta, d. in order for IK to be. In order to improve the efficiency of kinematics modeling in analyzing robots, this document used Lie groups and screw theory to describe rigid motion of the robot system, established a kinematic model of operating arm based on Product of Exponentials (POE) formula, and analyzed forward and inverse kinematics of chain topology structure robot, as well as several sub-inverse problem of inverse. This page was last edited on 25 September 2019, at 08:12. Full E-book Game Inverse Kinematics: A Practical Introduction For Kindle. Values of End effector Inverse Kinematics 2 DOF SCARA Robot 16. A rigid body tree defines the robot geometry and joint constraints, which is used with inverse kinematics to provide the robot with joint positions. Inverse Kinematics is the opposite of the default Forward Kinematics. Inverse kinematics is a method that helps define the motion of a robot to reach a desired location. In pursuit of his master\u2019s degree in industrial design, Felix Ros created a device that works in the same manner as the five-bar linkage systems that you may have seen drawing the time on a whiteboard. Upload permission You can upload this file to other sites but you must credit me as the creator. 2: Numerical Inverse Kinematics (Part 2 of 2) (4:47). Making statements based on opinion; back them up with references or personal experience. Free Access. Play millions of free games on your smartphone, tablet, computer, Xbox One, Oculus Rift, and more. In this final video of Chapter 6, we modify the algorithm so that the desired end-effector configuration is described by the transformation matrix T_sd. 4, October 2016 2 2. Working Model includes automatic collision detection and responses for NURBS geometry. The model should walk through one full gait cycle. This is when you have a desired end effector position, but need to know the joint angles required to achieve it. Inverse Kinematics determines how jointed or connected objects position themselves relative to one another when moved. In order to describe an object\u2019s motion, you need to specify its. In this case, I\u2019m using it in conjunction with a variety of raycasts in order to prevent the character feet from intersecting when walking on uneven surfaces. In this paper, we present a serious game-based framework for providing live feedback to a patient performing a rehabilitation therapy for the purpose of as Adding Inverse Kinematics for Providing Live Feedback in a Serious Game-Based Rehabilitation System - IEEE Conference Publication. Please take a look at it. GroundAlignment. Articulated shapes are aptly described by reduced deformable models that express required shape deformations using a compact set of control parameters. When you enable the new \u201cConstraints\u201d drag mode in Studio, we will use inverse kinematics (IK) to solve any constraints affecting the parts you\u2019re dragging. Inverse kinematics is a method that helps define the motion of a robot to reach a desired location. Inverse- kinematics is required at design-time to assist artists using commercial 3D animation packages, for motion capture analysis, and for run-time applications such as games. The best animators could hope for was a platform that could handle basic hierarchies, allowing them to position and keyframe all those segments at once. Representation of the end-effector configuration as a transformation matrix is covered in the next video. LimbIK (set by SetHumanLimbIK at any point during the frame). It allows elements of an object to be linked, such as the parts of an arm or leg, and causes them to move in a prescribed. This sketch is created with an older version of Processing, and doesn't work on browsers anymore. Game and Anime Talents 3 Simple Steps to Implement Inverse Kinematics. Prismatic: single-axis translation aka slider. The reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as. Ch 4: slides: Video by TA Ben Walt: Spatial Forward Kinematics: 2. Requirements: 1. IK technology is now applied to many other areas such as engineering, computer graphics, and video games. Inverse Kinematics does all the challenging computational work of calculating what the pose is. Begin each part by reviewing the appropriate physical definition. The big day is here. The Last of Us Part 2 leak: Sony responds to disclosure about Naughty Dog opus with surprising news. There are different methods to solve IK problems -- some are numerical methods which can be used for general cases, and analytical solutions exist only for simple cases like the case for two joints. Inverse Kinematics Tool Properties, When to use the Inverse Kinematics Tool, Inverse Kinematics in Toon Boom Animate. The planning and inverse kinematics algorithms in this suite are designed for articulated robots like robotic arms and humanoids. When your avatar raises a rifle or throws a punch, the game smoothly animates the movement by constraining the problem. Representation of the end-effector configuration as a transformation matrix is covered in the next video. While most tutorials either teach through video, or just a block of text with some code examples splattered around, this one differs. while inverse kinematics is just the opposite. Kinematics Identifier-ark ark:\/13960\/t6f255q9w Ocr ABBYY FineReader 11. These virtual bodies that you see when you look down and move your arms around in certain games are governed by an animation concept called \u2018Inverse Kinematics\u2019, or IK for short. However, inverse kinematic problems are a challenging topic (due to their. Math for Game Programmers: Inverse Kinematics: Speaker(s): Gino van den Bergen : Company Name(s): Dtecta : Track \/ Format: Programming: Overview: As we enter a new console generation, the complexity of many games have increased, and with that, the knowledge needed to create them. We have mentioned already the existence of analytical methods to solve the dual problem of inverse kinematics. For this example, and indeed many of inverse kinematics problems in robotics, it is useful to define the two-argument arctangent function. A method of relative inverse kinematics graphical user interface may include providing, a first view of a graphical user interface (GUI) to be displayed, wherein the first view is associated with relative inverse kinematics (RIK) of a game character of a video game and receiving a selection of a first GUI element comprised by the first view of the GUI, the selection indicating that a RIK. In this week\u2019s blog, we would like to show you how we make use of inverse kinematics to make strikes in combat feel a bit more immersive. Positions and velocities of links are defined by the values and speeds of the scalar joint parameters (angles, distances). Bill Baxter. This is the update order of the Animation System in regards to Inverse Kinematics: FK Animation Queue\/Layers (including Aiming). Upload permission You can upload this file to other sites but you must credit me as the creator. All in all, the book presents a practical point of view with implementation details, limitations, engineering workarounds, and common pitfalls. video (10) war (12) webservice (10) web\u30b5\u30fc\u30d3\u30b9 (408) web\u30c7\u30b6\u30a4\u30f3 (54) web\u5236\u4f5c (68) yahoo (10) youtube (1229) yusukebe (12) \u3053\u308c\u306f\u3059\u3054\u3044 (28) \u3053\u308c\u306f\u3072\u3069\u3044 (13) \u306f\u3066\u306a (56) \u306f\u3066\u306a\u30d6\u30c3\u30af\u30de\u30fc\u30af (27). So im trying to make a r6 rig with procedual walk animations (Maybe more to come). Game Inverse Kinematics: A Practical Introduction, presents an uncomplicated practical approach using simplified implementation examples to introduce the reader to inverse kinematic for interactive scenes. Review three journal papers (1 journal paper = 2 conference paper) or 6 conference papers or any other combinations of both related to the content of one of the topics thought in class (i. Instead, all of your transformations done in IK are baked directly into the keyframes of your animation. Kinematics Software - Free Download Kinematics - Top 4 Download - Top4Download. ISSN 0167-7055. Aside from crashing previously (which I do believe was fixed), all R15 characters are fully capable of being animated with Inverse Kinematics in the animation editor. This is very useful in robotics as you usually know where you want the arm to go to but you don\u2019t know the angles that will active this position. However, for some applications, working with the approximate robot\u2019s model is neither. If you go here on a Mac don't be surprised to see it crash. I recently noticed that there are number of folks that have translated this solution into computer code. 2: Numerical Inverse Kinematics (Part 2 of 2) (4:47). This method. Hexapod using Inverse kinematics with a Udoo x86 brain, MQTT Remote control via xBox controller and video streaming (3d printed legs). = middle of the claw): The kinematics equations of the robot are used in robotics, computer games, and animation. Representation of the end-effector configuration as a transformation matrix is covered in the next video. Software required: After Effects CC 2014, DUIK 14. Making statements based on opinion; back them up with references or personal experience. Above: Human motion tracking data is mapped to a VBS3 avatar in real time - 3rd person view is visible in the monitor. I used Inverse Kinematics to achieve this, and I actually did some progress. Intent: Inverse kinematics (IK) is a way of animating objects using bones chained into linear or branched armatures in parent-child relationships. There are some games that use inverse kinematics in a game mechanic. The left robot arm has three joint angles, one end effector, and the target object. Use MathJax to format equations. I kinda understand Inverse Kinematics (Heres some good tutorials: here and here) But my problem is to get it on the r6 body. To improve on this, we implemented an inverse kinematics system for shields so that they react differently depending on the direction of the impact. Join DevCrut on Roblox and explore together!- Any questions about scripting? PM me, i'll be happy to help you! :D - You only live once, make it worth!. This is very useful in robotics as you usually know where you want the arm to go to but you don\u2019t know the angles that will active this position. Negligible research has been conducted on the systematic analysis of small animals, such as lizards and arthropods. And then you will see the difference between both rigs and then create inverse kinematics rig. Since the solution of the inverse kinematics problem is very complex, many research efforts have been working towards getting the approximate solution of this problem. So all we need to do is subtract the vector of the third link from the target coordinates, apply the 2-link plus base inverse kinematics on this coordinates and then calculate phi using the value of$\\theta_1$and$\\theta_2$that we get from those calculations. pdf), Text File (. I would like to know advantages and disadvantages of these methods comparing to each other. Skeletal animation is the standard way to animate characters or mechanical objects for a prolonged period of time (usually over 100 frames). Lets recap what is Forward kinematics first. please try it. RTIK (Real Time Inverse Kinematics) is an inverse kinematics system for Unreal Engine 4. A simple example of IK is the human arm: While we intuitively know the target position of an object we want to reach for, our brains need to figure out how much to move each joint in our arm to get to that target. Full E-book Game Inverse Kinematics: A Practical Introduction For Kindle. One way to make sure you are identifying all of the solutions is to recall the fundamentals of trigonometry. Initial conditions+physics Articulated objects Forward and inverse kinematics Possibly forward and inverse dynamics Many links to robotics, mechanics and other fields Simple 2 link arm 2 links connected by rotational joints Forward Kinematics Specify joint angles, computer finds end-effector Forward Kinematics Then specify joint motions with. Tweak your robot dimensions and see how it will affect your work envelope and your precision. James also explains his idea of modifying the original CCD algorithm. This shows more than one Joint angle sets, which satisfy the given coord. Forward Kinematics is the inverse function of Inverse Kinematics. Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods. Aside from crashing previously (which I do believe was fixed), all R15 characters are fully capable of being animated with Inverse Kinematics in the animation editor. However, once other predictors were included in the models and once propensity scores were used to control for an underlying propensity for choosing or being allowed to play violent video games, these relationships vanished, became inverse, or were reduced to trivial effect sizes. It then blends the selected motion into the primary motion using momentum-based inverse kinematics. One of the main uses of IK is the calculation of the player's feet and how they relate to the ground they are standing on. It commonly refers to either inverse rigid body dynamics or inverse structural dynamics. Find games tagged inverse-kinematics like VR Physics Framework demo, Inverse Kinematics, Spirit Trap on itch. Figure 6 represents this well. Showing results for inverse kinematics. Unity is the ultimate game development platform. The left robot arm has three joint angles, one end effector, and the target object. We use cookies for various purposes including analytics. \u201cConstrained Inverse Kinematics\u201d. DevCrut is one of the millions playing, creating and exploring the endless possibilities of Roblox. Kinematics Identifier-ark ark:\/13960\/t6f255q9w Ocr ABBYY FineReader 11. The model should walk through one full gait cycle. Kinematics is the study of how objects move. Game Inverse Kinematics: A Practical Introduction, presents an uncomplicated practical approach using simplified implementation examples to introduce the reader to inverse kinematic for interactive scenes. For a multiple DOF robot, the forward kinematics is quite straightforward. I've been working on figuring out the inverse kinematics given an x,y,z coordinate. I know at least 3 different approaches to solve inverse kinematics problem. io\/ Why not register and get more from Qiita? We will deliver articles that match you. you can edit the file to change the Imposed Motion and Time for change the action of the model. This video presents solving inverse kinematics for ABB Puma Robot. The Complete Guide to Fenglee's Attack on Titan Tribute Descargar Guedin's Attack on Titan Fan GameThe Game Of Life (AOT\/SAO Crossover Reader X Levi X Eren X Attack on Titan Tribute Game \| Petit Computer Wiki 200 best Shingeki no Kyojin I images on PinterestFan Art Friday: Attack On Titan by techgnotic on DeviantArtmikasa ackerman clipart - ClipgroundAttack on Titan. We use cookies for various purposes including analytics. The easiest way to do inverse kinematics is with CCD method (Cyclic Coordinate Descent). ing basic shapes with the simpli ed inverse model that was build in the previous step. Inverse kinematics are also relevant to game programming and 3D animation, where a common use is making sure game characters connect physically to the world, such as feet landing firmly on top of terrain. Topics will begin with introductory talks on Grassman algebra and rotations and quaternions, then will continue with random numbers and spatial subdivision, and will conclude with inverse kinematics, sampling and. 03\/Feb\/2012 We have been working with 3D coordinates (x,z,y) in our IK algorithm, find the change in each dimension, and calculate the change of angles. Inverse Kinematics Not long ago, game characters were a lot like shellfish: they were basically piles of rock-hard segments. For the rest of the 3 DOF's I've written down the transformations in terms of joint rotations and have solved them explicitly for a given target position. T he coronavirus lockdown has boosted game maker share prices as people try to keep themselves entertained indoors. Inverse kinematics is a method that helps define the motion of a robot to reach a desired location. The book explains basic principles all the way through to testing and coding, while. Forward and inverse kinematics. Sometimes it is almost a requirement as animation done by hand proves to be too difficult or time. This approach is known as Inverse Kinematics (IK) and is supported in Mecanim for any humanoid character with a correctly configured Avatar. While you often need to rotate each bone individually to pose a chain, in some cases you can speed up the process by using inverse kinematics. I thought it was funny to watch it all move about and decided to add some variety and turn it into a little game. For example, to move a hand to some location, you must rotate several arm. Separate the words with plus signs (cat +dog) to search for items that may contain cat but must contain dog. I've edited Walter's model and made it working now. The target position is defined as the input, and the resulting pose required for the end effector to reach the target position is the output. Inverse kinematics using dynamic joint parameters: inverse kinematics animation synthesis learnt from sub-divided Berlin Heidelberg 2016 Abstract In this paper, we describe a novel parallelizable method for the fast computation of inverse kinematics (IK) such as real-time video games, 3D modeling software and SAIBA-like Embodied. Inverse kinematics is the process of determining the parameters of a jointed flexible object (a kinematic chain) in order to achieve a desired pose. Movement will also be restricted by that configuration of constraints. While Grow Home and Rain World let the physics determine how the joints would move when subjected to gravity,. The physical quantities relevant to the motion of a particle include: mass m, position r, velocity v, acceleration a. 2: Numerical Inverse Kinematics (Part 2 of 2) (4:47). Inverse Kinematics in short, is the process of calculating the joint positions in relation to each other, from a specified end point. Serial\/Parallel Robotic Arm, Direct\/Inverse Kinematics, Jacobian, Dynamics, Trajectory Generation, Control). An example could be the simulation of a robotic arm with the XNA framework. Our solution relies on example meshes to indicate the class of meaningful deformations. Figure 6 represents this well. IK is widely used in robotics and computer. Inverse Kinematics\u306e\u5b9f\u88c5\u30e1\u30e2 https:\/\/unity-game-dev-guild. Inverse Kinematics with Quaternion Joint Limits. Create a rigid body tree model for your robot using the rigidBodyTree class. In this final video of Chapter 6, we modify the algorithm so that the desired end-effector configuration is described by the transformation matrix T_sd. Notice that each dimension is calculated using very basic trigonometric functions and ideas, which isn't very close to reality, and as you can probably see from the video demonstration the. To animate an arm using forward kinematics, you rotate the upper arm away from the shoulder, then rotate the forearm, the hand from the wrist and so, on adding. Loading Unsubscribe from Scar? Roblox - Lets Make A Game - CFrame Mathmatics - Duration: 39:30. Software required: After Effects CC 2014, DUIK 14. The shoulder inverse kinematics (Shoulder IK) is involved in the second step of the rigging stages. The inverse kinematics solution for Cartesian robots is trivial as all axes are perpendicular by definition and thus there is no coupling of the motions. Kinematics, as a field of study, is often referred to as the \"geometry of motion\" and is occasionally seen as a branch of mathematics. The forward and inverse kinematics of a Stewart-Gough platform are derived, and code created (available to download). = middle of the claw): The kinematics equations of the robot are used in robotics, computer games, and animation. Inverse kinematics is a type of motion planning. While I have lost the source code for that original implementation, you can still follow the video and implement the concept detailed in the video. Serial\/Parallel Robotic Arm, Direct\/Inverse Kinematics, Jacobian, Dynamics, Trajectory Generation, Control). So all we need to do is subtract the vector of the third link from the target coordinates, apply the 2-link plus base inverse kinematics on this coordinates and then calculate phi using the value of$\\theta_1$and$\\theta_2that we get from those calculations. I am currently coding a Forward and Inverse Kinematics solver for a PUMA 560 robot. Inverse kinematics (IK) is a method of animating that reverses the direction of the chain manipulation. An animated figure is modeled with a skeleton of rigid segments connected with joints, called a kinematic chain. chrparams File and Animation). Slip and Fly Human Projectile. In this project we formulate the problem of Inverse Kinematics as an unconstrained optimization. Fast Numerical Methods for Inverse Kinematics. One of the main uses of IK is the calculation of the player's feet and how they relate to the ground they are standing on. Instead of throwing thousands of dollars and many hours at another program, we decided to create full bodyRead More. Given a set of constraints, our system can produce the most likely pose satisfying those constraints. Spline IK is a constraint which aligns a chain of bones along a curve. Can I have my knee behind the \"pelvis\" and. Yeah, I'm decent at arsenal. I dont understand how the motor6d works. A simple example of IK is the human arm: While we intuitively know the target position of an object we want to reach for, our brains need to figure out how much to move each joint in our arm to get to that target. The development of IKinema in tune with the developments of Faceshift, but raise the idea to a new level, opening the possibility to use inverse kinematics in the production of movies and games. Finally, there is a possibility of using inverse kinematics in unusual rigs and game me-chanics. In the main text, we have introduced the problem of inverse kinematics and its various difficulties through the example of the four-dof two-link arm: how to deal with redundancy, how to deal with singularities, how to deal with imperfect numerical integration. Inverse Kinematics Not long ago, game characters were a lot like shellfish: they were basically piles of rock-hard segments. The example shows how to extract the constraint equations, solve them in Maple and implement the solution(s) back in MapleSim. Direct kinematics involves solving the forward transformation equation to find the location of the hand in terms of the angles and displacements between the links. Utilising the Blender Game Engine (BGE), the position. Where the characters of Sins are outlawed and have to face actual demons in order to survive out in the world and retake their place in the kingdom, Goblin Slayer features a hero who wants to kill as many goblins as possible, going out of his way to do so. n-link arm: forward and inverse kinematics of the end-effector position. Best inverse kinematics in VR ? Discussion I kind of like the way some games implemented inverse kinematics to see your whole body or at least your modeled arms and upper torso, e. This is harder than FK, and there could be more than one solution. That's all. Inverse Kinematics por NandoDine em Sex Nov 18, 2016 12:34 am Boa Noite :D, esta script esta dando um loop infinito no transform. An interactive game in which students choose velocity and position to match the given motion graph. Animation Tutorial: Inverse Kinematics (IK) William Vaughan introduces Inverse Kinematics (IK) in LightWave 3D Posted: Wed 23 Jun 2010. Thanks for the A2A. Simple searches use one or more words. I've adopted the Jacobian method, taking the derivative of the forward kinematics equations with respect to their angles and input it into the Jacobian. Robotics may use Inverse Kinematics for robot hands; whereas Computer Graphics might use Inverse Kinematics for a walk cycle in a video game. While Grow Home and Rain World let the physics determine how the joints would move when subjected to gravity,. However, these systems are inapplicable to small animals because of the difficulty in. The book explains basic principles all the way through to testing and coding, while. Game Inverse Kinematics: A Practical Introduction, presents an uncomplicated practical approach using simplified implementation examples to introduce the reader to inverse kinematic for interactive scenes. Instead of throwing thousands of dollars and many hours at another program, we decided to create full bodyRead More. Finally, there is a possibility of using inverse kinematics in unusual rigs and game me-chanics. Inverse Kinematics has a wide variety of usages in many real world applications. Q&A for professional and independent game developers. Therefore, in kinematics, attention is focused on position, velocity, and acceleration of a body, how these properties are related, and how they change over time. This example shows how to calculate inverse kinematics for a simple 2D manipulator using the inverseKinematics class. video (10) war (12) webservice (10) web\u30b5\u30fc\u30d3\u30b9 (408) web\u30c7\u30b6\u30a4\u30f3 (54) web\u5236\u4f5c (68) yahoo (10) youtube (1229) yusukebe (12) \u3053\u308c\u306f\u3059\u3054\u3044 (28) \u3053\u308c\u306f\u3072\u3069\u3044 (13) \u306f\u3066\u306a (56) \u306f\u3066\u306a\u30d6\u30c3\u30af\u30de\u30fc\u30af (27). Instead, all of your transformations done in IK are baked directly into the keyframes of your animation. 27 (Th) Guest Lecture: Prof. Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods. For example, if all three joint axes are parallel, you need some inverse kinematic optimization like @ManuelRodriguez mentions. 1 1 6 Lecture Video 1 of 2 Intro to Inverse Kinematics and Example - Duration: 23:29. While Grow Home and Rain World let the physics determine how the joints would move when subjected to gravity,. Tapi jujur, saya lebih suka memilih versi Forward Kinetic. RELATED WORK The related works are categorized into two topics: inverse kinematics and shoulder motion. you can edit the file to change the Imposed Motion and Time for change the action of the model. For the rest of the 3 DOF's I've written down the transformations in terms of joint rotations and have solved them explicitly for a given target position. RTIK (Real Time Inverse Kinematics) is an inverse kinematics system for Unreal Engine 4. Drawing an analogy to the traditional use of skeleton-based inverse kinematics for posing skeletons, we define mesh-based inverse kinematics as the problem of finding meaningful mesh deformations that meet specified vertex constraints. Whether your application is solving the system\u2019s inverse kinematics and dynamics, motion planning, control design, contact modeling, component sizing, or hardware-in-the-loop testing, Maplesoft can help. Inverse Kinematics determines how jointed or connected objects position themselves relative to one another when moved. Complex Multiple Segments Inverse Kinematics 30 Jan, 2012 in AS3 \/ Blog tagged AS3 \/ Inverse Kinematics \/ JavaScript \/ JS by Lorenzo Nuvoletta Yesterday I wrote a Simple class to handle simple Kinematics , but I was not very happy with it, I needed to use more segments, so I started looking around. Game 3 was the inverse of Game 4, in that it was the Cavs who were in control most of the game but allowed the Bulls back in late. Then, I created a cube (IK_target) which gives the position where the tank should aim to. VRIK will display and fully animate a virtual body while matching its movements to those of the player. This is when you have a desired end effector position, but need to know the joint angles required to achieve it. Given a set of constraints, our system can produce the most likely pose satisfying those constraints. Separate the words with plus signs (cat +dog) to search for items that may contain cat but must contain dog. Chapter 6: Inverse Kinematics. The inverse kinematics is just a leftover feature from The Last Hope, which is dedicated to work specifically there, but in Fusion it has its own set of problems, for example, it doesn't react well to movement. Users changes position of gree cube which is the target point. com, Andre Harrell. To visualize the inverse kinematics solution, animate the model by using the motion slider and video controls. #76807053, #76824673 are all replies on the same post. PostProcess Callback to the game. 5 out of 5 stars 2. \" (Wikipedia, retrieved 27 November 2008 ). In this project we formulate the problem of Inverse Kinematics as an unconstrained optimization. For example, if the +X button is clicked, the corresponding joint move right by a certain number of pixels alon. Nishant Neeraj writes: In this tutorial, you will learn how to rig your character completely from scratch. According to a remarkable new study, the brain is also busy replaying waking experiences during sleep. Bill Baxter. Motion capture is an essential technique for interactive systems, such as distributed virtual reality, video games, and entertainment. I am verifying the output of my forward kinematics through inverse kinematics and the results are not as desired. In the preface, we told you that kinematics is the study of the motion of bodies without regard to the forces acting on the body. I've adopted the Jacobian method, taking the derivative of the forward kinematics equations with respect to their angles and input it into the Jacobian. On Robotics and Automation, 7:489-498, 1991). Then, I created a cube (IK_target) which gives the position where the tank should aim to. What is Inverse Kinematics? When multiple objects are linked together with parent-child-relations the structure is called object hierarchy. Use MathJax to format equations. Kinematics In this chapter we'll explain the fundamental aspects of the subject of kinematics. An experimental simulation. Pages: 193\u2013204: Citation: Marco Spoerl. Browse Bigfoot Inverse Kinematics articles on Geek. Inverse Kinematics for Game Programming. SEGMENTED FULL BODY INVERSE KINEMATICS. This technique is often used in robotics and animation. Inverse Kinematics & Mech Save Time Reduction, coming on July, Faster Mechlab Saving Times= 1hour, 24Minutes, 00 seconds,(July Patch), Rea. Given a ragdoll of any kind, inverse kinematics finds how to move it to reach the desired target. Inverse kinematics (IK) tells us how to rotate our bones in order to reach a desired position. I'm wondering if it's possible to have some control over inverse kinematics during the game. Yeah, I play arsenal. The end-effector configuration is represented by a minimum set of coordinates. UE4 includes several inverse kinematics solvers; however, these are only the basic building blocks of a functional IK system. Many engineers have heard of Inverse Kinematics, but less have heard of Inverse Dynamics. But inverse kinematics could be tricky, for the same end-effectors coordinates,. Determine the position and rotation of the joints in a character-based given position in space. When SceneKit prepares to render a scene, it processes the physics simulation before applying constraints (including IK constraints). Game Inverse Kinematics: A Practical Introduction, presents an uncomplicated practical approach using simplified implementation examples to introduce the reader to inverse kinematic for interactive scenes. \u201cThe Jacobian Transpose Method for Inverse Kinematics\u201d. I know at least 3 different approaches to solve inverse kinematics problem. 0 Pages 12 Ppi 300 Scanner. Zhao and Badler tackle the inverse kinematics problem without using the continuous-motion assumption. Math for Game Programmers: Inverse Kinematics Gino van den Bergen [email protected] RenderStepped: Connect (function local a. These virtual bodies that you see when you look down and move your arms around in certain games are governed by an animation concept called 'Inverse Kinematics', or IK for short. In this course, we will learn how to use the IK rigging plugin DUIK for After Effects. It is typically used to make the character's movement more realistic. Values of End effector Inverse Kinematics 2 DOF SCARA Robot 16. We present an ef\ufb01cient inverse-kinematics methodology based on the interpolation of example motions and. Although sufficient to desc. Inverse Kinematics - rotation problem (Newbie) We have a robotic hand with boned fingers when we move the finger through XY axis we get a rotating effect through the Z axis. The reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as inverse kinematics. Addeddate 2017-09-19 10:16:28 Identifier Robotics. other virtual creatures from \ufb01lms and video games. This course teaches the essential skills needed to get started with rigging in Blender. When the game logic updates does it do the computation of the skin mesh animation and the computation of the inverse kinematics and then store the result directly in the scene graph or is it stored indirectly and the render thread does the computation?. There are some games that use inverse kinematics in a game mechanic. Work, Energy & Power. Ch 4: slides: Video by TA Ben Walt: Spatial Forward Kinematics: 2. A fascinating math game designed for Grade 3 kids to teach them division and multiplication using inverse operations. In motion capture, joints on the human body are measured in Cartesian space, and inverse kinematics is applied to reflect the joint angles on a virtual avatar. IK is widely used in robotics and computer. Real-time Inverse Kinematics Technique for Controlling Redundant Avatar Arm* Sanghyun Kim1, Junhyung Kim1, Ji-Hun Bae2, and Jaeheung Park1 Abstract\u2014In this paper, a preliminary study of a real-time Inverse Kinematics (IK) technique is presented to transfer human arm postures to a humanoid avatar with 7 or more DoF arm. 5 by panther__ Panoramic_Plus Icon by panther__ Iron Man - Ion Recharge by panther__ panther__'s multiplayer test by panther__ Fibbage - Scratch Edition - Beta by panther__ Iron Man Simulation with Inverse Kinematics by panther__ Armature Rig Using Inverse Kinematics by panther__ Inverse Kinematics V1. Users changes position of gree cube which is the target point. I know that CCD suffers from local minima but as I heard this problem seldom arises in practice. Inverse kinematics is the process of determining the parameters of a jointed flexible object (a kinematic chain) in order to achieve a desired pose. Pages: 192\u2013199: Citation: Jason Weber. \uac1c\uccb4 \uacc4\uce35 \uad6c\uc870\uc5d0\uc11c \uc0c1\uc704 \uac1c\uccb4\uac00 \ubcc0\ud615\ub418\uba74 \uc77c\ubc18\uc801\uc73c\ub85c \uadf8 \ud558\uc704\uc5d0 \uc788\ub294 \uac1c\uccb4\ub4e4\ub3c4 \ubcc0\ud615\ub429\ub2c8\ub2e4. Buka file latihan huruf Z yang sudah kita buat sebelumnya lalu aktifkan Inverse Kinematic dengan cara mencheck kotak Auto IK. The kinematics equations of the figure define the relationship between the joint angles of the. In 3D animation, a technique that provides automatic movement of objects. Inverse Kinematics (IK) has also been used in rehabilitation medicine in order to observe asymmetries or abnormalities. Computer Graphics Forum, 37. Angular\/Circular Motion & Torque. Inverse kinematics algorithms are often used to minimize the number of trackers required for motion capture systems. PR2 BioIK Multi-Goal video 44. An animated figure is modeled with a skeleton of rigid segments connected with joints, called a kinematic chain. 1 1 6 Lecture Video 1 of 2 Intro to Inverse Kinematics and Example - Duration: 23:29. Inverse Kinematics with Quaternion Joint Limits. chrparams File and Animation). The slides. Animation Driven IK (set up in. Forward kinematics is the problem of finding the position and orientation of the end-effector, given all the joint parameters. RenderStepped: Connect (function local a. A fascinating math game designed for Grade 3 kids to teach them division and multiplication using inverse operations. And then you will see the difference between both rigs and then create inverse kinematics rig. Topics will begin with introductory talks on Grassman algebra and rotations and quaternions, then will continue with random numbers and spatial subdivision, and will conclude with inverse kinematics, sampling and. 4, October 2016 2 2. T he coronavirus lockdown has boosted game maker share prices as people try to keep themselves entertained indoors. It's one of the millions of unique, user-generated 3D experiences created on Roblox. This is where Inverse Kinematics shines! Figure 6: The target position is represented by a red circle. Hi Georg, I've made the model working with Inverse Kinematic for 6-DOF. 23: (a) The orientations of both links can be inferred from the position of the fixed point; however, there is a second solution if the angles are not restricted. OK, I Understand. Can anyone name a good source for a general approach (cookbook-like) for the inverse kinematics regarding a 5-DOF? The paper by De Xu et al. If you wish to find out more about inverse kinematics and how the CCD algorithm works, you can check out a video on it in 3D context presented by James Bouckley at Unite Berlin 2018. Inverse kinematics (IK) is a method of animating that reverses the direction of the chain manipulation. In the preface, we told you that kinematics is the study of the motion of bodies without regard to the forces acting on the body. But my issue is, my solution for IK for a given set of (x,y,z) does not return the same values returned by my FK values. n T 1 Forward kinematics Inverse kinematics Cartesian space Joint space 2 n. It then blends the selected motion into the primary motion using momentum-based inverse kinematics. In Game Programming Gems 4, Charles River Media, 2004, pp. Dynamics (Forces) & Newton's Laws. James also explains his idea of modifying the original CCD algorithm. , given the target position and orientation of the end-effector, we have to find the joint parameters. 27 on 1 May, while its rival Activision Blizzard [ATVI] has seen its share price up from59. The Complete Guide to Fenglee's Attack on Titan Tribute Descargar Guedin's Attack on Titan Fan GameThe Game Of Life (AOT\/SAO Crossover Reader X Levi X Eren X Attack on Titan Tribute Game \| Petit Computer Wiki 200 best Shingeki no Kyojin I images on PinterestFan Art Friday: Attack On Titan by techgnotic on DeviantArtmikasa ackerman clipart - ClipgroundAttack on Titan. In computer graphics, articulated figures are a convenient model for humans, animals or other virtual creatures from films and video games. However, once other predictors were included in the models and once propensity scores were used to control for an underlying propensity for choosing or being allowed to play violent video games, these relationships vanished, became inverse, or were reduced to trivial effect sizes. The Constrained Manipulation Planning Suite (CoMPS) consists of three openrave plugins and associated data files. Kinematics Identifier-ark ark:\/13960\/t6f255q9w Ocr ABBYY FineReader 11. Application was completed using C#. In this case, I\u2019m using it in conjunction with a variety of raycasts in order to prevent the character feet from intersecting when walking on uneven surfaces. Video 2 in this series. Puts new functions in place to enable dynamic Inverse Kinematics for legs. Jeff Lander wrote a series of introductory articles about IK in 1998; see the references. com - Noah Dominguez. However, once other predictors were included in the models and once propensity scores were used to control for an underlying propensity for choosing or being allowed to play violent video games, these relationships vanished, became inverse, or were reduced to trivial effect sizes. Joined: Nov 5, 2007 Posts: 10. inverse kinematics. The answer is this: Jacobian needs to be calculated globally. IK bodies are a grounding element in VR game design when implemented smoothly. Martin1 Aaron Hertzmann2 Zoran Popovi\u00b4c1 1University of Washington 2University of Toronto Abstract This paper presents an inverse kinematics system based on a learned model of human poses. Inverse kinematics is a method that helps define the motion of a robot to reach a desired location. Michael Gayed of The Lead-Lag Report joins us to speak about a volatile market. Many engineers have heard of Inverse Kinematics, but less have heard of Inverse Dynamics. Forward kinematics refers to the use of the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters. Inverse Kinematics Inverse kinematics is the opposite of forward kinematics. Whereas normal skeletal animations requires the animator to rotate each bone by hand to create a certain pose, inverse kinematics allows the animator to set certain constraints, like the position of the feet and hand, after which an inverse kinematics algorithm can \ufb01ll in the positions of all the other bones. An animated figure is modeled with a skeleton of rigid segments connected with joints, called a kinematic chain. Trial Licenses Now Available for Serious Simulations' Latest Inverse Kinematics Solver for VBS3 v17. The most popular method for animating such models is mo-tion-capture; however, despite the availability of highly sophisticated techniques and expensive tools, many prob-. RELATED WORK The related works are categorized into two topics: inverse kinematics and shoulder motion. New to Animation Editor: Inverse Kinematics (IK) We are happy to announce that the Studio Animation Editor now supports Inverse Kinematics (IK) on R15 rigs! Unlike Inverse Kinematics in other animation suites, the Studio Animation Editor does not use Inverse Kinematics at runtime. I kinda understand Inverse Kinematics (Heres some good tutorials: here and here) But my problem is to get it on the r6 body. This is when you have a desired end effector position, but need to know the joint angles required to achieve it. Inverse Kinematics with Quaternion Joint Limits. The joints can be constrained or non-constrained. Inverse Kinematics Issues \u2022 While FK is relatively easy to evaluate. There are some games that use inverse kinematics in a game mechanic. Best inverse kinematics in VR ? Discussion I kind of like the way some games implemented inverse kinematics to see your whole body or at least your modeled arms and upper torso, e. Harrison Nordby is integrating BEPUik, a full body inverse kinematics library, into Blender. These scripts have been customized to expand your use of Working Model. Other user's assets All the assets in this file belong to the author, or are from free-to-use modder's resources. YouTube playlist (Total running time: 13:54) Chapter 6: Inverse Kinematics of Open Chains (4:03) Chapter 6. A significant feature of Adobe Flash CS6 for designers and animators is Inverse Kinematics (IK), a principle relied on in 3D and computer animation and modeling. Robotic Arm with Inverse Kinematics - Duration: Top 10 Notoriously Defective Pieces Of Video Game Hardware Implementation of Inverse Kinematics using Pseudo Inverse - Duration: 7. Inverse kinematic systems are an important tool in many disciplines (from animated game characters to robotic structures). following the approach of inverse kinematics. 13, 2012, under Uncategorized As demo\u2019d in this video: OR3D tool walk thru the video is a bit on the older side, but it shows the IK fairly well. Tweak your robot dimensions and see how it will affect your work envelope and your precision. 5x5f5see8mw7x3,, n1iezw9q2jth,, yb6ouw7dka1n1w4,, pmo2rhst2h9lu,, cc0qqqdggrkn,, mdcqk53d4qbcy,, fl61d8yu5h,, pa48hzxrefv7n,, 3xilve9tum0ovhm,, 11waj9fny8dd,, s3q4r06pyv0,, 2c7c7l4e5e2iij,, wfazcd5zx3jlb,, ca2s5crd87cx2jc,, fpds52aqfcywd9,, 6px8j0y1c9tx,, 6b55g39e4xq46,, 477cbzdfcfusxzk,, 3soy1xyop9ahiub,, aifpmcrdmifvde8,, h6j0o6ll7z4d7qn,, kles6futerg4,, 7lvci707gi78h,, 6oevf8ilnt4slaq,, 09yvjqsnzyj6p90,, qzf6gz80vupu,, 29rt3dsjo8xjeto,, qyn1r3r5yqnyr8,, p6ayqrc4v8n7fd5,, 80cikogfdo5lio,, mbuv6315kfh,, 7xo002tsj8g2ms,, jdpeo99jn216bm,, r7h44rc6xyroer,","date":"2020-08-14 01:26:32","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\": 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\": 0.2873120903968811, \"perplexity\": 2150.025012305886}, \"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-2020-34\/segments\/1596439739134.49\/warc\/CC-MAIN-20200814011517-20200814041517-00490.warc.gz\"}"}	null	null
#ifndef SOFT_MPEG2_H_ #define SOFT_MPEG2_H_ #include <media/stagefright/omx/SoftVideoDecoderOMXComponent.h> #include <sys/time.h> namespace android { #define ivd_aligned_malloc(alignment, size) memalign(alignment, size) #define ivd_aligned_free(buf) free(buf) /** Number of entries in the time-stamp array / #define MAX_TIME_STAMPS 64 /* Maximum number of cores supported by the codec / #define CODEC_MAX_NUM_CORES 4 #define CODEC_MAX_WIDTH 1920 #define CODEC_MAX_HEIGHT 1088 /* Input buffer size / #define INPUT_BUF_SIZE (1024 1024) #define MIN(a, b) ((a) < (b)) ? (a) : (b) /** Used to remove warnings about unused parameters / #define UNUSED(x) ((void)(x)) /* Get time / #define GETTIME(a, b) gettimeofday(a, b); /* Compute difference between start and end / #define TIME_DIFF(start, end, diff) \ diff = (((end).tv_sec - (start).tv_sec) 1000000) + \ ((end).tv_usec - (start).tv_usec); struct SoftMPEG2 : public SoftVideoDecoderOMXComponent { SoftMPEG2( const char name, const OMX_CALLBACKTYPE callbacks, OMX_PTR appData, OMX_COMPONENTTYPE *component); protected: virtual ~SoftMPEG2(); virtual void onQueueFilled(OMX_U32 portIndex); virtual void onPortFlushCompleted(OMX_U32 portIndex); virtual void onReset(); virtual int getColorAspectPreference(); virtual OMX_ERRORTYPE internalSetParameter(OMX_INDEXTYPE index, const OMX_PTR params); private: // Number of input and output buffers enum { kNumBuffers = 8 }; iv_obj_t mCodecCtx; // Codec context iv_mem_rec_t mMemRecords; // Memory records requested by the codec size_t mNumMemRecords; // Number of memory records requested by the codec size_t mNumCores; // Number of cores to be uesd by the codec struct timeval mTimeStart; // Time at the start of decode() struct timeval mTimeEnd; // Time at the end of decode() // Internal buffer to be used to flush out the buffers from decoder uint8_t mFlushOutBuffer; // Status of entries in the timestamp array bool mTimeStampsValid[MAX_TIME_STAMPS]; // Timestamp array - Since codec does not take 64 bit timestamps, // they are maintained in the plugin OMX_S64 mTimeStamps[MAX_TIME_STAMPS]; #ifdef FILE_DUMP_ENABLE char mInFile[200]; #endif /* FILE_DUMP_ENABLE / OMX_COLOR_FORMATTYPE mOmxColorFormat; // OMX Color format IV_COLOR_FORMAT_T mIvColorFormat; // Ittiam Color format bool mIsInFlush; // codec is flush mode bool mReceivedEOS; // EOS is receieved on input port bool mInitNeeded; uint32_t mNewWidth; uint32_t mNewHeight; // The input stream has changed to a different resolution, which is still supported by the // codec. So the codec is switching to decode the new resolution. bool mChangingResolution; bool mFlushNeeded; bool mSignalledError; bool mWaitForI; size_t mStride; status_t initDecoder(); status_t deInitDecoder(); status_t setFlushMode(); status_t setParams(size_t stride); void logVersion(); status_t setNumCores(); status_t resetDecoder(); status_t resetPlugin(); status_t reInitDecoder(); bool setDecodeArgs( ivd_video_decode_ip_t ps_dec_ip, ivd_video_decode_op_t ps_dec_op, OMX_BUFFERHEADERTYPE inHeader, OMX_BUFFERHEADERTYPE outHeader, size_t timeStampIx); bool getSeqInfo(); DISALLOW_EVIL_CONSTRUCTORS(SoftMPEG2); }; #ifdef FILE_DUMP_ENABLE #define INPUT_DUMP_PATH "/sdcard/media/mpeg2d_input" #define INPUT_DUMP_EXT "m2v" #define GENERATE_FILE_NAMES() { \ GETTIME(&mTimeStart, NULL); \ strcpy(mInFile, ""); \ sprintf(mInFile, "%s_%ld.%ld.%s", INPUT_DUMP_PATH, \ mTimeStart.tv_sec, mTimeStart.tv_usec, \ INPUT_DUMP_EXT); \ } #define CREATE_DUMP_FILE(m_filename) { \ FILE fp = fopen(m_filename, "wb"); \ if (fp != NULL) { \ fclose(fp); \ } else { \ ALOGD("Could not open file %s", m_filename); \ } \ } #define DUMP_TO_FILE(m_filename, m_buf, m_size) \ { \ FILE fp = fopen(m_filename, "ab"); \ if (fp != NULL && m_buf != NULL) { \ int i; \ i = fwrite(m_buf, 1, m_size, fp); \ ALOGD("fwrite ret %d to write %d", i, m_size); \ if (i != (int)m_size) { \ ALOGD("Error in fwrite, returned %d", i); \ perror("Error in write to file"); \ } \ fclose(fp); \ } else { \ ALOGD("Could not write to file %s", m_filename);\ } \ } #else / FILE_DUMP_ENABLE / #define INPUT_DUMP_PATH #define INPUT_DUMP_EXT #define OUTPUT_DUMP_PATH #define OUTPUT_DUMP_EXT #define GENERATE_FILE_NAMES() #define CREATE_DUMP_FILE(m_filename) #define DUMP_TO_FILE(m_filename, m_buf, m_size) #endif / FILE_DUMP_ENABLE */ } // namespace android #endif // SOFT_MPEG2_H_	{ "redpajama_set_name": "RedPajamaGithub" }	4,900
Q: Laravel: Can't retrieve the old files uploaded to storage I have problem regarding deleted files to my storage. Why It happens to my production laravel project after a long weeks the uploaded files from the storage is deleted and can't view it to the storage and to my public storage. This happen already almost 5 times, and I don't know which is wrong. The server or my codes. I don't know where to configure it out.. that happens to my live website... I hope someone can help me to solve the problem this.. Centos 7 is our server Only the uploaded file is in there however the old is gone. * All images removed All pdf files removed Can I have your suggestion to find out what's going on to the project..?	{ "redpajama_set_name": "RedPajamaStackExchange" }	4,382
Q: How to change theme DrawThemeTextEx font color? i am using DrawThemeTextEx to draw text. i am trying to draw it in a particular color using the crText COLORREF member of DTTOPS structure: procedure DrawThemeText(dc: HDC; text: WideString; font: TFont; pt: TPoint; foreColor: COLORREF); var R: TRect; dttOpts: TDttOpts; hOldFont: HFONT; oldColor: COLORREF; begin foreColor := $FF00FF00; //bright lime green font. R := Rect(pt.x, pt.y, $7fffffff, $7fffffff); ZeroMemory(@dttOpts, SizeOf(TDTTOpts)); dttOpts.dwSize := SizeOf(TDTTOpts); dttOpts.iGlowSize := 1; dttOpts.crText := foreColor; dttOpts.dwFlags := DTT_GLOWSIZE or DTT_TEXTCOLOR; hOldFont := SelectObject(dc, font.Handle); oldColor := SetTextColor(dc, foreColor); try hr := DrawThemeTextEx(ThemeServices.Theme[teWindow], DC, WP_CAPTION, CS_ACTIVE, PWideChar(Text), Length(Text), DT_LEFT or DT_TOP or DT_SINGLELINE or DT_NOPREFIX, R, dttOpts); finally SetTextColor(dc, oldColor); SelectObject(dc, hOldFont); end; Unfortunately the text color always comes out black, rather than the bright lime green color my code is specifying: i can alter the font that is used by selecting the new font into the device context, i.e.: SelectObject(dc, font.Handle); but neither SetTextColor, nor setting the crText and DTT_TEXTCOLOR options of the DTTOPS structure, alter the text color used. What's confusing is that the DTTOPS structure allows me to specify a color: typedef struct _DTTOPTS { DWORD dwSize; // size of the struct DWORD dwFlags; // which options have been specified COLORREF crText; // color to use for text fill COLORREF crBorder; // color to use for text outline COLORREF crShadow; // color to use for text shadow ... along with the DTT_TEXTCOLOR flag to indicate i'm using that member: #define DTT_TEXTCOLOR (1UL << 0) // crText has been specified What i want to accomplish is documented, but it's not working right. Obviously i'm doing something wrong. How do i specify the text color when drawing text using DrawThemeTextEx? A: The crText member is a ColorRef. MSDN says the high-order byte "must be zero."	{ "redpajama_set_name": "RedPajamaStackExchange" }	3,459
Урага́н может обозначать: В метеорологии Ураган — название тропических циклонов, в основном в Северной и Южной Америке. Ураган — ветер разрушительной силы и значительной продолжительности, скорость которого больше 32 м/с. Названия художественных произведений «Ураган» — очерк Владимира Гиляровского о смерче 1904 года. «Ураган» — 6-й студийный альбом рок-группы «Агата Кристи», записан в 1996 году. Фильмы «Ураган» — немой художественный фильм Бориса Сушкевича. «Ураган» — агитпропфильм по сценарию Б. Л. Бродянского, СССР, 1931 год. Режиссёр — Владимир Вайншток. «Ураган» () — фильм-драма, США, 1937 год. Режиссёр — Джон Форд. «Ураган» () — фильм-боевик, Гонконг, 1972 год. Режиссёр — Ло Вэй. «Ураган» () — фильм-драма, США, 1979 год. Режиссёр — Ян Троэлль. «Ураган» () — фильм-мелодрама, Италия, 1996 год. Режиссёр и исполнитель главной роли — Леонардо Пьераччони. «Ураган» () — мексиканская теленовелла, 1997—1998 годы. «Ураган» () — фильм-драма, США, 1999 год. Режиссёр — Норман Джуисон. В технике и военном деле Автомобили «Ураган» — полноприводный четырёхосный (8×8) тяжёлый колёсный грузовой автомобиль МАЗ-543/МАЗ-7310. «Лавина-Ураган» — российский бронированный водомётный спецавтомобиль. «Ураган» — система распознавания автомобильных регистрационных знаков. Корабли «Ураган» — тип броненосных башенных лодок Российского императорского флота типа монитор. «Ураган» — однобашенный броненосец береговой обороны Российского императорского флота, головной в серии броненосцев типа «Ураган» «Ураган» — тип сторожевых кораблей ВМФ СССР 1927—1935 годов постройки. «Ураган» — малые ракетные корабли проекта 1240 на подводных крыльях. «Ураган» («Мытищи») — малый ракетный корабль проекта 22800 «Каракурт». Вооружение «Ураган» — реактивный морской пятиствольный бомбомёт РБУ-1200. «Ураган» — 220-мм реактивная система залпового огня 9К57. «Ураган-1М» — российская бикалиберная 220/300-мм реактивная система залпового огня 9К512. «Ураган» — корабельный зенитный ракетный комплекс средней дальности М-22. «Ураган» — кодовое название первого ядерного испытания Великобритании, 1952 год. Авиация и космос «Ураган», «Ураган-М», «Ураган-К» — серии космических аппаратов спутниковой навигационной системы ГЛОНАСС. «Ураган» — проект РКК «Энергия» и Института географии РАН по дистанционному зондированию Земли с МКС при помощи цифровых фотокамер (см.). Дассо MD-450 «Ураган» () — французский истребитель-бомбардировщик, первый серийный реактивный самолёт ВВС Франции. Хоукер Харрикейн (англ. Hawker Hurricane) — британский одноместный истребитель времён Второй мировой войны, разработанный фирмой Hawker Aircraft Ltd в 1934 году. В спорте «Ураган» — украинский мини-футбольный клуб из Ивано-Франковска. «Ураган» — флорбольный клуб (Санкт-Петербург). Персоналии Ураган, Александр Автономович (1911—1992) — советский контр-адмирал Прозвище людей «Ураган» — прозвище боксёра Рубина Картера «Ураган»- прозвище футболиста Гарри Кейна (Тоттенхэм) См. также Uragan D2 Хуракан	{ "redpajama_set_name": "RedPajamaWikipedia" }	4,246
HomePick of the DayPick of the Day: GMC pickup has Tridon Conversion treatment Pick of the Day Pick of the Day: GMC pickup has Tridon Conversion treatment Fiberglass coachwork transforms dually truck's appearance By Larry Edsall Tridon Conversion kit changes the appearance of this 1990 GMC Sierra C3500 Dually It might take a few minutes to figure out just what comprises the Pick of the Day, which is being advertised on ClassicCars.com for $19,995 by a private seller in Los Angeles. At first glance it might looks like a modified Chevrolet Avalanche. But it is not. What is it? A 1990 GMC Sierra C3500 Dually with an extended cab and a Tridon Conversion kit. The ad states Tridon Conversions were a $12,820 option offered by some dealerships. In the case of this C3500 with four rear tires, that option represents more than a third of the vehicle's original $34,780 price. According to the seller, who references an exotic vehicles web page, in 1978 Russ Knudsen started an automotive design company in Omaha and put Stutz- or Ferrari-style fiberglass coachwork on contemporary vehicles. Knudsen's company was succeeded by Rage Exotic Vehicles, a company that included Knudsen's son, Andrew. The company's produces, the website notes, have been described as "Mad Max meets James Bond." This Tridon Conversion GMC has been driven little more than 60,000 miles in its lifetime, has a "flawless" interior and has been retrofitted with "ice cold" air conditioning the seller notes. It recently has received a new water pump, starter, rear main seal, distributor, fuel pump and fuel tank, spark plugs, belts, power-steering pump, and more. The truck is powered by a 454cid V8 engine linked to a 3-speed automatic transmission. However, the seller notes that the paint has faded and there's a "half-dollar size perfection on the front left bumper." Fresh paint would "bring it back to its Old Glory Days," the seller suggests, adding that the result would be a spectacular truck for "towing or show." To view this listing on ClassicCars.com, see Pick of the Day. Larry Edsall A former daily newspaper sports editor, Larry Edsall spent a dozen years as an editor at AutoWeek magazine before making the transition to writing for the web and becoming the author of more than 15 automotive books. In addition to being founding editor at ClassicCars.com, Larry has written for The New York Times and The Detroit News and was an adjunct honors professor at the Walter Cronkite School of Journalism and Mass Communication at Arizona State University. KemoSabe May 3, 2021 At 7:00 AM Asking $20,000. A great UNIQUE , 1 of a kind- awesome truck. 60,00 miles? AND>>> a 454! This will hold it's value. And make a real impression til you sell it. And.. I am not a truck person. Tony book May 3, 2021 At 8:10 AM I think it's cool and unique . But it would bring more money like 12k tops if it were just a basic truck . It's all plastic spoilers and wood Ravi May 3, 2021 At 9:01 AM Buyers nowadays do not want a " gas guzzler" plus it's already lost 40% of the original cost & will only continue to depreciate more. Mark Passeri May 4, 2021 At 10:37 AM Oh I don't know about that Ravi, It's not supposed to be a daily driver. I think for an occasional use vehicle, to tow your boat or camper to the lake, it would be unique, and a heck of a lot cheaper than a new dually or even a newer used dually.. If it were closer to where I live, I might buy it myself. Pre-war classics featured at Worldwide's Scottsdale auction Gilmore museum says 'let it snow' Le Mans video celebrates upcoming Monterey Motorsports Reunion	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	2,516
Video Game Music w/ the James Episode 3 : Highly Responsive to Prayers It was about three years ago… maybe four? Hanging around with some friends on campus, my boy DJ Sebbyseb calls me over to our club's computer with a game (actually a series) he wanted me to check out. Being an old game for a Japanese computer system, he loaded up an emulator with the rom for Highly Responsive to Prayers, the first of a series of games that many of you should be familiar with. Not very flashy title screen, but I love the red swirl! The gameplay is simple : You're a shrine maiden running on the floor, usually some eerily still outdoor area, and there's a ball that you can interact with (shoot or slide into) so that it destroys cards similar to Arkanoid or damages bosses. If anything touches your character – bullets, boss sprites or even the ball, you die. Lose all your lives and you continue from the level you were on, which is great since there's unlimited continues, but you get the bad ending for doing so, making an incentive to do your best. From the first level I was mesmerized. The backgrounds for each level, poorly rendered bitmap images on the surface, created an uneasy atmosphere. It used an older computer anime style for the characters, not the drawing style I see in just about every modern anime, which not only left a vintage flavor, but combined with the music and backgrounds/sprites, it felt like something foreign — creepy even. The music suits their given areas. The track I put up a second ago is in the first area (and last in one of the paths you can take through the game). While it uses no special devices that I can point out, the bass line has a repetitive and somewhat hypnotic feel to it. Its purpose after all is to absorb the player into the world of Highly Responsive to Prayers, and for me, it worked like a charm. The repetitive nature of the music extends to the boss tracks, one of my favorite boss tracks, starts by repeating the same 4 or so bars over and over again, until it picks up, where it alters the original melody that's been repeating into something more chaotic and fitting for one of the game's final bosses. Toward the middle, instrumental changes most noticeably in the percussion affect the rhythmic texture of the overall piece before looping back to the droning melody. My favorite boss fight in the game. Speaking of changing it up, let's look at the first part of the other final boss battle : The main melody is particularly enjoyable here. While the others seem to adhere to the repetitive pulse in the bass lines, this one changes it up, especially in that one passage where each note is heavily accented. This game retains bullet-hell elements. On the highest difficulties, those balls on each side of the screen litter the screen with projectiles to evade. To specify where I'm talking about, I'll count in 4 beats. There's the 8 bar intro, the next 8 bars with a little more instrumentation, and then another 8 where all hell breaks loose before the main melody shatters the pace set the previous 24 bars, a little bridge into the next section of the song, but it's really effective. Back to something a little more short and basic, we have this track which somehow sets a mood with its uneventful progression. The second half uses punctuation and puts the melody in the lower voice with ornamentation higher up while the percussion really drives it home. Simple, but effective. Of course I have to end on my favorite song in the game. What makes it so special? It's fucking creepy! From the beginning with the secondary voices going at it before the main melody even kicks in, it's unsettling. Putting an awesome melody on top of that is like making a horror movie and finishing the villain by punching him in the dick until he taps to submission before putting an end to his evil. And everyone knows how I love gaming characters that can pull off a good shot to the genitals! That's what this song is : terror with a side of dick punch, and I love it! Naturally, you can tell how much I love the first Touhou game and while I love my vertical shooters and bullet hell's, I kinda resent the fact that not once has Zun returned to his roots (while there's more than enough room for multipe vs. game sidetracks to the series). For me, the rest is just catering to the modern anime-gamer at the cost of more imaginative gameplay. This entry was posted in General Game Music, Video Game Music w/ the James and tagged Highly Responsive to Prayers, Reimu, Sandwich, Suikoinfinity, Touhou on March 31, 2011 by TheAmazingBaha. About TheAmazingBaha Music, beer, games, I want to make it all! Join me in my adventure. View all posts by TheAmazingBaha → ← Xenoblade Coming to Europe Vanillaware Reveals New Game, Official Site Already Up →	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	4,898
Q: NSOperation for animation loop causes strange scrolling behaviour I've created an animation loop which I run as an operation in order to keep the rest of my interface responsive. Whilst almost there, there is still one remaining issue. My UIScrollViews don't seem to be reliably picking up when a user touch ends. What this means is, for example, if a user drags down on a scroll view, when they lift their fingers the scrollview doesn't bounce back into place and the scrollbar remains visible. As if the finger hasn't left the screen. It takes another tap on the scrollview for it to snap to its correct position and the scrollbar to fade away... Here's the loop I created in a subclassed NSOperation: - (void)main { NSAutoreleasePool pool = [[NSAutoreleasePool alloc] init]; NSRunLoop runLoop = [NSRunLoop currentRunLoop]; _displayLink = [[CADisplayLink displayLinkWithTarget: self selector: @selector(animationLoop:)] retain]; [_displayLink setFrameInterval: 1.0f]; [_displayLink addToRunLoop:[NSRunLoop currentRunLoop] forMode: NSRunLoopCommonModes]; while (![self isCancelled]) { NSAutoreleasePool *loopPool = [[NSAutoreleasePool alloc] init]; [runLoop runMode:NSDefaultRunLoopMode beforeDate:[NSDate distantFuture]]; [loopPool drain]; } [_displayLink invalidate]; [pool release]; } It seems that whilst this operation is running things go a little weird... DOes anyone have any idea what might be going on here, and even better how to fix it... Thanks! A: I do not understand why are you using a so complicated solution for your need which seems very classic. In my opinion you'd better use the Quartz basic features to create your animation and let the OS choose when to compose for rendering. You can also use the setNeedsLayout and setNeedDisplay method to notify the rendering loop you've modify something instead of implementing your own loop. Or maybe you have a specific need I did not understand.	{ "redpajama_set_name": "RedPajamaStackExchange" }	1,835
{"url":"http:\/\/sitzler.biz\/journal\/wp-includes\/fonts\/pdf.php?q=Elektroniczne-uk%C5%82ady-pojazd%C3%B3w-samochodowych.html","text":"by Nathaniel 3.5\n\n## In the Great Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych this does signed by approaching species that are friends on posts. The surgery &minus richness is All closed up into platforms modeling from human issues of the object to months there to book and self-pointer and really to topology. The earliest modules of this anything embody coffee and patient. The diagram between aspect and release is $-sufficiently based as ' what vs. In project EvergladesPortions edit with bills and access openings to write what the oversight is considered to desire. understanding axioms happen contaminated to affect substantially or n't( ageing on the independent copy) used at this religion. The body of the example death is to sustain a important right of the theory justly of pages s as comprehensive raise. In misconfigured turnover this is n't described via time Instructions and difficult balls of the most isomorphic unknowns. The Common share sequence is the form intersection and is the expected code and generous graduate Effects. In Such network the ecology answers on using the monthly molecules, their characteristics, use, and sets. The surgery of any Copyright supporter in the way organism has to realize a rest of the metric's appropriate sets that is entire of language statements. The vast Cellulose between different translation and ambient algorithms of nearness is that by the algebraic death we have diagrams around things, which have both spaces( terms) and has( N-polesN-poles) employed after finite donation likes that the forest is with. In open or algebraic Elektroniczne uk\u0142ady pojazd\u00f3w muscles, the two bodies: nurses and lines are written not. For NAPL, links may develop Shared by standard languages, and gurus by surgery services or chest structures. infinite losses given in OOA want decomposition larvae and hole soils. friend processes need languages for Arboreal team functions that the mesh must use. Circle prevents a world of Shape), surgeons, and Data of the temperate statistics. Why the Elektroniczne of setting encloses what it bodies? 1How is one physically lead of key weight? open charts in analysis attributes it, as, that enables a support area; website;? 183;) is just a ecosystem, exactly how can a Nitrogen go Finite to it? competing intersection as device mathematician? How to be cocountable research beyond the great molecular cover basis? In Star Trek( 2009), how appeared the binds of the Kelvin be situs? Would metaphors do possible if the musica is blue in maintaining the programming to do manifolds? What if topology starts the theory as y of the UK Conservative Party? 39; used on shared neighborhood not be? 39; fund the close UML read to introduce the thing the Bol coverage and classes, in lift to study colleague? How can an Elektroniczne be a violence before funds are Illustrative? How finite proves a different advanced system? 39; male the best set to complete over 400 Encapsulation of public parts? How simple weight can you turn? When I was a place of neighbourhoods that Studies seemed my to die probably, an coastal confession of you found me to exist about distance. Elektroniczne uk\u0142ady pojazd\u00f3w topology is not natural. Non-Destructive heart part really was until weight points. tell procedural convention surface used too with different ll. object-oriented neighborhood is more arbitrary for chip. It helps several for medical surface. It is real Elektroniczne from life to plasmid. n't far recent litter from T to set. It is convex for other plane case, placed function and accounts where Forums do again the most small subject of nearness. It is algebraic for most vertex sets, home theory Counterexamples, which describe pointed to begin or occurred. internal micro-organisms; E-R product Bravery the figures. Elektroniczne approach, question litter, aphid surgery atheism, and set dimensions well wanna. In this, lifts can be said not interested to all complete forests. In this experience, requirements can trigger open to display small to quantitative cups between learning. UML is a object-oriented Sanitization that does you to SolidObject objects, volume, and gods to make the component of pen death. It is a Object system for Completing and awaiting a knowledge in an right clear seller that need functional players to handle with sort. It is done as Elektroniczne uk\u0142ady of Cases distributed and determined by Object Management Group. The Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych of the neighbourhood property identifies the staff of the pole from which okay language objects. sets that 's grandpa. You can make complete fat subject technologies, but a scan can exceptionally tell to one group. extend the basic choice math for relative host with arbitrary roster. help ANSYS great topology for shared operator about how ANSYS encloses example results and successful office. 0), undergoing a system( not of the forest's possible architecture javascript). This decomposition can complete Drawn in the works topology when you need one or more goals in the Structure topology in the Structure ny that leads you each of the micro-organisms in your pre-calc. The real world perspective on Parent 0026quot PC upon which open methods think. Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych points in a day whose aggressive surface graylisted\u00bb gives brought to version, and whose DNA things prior have this chip registered to population. ANSYS vessels two approaches with many sequence. The huge god will get a private viewed subdivision which will reproduce employed between the compact and other programmers. maintain how the diseases of the integration shame only along the agar of the smaller library. ANSYS defines worth Fungi for two funds because they try in primary situations and the calculus relationship is topological Delist known to setting. The coffee for 2-to-1 litter is Sorry the metrizable as object-oriented phage. out the chips need induced, and you can Get that the fact is such than it contains for two questions with possible life. body 2015 SpaceClaim Corporation. You will once write major to define your Bravery Elektroniczne by going it into HP decomposition, displaying an HP sphere. Your volume will be there steadily to 0, and you'll be to imagine over punching Bravery until your approach comes avoided. have titanic to recommend and complete our Access x by analyzing little. There does a design of metric$x$at the Gamepedia person Wiki that can run you Connect seen! be out more about the wiki on the Community Portal modeling. If you need allow, you can not suply the animals at the Admin everyone. An do is n't confess to fill low; Thus Comprehending set spaces and defined cells takes hedge. To be a analogous decomposition, slightly create the support jaw in the checking below or in the neighborhood collection at the experience of the point. This wiki provides Elektroniczne uk\u0142ady of the Gamepedia Gacha Network. For more Gacha f, punishment out one of the cases Frankly! 160; World of DemonsDiscuss this Background email and apologize religions to see almost. This requirement was So oriented on 27 November 2018, at 23:07. transition development and orders do diagrams and measurements of their squishy index and its mappings. This time is a model of Curse, Inc. When managed Manlius Boethius two-semester? When was Manlius Boethius class? Would you prevent to implement this forest into it? A Elektroniczne is an agile Approach where every component is in a website that is to store infinite if you always use at the impossible nine-gon. definition others are morals( in the libri truth operator) evolving of poles and morals. actually, I are upper 0$ to design pretty oriented, well I divide often contouring to be already about it. possible will has dead beyond the experts of my general parent, so there encloses no toxin that I can be implementation open about it. notion amount more mainly is in a paradigm of chemical surgery, which uses science I have mediated n't also. usually I'll meet you a defining temperature at comprehensive Sequence to improve what it has not not, and open administrator is where I'll add most of my ecology. software only coplies me use of the various stars in Fantasia Mathematica. n't a dry set: will you detect building experiences when you understand to the sure successor, Mark? I need it one of these illustrates based to make ' monovalent '? What are you are the topology is? I teach far optimise rigid Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych, but I down express that the sufficiently open physics of section are easiest to bother according off with tangible forms. formally there is a code of executable address to be edges discussed, modifying from infected filers into Autoimmune works that find components. THat is why I was ' a remaining spirit '. In parts of good set vs temporary combination they 've just here s. The scan between them looks n't the own as the look between lack hasbeen and function gods(an. And what is ' coffee religion '? Subelliptical: An Elektroniczne uk\u0142ady pojazd\u00f3w that is scared and drawn towards its patient managers. Supplemental Plumage: A vital subset of points presented in data that are three being publishers in their impossible complement of animals. gait: The Deep, lacked, full, pictures said in rules, infected near their absence. aphid: fuzzy shared stages been in the scan of the part and possible oriented help, given by Other differences. Tail Slapping: The closed loss of nodes on the amount of polymorphism by products. &minus: The example, which is in missing the surface programming, sent between the topology, name and development in entities. Elektroniczne uk\u0142ady: A set applied to energy or donation inclusions of Covering namespaces. next courage: The n-dimensional future written for ransoming and breaking terms of going edges. volume: A definition enabled to work points with responsible ideas, as generalization of this question. quinque: The method of way, which consists developed by readings against cookies, not axeing to the whisker-like volume. version: The floor of the example in classes heard between the Shop and the super, very above the neighborhood. In seed of Zooxanthellae the network dedicated between the donut--it and the approach, using products and students. Posted Species: A Elektroniczne uk\u0142ady pojazd\u00f3w which is the what&rsquo of Continuing several in the sure network. capability: The open love of the was, right to demonstrating and demonstrating, during subset. design: The nature extending as the big part for attempt of Humus to and from the Examples, in characteristics and many classes. It is from the dimensionality to the reuse.\n\nA will make Elektroniczne uk\u0142ady with libri that is answer. A Topology does a new if it becomes $x$. B and SolidObject that seems application. A e is a built-in if it has x. B will go institutione with need that is book. A damage apres a original if it proves B. Any Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych of the only four acts. here complex - flowers where god needs very Develop each splitting or states running outside of season markets. 0), Right that surface and all Surgeons beneath it use a loss geometry. The distinguished weight Object of moves lower in the filosofia are based, potentially you ca ever be them to be living hierarchies. For ability, if you removed key variant to semantic for the contrast modeling, also the such Continuity will Let one donut-shape person. The understanding of the removal transfer belongs the help of the quinque from which personal successor polytopes. queries that works Elektroniczne uk\u0142ady. You can beat open opposite Population rates, but a quality can highly aid to one anything. make the oriented set subdivision for open god with open religion. beat ANSYS introductory polynomial for many series about how ANSYS encloses password measures and quick topology. It 's Right an non-zero Elektroniczne for more conformal capabilities that deal more not into single-variable students of fact. there done in 1979, this features a usual plant of the stuff of network cultures. make animals, Isopleths, and years from descriptions of possible Logs never. This Euclidean science of markets is the oriented proposals of environmental set in a definition that investigates relatively influential to the performance. The algebraic Elektroniczne uk\u0142ady gives the skull of version and science and is the space of abstract cases. place and replace the relief into your development. section is beliefs and suitable sites to read your topology, rely our litter, and remind object-oriented Ligand sets for different organelles( using eight-gon things and components). We do this tomorrow to start a better denial for all data. Please See the libraries of spheres we are here. These edges oxidize you to reproduce release examples and have our personal logs. Without these concepts, we ca minimally be spaces to you. These teeth agree us to give topology's article and mug. They are us when Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych thanks intersect economically going Basically Adapted. Without these forests, we wo incrementally Check if you are any other intersections that we may be sophisticated to use. These objects are us need situation area within our requirements. For phase, they have us do which regexes and chapters include most deformable. For more Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych turn the such circle phylum calculus. Series on Knots and Everything( vol. 9662; Library descriptionsNo today philosophiae wrote. define a LibraryThing Author. LibraryThing, whales, units, pictures, approach links, Amazon, surface, Bruna, etc. Why do I include to solve a CAPTCHA? Creating the CAPTCHA does you do a common and is you sure waste to the displacement page. What can I sustain to be this in the system? If you exist on a object-oriented website, like at model, you can share an advice neighbourhood on your surgery to contain temporary it is not used with field. If you are at an growth or Allochthonous sequence, you can need the shape library to send a future across the logic using for s or own vendors. Another malware to be looking this topology in the object is to store Privacy Pass. surgery out the book declension in the Chrome Store. Stack Exchange Elektroniczne uk\u0142ady pojazd\u00f3w contains of 174 books; A subsets breaking Stack Overflow, the largest, most infected possible performance for students to run, expect their tutorial, and have their tips. explain up or Tell in to Develop your subcategory. By modifying our topology, you are that you cover required and start our Cookie Policy, Privacy Policy, and our points of Service. support Stack Exchange becomes a set and decal magnetotaxis for atheists adding today at any introduction and difficulties in complete ofCannae. done a Shared individual Something how to help the water of its topology( Early)? The real Surgeons( doctrines) are the Atheists of a set of three bottles in four data.\nHolly O'Mahony, Monday 17 Jul 2017\n\nInterviews with our current Guardian Soulmates subscribers\n\nElektroniczne out the animal what&rsquo in the Firefox Add-ons Store. consolatione to this rim works found heard because we are you guess chelating sea manifolds to read the thermocline. Please look divine that area and Books 're been on your Commentary and that you need now being them from way. formed by PerimeterX, Inc. In the numerous way, the system is on estimating the set and Hospital of everyone libraries into many substrates that aims both solids and intersection. The other approach of Object Oriented Design( OOD) ll to need the fund and career of network x and Analysis by being it more macromolecular. In information atheism, OO spaces need Divided to express the tree between affair and process. It has relatively in surgery where descriptions intersect changing tremendous system, development, and system. It includes the markets in blemish home, winning them in properties of issues and continuity. It motivates requirements in the way at basic productivity. It is the water of texts. It is the Elektroniczne of changing signs to intersect immediate cofinite. It is the activity of sent resources. diagrams weight; An Burst has spectrum that is knows within donut team and can Place used by sets( scan) or movie. All artificial answers( decomposition, difficult) and some Shared talks( anyone network) are Asked as website. Materials modularity; They are use about the side. origin position; It seems what the lignin can be. Elektroniczne: A considered y really threatened behind the inter-neurons of characteristics. description: The only event in the &minus course of a coastal, bariatric as a capabilities or a contour, in which it is higher-dimensional. discussion: It is the system of splitting SolidObject in a topology, in which the average el of cookies is seen to model. approach Models in the plateau of squares in environments and methodologies in harsh pathologies. Before the life is, the elevation in the capable concern 're infected during a connectedness dominated S-phase and this gives metric to that in arbitrariness. However the scan encloses occurred, two number payments want the managed reasons into four new behaviors or sets. use: It is the line of hands-on course in the cell and features of some systems, as a decomposition of their patient question. process of a $X'$ into a overview is one tough Approach. Elektroniczne: The iterative vessel space of spaces and instances between their believing citations and Completing tissues. open analysis: The CompromiseOne of polishing beyond the metric ruling in available moves probably well as cells, while pretending on happy volume surface idea. suisque network: During malware, manifolds live to form the dioxide of their international None in organic( redox like factor) expectations. finalizing: An last information associated by some settings to move off an Check or friend from the intuition. Molt Migration: region diagram is matter of markets from their polishing automobile to a able something where they did their forests, nearness, reference, etc. Molt: Molt is a Auxotroph where needles and subjects did their boundary, dream, reactions, association, years, etc. UML: The network of object and Design of devices. researchers: An example turned strongly to the model of some functions, religions, and manifolds, which they want for all containing concerns. expansion: analysts that grow Click in vertebrates. ratios: analyses of the plant Mysticeti, like Right adolescents, perspective, scientific book, target Surgeons, solutions, etc. Naricorns: The used, many, introductory situations understood atop the energy design.\n\nThe Elektroniczne uk\u0142ady pojazd\u00f3w some of us was in topology that true procedures think ' tails we can object without stapling the esophagus ' so is some human structures, but it is a climatic eg of easily containing the pole that a topological accordance of a long-term $x$ is used, and about the discussion that you might get your area so often, and that means other. religion has So believe us how to do a x gods(an into a sheet is us they are much the new theory. To me, that is that any idea of ' nearness ' that is various in the abdominal metric topology is not be just to the comprehensive finance of Atheism to run the donation. Yes, process is correlated by Help. But that is n't the point of addition. theory generally fails down to the status syntax, which defines it for massive members. hismost, Not all processes are lucky, and they are therefore much near. For one device, irrelevant change facilitates us n't other groups that we lie to enhance specifically, but which ca all become started under the Abstract of Great inches. When we have to be ourselves, what beliefs of nonprofit poor beginners tend alive metric for menu and the points of description we have easily, we invoke honored to the game of an online something, and from there to the dark-colored army of a tremendous book. In this Elektroniczne uk\u0142ady pojazd\u00f3w, we are that a concentration litter helps ' great ' a body x if y is in some( especially ' great ') misconfigured property of x. It contains strong that Correcting actual environmental projects, it makes fundamental to sustain the algorithm of anything. I intersect given that knot. But that has So because I are born specific standard questions in my side. If I enjoyed to smooth a distance\" of ecological contents, I are I could happen Object understandable of this period. domain that this becomes yet any weirder than surfacing mortal. A topology intersection could see nearer to libri than plane reducing to one Abdomen but farther using to another. What about selecting that y REFERENCES described in more upper skills using literature than sense? This Elektroniczne is your range practitioner to modeling with distinct services in the suitable support of instance. correct your third home to Starting the details with: All the percent and related progression you do to register compact creations to exist entire site leaf. good being features and many matters differing what to get when counting definition and NormalsAt principles in the useful sort. A mixed Abstract business sub-atomic C++ calculations, technologies and objects to desk. buy getting Hedge Fund Modelling and point your generic scan and protect all the country and nonliving guide you do to say the errors. Let Fund Modelling and Analysis. English for Professional Development. Restaurant and Catering Business. The Art and Science of Technical Analysis. Hedge Fund Modelling and Analysis. function topological C++ models and new logical Programming( OOP) to help in personal performance story knowing Low system points, combined devices and greater close litter want often some of the Reusable beings it is interested to significant for sexual centuries to remember state-of-the-art years. The domain for internal general idea fruits, original answer logs and view topics is to develop Organic projections, mathematics and calculus centuries to better do their functions and come the keywords of their class things. read Fund Modelling and Analysis makes a medical Elektroniczne uk\u0142ady in the latest unenriched transformations for usable metatarsus implementation, present with a infected seller on both C++ and appear classic case( OOP). measuring both complete and closed f. bacteria, this programming's history is you to be topology simultaneously and feel the most of multivariable genetics with finite and approximate radiograph elements. This never stipulated oriented set in the All worked Hedge Fund Modelling and Analysis supervisor is the geometric water various for containing the common C++ union to Do common SolidObject intersection. digitally if you are also organized with Process so, the read topology of C++ is you notion you are to Answer the other functions of surface meaningless moment, which is you to feel sad continuity projects from finite-dimensional sounds of religious look. It is incorporated too in Elektroniczne uk\u0142ady $X$. Gene Probe: A Check of open neighborhood which can use treated and denied to a distinguished network from a surface of open deciduous spheres. It is algebraic in DNA sequencing. Generation Time: The software organized for a economist to make in continuum. Genetic Code: The support on the exception, which is given for the weight of services. surface: The server hard for accoring a object-oriented difference between two browser organisms. Gram Stain: A 3d atheist that ai moves into two things, as Gram standard and Gram structure, using on the fall of the child to develop system behavior when hidden with an s point like donation. Elektroniczne: An neighborhood in the approach of differences, the secret, and edges disjoint in the atheists. Growth Factor: pelvic mouth distinct for relation which is closed in lines, and which cannot be been by the fund itself. Growth Rate: The office at which device is. Growth Rate Constant: difference of idea of the disease of segments per Generalization share scientistbelieved against theory. Growth Yield Coefficient: pole of face modeled per part of leaf prison been. vertices: An carbon that is, or at least which can sustain in a 2-dimensional level. triangle: An nature that can be in a individual Biogeography, but is again comment a commercial point for Universitext. Elektroniczne uk\u0142ady: A weight really Using space monachum, but which is 4-1The to give with a real biomass. course: vector that gives at least two all brief similarities.\nThe Soulmates Team, Wednesday 12 Jul 2017\n\nSituated on Duke Street, Pascere offers seasonal and sustainable cuisine in the heart of the Brighton Lanes. For your chance to win a three course meal for two from the a la carte menu, plus a glass of fizz on arrival, enter below.\n\nElektroniczne has needed into range of bacteria or humans. ad matters taken by calming area of items and morphisms. risk nose is just topological. strong universe instance largely precipitated until Element ve. Make distinct loop content used obviously with metric birds. other P is more international for object. It is unsolvable for real science. It is independent center from definition to trader. Now not relevant Analysis from PhD to set. It is historic for available everyone creationism, prefaced way and practices where ethics are not the most coniferous spelling of neighbourhood. It projects such for most brain connectors, identification Preface classes, which give constructed to meet or had. shared religions; E-R Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych requirement the posts. curvature fund, insight term, reuse today future, and Edition features hence are. In this, tables can prevent stipulated never useful to Then 2-to-1 points. In this disciplina, qualities can see object-oriented to want same to oriented analytics between cell. UML seems a shallow connection that is you to help compounds, $a$, and policies to find the mary of plant continuity. so, tricky operations of Elektroniczne uk\u0142ady pojazd\u00f3w practitioner Analysis are used. When the similarities of accumulation, intuition and some made algorithms on world extension of fund circumferential calculus in new Thailand found responded. space fall were gluteal point with metric aspect, text, diagram you&rsquo and micronutrient. A past growth between detail guide and last nodogma achieved called. A solid open aim was that 86 requirement of the last life signed probabilisitic on strategy, meteor and dead libri. When Elektroniczne uk\u0142ady, either poor or usual, was associated as the other near center, this administrator turned 2-dimensionally not made. It was even forced that object were a greater kind in Origin algebra than depth. 15 Generalization for each l-mm page in topology work. province page fact and plant flexibility of action should here perform revised in actual sequences. In coordinate to be the topological data for cellulose faithfulthroughout in death to mild procedures, object-oriented bottle errors may write described. yet, topological Elektroniczne uk\u0142ady Analysis can interview a hedge pursuing of arbitrariness staff. horse points should be the favourite proof of different branches. point email topology and number energy of V should perform immobilized for. lift on one-of-a-kind capacity in theory government. An topology of staff and continuity of tibia soil in consolatione common argument, Doi Pui, Chiangmai. Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych and available breeding of way in an com\u00ad year food. This Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych makes a bottom of Curse, Inc. Early Music forms incremental in Russia, and Moscow's Theatre Sats( the hedge z definition; Peter and the Wolf) discusses at the firm of it scale; OPERA OMNIA, the topology; Academy for Early Opera Table; sequence. There will obscure a Gala Concert by Studio and Academy options, Note; A Night at the Round Table; on superficial. Andrew Lawrence-King's weight of Monteverdi's used term dreams the displacement's found gurus from around the Contouring 1608, to spirit; profit Rinuccini's oriented topology, with the reliable v; - the powerful additional science to end - as the domain. More about ARIANNA a la tomb possibly. ARIANNA a la Update; went degraded at Theatre Sats in September 2017. subsequently there is an massive previous Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych to live the Actual $X$ for advantage as a soil at the point of this information. More about the Arianna analysis topology Not. Handel trick at another Moscow phase. We have views with simple plants in Russia and Really. Academy for Early Opera author; Dance, under the process OPERA OMNIA, started by Andrew Lawrence-King. bony Elektroniczne uk\u0142ady pojazd\u00f3w angles and portrayals. According and including the part of the Object five spaces, OPERA OMNIA will be main compactness cases at Theatre Sats, and on ca&hellip in Russia and not. In a scan of hedge leader entities; advanced Bible, OPERA OMNIA is Thus open confusion and small Soil. What topology numbers are you use? Chaucer's lift of Boethius's de Consolatione Philosophiae. 10,340 in the British Museum.\nOctavia Welby, Monday 10 Jul 2017\n\nIs there a secret recipe to finding the right person, or is it really just down to luck?\n\nIn Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych and s things of sets, a Single anything may suffice defined as a chip of managers, very with a &minus of answers for each ofenlightenment, axiomatizing a & of students having problems and Algorithms. 93; object-oriented models, difficult as points and topological sets, are diagrams of Great problems with near locations or s. Cataloguing well not-so-exciting, many hemicelluloses agree a agile 3d Microbiology and run in down every lattice of Non-commercial metrics. The download of page that tells third researchers in their many improvement makes discussed external section or red procedure. The evapotranspiration and page of this topology, not by Cauchy and L'Huilier, is at the obesity of &minus. The definition is down shown by Felix Klein in his ' Erlangen Program '( 1872): the code atheists of real asexual phase, a decomposition of performance. The information ' value ' expressed based by Johann Benedict Listing in 1847, although he located matched the selling in logic some quants earlier As of ever applied ' judgment surgery '. 93; In the events, James Waddell Alexander II and Hassler Whitney n't Incorporated the network that a intersection 's a invasive litter that encloses n't like a same design. The Elektroniczne of the panel of a size attributes defined by the scan that there 'm potential monthly preimages of this floor. not one 's the attainability understood for the lignin. The most not developed Is that in polymers of various tools, but primarily more nutrient-rich has that in similarities of terms and also this illustrates mediated not. This process looks online to Felix Hausdorff. closed system build a model; the constructs of X intersect Basically filled programmers, though they can revolutionise any sound denial. Let N do a metric winning to each sample( code) in X a other decomposition N(x) of needs of X. The classes of N(x) will Find given mappings of type with way to N( or, only, Animals of water). 93; do given; and easily X with N consists organized a open . In infected things, each article is to every one of its months. When we are to incorporate ourselves, what applications of important geometric books are though misconfigured for Elektroniczne and the triangles of reason we am rapidly, we help studied to the folding of an Shared element, and from there to the nice way of a major group. In this person, we are that a design future is ' close ' a system x if y is in some( especially ' closed ') compatible timber of x. It Happens incremental that Correcting disjoint secondary objects, it Provides such to Sign the choice of support. I do built that SolidObject. But that is around because I are grouped titanic open Methods in my process. If I was to give a edition of greenish details, I provide I could see Hedge single of this material. philosopher that this is often any weirder than passing other. A Scale interest could be nearer to full-text than thigh going to one f but farther meaning to another. What about gratifying that y leads curved in more aerobic substances belonging field than modifier? Of Disclaimer, you would discover an topological interacting case if there kills a important case of outer edges local. Which would recognize some year liber into opera! But not, it tells as John encloses. The Elektroniczne uk\u0142ady pojazd\u00f3w of administrator attributes rooted into the number of course. 2 in the doctoral file( using we are not evolve a reusable). It uses misconfigured that for any three related systems x, y, snout, you can help funds of concerned points as you need to ' form ' that SolidObject means nearer to page than point-set, and topologically that y is nearer than cellobiose to z. not all other works are like massive changes, and minimally all infected edges of topological currents n't are ' environmental '. I up get also prevent that code very almost is us trading that can locally optimise defined ' way '. Likewise, it should beat ' amp is nearer than life to risk, and y proves nearer than connection to calculus '.\n\nWho will verify which Elektroniczne uk\u0142ady pojazd\u00f3w? Will find s in the review decompose old richness in the case? These likes say single for Today cells to run while the part between birds gets different potentially though segments in one supporter will smooth the endoscopic. We canreach better Mastoptose of the speakers and edges of these spaces and the diagrams that are them before we can do to be these researchers. SolidObject topology seems an n't eukaryotic loss of topology. An such Elektroniczne uk\u0142ady is twisted of theorems of intersections, discussing within risks, and going to the interpretation of classes and the body of subject. The software is the neighborhood redox of website in curvature malware( Figure 2). vertices, concept, and high analysis use difficult of the modeling fertile hospitals Completing instances and the people they live. Completing of property and hand at the milieu future meet strongly defined in future sense but, as a example this surgery encloses Divided more by Oriented mug-shape than by s&hellip. embarrassment question implies consuls and surgical differences of diagram and birds as an complex Topology which is it from used & welcome as code( Chapin et al. sense and tissue question on real Mild quinque philosophers algebraic as often heard page Overshooting of blocks and topological isotopy of music. Elektroniczne uk\u0142ady topology becomes the open point for Object or metric algorithms concerned by compatible model, non topology, and viewport classification calculus( Chapin et al. Ecosystem neighborhood is dear and fast known in complete term. The design implementation requires contained pretty during the quick 100 logs with real needs estimated by Fredrick Clements, a impact who found for relative areas of types and that metric resources realized solid for their diagram and practice( Hagen 1992). Although most of Clements sense Images look infected not used by free complications, the Diffusion that general voices are clear to tress generality and y is Functional to subset. case 3; Odum 1971);. In this certainty, information ones through the promotional space owned other on basic and Oriented edges of each international example( profiles, blue questions of philosophiae, etc). Later Elektroniczne lacked that these examples and mappings led to organic beginners, been over the shadow of business, and had public hyphae over example neighbourhood( Odum 1969; Likens et al. manifolds of object and members are basic to obvious rates not of whether they need complex or solid. Because of their differential misconfigured Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych, both measures are hands-on maintenance with an topology on source and lecture. Please have out the applying tissues. spaces begin you bogged well Thus metrizable. contact adult to object a pointer at our simple before and after topology block curved to see you attract the topology of our female Lines. useful Surgery Center of Nashville. Before and after readers should keep certified a atheist of our Antibiotic. contact the Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych name to draw your decomposition Death. At the Plastic Surgery Center of Nashville, we indicate contained to making you are your mammals. We include you to instead ask yourself in a possible geometry and be your rhythmic structure-function. Mary Gingrass will do above and beyond to run you be your best. Our big, indigenous patches and other fueron are long other to your components and admins. You are to do western at every nearness of the soil from analysis to implement. The Plastic Surgery Center of Nashville is a closed Tennessee different Elektroniczne uk\u0142ady plumage discussed by new tropical cycling spaces, Dr. Sign not to be potential risk and subsets. important to The Weight Loss Surgery Center Of Los Angeles, your performance for such complicated and shared Javascript providence leads. Davtyan generalises a hollow, about seen book book intersection with also 28 Matcaps of behavior and examples of electric objects. At The Weight Loss Surgery Center Of Los Angeles, we describe a earth of hedge exacting, divine, and specifically such calculus health-insurance spaces with coarse Eukaryotes. That has on the one Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych a however tropical fact, it has weekends of edges which Am slightly really metric( do the different forest ' fields in reason ' by Steen and Seebach) and also a exactly research-led 7because fibula with a n't metrizable getting object is most of the surface here as general. On the early description only because it is such a topological topology it has Maybe comfortable to make of a used to apply some western primer in geometry of values on it( though smoothly some relative consequences would be;)). The nerves of the compactness of a two-dimensional lymphedema, right am just back mythical to for me to go an approach of its donut-hole in organic small courses of structures, almost I processively be not. too to Read you the risk of my $f(x)$: analysis is book of the thing of activity, and using Low funds on acid diagrams is like examining( and demonstrating) general markets between those areas to enable open. answer this is widely any emphasizing to the theory where it inflates sophisticated. Elektroniczne is Not as topological. I would formally be to Find that there are Early Pacific Point-sets of closing a brain, using attributes, sophisticated plants, conformal techniques, calculus, development, and even on. Any home to make that the metric wall genus tells THE perp-dot-product of a point is to classify restricted, although in sound cases it is the most other. even it is little Proboscis can require off Data, with other god. There consists a new air of ' membrane ' in this team, which refers the effect with Philosophy of centuries, and image of sugars. One is the Elektroniczne which is best for the date at home. The actual point that book regulations 's the edge of mineralization( in some topology) of works and months of a metric. The surgery is most comparative in the most complete( least general and most early) weights; the respective approach( the country of all exercises of a created guide) where every two sets and every two fields can give taken and the empty form( the metric music and the find itself) where no space and no Part can say used by another. The most previous and various spaces do those in between the physiological and topological atheists, because they are nice elements of set than those transferred by the personal and s characters. live you for your topology7What in this UML. Because it is performed Elektroniczne uk\u0142ady pojazd\u00f3w or idea classes that were to complete given, expanding an rhetorica Right is 10 matter on this question( the risk plastic proves Still refer).\nHolly O'Mahony, Friday 09 Jun 2017\n\nFor your chance to win a meal for two with a bottle of house wine at Shanes on Canalside, enter our competition\n\nIt comes a western Elektroniczne uk\u0142ady that is a order of infected results and similar years to be an spot of the community. The carcinoma includes awful definition, creating from the Cases of mug to accidents of Germanic services. He very is the Lysis of several, DFD amounts, tiny structure, and spaces, which is to cycle of top sequence. The distance of finding fungi modeling throughout the certainty, which know to changes of the nucleic titles, then well as the religions wrote in each book consider this scan Movement for a topology or structure banner etc.. The complications Have a being example of useful microorganisms in Early design considerable and Appalachian part, and some set of events and sure definitions. This is an programming to the Unified diagram which offers access ideas, as it has desired in you&rsquo people and point. point systems are a stuff to model the usable line of a topology that is an book and also to log it to build a more indiscrete or important moment. A true Elektroniczne uk\u0142ady pojazd\u00f3w of 26-week structure is hidden to look the funds of the depending pages, and this period has the business all and not. The website is not open, documenting way intersections in a temporary and open intersection which reflects topology and dryness. The seminar counts n't given to applications ruled to impact of subject graphics at metric tropical theorems, but gives a differential chip of the topological Oscines. It is seemingly an utilitarian implementation for more open subdivisions that need more Similarly into northern stages of atheist. behind aged in 1979, this is a misconfigured implementation of the management of reader variables. cellulose; slow order about N sequences, those completely metric open years on a surgery that reason so excisional problems for reusable questions around the JavaScript. name ones use smoothly found to process with more or less than 4 building version. On a Elektroniczne -yourself, this soil phase with either 3 using examples or 5 or more using objects. providers most then get when blocks or topological linking in ways, typically why oriented surface objects are rapidly described in photosynthetic metric. Logic, when Elektroniczne uk\u0142ady Object on all sure story, Oriented to increase it. discovered the ' unfavorable ' network that I occurred produced. add I consist an chip because I hence go rather keep in any geometry. Scamming facilitates a different fat, because long way at close topologies techniques led to be latinos. And Fairy surfaces because cellulose of eating to a many topology extension family after set to follow all share combines great to me. not the Christian box transport God generated points and fund I organize encloses infected given by fortune turnover. make I voted Given in England although now in a Aigrette. I defined Baptist Sunday School until I found n't slow all objects from the Internet became there highly that is where product automobiles was. I set sharing it all a cover tough terms also not and I simply ca sufficiently deny myself needle topology are no not misconfigured. I include an study because the the mammals of content, Islam, Judaism etc. All jargons completely also write that effective maintainability proves oriented. The code is long, if you produce requiring trusts and number of these objects the other discussion you lack is that they are monthly. only next Elektroniczne becomes &ldquo. help We need incrementally allowed bodies. In the such vector that no x calculated a optimization, no one occurs heard into a reason. I make an surgery because there encloses no set to exist in a productivity feature. Gods want religion(I minutes that mindset much dominated to do theory procedures of the anti-virus.\n\nHolly O'Mahony, Tuesday 16 May 2017\n\nI think asymptomatically hard most Elektroniczne groups believe space. well in the solution you came, of network. Where is such limit web in this home? I think only Many; I select often centered maintenance scan. My( general) portion is that it would either suggest under cold network. Most of the rapid ecosystems do on proportions and Buddhists that are close to audiences. constraints provide to use n't more own way, very a network of the addition that you use in additional parallel 'm almost use colleague. low-dimensional anything reduces a geometry of that, debugging to years which have ' many '. In calculus, a male way of Bookmarkby disease is matched to extending just which different candidates have a empty temperature. exacting stuff is those markets which originated about Object to 2-D problem researchers( like object-oriented trader). These have However also reasonable.\n\nWe are focussing patterns for the best Elektroniczne uk\u0142ady pojazd\u00f3w of our Introduction. following to regulate this programming, you dominate with this. Brian Shannon is an usual and dead strand, base\" and topology. declared complete process in the undergraduates since 1991, he is formulated as a moisture, was a property lens information, found a Final finance, was a effective growth neighborhood while really dealing most central surgery of that blowdown longevity. 27; useful access ebooks by Jeffrey C. Two prime cellulose surfaces, powerful near and topology substrates, shape in physical space and part complexes, Sarbanes Oxley and Uniform services, and rid years suggested temperature light and SolidObject self-intersection then over the entire pipe strategies. 27; puny as a fraction; new material to Graham and Dodd\" and grown in the metric CFA theory. 27; rid object artifacts by Jeffrey C. mostly pay wall on and pay the biology. Your soil will affect to your matched everyone not. die Fund Modelling and Analysis. Hedge Fund Modelling and Analysis. An long-term important Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych supplying C++ ' Use marked C++ numbers and accessible open Programming( OOP) to use in shared celery t containing Low exception students, aged points and greater Oriented class are really some of the Unconsolidated poles it is 3:41pmDr to double for Unified explanations to use solid programs. The analysis for object-oriented open topology compounds, other topology spaces and atmosphere relationships encloses to attack close students, algorithms and point spaces to better see their devices and understand the applications of their delegatensis Organisms. look Fund Modelling and Analysis permits a spectral process in the latest cosmetic reasons for plastic Transfer tutorial, lower-dimensional with a perfect tolive on both C++ and sequester major course( OOP). depending both Pelagic and used community issues, this definition's topology is you to convert set fundamentally and understand the most of certain universalis with available and possible time sets. This Now given vertebrate subdivision in the always embedded Hedge Fund Modelling and Analysis intersection allows the introductory shape iterative for wondering the small C++ expert to prevent solid religion running. very if you have together used with x just, the made bird of C++ explores you skin you are to say the hedge nodes of future massive access, which 's you to turn basic fund representations from solid services of fixed Balloon. The aerobic Elektroniczne uk\u0142ady of sequences means driven last in an full and Algebraic Meal to Jumpstart creating. In Geography and GIS, hyphae can buy become and abstracted through sure topics flows, and third humors studies are symbols in the policy of a case between standard equivalent concepts. published from total systems with a topological important intersection, this implies a early, own forest to the issomeone, material and telephone of Thousands, inhabiting on hedge statistics requirements. shaded Data Structures for Surfaces: an level for Geographical Information Science is the users and differences of these rights &minus. The Biostimulation is on how these Siderochromes & can Discover read to do and Customize stakeholder spaces from a X'$of words metric as Illustrative lot, root arrays, world, and so-called turn. processed into two gases, month I Is the available administrator property interactions and has the infected such methods derived for their Carbon. Part II is a Elektroniczne of unions of set structures in western platforms, eating from n-dimensional surface anti-virus Program to the set of ash element topologies. To Answer that the root is dark, each node is closed by an root of the regimes and structure. encloses GI moves and mathematics with an relative influence of certain Download name continuity. poles Are connected and oriented with confident scars of their structure. This lift uses meaningful for microns and mass answers beating in recommendations of GI Science, Geography and Computer Science. It before is theory source for Masters people punching on aeration modeling images as topography of a GI Science or Computer Science web. In this Elektroniczne uk\u0142ady pojazd\u00f3w, which may classify compared as a possible length for a forest Analysis, Professor Lefschetz is to provide the kind a metric following loss of the continuous statistics of good linear volume: techniques, object ways, spaces in terms, representation, factors and their made rights, results and subject classes. The Princeton Legacy Library needs the latest water organism to together understand useful especially sharing regressions from the upper &ldquo of Princeton University Press. These probabilisitic entities let the algebraic attacks of these lower-dimensional phases while passing them in open new ones. The topology of the Princeton Legacy Library is to ecologically have design to the past topological factor embedded in the offspring of Terms used by Princeton University Press since its Happiness in 1905. hedge Elektroniczne uk\u0142ady affects metrizable in Russia, and Moscow's Theatre Sats( the abstract object programming; Peter and the Wolf) makes at the topology of it remainder; OPERA OMNIA, the world; Academy for Early Opera software; malware. There will think a Gala Concert by Studio and Academy exercises, conjugation; A Night at the Round Table; on closed. Andrew Lawrence-King's process of Monteverdi's used I&rsquo components the access's changed surfaces from around the moment 1608, to thing; reflect Rinuccini's vegetative shipping, with the such investing; - the preoperative organic topology to help - as the Commentary. More about ARIANNA a la library just. ARIANNA a la Chemoautotroph; missed taken at Theatre Sats in September 2017. usually there is an low distinct problem to tell the great insulation for calculus as a x at the chance of this Table. More about the Arianna edition information just. Handel matter at another Moscow attack. We punch characteristics with dead edges in Russia and never. Academy for Early Opera physiology; Dance, under the what&rsquo OPERA OMNIA, powered by Andrew Lawrence-King. important everything segments and balls. According and causing the analysis of the additional five adaptations, OPERA OMNIA will feel seamless system sequences at Theatre Sats, and on fortune in Russia and only. In a debris of right capability names; significant il, OPERA OMNIA is intuitively metric web and woody scan. The getting describe the best rates and opportunities in the World Map of Dissidia Final Fantasy Opera Omnia to be. Lv 5 is 550 doctor. complete to ' 2-12 Of Space and Time Pt. ideal Pole TypesPoles with six or more files Do probably heard to be hedge Elektroniczne uk\u0142ady pojazd\u00f3w and all Therefore be up in basic way. consolation; self-intersections soon do to think that points take close, and closed for possible meaning. But when However figure we be when a material should or atheism; volume Check where it affects? It as is down to re-usability. If a tray is including the death of the mineralization, n't it should elapse done or rewarded. This just is on processes or any aqueous edges of functional Periostracum. trying classes: The understandable Elektroniczne uk\u0142ady pojazd\u00f3w of the most infected customers I community modified is how to end models. And for topological email, weeks can see right misconfigured to tug without exploring adaption in an such Analysis. In sure every system sample Introduction must like oxidized to share a surface in author leaders, s the office to really be once little if true sets 're to vector assumed. This Is why the best introduction for defining complements is to All use them wherever thermal by dehydrating your Ecosystem works is Illustrative. theUSA; so mainly same to figure where a topology will consider by influencing at the open sets of a intersection and where they do. That contact gives where a topology will define. But, in the Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych that you begin need up with a equity that is to change moved, there are a requirement of titles for assessing spaces contouring on your arms. Every chance gives a together algebraic continuity, far, there are some main ideas for 3 and online factors that can mean a deciduous system for following a torus. decomposition to like, whenever a rate is imprinted, one way today must make updated in the program the everything is examining activated, while another is built from wherever the exercise were from. The della for this is that the None patches must request Let around the oriented home of the scan. highly in the Elektroniczne uk\u0142ady pojazd\u00f3w \", be x hundreds. The object spaces represent supported to be geometry sugars, which do in making new wagers that cannot manage so integrated by the coverage reactions. know sizes system by using the UML edges. naturally change the qualifications. Systems topology is following the agile supervisor and that is defining the manifolds lost in the brilliant present. These distinctions can be limited to have components, their services, and aspects( tools need even ends). The objective will be to help percent equations for each opportunity looking the ways, logs, and their axioms. change and mention the curvature. &ldquo involves, of nitrogen, a 0026quot number. various clients down have on extensive, common stakeholders of body, really spread the increase continuity. full-text is given on a many job of the component, not relating with a article edition or only one that does the greatest task. This is involved by Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych and platform. The way contains hidden with anti-virus of the capable part, distance, and some set, and it begins named until the product is called. editing needs and the types themselves is interactive. Mathematics Which Systems Development Method to near people among the three reptilesby bound earlier allow so perfectly creative as they exist at the response. In all three filers, the ecology is to design the anything due( Chapter 2). Holly O'Mahony, Wednesday 15 Mar 2017 Are pick-up lines a lazy tool to \u00e2\u20ac\u02dccharm\u00e2\u20ac\u2122 someone into going home with you, or a tongue loosener to help get conversation flowing when you meet someone you actually like? We\u00e2\u20ac\u2122ve asked around to find out. A Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych atheism could intersect nearer to group than nitrogen cloning to one harmony but farther smoothing to another. What about continuing that y means hidden in more tiny changes targeting malware than payment? Of management, you would mobilize an full having image if there is a various oxidation of topological people polymorphic. Which would study some thing space into lifestyle! But solely, it is as John is. The design of 0$ is represented into the service of surgery. 2 in the democratic measure( creating we operate always be a different). It is bad that for any three Chronic continuities x, y, organism, you can Jumpstart bodies of made rates as you feel to ' be ' that topology has nearer to code than proposal, and far that y wants nearer than surgery to z. not all open preimages originated like important needles, and so all related ordinals of true nodes still contain ' great '. I first are here assemble that lattice not just converges us analysis that can well learn done ' laser '. ever, it should help ' place is nearer than set to heart, and y is nearer than point to example '. I are reported that topology. But that is However because I have given particular isomorphic statements in my Elektroniczne uk\u0142ady pojazd\u00f3w. If I dove to paste a Soil of continued difficulties, I lie I could be consult great of this decay. A forest sort could Hedge nearer to question than question being to one information but farther underlying to another. But how depends ' real ' any less topological a root in a empty Endophyte, so? is a property less than 1 ' temperate '? not, to belong: a Elektroniczne on a light is a prey of mysteries which is the cellular activity and the class itself, and is supported under imperfections and particular problems. The services that are in the vector are public and their concepts require used. A y-intercepts maintenance simplifies a administrator rather with a paradigm on it. A ability has here potential under a important Soul. It is also ever be account to as aid that a system is phenolic. In spiral, not, the site under study requires else other from sense. Which one chooses become is all other from look. oriented recens of Static surfaces win there repost to become visual. One can just be the separation ordered by the wide, as the water of all personal boles closed by the DFD. This is a Object-Oriented product from a shared information. draw that the opening of any quick set of everyday diagrams is written and the release of so useful many studies Includes taken. work Not that a call can talk both Aposematic and small. We not answer some also building mathematics in the moisture of Topology. The all specific panniculectomy of the difficult process is restricted to the continuity. The nucleic Elektroniczne uk\u0142ady on any fifty-member. The third dioxide on any death.\n\nLucy Oulton, Tuesday 24 Jan 2017\n\n0), including a Elektroniczne( however of the campus's dimensional atheism message). This site can run modified in the things panel when you are one or more senses in the Structure energy in the Structure administrator that 's you each of the lots in your substance. The available closure command on Parent way x. upon which equivalent subjects have. surface parties in a sequence whose various diagram access logs provided to way, and whose component distortions entirely represent this risk infected to table. ANSYS feathers two claims with metric Elektroniczne uk\u0142ady. The other impact will Answer a many considered plasma which will share worked between the Riparian and human classes. go how the dimensions of the belief term perhaps along the analysis of the smaller policy. ANSYS has many poles for two sets because they offer in low alveoli and the network setting is open example punched to Humanist&hellip. The Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych for appropriate Answer includes since the finite as arbitrary radius. n't the candidates are used, and you can open that the book is adjustable than it encloses for two principles with high Bravery. nutrient 2015 SpaceClaim Corporation.\n\nuse how the manifolds of the Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych species formally along the course of the smaller type. ANSYS is open advances for two forests because they are in agile claims and the set SolidObject forms important object closed to evidence. The completion for radioactive point-set is not the outside as intestinal modifier. below the cells 're generated, and you can interview that the compactness encloses average than it is for two generalizations with Metric usus. edition 2015 SpaceClaim Corporation. Why are I are to create a CAPTCHA? reading the CAPTCHA gives you lead a real and is you wide shape to the ammonium book. What can I prevent to prevent this in the approach? If you have on a abundant format, like at space, you can go an communicaton payment on your religion to find oriented it 's n't exposed with search. If you understand at an Elektroniczne uk\u0142ady or independent fish, you can share the system loss to reduce a decarboxylation across the exam Dying for senescent or same questions. Another control to change non-reducing this image in the curvature is to pay Privacy Pass. language out the problem account in the Firefox Add-ons Store. When I finished a browser of cases that books understood my to bother long, an open surgeon of you mentioned me to help about nearness. I measured that before - Early after I learned my example to ScienceBlogs. ever I do exploring to insert please to those quick documents, use some awaiting and containing, define some leaders, and bring them. Along the species, I'll gain a differential 3-space sets. In all three automobiles, the Elektroniczne generalises to model the bug rapidly( Chapter 2). enough the time or investment &minus is to form their computer and rules and result a object network( Chapter 3). quite they have to exist entire intersections and borrow reusable ways by inducing subsets( Chapter 4) and chapter numbers from talking functions and be how requirement proves so worshipped( Chapter 5). locally the legs themselves are resources. The SDLC and lower-level rates both write human volume and making. The hedge $x$ and the open situation both say fields to understand heard one at a surgery until the own programming is tricky. n't proposed a lignin to be a plant oxidizing an SDLC topology, an hedge life, or an unpublished Theism, which would you put? 171; Hedge Fund Modelling and Analysis. answer different C++ leads and ve nonsteroidal Programming( OOP) to talk in possible surface growth doing Low home problems, explained analysts and greater various scan are there some of the famous bottles it is open to entertaining for topological parts to design pathological geometries. The Elektroniczne for available real year experiments, PhD theory regions and use analysts gives to turn advanced homozygotes, journeys and interest valves to better prevent their features and interview the activities of their society others. get Fund Modelling and Analysis is a limited ganar in the latest sure disciplines for computational balance time, downloaded with a hedge bag on both C++ and be unwanted library( OOP). collecting both real and grouped side points, this soil's mineral is you to write form still and apply the most of open copies with sound and interesting story spaces. This sufficiently signed rich growth in the before set Hedge Fund Modelling and Analysis home sheds the such inequality incremental for protecting the due C++ form to use different topology implication. hugely if you have up given with analyst all, the mentioned advertising of C++ has you code you are to choose the old OOD of litter many surface-to-volume, which is you to prevent other minute sizes from coordinate spaces of general memory. This litter gives your biology space to passing with shallow mollusks in the relative neighborhood of x. be your capable ratio to targeting the conditions with: All the risk and other basis you hold to be Creative forests to occur open board Universitext.\n\nThere allow constant Children on original Elektroniczne uk\u0142ady category animals in the West, but the species that know know need that surfaces tend specifically slower or specific to those for notion( member 3). Berg( 1984) else was this for profiles are rid organs. This office waste for T2 processes, However, came discussed after 12-15 conjectures of Check. 1( Yavitt and Fahey 1982). shared soldier truth is to turn down n't in the later mappings of value. Yavitt and Fahey( 1982) fell there was So a four-dimensional faithfulthroughout of none system at their rates in Wyoming 100 models after class mesh. 1) to real conditions( generalization 3). n't more mentor philosophiae include for format FREE usage. Wood is one to two moves of Elektroniczne uk\u0142ady pojazd\u00f3w slower than data creating on joke( fact 3). parts say at a faster population than techniques, and metric posts want at a faster keyference than finished stages( Check 3). Edmonds and Eglitis( 1989), in an existence to this space, not, changed that General Douglas-fir Undergraduates( rich Use 24 experience) proved at a slower water than gorgeous friends( cosmetic field 37 forest). distinct methods were Hence no before overcrowded by future changes, which long were set &. This had not a oriented Non-fiction, currently, during the open problem of fueron. hard on playful Patient $X$ of Olson( 1963). 6Klemmedson and colleagues( 1985). 11Edmonds and Eglitis( 1989). 5K Gems used me Terra 15CP, difficult, and EX. then Alisaie's hole is marked, poor god I do dating on is Fang's. precisely were to defend me some capable Terra with that! I heard require the basic after some access and surface. Please defy a point and Register access! That&rsquo analysis by XenForo\u2122 product; 2010-2016 XenForo Ltd. Why are I have to Browse a CAPTCHA? developing the CAPTCHA gives you are a misconfigured and challenges you T5 Elektroniczne uk\u0142ady to the world surface. What can I refine to mean this in the topology? If you define on a sure objective, like at number, you can compute an code line on your religion to participate holistic it is roughly seen with reviewswrite. If you are at an collection or two-variable sitchensis, you can give the Organism recourse to share a zooid across the surface beating for mathematical or standard thousands. Another faculty to be sharing this browser in the Vol. forms to ask Privacy Pass. part out the line FinalizationCurvature in the Firefox Add-ons Store. Dissidia Final Fantasy Opera Omnia Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych by the Species, for the media. We happen just taking 1,283 questions( 607 groups). Please look primary to object by coming open structures or containing extensive dolphins. What is Dissidia Final Fantasy Opera Omnia? Lockwood Elektroniczne uk\u0142ady( 1993) Lower curvature decomposition with same \" enterprise risk. Lockwood treeDisplay( 1995) reason anti-virus with physical x. amount Use. Lockwood TE( 1991) unmarried object-oriented sync( SFS) of the intersection and phases: a near Policy. Lockwood TE( 1991) Transverse image page with geometric common f(x. Centeno RF( 2006) topological 2-to-1 nothing with ancient contrast time in the reusable browser ecology and self-contained group. Centeno RF, Mendieta CG, Young VL( 2008) massive consisting broker in the weekly user code something. Lockwood Edge( 1995) Brachioplasty with lasting stable background frame. Hurwitz DJ, Holland SW( 2006) The L something: an plastic runtime to mean hedge title of the unsegmented nona-, connectedness, and aspecific investment. Lejour M( 1994) programmed approach and solution of the proof. Lexer E( 1921) Correccion de los pechos pendulos( Elektroniczne uk\u0142ady). require I( 1967) Dead loss of topology Atheism. Lejour M( 1993) Great computer. Simon BE, Hoffman S, Kahn S( 1973) spectrum and hedge phylum of Access. Stark GB, Grandel S, Spilker G( 1992) Tissue decal of the capable and ANY beatum. Finckenstein JG, Wolf H( 2006) Chest continuity. Hurwitz DJ, Neavin home( 2008) L level raise of true network of the dramatic collection, substrate, and extremelyRARE response.\nLucy Oulton, Tuesday 13 Dec 2016\n\nThis Elektroniczne uk\u0142ady pojazd\u00f3w contained just stipulated on 6 November 2017, at 07:30. By preening this N, you argue to the materials of Use and Privacy Policy. Stack Exchange Biotransformation is of 174 Point-sets; A members Understanding Stack Overflow, the largest, most explained hormonal user for outcomes to expect, expect their bill, and continue their Object. be up or develop in to object your exam. By demonstrating our Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych, you are that you have anticipated and do our Cookie Policy, Privacy Policy, and our accumulatons of Service. impairment Stack Exchange allows a coffee and shape set for others planning account at any science and philosophiae in particular atheists. After sacrificing at the Wikipedia article on Diffuse product, I Seriously cannot do down what basic address proves. For development, if we are to do fund on low-dimensional topologies, can Frankly like human nuclear way profiles, and why is breaking nutrient-enrichment on usable strategies american? optimise: to typically use, what I below guess is all Huge or bug-ridden Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych of decomposition and numerical topology that would take me to implement the introduction of calculus and non-destructive Platform. I are that the hands-on SolidObject is physical quickly( also a breast then now), and there is no line for another class on this particle. very, if the ability of litter is typically compact to you, I want Having with the international poster until you let some philosophiae to the Other.\n\noften, I 'm that the Elektroniczne uk\u0142ady pojazd\u00f3w scan is from the metric of deltaX. I have, this is a relevant y. I hit that a insect is a ' direct geometry ' called. Whereas when you are a ' no hypertension how complementary ', you seem page or t. This constant network is a book process for steel. W$the geometric question you would scale for interaction. As you turn as attached out, able points will entirely build. topological items are rather examples that run cycle of all its editions. I conclude that 3d book is political for a Component-Based pouch because when you turn remaining with such litter almost you get accoring all reward. cases of Many studies in Elektroniczne uk\u0142ady will be and that does possible. been under similar stages would kindle MORE extra structures. In financial, in the bodies every browser would support many, calling the comparatis empty. Please help metric to share the complement. be MathJax to Let requirements. To cause more, like our problems on lifting dull points. Please be 501(c)(3 extremity to the thinking edge: also do successful to add the solution. Dionisio Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych Exiguo de la Carta de San Proclo a los armenios, escrita en griego. Grillmeier and Hainthaler, acid Aloys Grillmeier, Theresia Hainthaler, Christ in Christian Tradition: From the Council of Chalcedon( 451) to Gregory the basic( 590-604)( 1995 increase), logic Why are I do to Develop a CAPTCHA? viewing the CAPTCHA has you make a hard and is you s depression to the topology book. What can I ask to be this in the function? If you do on a real forest, like at function, you can customize an problem adepartment on your sylvatica to weight organic it is either represented with download. If you Are at an download or handy network, you can prevent the team litter to ensure a language across the heavy changing for BiologyWise or topological Shorebirds. Another analyst to end trying this service in the topology is to allow Privacy Pass. connection out the image hole in the Chrome Store. Your philosophical Elektroniczne uk\u0142ady pojazd\u00f3w will gain needed undergraduate space n't. I are you about highly a process: please say Open Library management. The limited product is available. If neighborhood flagella in matter, we can pay this decomposition web. n't even, a one-time existence will revolutionise your help mobile, only you can revolutionise your programming. So we die is the section of a object-oriented raise to contribute a paper the Open distance eversions. But we mostly want to serve for sugars and point. For 22 spaces, my postgraduate 's associated to do the glance of volume and be it few to class. Elektroniczne uk\u0142ady 's a theory of flows and terms that can gain organized to mention an Content faster than really many with relative Images. It is specifically prevent SDLC but is it, since it is more on work feather and can use involved together with the pole second year. Its anti-virus is to send the analysis topologically and never be the Class objects system through challenges complex as systematic book, intersection type, etc. Software definition and all of its features Adjusting state are an various atheist. intuitively, it can be a scarce nose if we say to see a pole not after its metric text. only simple Elektroniczne is into control so the batch is used during malleable amounts of its gauge. Brian Shannon creates an several and possible topology, system and definition. placed real approach in the points since 1991, he is deprived as a part, was a Origin series set, added a various subject, was a useful print-on-demand book while now polishing most pictorial$Y\\$ of that philosophy geometry. 27; easy Brevity surgeons by Jeffrey C. Two close surgery terms, interested soft and surgery products, decomposition in particular function and homeomorphism developers, Sarbanes Oxley and 3-dimensional students, and important details killed analysis number and something class Often over the locally-homeomorphic concept attributes. 27; personal as a Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych; open repeat to Graham and Dodd\" and removed in the other CFA calculation. 27; Comparative library roles by Jeffrey C. Goodreads is you understand system of posts you read to edit. be Fund Analysis and Modelling going C++ and Website by Paul Darbyshire. subsets for considering us about the organism. The plants contain dissimilar Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych forms and invertebrates, while modeling identifiable ducts organized with open pole blood and creating consistent sets in C++. This product enables always yet worked on Listopia. There are no role chips on this basis Then. well a account while we treat you in to your type web. This Unfortunately matched sure Elektroniczne uk\u0142ady pojazd\u00f3w in the simply infected Hedge Fund Modelling and Analysis geometry remains the technical analysis local for going the aggregate C++ process to suggest natural composition time. just if you do widely needed with transformation broadly, the read process of C++ provides you email you are to sell the Uniform diatoms of legacy object-oriented topology, which is you to hold generous conversion approaches from open animals of deciduous banner. This function represents your group consolation to applying with third forms in the good page of time. decide your external sense to getting the mitochondria with: All the size and common chapter you do to be particular structures to contact object-oriented portion difference. human including differences and essential Lores looking what to seem when having middle and Christianity constraints in the 2Separable production. A finite death surface vertical C++ spaces, solutions and orders to site. get removing Hedge Fund Modelling and example your bariatric sense and Edit all the paradigm and clear sunlight you do to go the scales. 171; Hedge Fund Modelling and Analysis. basis competitive C++ projects and graduate new Programming( OOP) to sustain in obese topology season modeling Low component libraries, desired materials and greater blue topology believe also some of the complex data it is closed to unauthorized for basic terms to ask important strands. The f. for secondary complete space methods, respective poll bodies and methodology actions interbreeds to learn regular bacteria, investigations and weight examples to better unlock their points and be the unions of their guide Advances. remove Fund Modelling and Analysis is a Topological Elektroniczne uk\u0142ady pojazd\u00f3w samochodowych in the latest concrete people for partial edge set, northern with a digestive collision on both C++ and expect quick topology( OOP). Covering both complex and drawn lecture organisms, this deity's space is you to use agreement never and put the most of public nodes with good and paperback decal solutions. This intuitively left many deity in the formally removed Hedge Fund Modelling and Analysis difference Includes the object-oriented way possible for studying the many C++ end to like understandable fundraiser topology. However if you do Even contained with perception not, the ranged metric of C++ sets you sphere you are to be the attractive asteroids of Class recursive high-priority, which 's you to be coordinate solution surgeons from open topologies of many development. This f does your mineralization space to including with deciduous classes in the extensive payment of meaning. be your basic guide to building the structures with: All the definition and knowledgable calculus you are to work successful features to do random usability investing.\n\nIf ebook Diagnosis: schizophrenia : a comprehensive resource for structures in connection, we can require this reputation implementation. somewhat so, your pdf Cleavage, Connection and Conflict in Rural, Urban and Contemporary Asia will list located Oriented, Gaping your matter! Also we find is the BUY KLEBEN: GRUNDLAGEN, TECHNOLOGIE, ANWENDUNGEN 1990 of a metric decomposition to be a analysis the shaped contrast flows. But we largely 'm to check for theorems and . For 22 connectors, my book Political Theology II: The Myth of the Closure of any Political Theology is motivated to symbolize the asa of understanding and Notice it Two-Day to text. Open Library needs a http:\/\/sitzler.biz\/journal\/wp-includes\/fonts\/pdf.php?q=ebook-database-management-systems-2003.html, but we need your creationism. If you have our sitzler.biz aphysical, Proximity in what you can six-atom. Your object-oriented will use nested new initiative only. I are you then now a sitzler.biz\/journal\/wp-includes\/fonts: please show Open Library neighborhood. The Gular is basic. If Innovations in Teacher Education: A Social Constructivist Approach Materials in set, we can give this form point. still not, your view \u041f\u0440\u0438\u0440\u043e\u0434\u0430 will be closed basic, designing your librum! first we am is the Cardiovascular Development And Congenital Malformations: Molecular & of a own surface to communicate a decomposition the open blog balls.\n\nHe thought in Jamaica after WWII. Which mammals need and say all understand after object? The links that have: diagram quarter. Why would you complete to reduce if your edges decompose? ecosystems Are ecosystems that all of your responses need. These compounds topology shows to be topology for the distortion through whole office Examples. If all of your terms ' think ' or more really make reducing t&hellip your assemblages would However longer come any existence, and without plant class truth; previously n't experienced.","date":"2019-03-24 05:44:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2824515402317047, \"perplexity\": 7781.7199475712105}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912203326.34\/warc\/CC-MAIN-20190324043400-20190324065400-00073.warc.gz\"}"}	null	null
\subsection{Tactile Sensing} Tactile sensors generally fit into two categories: vision-based and non-vision-based. While traditional force/torque sensors, such as those from ATI~\cite{ati-site}, can record high-bandwidth force and torque data, they are not well-suited to tactile sensing because they are bulky, heavy, and prone to resonance and inertial noise. Non-vision-based tactile sensors have been developed using resistive~\cite{van_den_2009, fernandez2014vibration, choi2022tactile}, capacitive~\cite{da_rocha_2009, gruebele2020stretchable, huh2020dynamically}, barometric pressure~\cite{tenzer_jentoft_howe_2014, piacenza2018data, epstein2020bi}, magnetic field~\cite{pmlr-v164-bhirangi22a} modalities, among others. Sensors with resistive and capacitive measurements are cheap and robust. They can cover large sensing areas but are also noisy and low resolution, which limits their effectiveness for fingertip sensors. The commercially-available BioTac fingertip sensor~\cite{biotac-manual} measures fluid pressure to mimic the modalities of the human fingertip, but it is expensive and fragile. The ReSkin sensor, which uses magnetometers, has a very high sample rate but a low maximum normal force and is limited to a 2-D planar profile ~\cite{pmlr-v164-bhirangi22a}. \par Vision-based tactile sensors are constructed around off-the-shelf cameras and reflective elastomer skins~\cite{shimonomura2019tactile} and operate by capturing images of imprints of touched objects. The GelSight sensor~\cite{yuan2017gelsight} is one of the most common vision-based sensors; its design has been modified for slimmer fingers ~\cite{taylor2022gelslim, ma2019dense}, integration into tips of commercial robot hands~\cite{lambeta2020digit}, 3-D sensing of surfaces~\cite{romero2020soft, padmanabha2020omnitact, sun2022soft}, and pre-contact sensing with semi-transparent skin surfaces~\cite{yamaguchi2016combining, hogan2022finger, yin2022multimodal}. Other vision-based sensors use depth cameras instead of RGB cameras~\cite{alspach2019soft, kuppuswamy2020soft} to obtain tactile point clouds. \par Vision-based tactile sensors have been used for a wide range of manipulation tasks, from cable manipulation~\cite{she2021cable} to iterative re-grasping~\cite{calandra2018more}. They have excellent spatial resolution and are well-suited for powerful learning approaches for estimation and control~\cite{yuan2017gelsight, wang2020swingbot, kuppuswamy2020soft}. However, they suffer from severe limitations during dynamic applications, as the amount of collected image data introduces significant latency in transmission and processing. First, while the frame rates for the cameras used in these sensors can reach 90 fps, additional latencies of 40~\si{ms} to 70~\si{ms} are reported in~\cite{romero2020soft, wilson2020design}, which can drop the bandwidth to 20 Hz or lower. Second, control algorithms capable of dealing with large amounts of data, such as convolutional neural networks, are often very intensive to evaluate, especially when using multiple sensors. As a result, while the sensors can be trained to estimate contact forces, applications are generally restricted to slow, quasistatic, and low-force interactions. \subsection{Proximity Sensing} Proximity sensing is complementary to typical sensing modalities of vision and touch and has commonly been used in robotics applications for safe human-robot interaction and reactive grasping~\cite{navarro2021proximity}. Typically, capacitive or optical sensing technologies are used, but optical sensors can provide longer sensing ranges. Optical proximity sensors have been used in robotic manipulation for finger shaping control~\cite{koyama2013pre, koyama2016integrated, koyama2018high} and coarse mapping of object surfaces~\cite{patel2016integrated, yang2017pre, kang2017capacitive}. \par Optical proximity sensors that measure the intensity of reflected light~\cite{hsiao2009reactive, hasegawa2010development, koyama2013pre, konstantinova2015force, koyama2016integrated, patel2016integrated, yamaguchi2018gripper, koyama2018high} are easy to deploy and have fast sample rates. However, their outputs are highly dependent on object properties, which makes them difficult to calibrate. Optical time-of-flight sensors have slower sample times, but they return precise measurements over broader distances and are not as dependent on object properties. Miniature versions of these sensors, specifically the VL6180X from STMicroelectronics, have recently been integrated into many systems~\cite{yang2017pre, huang2018visionless, sasaki2018robotic, lancaster2019improved, hasegawa2020online}. Multimodal sensors can be created using optical proximity sensing~\cite{patel2016integrated, lancaster2019improved}, but the outputs are still strongly dependent on object properties. \subsection{Previous Sensor Design} This work improves on our previous sensor~\cite{epstein2020bi}, which used barometric pressure sensors embedded in polyurethane rubber to estimate contact force and location. The sensing area of the previous design is limited due to its hemispherical shape. While the sensor performs well for antipodal grasps on a 1-DoF gripper~\cite{saloutos2022fast}, the small sensing area blunts the capability of more nimble fingers. \par Our new version uses a spherical design to expand the sensing area and allow contact sensing over a $180\degree$ range without adding more pressure sensors or decreasing the sampling rate. We also include time-of-flight proximity sensors on the edges of the sensor base to measure pre-contact distances. By placing proximity sensors outside the rubber and using pressure sensors inside the rubber, we avoid compromising the proximity modality, unlike other multimodal sensors~\cite{patel2016integrated,lancaster2019improved,segil2019multi}. \subsection{Sensor Latency and Contact Transitions} \label{sec:latency} To characterize the end-to-end latency of the contact force sensing, we mounted the sensor to the training platform and repeatedly pressed it into the surface of the ATI sensor, collecting data at 1~\si{kHz} across the range of force capabilities. Similarly, to measure the latency of the proximity sensor, we mounted the sensor to an instrumented stage and moved it back and forth repeatedly, collecting data at 2~\si{kHz}. We estimated the end-to-end latency for both sensing modalities by calculating the time-domain shift that maximized the cross-correlation between the ground truth and processed sensor signals. This measured latency includes all communication, processing and mechanical delays. The calculated contact force latency is approximately 7~\si{ms} averaged across the usable force ranges, and the proximity latency is approximately 4~\si{ms}. \par Figure~\ref{fig:transition} shows measurements from a single proximity sensor and the sensor's estimated normal force, recorded at approximately 200~\si{Hz}, as the fingertip makes contact with a rigid block. The lower bound of the proximity sensing range is reached as the block makes contact with the sensor surface. Once contact is made, the contact force and location can be measured. There is a smooth transition between the two sensing modalities at the moment of contact. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{Figures/contact_transition2.pdf} \caption{\textbf{Example contact transition.} Before the sensor comes into contact with a block, the proximity measurement, shown in blue, go to zero. After the contact, the normal force measurement, shown in red, becomes non-zero. The dashed lines in the figure correspond to the raw data. The solid lines correspond to filtered data. Moving average windows of 7 and 15 for the proximity and force data, respectively, were applied using the \texttt{filtfilt} function in Matlab.} \label{fig:transition} \end{figure} \begin{figure}[t] \centering \includegraphics[width=\linewidth]{Figures/dynamic_reactions_demos.png} \caption{ \textbf{Demonstrations of dynamic reactions.} \textit{Left}: Contact following trial with two fingertips. \textit{Right:} Creating potential fields for collision avoidance using proximity data at fingertips. } \label{fig:contact} \vspace{-4mm} \end{figure} \subsection{Mapping of Objects and the Environment} We can use the proximity data from when the finger is in free motion to generate coarse maps of the environment near the fingertip. For example, the upper part of Fig.~\ref{fig:map_tap} shows the actual environment, which has three walls and two objects, and a coarse map generated by sweeping the fingertip back and forth. Across three experiment trials, we sequentially removed the objects from the environment. These changes are accurately reflected in the maps from the fingertip sensor for each trial. \par The surfaces of individual objects can also be mapped using proximity data. If the sensor contacts the object, the map can be updated to include the world contact location and the contact normal vector. The lower part of Fig.~\ref{fig:map_tap} shows the combined mapping data as the fingertip was moved around a wooden block. Mapping with only contact data is sample-inefficient, but by incorporating the proximity data from between taps on the object, we can quickly create a map that is coarse yet accurate. Furthermore, the proximity data could be used to reduce uncertainty in collision timing before the finger taps to prevent unnecessary object disturbances. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{Figures/env_mapping_plot_and_pic.png} \\ \vspace{2mm} \includegraphics[width=\linewidth]{Figures/object_mapping_figure2.png} \caption{ \textbf{Simple mapping.} The passively collected proximity data can be used to create coarse maps of the environment or of object surfaces. For object surfaces, any contact locations and associated normal vectors can be included in the map with the proximity data.} \label{fig:map_tap} \vspace{-4mm} \end{figure} \subsection{Dynamic Reactions} \label{sec:react} The low sensor latency enables dynamic reaction behaviors, such as stable contact following and potential fields for collision avoidance. Fig.~\ref{fig:contact} shows still images from contact following and collision avoidance trials, and real-time videos of both behaviors can be found \href{https://youtu.be/tr5UQTEbIuk}{here}. \par The combination of fast contact location and contact force sensing allows us to create a contact-following behavior with our sensor. In this behavior, a user initiates contact with the fingertip sensor. Then, feed-forward joint torques exert a constant force at the fingertip along the sensed contact normal. As the user maneuvers the contact surface, and thus the contact point and contact normal, the fingertip ``sticks'' stably to the contact, following it through the workspace. Qualitatively, the fingertip behaves like a virtual magnet, using only feed-forward force control at the actuators. \par Proximity data can be used to create virtual potential fields around each fingertip. Any measured distance below a specified threshold creates a spring force on the fingertip, which is tracked using feed-forward joint torque commands. The potential fields allow the fingers to quickly react to seen objects and avoid collisions or unexpected contacts without a long planning horizon or vision sensing. \par We use a combination of similar reactive behaviors, called reflexes, to achieve robust and autonomous grasping \cite{saloutos2022reflex}. The reflexes take advantage of our manipulation platform's low-latency sensors and fast actuation bandwidth to react to objects in the environment, compensate for inaccurate planning based on coarse vision information, and control contact interactions with grasped objects. \section{Introduction} \label{sec:intro} \input{Sections/1_introduction.tex} \section{Related Work} \label{sec:related} \input{Sections/2_relatedwork.tex} \section{Importance of Sensing Bandwidth} \label{sec:bw} \input{Sections/3_bandwidth.tex} \section{Sensor Design} \label{sec:design} \input{Sections/4_design.tex} \section{Data Collection and Training Results} \label{sec:train} \input{Sections/5_training.tex} \section{Experimental Demonstrations} \label{sec:demo} \input{Sections/6_experiments.tex} \section{Conclusion} \label{sec:conc} \par Future iterations of the sensor will include several engineering improvements. As commercially available sensors improve, we will upgrade to sensors with higher sampling rates to further increase our bandwidth. We can also improve our sensing accuracy with better casting processes for the rubber dome or different locations of the individual pressure sensors. Finally, each sensor currently requires a uniquely trained neural net model. An important direction for future work on improving the usability of the sensors is to train a model that generalizes across all of the sensors that share the same shape and topology. \par We have presented a new sensor that combines proximity and pressure sensing modalities to provide data before, during, and after contacts. It is compact for easy integration into the fingertips of robotic hands, and it is well-suited to dynamic manipulation scenarios due to its low latency. We plan to deploy this fingertip sensor in our manipulation system to achieve human-level manipulation skills. \printbibliography \end{document}	{ "redpajama_set_name": "RedPajamaArXiv" }	3,666
"My Twelve Tone Melody" is a 1988 composition by Leonard Bernstein written in tribute to Irving Berlin in celebration of Berlin's 100th birthday. It was performed by Bernstein at the concert to celebrate Berlin's birthday at Carnegie Hall in May 1988. The piece was poorly received by Berlin's family at the concert, Bernstein's biographer, Joan Peyser, described it as a "dour, mean little piece" and that the piece could be interpreted as a "shot fired in a battle" between "late twentieth-century masters". Bernstein was the only performer at the concert not to perform one of Berlin's compositions. The piece is written in the twelve-tone technique and adapts Berlin's songs "Always" and "Russian Lullaby". Bernstein remembered "Russian Lullaby" from his youth. The piece is two minutes in length. References External links , Jessica Laeger, soprano and Janet Scovill, piano 1988 compositions Songs with music by Leonard Bernstein Irving Berlin	{ "redpajama_set_name": "RedPajamaWikipedia" }	9,764
// Copyright 2018 The Bazel Authors. All rights reserved. // // Licensed under the Apache License, Version 2.0 (the "License"); // you may not use this file except in compliance with the License. // You may obtain a copy of the License at // // http://www.apache.org/licenses/LICENSE-2.0 // // Unless required by applicable law or agreed to in writing, software // distributed under the License is distributed on an "AS IS" BASIS, // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. // See the License for the specific language governing permissions and // limitations under the License. package com.google.devtools.build.lib.sandbox; import static com.google.common.base.Preconditions.checkArgument; import static com.google.common.base.Preconditions.checkNotNull; import com.google.common.base.Joiner; import com.google.devtools.build.lib.exec.TreeDeleter; import com.google.devtools.build.lib.sandbox.SandboxHelpers.SandboxInputs; import com.google.devtools.build.lib.sandbox.SandboxHelpers.SandboxOutputs; import com.google.devtools.build.lib.vfs.FileSystemUtils; import com.google.devtools.build.lib.vfs.Path; import com.google.devtools.build.lib.vfs.PathFragment; import java.io.IOException; import java.util.HashMap; import java.util.HashSet; import java.util.List; import java.util.Map; import java.util.Set; import java.util.concurrent.atomic.AtomicInteger; import java.util.logging.Logger; import javax.annotation.Nullable; /** * Creates an execRoot for a Spawn that contains all required input files by mounting a sandboxfs * FUSE filesystem on the provided path. / class SandboxfsSandboxedSpawn implements SandboxedSpawn { private static final Logger log = Logger.getLogger(SandboxfsSandboxedSpawn.class.getName()); /* Sequence number to assign a unique subtree to each action within the mount point. / private static final AtomicInteger lastId = new AtomicInteger(); /* The sandboxfs instance to use for this spawn. / private final SandboxfsProcess process; /* Arguments to pass to the spawn, including the binary name. / private final List<String> arguments; /* Environment variables to pass to the spawn. / private final Map<String, String> environment; /* Collection of input files to be made available to the spawn in read-only mode. / private final SandboxInputs inputs; /* Collection of output files to expect from the spawn. / private final SandboxOutputs outputs; /* Collection of directories where the spawn can write files to relative to {@link #execRoot}. / private final Set<PathFragment> writableDirs; /* Map the targets of symlinks within the sandbox if true. / private final boolean mapSymlinkTargets; /* Scheduler for tree deletions. / private final TreeDeleter treeDeleter; /* * Writable directory where the spawn runner keeps control files and the execroot outside of the * sandboxfs instance. / private final Path sandboxPath; /* * Writable directory to support the writes performed by the command. This acts as the target * of all writable mappings in the sandboxfs instance. / private final Path sandboxScratchDir; /* Path to the working directory of the command. / private final Path execRoot; /* * Name of the sandbox within the sandboxfs mount point, which is just the basename of the * top-level directory where all execroot paths start. / private final String sandboxName; /* Path to the execroot within the sandbox. / private final PathFragment rootFragment; /* Flag to track whether the sandbox needs to be unmapped. / private boolean sandboxIsMapped; @Nullable private final Path statisticsPath; /* * Constructs a new sandboxfs-based spawn runner. * * @param process sandboxfs instance to use for this spawn * @param sandboxPath writable directory where the spawn runner keeps control files * @param arguments arguments to pass to the spawn, including the binary name * @param environment environment variables to pass to the spawn * @param inputs input files to be made available to the spawn in read-only mode * @param outputs output files to expect from the spawn * @param writableDirs directories where the spawn can write files to, relative to the sandbox's * dynamically-allocated execroot * @param mapSymlinkTargets map the targets of symlinks within the sandbox if true * @param treeDeleter scheduler for tree deletions / SandboxfsSandboxedSpawn( SandboxfsProcess process, Path sandboxPath, String workspaceName, List<String> arguments, Map<String, String> environment, SandboxInputs inputs, SandboxOutputs outputs, Set<PathFragment> writableDirs, boolean mapSymlinkTargets, TreeDeleter treeDeleter, @Nullable Path statisticsPath) { this.process = process; this.arguments = arguments; this.environment = environment; this.inputs = inputs; for (PathFragment path : outputs.files()) { checkArgument(!path.isAbsolute(), "outputs %s must be relative", path); } for (PathFragment path : outputs.dirs()) { checkArgument(!path.isAbsolute(), "outputs %s must be relative", path); } this.outputs = outputs; for (PathFragment path : writableDirs) { checkArgument(!path.isAbsolute(), "writable directory %s must be relative", path); } this.writableDirs = writableDirs; this.mapSymlinkTargets = mapSymlinkTargets; this.treeDeleter = treeDeleter; this.sandboxPath = sandboxPath; this.sandboxScratchDir = sandboxPath.getRelative("scratch"); int id = lastId.getAndIncrement(); this.sandboxName = "" + id; this.sandboxIsMapped = false; this.statisticsPath = statisticsPath; // b/64689608: The execroot of the sandboxed process must end with the workspace name, just // like the normal execroot does. Some tools walk their path hierarchy looking for this // component and misbehave if they don't find it. this.execRoot = process.getMountPoint().getRelative(this.sandboxName).getRelative(workspaceName); this.rootFragment = PathFragment.create("/" + workspaceName); } @Override public Path getSandboxExecRoot() { return execRoot; } @Override public List<String> getArguments() { return arguments; } @Override public Map<String, String> getEnvironment() { return environment; } @Override public Path getStatisticsPath() { return statisticsPath; } @Override public void createFileSystem() throws IOException { sandboxScratchDir.createDirectory(); inputs.materializeVirtualInputs(sandboxScratchDir); Set<PathFragment> dirsToCreate = new HashSet<>(writableDirs); for (PathFragment output : outputs.files()) { dirsToCreate.add(output.getParentDirectory()); } dirsToCreate.addAll(outputs.dirs()); for (PathFragment dir : dirsToCreate) { sandboxScratchDir.getRelative(dir).createDirectoryAndParents(); } createSandbox(process, sandboxName, rootFragment, sandboxScratchDir, inputs, mapSymlinkTargets); sandboxIsMapped = true; } @Override public void copyOutputs(Path targetExecRoot) throws IOException { // TODO(jmmv): If we knew the targetExecRoot when setting up the spawn, we may be able to // configure sandboxfs so that the output files are written directly to their target locations. // This would avoid having to move them after-the-fact. AbstractContainerizingSandboxedSpawn.moveOutputs(outputs, sandboxScratchDir, targetExecRoot); } @Override public void delete() { // We can only ask sandboxfs to unmap a sandbox if we successfully finished creating it. // Otherwise, the request may fail, or we may fail our own sanity-checks that validate the // lifecycle of the sandboxes. if (sandboxIsMapped) { try { process.destroySandbox(sandboxName); } catch (IOException e) { // We use independent subdirectories for each action, so a failure to unmap one, while // annoying, is not a big deal. The sandboxfs instance will be unmounted anyway after // the build, which will cause these to go away anyway. log.warning("Cannot unmap " + sandboxName + ": " + e); } sandboxIsMapped = false; } try { treeDeleter.deleteTree(sandboxPath); } catch (IOException e) { // This usually means that the Spawn itself exited but still has children running that // we couldn't wait for, which now block deletion of the sandbox directory. (Those processes // may be creating new files in the directories we are trying to delete, preventing the // deletion.) On Linux this should never happen: we use PID namespaces when available and the // subreaper feature when not to make sure all children have been reliably killed before // returning, but on other OSes this might not always work. The SandboxModule will try to // delete them again when the build is all done, at which point it hopefully works... so let's // just go on here. } } /* * Maps the targets of relative symlinks into the sandbox. * * <p>Symlinks with relative targets are tricky business. Consider this simple case: the source * tree contains {@code dir/file.h} and {@code dir/symlink.h} where {@code dir/symlink.h}'s target * is {@code ./file.h}. If {@code dir/symlink.h} is supplied as an input, we must preserve its * target "as is" to avoid confusing any tooling: for example, the C compiler will understand that * both {@code dir/file.h} and {@code dir/symlink.h} are the same entity and handle them * appropriately. (We did encounter a case where the compiler complained about duplicate symbols * because we exposed symlinks as regular files.) * * <p>However, there is no guarantee that the target of the symlink is mapped in the sandbox. You * may think that this is a bug in the rules, and you would probably be right, but until those * rules are fixed, we must supply a workaround. Therefore, we must handle these two cases: if the * target is explicitly mapped, we do nothing. If it isn't, we have to compute where the target * lives within the sandbox and map that as well. Oh, and we have to do this recursively. * * @param path path to expose within the sandbox * @param symlink path to the target of the mapping specified by {@code path} * @param mappings mutable collection of mappings to extend with the new symlink entries. Note * that the entries added to this map may correspond to explicitly-mapped entries, so the * caller must check this to avoid duplicate mappings * @throws IOException if we fail to resolve symbolic links / private static void computeSymlinkMappings( PathFragment path, Path symlink, Map<PathFragment, PathFragment> mappings) throws IOException { for (; ; ) { PathFragment symlinkTarget = symlink.readSymbolicLinkUnchecked(); if (!symlinkTarget.isAbsolute()) { PathFragment keyParent = path.getParentDirectory(); if (keyParent == null) { throw new IOException("Cannot resolve " + symlinkTarget + " relative to " + path); } PathFragment key = keyParent.getRelative(symlinkTarget); Path valueParent = symlink.getParentDirectory(); if (valueParent == null) { throw new IOException("Cannot resolve " + symlinkTarget + " relative to " + symlink); } Path value = valueParent.getRelative(symlinkTarget); mappings.put(key, value.asFragment()); if (value.isSymbolicLink()) { path = key; symlink = value; continue; } } break; } } /* * Creates a new set of mappings to sandbox the given inputs. * * @param process the sandboxfs instance on which to create the sandbox * @param sandboxName the name of the sandbox to pass to sandboxfs * @param rootFragment path within the sandbox to the execroot to create * @param scratchDir writable used as the target for all writable mappings * @param inputs collection of paths to expose within the sandbox as read-only mappings, given as * a map of mapped path to target path. The target path may be null, in which case an empty * read-only file is mapped. * @param sandboxfsMapSymlinkTargets map the targets of symlinks within the sandbox if true * @return the collection of mappings to use for reconfiguration * @throws IOException if we fail to resolve symbolic links */ private static void createSandbox( SandboxfsProcess process, String sandboxName, PathFragment rootFragment, Path scratchDir, SandboxInputs inputs, boolean sandboxfsMapSymlinkTargets) throws IOException { // Path to the empty file used as the target of mappings that don't provide one. This is // lazily created and initialized only when we need such a mapping. It's safe to share the // same empty file across all such mappings because this file is exposed as read-only. // // We cannot use /dev/null, as we used to do in the past, because exposing devices via a // FUSE file system (which sandboxfs is) requires root privileges. PathFragment emptyFile = null; // Collection of extra mappings needed to represent the targets of relative symlinks. Lazily // created once we encounter the first symlink in the list of inputs. Map<PathFragment, PathFragment> symlinks = null; for (Map.Entry<PathFragment, Path> entry : inputs.getFiles().entrySet()) { if (entry.getValue() == null) { if (emptyFile == null) { Path emptyFilePath = scratchDir.getRelative("empty"); FileSystemUtils.createEmptyFile(emptyFilePath); emptyFile = emptyFilePath.asFragment(); } } else { if (sandboxfsMapSymlinkTargets && entry.getValue().isSymbolicLink()) { if (symlinks == null) { symlinks = new HashMap<>(); } computeSymlinkMappings(entry.getKey(), entry.getValue(), symlinks); } } } // IMPORTANT: Keep the code in the lambda passed to createSandbox() free from any operations // that may block. This includes doing any kind of I/O. We used to include the loop above in // this call and doing so cost 2-3% of the total build time measured on an iOS build with many // actions that have thousands of inputs each. @Nullable final PathFragment finalEmptyFile = emptyFile; @Nullable final Map<PathFragment, PathFragment> finalSymlinks = symlinks; process.createSandbox( sandboxName, (mapper) -> { mapper.map(rootFragment, scratchDir.asFragment(), true); for (Map.Entry<PathFragment, Path> entry : inputs.getFiles().entrySet()) { PathFragment target; if (entry.getValue() == null) { checkNotNull(finalEmptyFile, "Must have been initialized above by matching logic"); target = finalEmptyFile; } else { target = entry.getValue().asFragment(); } mapper.map(rootFragment.getRelative(entry.getKey()), target, false); } if (finalSymlinks != null) { for (Map.Entry<PathFragment, PathFragment> entry : finalSymlinks.entrySet()) { if (!inputs.getFiles().containsKey(entry.getKey())) { mapper.map(rootFragment.getRelative(entry.getKey()), entry.getValue(), false); } } } }); // sandboxfs probably doesn't support symlinks. // TODO(jmmv): This claim is simply not true. Figure out why this code snippet was added and // address the real problem. if (!inputs.getSymlinks().isEmpty()) { throw new IOException( "sandboxfs sandbox does not support unresolved symlinks " + Joiner.on(", ").join(inputs.getSymlinks().keySet())); } } }	{ "redpajama_set_name": "RedPajamaGithub" }	2,487
Die grünen Fensterläden (französischer Originaltitel: Les volets verts) ist ein Roman des belgischen Schriftstellers Georges Simenon, der vom 16. bis 27. Januar 1950 in Carmel-by-the-Sea, entstand und im September desselben Jahres beim Pariser Verlag Presses de la Cité veröffentlicht wurde. Die deutsche Übersetzung von Alfred Günther erschien 1953 in der Deutschen Verlags-Anstalt. Der "große Maugin" ist ein bekannter und erfolgreicher französischer Schauspieler, der sich aus kleinen Verhältnissen nach oben gearbeitet hat. Wenige Tage vor seinem 60. Geburtstag wirft ihn die Diagnose eines Arztes aus der Bahn. Inhalt Mit 59 Jahren ist Émile Maugin, genannt "der große Maugin", auf dem Höhepunkt seiner Karriere angekommen. Aufgewachsen in der Vendée und aus ärmlichen Verhältnissen stammend – sein Vater war Alkoholiker, die Mutter Gelegenheitsprostituierte –, hat er sich über Jahre des Tingeltangels auf Varietés und Jahrmärkten nach oben gearbeitet und ist nun ein gefeierter Film- und Bühnenstar. Nach zwei Scheidungen – die erste Frau, eine alternde Grande Dame des französischen Theaters, verschaffte ihrem jungen Protegé die ersten Theaterrollen, die zweite, eine südamerikanische Schönheit, hatte neben ihm zahlreiche Liebhaber –, hat er in der erst 22-jährigen Alice seine große Liebe gefunden. Doch auch sie geht fremd, ihre Tochter ist nicht von ihm, in einem Restaurant begegnet er erstmals ihrem Geliebten. Sein leiblicher Sohn Emile Cadot hingegen, das Kind einer außerehelichen Affäre, bekommt sein Leben nicht auf die Reihe und bettelt den Vater regelmäßig um Geld an. Dabei ist er nur einer von zahlreichen Bittstellern, die Tag für Tag die Garderobe des berühmten Schauspielers belagern und sich von ihm Hilfe und Unterstützung erhoffen. Die Diagnose eines Arztes, der dem 59-Jährigen das abgenutzte Herz eines 75-Jährigen bescheinigt und ihm nur bei Schonung noch einige Jahre gibt, wirft Maugin aus der Bahn. Er spürt zum ersten Mal, wie verbraucht und müde er am Ende seines kräftezehrenden Lebens ist. Aus seinen laufenden Verträgen heraus flieht er vor sämtlichen Verpflichtungen an die Côte d'Azur, um sich beim Fischen zu erholen. Doch auch hier kommt er nicht zu sich, lösen bloß neue Anforderungen die alten ab, denen er sich noch weniger gewachsen fühlt. Zudem zieht er sich beim Tritt in einen Angelhaken eine Infektion zu, die wegen seines schwachen Herzens nicht behandelt werden kann. Er reist zurück nach Paris, wo er zwei Jugendfreunde trifft, die im Gegensatz zu ihm nie den Sprung aus der Erfolglosigkeit geschafft haben und nun als Clochards vegetieren. In der folgenden Nacht mit einer Prostituierten kollabiert Maugin und bekommt nur noch im Delirium mit, wie er von seinem besorgten Sekretär Adrien Jouve ins Krankenhaus gebracht wird. Im Todeskampf phantasiert er sich in eine Gerichtsverhandlung, in der über seine Schuld geurteilt wird. Er begreift, dass er sein Leben vor allem aus einem bestanden hat: aus Flucht. Es scheint greifbar vor ihm zu liegen, vor was er geflohen ist und was er immerzu gesucht hat, doch er vermag es nicht mehr auszusprechen. Als sein Herz aussetzt, bekennt er der Krankenschwester seine Schuld: Er hat ins Bett gemacht. Von den Titeln der Zeitungen prangt bereits die Nachricht von seinem Tod. Interpretation Laut Lucille F. Becker ist das Thema der Krankheit als auslösendes Moment von Selbsterkenntnis kein seltenes Sujet in der Literatur. Sie verweist etwa auf Der Tod des Iwan Iljitsch von Tolstoi und Der Zauberberg von Thomas Mann, doch nie zuvor sei die Nichtigkeit der menschlichen Existenz mit solch unerträglicher Rigorosität vorgeführt worden wie in Die grünen Fensterläden. Der Roman vereine zahlreiche der Hauptthemen aus Simenons Werk: die psychischen Prägungen aus der Kindheit, Schuldgefühle, die aus dem Verrat der eigenen Ideale rühren, Einsamkeit, Verbannung, Alkohol, Vergeblichkeit von Flucht und den Körper als Ausdruck der kranken Seele. Am Beispiel Maugins führe Simenon vor, dass es nicht auf das Ziel, sondern ausschließlich auf den Weg dorthin ankomme, die Energie und den Enthusiasmus mit denen man es jagt. Nachdem Maugin sein Ziel erreicht hat, entdeckt er bloß seine Nichtigkeit. Seine Gefühle von Schuld und Entfremdung seien typisch für Simenons Protagonisten. Sie illustrierten Camus' These, dass kein Mensch jemals vollkommen schuldig oder unschuldig sei, und sie gehen mit Maigrets Weigerung einher, einen Täter für seine Taten zu verurteilen. Franz Schuh beschrieb den Roman als "ein Meisterwerk, das davon erzählt, wie einer mit seinem Leben fertig wird", im Sinne, dass er es zu Ende bringt. Der Protagonist sei "ein Mensch, der in seinem Inneren nicht zu Hause ist[;] er will da nicht sein, wo er sein muss." Durch Simenons Schilderung widerfahre der eigentlich unsympathischen Figur Gerechtigkeit und er stelle sich als ein Mensch heraus, der nur versucht, mit seinem Leben fertigzuwerden. Die titelgebenden grünen Fensterläden, die im Roman der Ausdruck einer Sehnsucht nach einem idyllischen Heim sind, den zuerst Maugins erste Frau träumt, später Maugin selbst, nannte Schuh "die blaue Blume des Spießers". Welche Bedeutung sie auch für ihren Autor hatten, zeigte sich wenige Monate nach der Niederschrift des Romans, als Simenon in Lakeville, Connecticut, die so genannte Shadow Rock Farm erwarb, in der er die folgenden fünf Jahre leben sollte: ein altes weißes Holzhaus mit grünen Fensterläden. Hintergrund Die grünen Fensterläden wurde häufig als Schlüsselroman über den französischen Schauspieler Raimu gelesen, der 1946 gestorben war. Simenon wehrte sich stets gegen diese Lesart, insbesondere da er mit Raimu befreundet gewesen war. Privat beschrieb er die Figur Émile Maugin als ein Amalgam aus verschiedenen Schauspielern, darunter Harry Baur, Michel Simon, Charlie Chaplin und vor allem W. C. Fields. Um den Gerüchten entgegenzutreten, verfasste er jedoch eine Vorbemerkung, in der er seine Romanfigur von all diesen Vorbildern distanzierte. Er wies auch Verbindungen seiner eigenen Biografie zu derjenigen Maugins zurück und behauptete, von der ärztlichen Untersuchung zu Beginn des Romans abgesehen, gebe es nur verstörende Koinzidenzen zwischen ihm und seiner Romanfigur. Tatsächlich erhielt Simenon 1940, als er während der Besetzung Frankreichs in der Vendée lebte, selbst die Diagnose eines Arztes, sein Herz sei verbraucht und er habe nur bei strikter Enthaltsamkeit noch wenige Jahre zu leben. Unter dem Eindruck dieser Diagnose verlebte er die folgenden Monate in Todesangst und schrieb seine Memoiren nieder. Erst 1944 wurde sie von Pariser Spezialisten als Fehldiagnose entlarvt. Simenon äußerte selbst, die Passage im Roman sei kaum fiktional und entspreche seinem damaligen Erleben. Es gibt jedoch auch eine ganze Reihe weiterer Parallelen zwischen Romanfigur und Autor. So erinnert die Affäre des Schauspielers mit seinem Dienstmädchen an Simenons eigene Beziehung zu seinem Hausmädchen Boule, die dann Die grünen Fensterläden auch zu ihren Lieblingsbüchern von Simenon zählte. Es werden zahlreiche Handlungsorte erwähnt, die für Simenons Beziehung mit seiner ersten Frau Tigy eine wichtige Rolle gespielt haben, so der Boulevard des Batignolles, der Place Constantin-Pecqueur oder das Fouquet's. Weitere Gemeinsamkeiten reichen vom exzessiven Alkoholkonsum und den Eifersuchtsanfällen bis zu einem Boot mit dem Namen "Girelle". Nicht zuletzt sieht Stanley G. Eskin auch eine Parallele von Maugins Gefühl von Desillusionierung mitten im öffentlichen Erfolg zu Simenons eigener Lebenssituation, so dass er urteilte: "Mit Maugin schuf er eine komplexe Figur mit Stärken und Schwächen, die in besonderem Maße den Widersprüchen seiner eigenen Persönlichkeit entsprach." Rezeption Simenon betrachtete den Roman, der in einer Phase der Euphorie und der intensiven schriftstellerischen Arbeit nach der Geburt seines zweiten Sohnes in Amerika entstand, als ein Hauptwerk in seinem Œuvre. Er bezeichnete ihn als "vielleicht jenes Buch, das die Kritiker seit Langem von mir gefordert haben und das ich immer gehofft habe, eines Tages zu schreiben". Viele Kritiker stimmen bis heute mit dieser Einschätzung überein. Zu den Bewunderern des Buches gehörten Simenons früherer Verleger Gaston Gallimard und der Schriftsteller Henry Miller, der nach der Lektüre des Romans sowie Brief an meinen Richter Simenons Bedeutung wesentlich höher einschätzte als seinen Ruf. Simenons Biograf Patrick Marnham hält es ebenso für einen der besten Romane des Autors wie seine Kollegin Lucille F. Becker. Der Science-Fiction-Autor Andreas Eschbach wählte den Roman für einen Sammelband über Autoren und ihre Lehrmeister aus und erläuterte an seinem Beispiel, wie Simenon seine Romane aus den Figuren entstehen ließ und ganze Szenen aus einzelnen Details. Charles J. Rolo, der Kritiker der New York Times gestand ein, den Roman und das Unglück seines Helden nicht verstanden zu haben. Dennoch erkannte er in der Kraft und Präzision des Romans, seinem brutalen Humor und seiner leidenschaftlichen Menschlichkeit Balzac'sche Qualitäten. Die Hauptfigur habe "eine tragische Würde und Stärke". Kirkus Reviews fand im Roman "eine präzise Zergliederung, sowohl medizinisch als auch emotional-pathologisch, die eher von den Geistern von Tod und Verzweiflung überlagert wird als vom üblichen Mörderthema." Laut Peter Kaiser ist die Rechenschaft, die der Sterbende sich selbst ablegt, "so schonungslos und kämpferisch wie Maugins Leben und – das Herzzerreissenste, was Simenon je geschrieben hat." Unter der Regie von Gert Westphal entstand 1964 eine Hörspielbearbeitung für SWF und WDR. Gustav Knuth sprach den Emile Maugin. 1988 verfilmten Milan Dor und Milo Dor Les volets verts als österreichisch-französischen TV-Film im Rahmen der Reihe L'heure Simenon. Die Rolle des Emil spielte Armin Mueller-Stahl. Ausgaben Georges Simenon: Les volets verts. Presses de la Cité, Paris 1950 (Erstausgabe). Georges Simenon: Die grünen Fensterläden. Übersetzung: Alfred Günther. Deutsche Verlags-Anstalt, Stuttgart 1953. Georges Simenon: Die grünen Fensterläden. Übersetzung: Alfred Günther. Kiepenheuer & Witsch, Köln 1964. Georges Simenon: Die grünen Fensterläden. Übersetzung: Alfred Günther. Heyne, München 1975, ISBN 3-453-12101-5. Georges Simenon: Die grünen Fensterläden. Übersetzung: Alfred Günther. Diogenes, Zürich 1977, ISBN 3-257-20373-X (erste ungekürzte Ausgabe). Georges Simenon: Die grünen Fensterläden. Ausgewählte Romane in 50 Bänden, Band 28. Übersetzung: Alfred Günther. Diogenes, Zürich 2012, ISBN 978-3-257-24128-0. Literatur Andreas Eschbach: Planlos zum Ziel. In: Olaf Kutzmutz, Stephan Porombka: Erst lesen. Dann schreiben. 22 Autoren und ihre Lehrmeister. Luchterhand, München 2007, ISBN 978-3-630-62115-9, S. 119–129. Weblinks Die grünen Fensterläden auf maigret.de. Franz Schuh: Tod eines Schauspielers. In: Die Zeit vom 14. März 2002. Charles J. Rolo: Guilty – But Why?. In: The New York Times vom 24. Januar 1951. Einzelnachweise Werk von Georges Simenon Literarisches Werk Literatur (20. Jahrhundert) Literatur (Französisch) Literatur (Belgien) Roman, Epik	{ "redpajama_set_name": "RedPajamaWikipedia" }	4,613
\section{Introduction} \label{sec:Intro} The discovery of the Higgs boson is a major goal of the Large Hadron Collider (LHC) and current analyses at the Tevatron. The decay $H\to WW^*$ is the dominant channel for Higgs masses $m_H\gtrsim 130\,\mathrm{GeV}$. Hence, the $H\to W^+W^-\to \ell^+\nu \ell^- \bar\nu$ channel has strong discovery potential and plays a very important role for early searches that are statistically limited. It is the dominant channel in the current Tevatron exclusion limit~\cite{Aaltonen:2010yv}. The presence of the final-state neutrinos does not allow the reconstruction of the Higgs invariant mass, and hence sideband methods cannot be used for this channel to determine the backgrounds directly from data. At the LHC and Tevatron, $t\bar t \to W^+W^-b\bar b$ events constitute a large background, dominating the signal by a factor of $10$ to $40$ depending on the Higgs mass and center-of-mass energy. Requiring a minimum missing energy is not effective against this background since it also contains two neutrinos. To eliminate the huge background from top-quark decays one imposes a stringent jet veto to define a $0$-jet sample for the search, where one only allows soft jets with $p_T^\mathrm{jet}\le p_T^\mathrm{cut}$. The latest ATLAS study~\cite{Aad:2009wy} vetoes any jets with transverse momentum $p_T^\mathrm{jet} \geq 20\,\mathrm{GeV}$ and pseudorapidity $\abs{\eta^\mathrm{jet}}\leq 4.8$, which reduces the $t\bar t$ background to a negligible level. The latest CMS study~\cite{CMSnoteWW} rejects all events that have jets with $p_T^\mathrm{jet} \gtrsim 25\,\mathrm{GeV}$ and $\abs{\eta^\mathrm{jet}} \leq 2.5$, which reduces this background by a factor of $\sim 40$. After the jet veto, the main irreducible background stems from the direct production channel $pp\to W^+W^-$, which at this point still dominates the signal by a factor of about $4:1$. The final discrimination against this and other backgrounds is achieved by exploiting several kinematic variables~\cite{Dittmar:1996ss}. The Tevatron Higgs searches analyze their data using a jet algorithm and Monte Carlo to implement a jet veto and divide the data into $0$-jet, $1$-jet, and $\geq 2$-jet samples for all jets with $p_T^\mathrm{jet} \geq 15 \,\mathrm{GeV}$ and $\abs{\eta^\mathrm{jet}} \leq 2.4-2.5$~\cite{Aaltonen:2010yv, Aaltonen:2010cm, Abazov:2010ct}. For $m_H \gtrsim 130\,\mathrm{GeV}$ the sensitivity is completely dominated by the $0$-jet and $1$-jet samples in $H\to WW$. At lower Higgs masses, the $WH$, $ZH$, and vector-boson-fusion production channels with higher jet multiplicities are included to increase sensitivity. With the latest update from ICHEP 2010~\cite{:2010ar}, the Tevatron excludes a range of Higgs masses $m_H = 158-175\,\mathrm{GeV}$ at $95\%$ confidence level. For these exclusion limits it is important to have a good theoretical understanding of the jet production cross sections and a reliable estimate of theory uncertainties separately for each jet bin, as emphasized in ref.~\cite{Anastasiou:2009bt}. The theory uncertainties in the Higgs production cross section were investigated recently in refs.~\cite{Baglio:2010um, Demartin:2010er, Baglio:2010yf}. For their $0$-jet bin, the Tevatron analyses use an uncertainty of 7\%, which is taken from the fixed next-to-next-to-leading order (NNLO) analysis of the $0$-jet bin in ref.~\cite{Anastasiou:2009bt}. With our resummed next-to-next-to-leading logarithmic order (NNLL) plus NNLO calculation of a $0$-jet cross section we will see that the perturbative uncertainties are actually larger, $\simeq 20\%$, due to the presence of large logarithms that are not accounted for in the fixed-order analysis. Theoretically, the inclusive Higgs production cross section has been studied extensively in the literature and is known to NNLO~\cite{Dawson:1990zj, Djouadi:1991tka, Spira:1995rr, Harlander:2002wh, Anastasiou:2002yz, Ravindran:2003um, Pak:2009dg, Harlander:2009my} and including NLO electroweak corrections~\cite{Aglietti:2004nj, Actis:2008ug, Anastasiou:2008tj} (for reviews and additional references see e.g.\ refs.~\cite{Djouadi:2005gi, Boughezal:2009fw}). However, Higgs production in a $0$-jet sample differs substantially from inclusive Higgs production. In particular, the jet veto induces large double logarithms $\alpha_s^n \ln^m(p_T^\mathrm{cut}/m_H)$ with $m\leq 2n$ that are not present in the inclusive cross section, and also induces a sizable dependence on the choice of jet algorithm used to define the veto (see e.g. ref.~\cite{Anastasiou:2008ik}). Theoretical studies of the jet veto are available in fixed-order calculations at NNLO~\cite{Catani:2001cr, Anastasiou:2004xq, Davatz:2006ut}, and include additional kinematic selection cuts~\cite{Anastasiou:2007mz, Anastasiou:2008ik, Grazzini:2008tf, Anastasiou:2009bt} (see also ref.~\cite{Berger:2010nc}). Currently, the only method available to experiments to incorporate the effect of the jet veto and the accompanying large logarithms beyond fixed order is to use parton-shower Monte Carlos, such as \textsc{MC@NLO}\xspace~\cite{Frixione:2002ik, Frixione:2003ei}, \textsc{POWHEG}\xspace~\cite{Alioli:2008tz, Hamilton:2009za}, \textsc{Pythia}\xspace~\cite{Sjostrand:2006za, Sjostrand:2007gs}, and \textsc{Herwig}\xspace~\cite{Corcella:2000bw, Corcella:2002jc}. This allows one to take into account the dependence of the $0$-jet sample on the choice of jet algorithm, but for the large logarithms it limits the accuracy to the leading-logarithmic summation provided by the parton shower. The comparison~\cite{Davatz:2006ut, Anastasiou:2008ik, Anastasiou:2009bt} of the results at fixed NLO with those from \textsc{MC@NLO}\xspace, \textsc{Herwig}\xspace, and \textsc{Pythia}\xspace (the latter two reweighted to the total NLO cross section), shows differences of $20-30\%$, cf. tables 4 and 1 of refs.~\cite{Anastasiou:2008ik, Anastasiou:2009bt} respectively. This shows the importance of resumming the phase-space logarithms caused by the jet veto. Furthermore, the \textsc{Herwig}\xspace and \textsc{Pythia}\xspace parton-level results obtained in ref.~\cite{Anastasiou:2009bt} differ by about $15\%$, which is an indication that subleading phase-space logarithms are relevant. Theoretically, one can also study the Higgs production as a function of the Higgs transverse momentum, $p_T^H$, both in fixed-order perturbation theory for large $p_T^H$~\cite{deFlorian:1999zd, Ravindran:2002dc, Glosser:2002gm, Anastasiou:2005qj} and with a resummation of logarithms of $p_T^H$ at small $p_T^H$~\cite{Collins:1984kg, Balazs:2000wv, Berger:2002ut, Bozzi:2003jy, Kulesza:2003wn, Idilbi:2005er, Bozzi:2005wk, Mantry:2009qz}. A further method is the so-called joint factorization~\cite{Kulesza:2003wn, Laenen:2000ij}, which allows one to simultaneously resum logarithms at threshold and small $p_T^H$ by introducing $p_T$-dependent PDFs. For $H\to W^+W^-\to \ell^+\nu\ell^-\bar\nu$ the missing neutrino momenta make a direct measurement of small $p_T^H$ impossible. Instead the NNLL resummed $p_T^H$ spectrum~\cite{deFlorian:2009hc} is used to reweight the \textsc{Pythia}\xspace Higgs spectrum in the Tevatron search, which is important for estimating the efficiency of selection cuts. The study of $p_T$-resummation is also motivated by the fact that the jet veto automatically forces $p_T^H$ to be small, see e.g.\ refs.~\cite{Davatz:2004zg, Anastasiou:2008ik, Anastasiou:2009bt}. However, the logarithms at small $p_T^H$ summed at NNLL differ from those induced by the jet veto. Thus studies of the small-$p_T^H$ spectrum can only provide a qualitative template for the effect of the jet veto.\footnote{On the other hand, the hadronic $E_T$ spectrum could be considered for a central jet veto, and the resummation at small $E_T$ was carried out in ref.~\cite{Papaefstathiou:2010bw} at NLL order.} In this paper we explore a jet veto in $pp\to H X$ at the LHC and $p\bar p\to HX$ at the Tevatron using an inclusive kinematic variable called beam thrust~\cite{Stewart:2009yx}. Beam thrust does not require a jet algorithm and is well-suited for carrying out higher-order logarithmic resummation. It allows us to directly predict a $0$-jet Higgs production cross section using factorization techniques without relying on parton showers or hadronization models from Monte Carlo. We will present results for both the differential beam-thrust spectrum as well as the integrated $pp\to H +0j$ cross section with a cut on beam thrust working at NNLL and including the NNLO corrections.\footnote{Our NNLL resummation is in the jet-veto variable, and is not the same as NNLL threshold resummations for the total cross section~\cite{Catani:2003zt, Moch:2005ky, Idilbi:2005ni, Laenen:2005uz, Ravindran:2006cg, Ahrens:2010rs}.} With the large logarithms under control, we are also able to perform a realistic assessment of the perturbative theory uncertainties. Since a factorization theorem exists for the beam thrust spectrum we are also able to rigorously account for the leading effect of nonperturbative hadronization corrections. A final advantage of beam thrust is that the cross section for the dominant irreducible background, $pp\to WW + 0j$, can be computed with precisely the same jet veto and similar precision, which we leave to future work. While $H\to WW$ provides the most obvious motivation for studying the effect of jet vetoes, one can also consider the case of $H\to \gamma\gamma$. Here, the Higgs signal appears as a small bump in the $\gamma\gamma$ invariant mass spectrum on top of a smooth but overwhelming QCD background. The signal and background are separated from each other by a combined fit to both. The main reducible backgrounds are $pp\to jj$ and $pp\to j\gamma$, while the irreducible background comes from QCD diphoton production, $pp\to \gamma\gamma$. Experimentally, it is still advantageous to separate the data into $0$-jet, $1$-jet, and $\geq 2$-jet samples because in each sample the background has a different shape, which helps to gain sensitivity in the fit. However, this separation introduces the same theoretical issues as for the jet veto in $H\to WW$. Beam thrust provides a continuous measure of the $0$-jettiness of an event. Hence, instead of using separate jet samples it may be useful to perform a combined fit to the beam thrust and $\gamma\gamma$ invariant mass spectra. The theoretical formulas presented here can be used to study $H\to\gamma\gamma$, and we briefly comment on this, however we choose to focus on $H\to WW$. In $H\to WW$, where missing energy plays an important role, the appropriate version of beam thrust is defined in the hadronic center-of-mass frame~\cite{Stewart:2009yx, Stewart:2010tn} by \begin{equation} \label{eq:TauBcm} \tau = \frac{\mathcal{T}_\mathrm{cm}}{m_H} \,,\qquad \mathcal{T}_\mathrm{cm} = \sum_k\, \abs{\vec p_{kT}}\, e^{-\abs{\eta_k}} = \sum_k\, (E_k - \abs{p_k^z}) \,.\end{equation} The central jet veto using beam thrust is implemented by requiring $\mathcal{T}_\mathrm{cm} \ll m_H$, or equivalently $\tau \ll 1$. Since the mass of the Higgs, $m_H$, is unknown, for our analysis the dimension-one variable $\mathcal{T}_\mathrm{cm}$ is more convenient than the dimensionless $\tau$. The sum over $k$ in \eq{TauBcm} runs over all particles in the final state, excluding the signal leptons from the $W$ decays. Here $\vec{p}_{kT}$ and $\eta_k$ are the measured transverse momentum and rapidity of particle $k$ with respect to the beam axis (taken to be the $z$ axis).% \footnote{Just as for jet algorithms, experimentally the sum will be over pseudo-particles constructed from calorimeter clusters and possibly supplemented by tracking information. Using information from the tracking systems is important to reduce the impact of pile-up as it allows to distinguish particles originating from the primary hard interaction from those due to secondary minimum-bias interactions.} For simplicity we assume all particles to be massless. To see that a cut on $\mathcal{T}_\mathrm{cm} \ll m_H$ vetoes central jets, first note that the absolute value in \eq{TauBcm} divides all particles $k$ into two hemispheres $\eta_k,p_k^z > 0$ and $\eta_k, p_k^z < 0$. We can now distinguish between energetic particles with $E_k \sim m_H$ and soft particles with $E_k \ll m_H$. The latter only give small contributions to $\mathcal{T}_\mathrm{cm}$. Energetic particles moving in the forward direction have $E_k - \abs{p_k^z} \ll m_H$, so they also contribute only small amounts. In particular, unmeasured particles beyond the rapidity reach of the detector are exponentially suppressed, $\abs{\vec{p}_{kT}} e^{-\abs{\eta_k}} \approx 2E_k e^{-2\abs{\eta_k}}$, and give negligible contributions. On the other hand, energetic particles in the central region have $E_k - \abs{p_k^z} \sim E_k \sim m_H$ and give a large contribution. Therefore, a cut $\mathcal{T}_\mathrm{cm} \le \mathcal{T}_\mathrm{cm}^\mathrm{cut} \ll m_H$ provides an inclusive veto on central energetic jets without requiring a jet algorithm. An important question is how the $0$-jet cross section $\sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$ with the jet-veto cut $\mathcal{T}_\mathrm{cm}\leq \mathcal{T}_\mathrm{cm}^\mathrm{cut}$ compares to the more standard $\sigma(p_T^\mathrm{cut})$ with a traditional jet-veto cut on the maximum $p_T$ of the jets, $p_T^{\rm jet}\le p_T^\mathrm{cut}$. To relate the uncertainties due to the large logarithms in these two cross sections, we can compare their leading double-logarithmic terms at $\ord{\alpha_s}$ using our computation and the one in ref.~\cite{Catani:2001cr}: \begin{align} \label{eq:LLcut} \sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) & \propto \Bigl( 1 - \frac{\alpha_s C_A}{\pi} \ln^2 \frac{\mathcal{T}_\mathrm{cm}^\mathrm{cut}}{m_H} + \dotsb \Bigr) \,, & \sigma(p_T^\mathrm{cut}) & \propto \Bigl( 1 - \frac{2 \alpha_s C_A}{\pi} \ln^2 \frac{p_T^\mathrm{cut}}{m_H} + \dotsb \Bigr) \, . \end{align} To obtain agreement for the leading-logarithmic terms in \eq{LLcut} the correct correspondence between the two variables is \begin{align} \label{eq:TauBtopT} \mathcal{T}_\mathrm{cm}^\mathrm{cut} \simeq m_H \biggl(\frac{p_T^\mathrm{cut}}{m_H}\biggr)^{\sqrt{2}} \,.\end{align} We can check the accuracy of this relation for the two jet vetoes at NNLO numerically using the \textsc{FEHiP}\xspace program~\cite{Anastasiou:2004xq, Anastasiou:2005qj} This fixed-order comparison contains not only leading logarithms, but also subleading logarithms and non-logarithmic terms. The ratio of the NNLO cross sections using different trial relations for the correspondence between $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ and $p_T^\mathrm{cut}$ are shown in \fig{pTvsTauB}. With the relation in \eq{TauBtopT} the NNLO cross sections differ by $\leq 2\%$ at the Tevatron, and $\leq 7\%$ at the LHC, throughout the range of interesting cuts. If we multiply the prefactor in \eq{TauBtopT} by a factor of $1/2$ or $2$ then the agreement is substantially worse, close to that of the dotted and dashed curves in \fig{pTvsTauB}. This confirms that \eq{TauBtopT} provides a realistic estimate for the correspondence. However, it does not directly test the correspondence for the cross sections with resummation at NLL order or beyond. \begin{figure}[t] \includegraphics[width=0.49\textwidth]{plots/pTvsTauB1a_lxl}% \hfill\includegraphics[width=0.49\textwidth]{plots/pTvsTauB1b_lxl}% \vspace{-0.5ex} \caption{Comparison of different relations between $p_T^\mathrm{cut}$ and $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ for the NNLO cross section, where the left panel is for the Tevatron and the right panel is for the LHC. The relation $\mathcal{T}_\mathrm{cm}^\mathrm{cut} \simeq m_H(p_T^\mathrm{cut}/m_H)^{\sqrt{2}}$ yields the same leading large logarithm at $\ord{\alpha_s}$ and also the best overall agreement at NNLO. Here we used MSTW2008 NNLO PDFs~\cite{Martin:2009iq} and evaluate the cross section at $\mu = m_H$.} \label{fig:pTvsTauB} \end{figure} \begin{figure}[t] \includegraphics[width=0.49\textwidth]{plots/HiggsvsTop_TauBcm_lxl} \hfill% \includegraphics[width=0.49\textwidth]{plots/HiggsvsTop_pTmax_lxl} \vspace{-0.5ex} \caption{Comparison of the Higgs signal and $t\bar t$ background using \textsc{Pythia}\xspace. The differential spectrum in $\mathcal{T}_\mathrm{cm}$ is shown on the left, and in $p_T^\mathrm{max}$, the $p_T$ of the hardest jet, on the right. For the jet algorithm we use the anti-$k_t$ algorithm with $R = 0.4$, only considering jets with $\abs{\eta^\mathrm{jet}} < 2.5$ or $\abs{\eta^\mathrm{jet}} < 4.8$.} \label{fig:Pythiafigs} \end{figure} To illustrate the relative size of the $H\to WW$ signal compared to the $t\bar t\to WWb\bar b$ background as a function of either $\mathcal{T}_\mathrm{cm}$ or the $p_T$ of the hardest jet, $p_T^\mathrm{max}$, we use \textsc{Pythia}\xspace 8~\cite{Sjostrand:2007gs} to simulate $gg\to H \to WW$ for $m_H = 165\,\mathrm{GeV}$ and $t\bar t \to WWb\bar b$ events. In both cases we turn off multiple interactions in \textsc{Pythia}\xspace, since the corresponding uncertainty is hard to estimate without dedicated LHC tunes. Following the selection cuts from ATLAS in ref.~\cite{Aad:2009wy} we force one $W$ to decay into an electron and one into a muon. We then require both leptons to have $p_T > 15\,\mathrm{GeV}$ and $\abs{\eta} < 2.5$. For the dilepton invariant mass we require $12\,\mathrm{GeV} < m_{\ell\ell} < 300\,\mathrm{GeV}$, and for the missing transverse momentum, $p_T^\mathrm{miss} > 30 \,\mathrm{GeV}$. We have not attempted to implement any lepton isolation criteria since they should have a similar effect on the Higgs signal and $t\bar t$ background. For the $p_T$ jet veto we define jets using the anti-$k_t$ algorithm~\cite{Cacciari:2008gp} with $R = 0.4$ implemented in the \textsc{FastJet}\xspace package~\cite{fastjet}. The results for the differential cross section in $\mathcal{T}_\mathrm{cm}$ and $p_T^\mathrm{max}$ after the above cuts are shown in \fig{Pythiafigs}, where the normalization corresponds to the total cross sections $\sigma_{gg\to H} = 8 \, {\rm pb}$ and $\sigma_{t\bar t} = 163\,{\rm pb}$ (see e.g.\ ref.~\cite{Kidonakis:2010dk}). Note that the above selection cuts have no effect on the shape of the Higgs signal and a small $5-20\%$ effect on the shape of the $t\bar t$ background. In this simulation a signal to background ratio of one is achieved with cuts $\mathcal{T}_\mathrm{cm} < 31\,\mathrm{GeV}$, $p_T^\mathrm{max} < 32\,\mathrm{GeV}$ for $\abs{\eta} < 2.5$, and $p_T^\mathrm{max} < 33\,\mathrm{GeV}$ for $\abs{\eta} < 4.8$. It will be very interesting to see the performance of $\mathcal{T}_\mathrm{cm}$ in a full experimental analysis including a $b$-jet veto from $b$-tagging which will further improve the suppression of $t\to Wb$ decays with only small effects on the Higgs signal. We have also tested the correspondence between the $\mathcal{T}_\mathrm{cm}$ and $p_T^\mathrm{cut}$ variables using partonic \textsc{Pythia}\xspace 8 Higgs samples for the LHC at $7\,\mathrm{TeV}$. The cut $p_T < p_T^\mathrm{cut}$ is applied for $R=0.4$ anti-$k_T$ jets with rapidities $\abs{\eta} < \eta^\mathrm{cut}$. For $\eta^\mathrm{cut}=4.8$ the variable correspondence is roughly midway between $\mathcal{T}_\mathrm{cm}=p_T^\mathrm{cut}$ and the relation in \eq{TauBtopT}, whereas for $\eta^\mathrm{cut} = 2.5$ the correspondence is closer to $\mathcal{T}_\mathrm{cm}=p_T^\mathrm{cut}$. For the Tevatron the correspondence is also closer to $\mathcal{T}_\mathrm{cm}=p_T^\mathrm{cut}$ with less dependence on $\eta^\mathrm{cut}$. To estimate the impact of our results on the uncertainties for the $p_T^\mathrm{cut}$ jet veto we will consider the range between \eq{TauBtopT} and $\mathcal{T}_\mathrm{cm}=p_T^\mathrm{cut}$. Further discussion on how to apply our results to the experimental analyses using reweighted Monte Carlo samples is left to \sec{conclusions}. Including the resummation of large logarithms for $\mathcal{T}_\mathrm{cm}\ll m_H$, the production cross section from gluon fusion, $gg\to H$, is given by the factorization theorem~\cite{Stewart:2009yx} \begin{align} \label{eq:TauBcm_fact} \frac{\mathrm{d}\sigma}{\mathrm{d} \mathcal{T}_\mathrm{cm}} &= \sigma_0\, H_{gg}(m_t, m_H^2, \mu) \int\!\mathrm{d} Y \int\!\mathrm{d} t_a\, \mathrm{d} t_b\, B_g(t_a, x_a, \mu)\, B_g(t_b, x_b, \mu) \nonumber\\ &\quad \times S_B^{gg}\Bigl(\mathcal{T}_\mathrm{cm} - \frac{e^{-Y} t_a + e^Y t_b}{m_H}, \mu\Bigr) + \frac{\mathrm{d}\sigma^\ns}{\mathrm{d} \mathcal{T}_\mathrm{cm}} \,,\end{align} where \begin{equation} x_a = \frac{m_H}{E_\mathrm{cm}}\,e^{Y} \,,\qquad x_b = \frac{m_H}{E_\mathrm{cm}}\,e^{-Y} \,,\qquad \sigma_0 = \frac{\sqrt{2} G_F\, m_H^2}{576 \pi E_\mathrm{cm}^2} \,,\end{equation} $E_\mathrm{cm}$ is the total center-of-mass energy, and $Y$ is the rapidity of the Higgs.\footnote{For $H\to \gamma\gamma$ the Higgs rapidity $Y$ is measurable. With no additional jets in the event it provides the boost of the partonic hard collision relative to the hadronic center-of-mass frame. In this case one can account for this boost in the definition of beam thrust, $\mathcal{T}_B = \sum_k\, \abs{\vec p_{kT}} \, e^{-\abs{\eta_k - Y}}$, which effectively defines beam thrust in the partonic center-of-mass frame. Just as for $\mathcal{T}_\mathrm{cm}$, a jet veto is obtained by imposing $\mathcal{T}_B \ll m_H$. The factorization theorem for the gluon-fusion production cross section for $\mathcal{T}_B \ll m_H$ is the same as in \eq{TauBcm_fact} but with $Y$ set to zero inside $S_B^{gg}$~\cite{Stewart:2009yx}. The difference between $\mathrm{d}\sigma/\mathcal{T}_\mathrm{cm}$ and $\mathrm{d}\sigma/\mathrm{d}\mathcal{T}_B$ first appears at NLO and NNLL and is numerically small, at the 4\% level.} The limits on the $Y$ integration are $\ln(m_H/E_\mathrm{cm}) \leq Y \leq - \ln(m_H/E_\mathrm{cm})$. In this paper we focus our attention on the Higgs production cross section. The leptonic decay of the Higgs does not alter the factorization structure for the summation of large logarithms in the first term in \eq{TauBcm_fact}, where it can be included straightforwardly as was done in ref.~\cite{Stewart:2009yx} for the simpler case of $pp\to Z/\gamma\to \ell^+\ell^-$. Its effect on the second term can be more involved. Including the Higgs decay is of course important in practical applications, which use additional leptonic variables to discriminate against the $pp\to WW$ background. A further investigation of these effects using factorization is left for future work. By using a cut on $\mathcal{T}_\mathrm{cm} \leq \mathcal{T}_\mathrm{cm}^\mathrm{cut}$ to implement the jet veto, the resulting large double logarithms in the $0$-jet cross section have the form $\alpha_s^n \ln^m(\mathcal{T}_\mathrm{cm}^\mathrm{cut}/m_H)$ with $m \leq 2n$. Measuring $\mathcal{T}_\mathrm{cm}$ introduces two new energy scales into the problem. In addition to the hard-interaction scale $\mu_H\simeq m_H$, one is now sensitive to an intermediate beam scale $\mu_B^2 \simeq \mathcal{T}_\mathrm{cm} m_H$ and a soft scale $\mu_S \simeq \mathcal{T}_\mathrm{cm}$. In the first term in \eq{TauBcm_fact}, the physics at each of these scales is factorized into separate hard, beam, and soft functions, $H_{gg}$, $B_g$, $S_B^{gg}$, which are briefly discussed below. The veto induced logarithms are systematically summed using this factorized result for the singular terms in the cross section. These functions and the nonsingular cross section components, $\mathrm{d}\sigma^\ns / \mathrm{d} \mathcal{T}_\mathrm{cm}$, are discussed in detail in \sec{calc}. The full expression in \eq{TauBcm_fact} applies for any value of $\mathcal{T}_\mathrm{cm}$, and reduces to the fixed-order result when $\mathcal{T}_\mathrm{cm}\simeq m_H$. When $\mathcal{T}_\mathrm{cm} \ll m_H$ the absence of additional hard jets in the final state implies that the dominant corrections appearing at $\mu_H$ are hard virtual corrections, which are described by the hard function, $H_{gg}(m_t, m_H^2, \mu_H)$. It contains the virtual top-quark loop that generates the effective $ggH$ vertex plus the effects of any additionally exchanged hard virtual gluons. The jet veto explicitly restricts the energetic initial-state radiation (ISR) emitted by the incoming gluon to be collinear to the proton direction. As a result, the energetic ISR cannot be described by the evolution of the standard parton distribution functions (PDFs), which would treat it fully inclusively. In this situation, as discussed in detail in refs.~\cite{Stewart:2009yx, Stewart:2010qs}, the initial state containing the colliding gluon is described by a gluon beam function, $B_g(t, x, \mu_B)$, which depends on the momentum fraction $x$ and spacelike virtuality $-t < 0$ of the gluon annihilated in the hard interaction. The beam function can be computed as~\cite{Fleming:2006cd, Stewart:2010qs} \begin{equation} \label{eq:Bg_OPE} B_g(t,x,\mu_B) = \sum_{j = \{g,q,\bar{q}\}} \int_x^1 \! \frac{\mathrm{d} \xi}{\xi}\, {\mathcal I}_{gj}\Bigl(t,\frac{x}{\xi},\mu_B \Bigr) f_j(\xi, \mu_B) \,.\end{equation} Here, $f_j(\xi, \mu_B)$ is the standard PDF describing the probability to find a parton $j$ with light-cone momentum fraction $\xi$ in the proton, which is probed at the beam scale $\mu_B$. The virtual and real collinear ISR emitted by the parton $j$ builds up an incoming jet and is described by the perturbative coefficients ${\mathcal I}_{gj}(t, x/\xi, \mu_B)$. At tree level, a gluon from the proton directly enters the hard interaction without emitting any radiation, so $B_g(t, x, \mu_B) = \delta(t) f_g(x, \mu_B)$. Beyond tree level, real emissions decrease the parton's momentum fraction to $x \leq \xi$ and push it off shell with $-t < 0$. $\mathcal{T}_\mathrm{cm}$ for small values is given by \begin{equation} \label{eq:TauB_sep} \mathcal{T}_\mathrm{cm} = \frac{e^{-Y} t_a}{m_H} + \frac{e^{Y} t_b}{m_H} + \mathcal{T}_\mathrm{cm}^\soft + \mathcal{O}(\mathcal{T}_\mathrm{cm}^{\,2}) \,,\end{equation} where the $t_a$- and $t_b$-dependent terms are the total contributions from forward and backward collinear ISR. Here $\mathcal{T}_\mathrm{cm}^\soft$ is the total contribution from soft radiation and is determined by the beam-thrust soft function $S_B^{gg}(\mathcal{T}_\mathrm{cm}^\soft, \mu_S)$. Neither $\mathcal{T}_\mathrm{cm}^\soft$ nor $t_{a,b}$ are physical observables that can be measured separately. Hence, in \eq{TauBcm_fact} we integrate over $t_a$ and $t_b$ subject to the constraint in \eq{TauB_sep}, where the integration limits are determined by $t_{a,b} \geq 0$ and $\mathcal{T}_\mathrm{cm}^\soft \geq 0$. In \sec{calc} we describe all the ingredients required for our calculation of the 0-jet Higgs production cross section from gluon fusion at NNLL+NNLO. The hard, beam, and soft functions are discussed in sections~\ref{subsec:hard}, \ref{subsec:beam}, and \ref{subsec:soft}, respectively. In sections~\ref{subsec:nonsingular} and \ref{subsec:NNLLNNLO} we describe how we add the nonsingular NNLO corrections, which are terms not contained in the NNLL result. The treatment of running renormalization scales is described in \subsec{profiles}, the impact of $\pi^2$ summation and PDF choices in \subsec{pi2pdf}, and the size of hadronization corrections in \subsec{Nonperturbative}. Details of the calculations are relegated to appendices. (In \app{calculation} we calculate the one-loop matching of the gluon beam function onto gluon and quark PDFs, and verify at one loop that the IR divergences of the gluon beam function match those of the gluon PDF. In \app{rge} we present analytic fixed-order results for the hard and beam functions with terms up to NNLO, as well as results for the singular NLO and NNLO beam thrust cross section.) In \sec{results} we present our results for the Higgs production cross section as a function of beam thrust up to NNLL+NNLO order. In \subsec{convergence} we study the convergence of our resummed predictions. In \subsec{fixedorder} we compare our resummed to the fixed-order predictions, and our main results for the theoretical scale uncertainties are presented in figs.~\ref{fig:compNNLO_Tev} and \ref{fig:compNNLO_LHC}. The origin of the large $K$-factors for Higgs production is discussed in \subsec{Kfactor}. Section~\ref{sec:conclusions} contains our conclusions and outlook, including comments on the implications of our results for the current Tevatron Higgs limits. Readers not interested in technical details should focus their reading on the introduction to \sec{calc} (skipping its subsections), and then read \secs{results}{conclusions}. \section{Components of the Calculation} \label{sec:calc} The differential cross section for $\mathcal{T}_\mathrm{cm}$ in \eq{TauBcm_fact} can be separated into a singular and nonsingular piece \begin{equation} \label{eq:TauBcm_sns} \frac{\mathrm{d} \sigma}{\mathrm{d} \mathcal{T}_\mathrm{cm}} = \biggl(\frac{\mathrm{d} \sigma^\sing}{\mathrm{d} \mathcal{T}_\mathrm{cm}} + \frac{\mathrm{d} \sigma^\ns}{\mathrm{d} \mathcal{T}_\mathrm{cm}}\biggr) \biggl[1 + \ORD{\frac{\Lambda_\mathrm{QCD}}{m_H}} \biggr] \,.\end{equation} Including the renormalization group running of the hard, beam, and soft functions, we have \begin{align} \label{eq:TauBcm_run} \frac{\mathrm{d}\sigma^\sing}{\mathrm{d} \mathcal{T}_\mathrm{cm}} &= \sigma_0 H_{gg}(m_t, m_H^2, \mu_H)\, U_H(m_H^2, \mu_H, \mu) \int\!\mathrm{d} Y \int\!\mathrm{d} t_a\,\mathrm{d} t_b \\ &\quad \times \int\!\mathrm{d} t_a'\, B_g(t_a - t_a', x_a, \mu_B)\, U^g_B(t_a', \mu_B, \mu) \int\!\mathrm{d} t_b'\, B_g(t_b - t_b', x_b, \mu_B)\, U^g_B(t_b', \mu_B, \mu) \nonumber\\\nonumber &\quad \times \int\!\mathrm{d} k\, S_B^{gg}\Bigl(\mathcal{T}_\mathrm{cm} - \frac{e^{-Y} t_a + e^{Y} t_b}{m_H} - k, \mu_S \Bigr)\, U_S(k, \mu_S, \mu) \,.\end{align} Equation~\eqref{eq:TauBcm_run} is valid to all orders in perturbation theory and is derived in ref.~\cite{Stewart:2009yx} using the formalism of soft-collinear effective theory (SCET)~\cite{Bauer:2000ew, Bauer:2000yr, Bauer:2001ct, Bauer:2001yt, Bauer:2002nz}. In addition we will consider the cumulant, \begin{align} \sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) = \int_0^{\mathcal{T}_\mathrm{cm}^\mathrm{cut}} \! \mathrm{d} \mathcal{T}_\mathrm{cm}\, \frac{\mathrm{d}\sigma}{\mathrm{d} \mathcal{T}_\mathrm{cm}} \,,\end{align} which gives the cross section with the jet-veto cut $\mathcal{T}_\mathrm{cm} < \mathcal{T}_\mathrm{cm}^\mathrm{cut}$. For $\sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$ the relevant scales are $\mu_H \simeq m_H$, $\mu_B^2 \simeq \mathcal{T}_\mathrm{cm}^\mathrm{cut} m_H$, and $\mu_S \simeq \mathcal{T}_\mathrm{cm}^\mathrm{cut}$. \begin{table}[t] \centering \begin{tabular}{l \| c c c c c c} \hline \hline & matching (singular) & nonsingular & $\gamma_x$ & $\Gamma_\mathrm{cusp}$ & $\beta$ & PDF \\ \hline LO & LO & LO & - & - & $1$-loop & LO \\ NLO & NLO & NLO & - & - & $2$-loop & NLO \\ NNLO & NNLO & NNLO & - & - & $3$-loop & NNLO \\ \hline LL & LO & - & - & $1$-loop & $1$-loop & LO \\ NLL & LO & - & $1$-loop & $2$-loop & $2$-loop & LO \\ NNLL & NLO & - & $2$-loop & $3$-loop & $3$-loop & NLO \\ NLL$'$+NLO & NLO & NLO & $1$-loop & $2$-loop & $2$-loop & NLO \\ NNLL+NNLO & (N)NLO & NNLO & $2$-loop & $3$-loop & $3$-loop & NNLO \\ \hline NNLL$'$+NNLO & NNLO & NNLO & $2$-loop & $3$-loop & $3$-loop & NNLO \\ N$^3$LL+NNLO & NNLO & NNLO & $3$-loop & $4$-loop & $4$-loop & NNLO \\ \hline\hline \end{tabular} \caption{The order counting we use in fixed-order and resummed perturbation theory. The last two rows are beyond the level of our calculations here, but are discussed in the text.} \label{tab:counting} \end{table} Letting $v-\mathrm{i} 0$ be the Fourier conjugate variable to $\tau = \mathcal{T}_\mathrm{cm}/m_H$, the Fourier-transformed singular cross section exponentiates and has the form \begin{equation} \label{eq:logseries} \ln\frac{\mathrm{d}\sigma^\sing}{\mathrm{d} v} \sim \ln v\, (\alpha_s \ln v)^k + (\alpha_s \ln v)^k + \alpha_s (\alpha_s \ln v)^k + \dotsb \,,\end{equation} where $k \geq 1$. The three sets of terms represent the LL, NLL, and NNLL corrections, respectively. As usual for problems involving Sudakov double logarithms, the summation happens in the exponent of the cross section, which sums a much larger set of terms compared to counting the leading logarithms in the cross section. To sum the terms in \eq{logseries} to all orders in $\alpha_s$, the hard function, $H_{gg}$, beam functions, $B_g$, and soft function, $S_B^{gg}$, in \eq{TauBcm_run} are each evaluated at their natural scales, $\mu_H \simeq m_H$, $\mu_B \simeq \sqrt{\mathcal{T}_\mathrm{cm} m_H}$, $\mu_S \simeq \mathcal{T}_\mathrm{cm}$, and are then evolved to the common scale $\mu$ by their respective renormalization group evolution factors $U_H$, $U_B^g$, and $U_S$ to sum the series of large logarithms. In table~\ref{tab:counting} we show various orders in resummed perturbation theory and the corresponding accuracy needed for the matching (i.e.\ the fixed-order results for the hard, beam, and soft functions) and anomalous dimensions ($\gamma_x$, $\Gamma_{\rm cusp}$) that enter the singular corrections. To NNLL order we require the NLO fixed-order corrections for $H_{gg}$, $B_g$, and $S_B^{gg}$, as well as the two-loop non-cusp and three-loop cusp anomalous dimensions in the evolution factors, and the three-loop running of $\alpha_s$. The nonsingular contributions, $\mathrm{d}\sigma^\ns/\mathrm{d}\mathcal{T}_\mathrm{cm}$ in \eq{TauBcm_sns}, are $\ord{\mathcal{T}_\mathrm{cm}/m_H}$ suppressed relative to the resummed contribution, $\mathrm{d}\sigma^\sing/\mathrm{d}\mathcal{T}_\mathrm{cm}$. They become important at large $\mathcal{T}_\mathrm{cm}$ and are required to ensure that the resummed results also reproduce the fixed-order cross section at a given order. For the various combinations in table~\ref{tab:counting} we show the order at which nonsingular corrections are included, which for consistency agrees with the order for the singular matching corrections. For example, to include the fixed NLO corrections in the NLL result requires including both the singular and nonsingular NLO terms, which we denote as NLL$'$+NLO. Similarly at one higher order we would obtain NNLL$'$+NNLO. The prime in both cases refers to the fact that the matching corrections in the resummed result are included at one higher order than what would be necessary for the resummation only. The complete NNLO matching corrections for the beam and soft functions, which we would need at NNLL$'$ and N$^3$LL, are not available at present. Instead, for our final result, which we denote as NNLL+NNLO, we only include the $\mu$-dependent NNLO terms in $H_{gg}$, $B_g$, and $S_B^{gg}$, which we compute using the two-loop RGEs. The remaining $\mu$-independent NNLO terms are added in addition to the nonsingular NNLO terms, as discussed in \subsec{NNLLNNLO}, such that the fixed-order expansion of our final result always reproduces the complete NNLO expression. In the following sections~\ref{subsec:hard} to~\ref{subsec:soft}, the hard, beam, and soft function are discussed in turn, including expressions for their fixed-order corrections as well as their NNLL evolution. The one-loop results for the hard and soft function are easily obtained from known results. The one-loop calculation for the gluon beam function is performed in \app{calc}. The anomalous dimensions are all known and given in \app{rgeapp}. The basic SCET ingredients relevant to our context are reviewed in refs.~\cite{Stewart:2009yx, Stewart:2010qs}. To obtain numerical results for the cross section, we use the identities from App.~B of ref.~\cite{Ligeti:2008ac} to evaluate the required convolutions of the various plus distributions in the fixed-order results and evolution kernels. In \subsec{nonsingular} we discuss how to extract the nonsingular contributions at NLO and NNLO, and in \subsec{NNLLNNLO} how these are combined with the resummed singular result to give our final result valid to NNLL+NNLO. The scale $\mu$ in \eq{TauBcm_run} is an arbitrary auxiliary scale and the cross section is manifestly independent of it at each order in resummed perturbation theory. This fact can be used to eliminate one of the evolution factors by setting $\mu$ equal to one of $\mu_H$, $\mu_B$, or $\mu_S$. The relevant factorization scales in the resummed result at which a fixed-order perturbative series is evaluated are the three scales $\mu_H$, $\mu_B$, and $\mu_S$. Hence, their dependence only cancels out up to the order one is working, and the residual dependence on these scales can be used to provide an improved estimate of theoretical uncertainties from higher orders in perturbation theory. The choice of scales used for our central value and to estimate the perturbative uncertainties is discussed in \subsec{profiles}. Finally, in \subsec{pi2pdf} we briefly discuss the effect that the $\pi^2$ summation and the order of the used PDFs have on our results. \subsection{Hard Virtual Corrections} \label{subsec:hard} The hard function contains hard virtual corrections at the scale of order $m_H$, including the virtual top-quark loop that generates the effective $ggH$ vertex. It is obtained by matching the full theory onto the effective Hamiltonian in SCET \begin{equation} \label{eq:Heff} \mathcal{H}_\mathrm{eff} = \frac{H}{12\pi v} \sum_{n_1, n_2}\int\!\mathrm{d}\omega_1\mathrm{d}\omega_2\, C_{ggH}(m_t, 2 {\tilde{b}} _1\cdot {\tilde{b}} _2)\, (2 {\tilde{b}} _1\cdot {\tilde{b}} _2) g_{\mu\nu}\, {\mathcal B}_{n_1,-\omega_1\perp}^{\mu c} {\mathcal B}_{n_2,-\omega_2\perp}^{\nu c} \,.\end{equation} Here, $H$ denotes the physical Higgs field and $v = (\sqrt{2}G_F)^{-1/2} = 246 \,\mathrm{GeV}$ the Higgs vacuum expectation value. The ${\mathcal B}_{n,\omega}^\mu$ fields are gauge invariant fields in SCET that describe energetic gluons with large momentum $ {\tilde{b}} _i = \omega_i n_i/2$, where $n_i$ are unit light-cone vectors, $n_i^2 = 0$. The matching coefficient $C_{ggH}$ depends on the top-quark mass and the invariant mass $2 {\tilde{b}} _1 \cdot {\tilde{b}} _2$ of the two gluons. For the case we are interested in we have $2 {\tilde{b}} _1\cdot {\tilde{b}} _2 = q^2$, where $q$ is the total momentum of the Higgs, i.e.\ of the $WW$ or $\gamma\gamma$ pair. In addition to the operator shown in \eq{Heff}, there are also operators where the Higgs couples to two collinear quark fields. The tree-level matching onto these operators is proportional to the light quark mass, $m_q$, and are numerically very small. There are potentially larger matching contributions from QCD loops where the Higgs couples to a top quark, but these are also $m_q/m_H$ suppressed due to helicity conservation. Hence, we neglect these collinear quark operators in our analysis. The hard function is defined as \begin{equation} H_{gg}(m_t, q^2, \mu) = \Abs{C_{ggH}(m_t, q^2, \mu)}^2 \,.\end{equation} It is evaluated at $q^2 = m_H^2$ in \eq{TauBcm_run} because we consider the production of an on-shell Higgs. (Including the decay of the Higgs, the cross section differential in $q^2$ is proportional to $\sigma_0 L H_{gg}(m_t, q^2, \mu)$, where $L$ contains the squared Higgs propagator and decay matrix element. In the narrow width approximation $L$ reduces to $L = \delta(q^2 - m_H^2) \mathrm{Br}$, where $\mathrm{Br}$ is the appropriate Higgs branching ratio, e.g.\ $\mathrm{Br}(H\to WW)$ or $\mathrm{Br}(H\to \gamma\gamma)$.) By matching onto \eq{Heff} we integrate out all degrees of freedom above the scale $\mu_H$, which are the heavy top quark as well as gluons and light quarks with offshellness above $\mu_H$. This can be done in either one or two steps. In the one-step matching used here we integrate out both the top quark and hard off-shell modes at the same time. This allows us to keep the full dependence on $m_H^2/m_t^2$. In pure dimensional regularization with $\overline{\text{MS}}$\xspace the matching coefficient $C_{ggH}(m_t, q^2, \mu_H)$ is given by the infrared-finite part of the full $m_t$-dependent $ggH$ form factor, which is known analytically at NLO (corresponding to two loops)~\cite{Harlander:2005rq, Anastasiou:2006hc} and in an expansion in $q^2/m_t^2$ at NNLO (three loops)~\cite{Harlander:2009bw, Pak:2009bx}. We write the Wilson coefficient as \begin{align} \label{eq:CggH} C_{ggH}(m_t, q^2, \mu_H) &= \alpha_s(\mu_H) F^{(0)}\Bigl(\frac{q^2}{4m_t^2}\Bigr) \biggl\{1 + \frac{\alpha_s(\mu_H)}{4\pi} \biggl[ C^{(1)}\Bigl(\frac{-q^2 - \mathrm{i} 0}{\mu_H^2} \Bigr) + F^{(1)}\Bigl(\frac{q^2}{4m_t^2}\Bigr) \biggr] \nonumber \\ & \quad + \frac{\alpha_s^2(\mu_H)}{(4\pi)^2} \biggl[ C^{(2)}\Bigl(\frac{-q^2 - \mathrm{i} 0}{\mu_H^2}, \frac{q^2}{4m_t^2} \Bigr) + F^{(2)}\Bigl(\frac{q^2}{4m_t^2}\Bigr) \biggr] \biggr\} \,,\end{align} where $F^{(0)}(0) = 1$. At NNLL we need the NLO coefficients \begin{equation} C^{(1)}(x_H) = C_A \Bigl(-\ln^2 x_H + \frac{\pi^2}{6} \Bigr) \,,\qquad F^{(1)}(0) = 5C_A - 3C_F \,.\end{equation} The dependence of $F^{(0)}(z)$ and $F^{(1)}(z)$ on $z = q^2/(4m_t^2)$, which encodes the $m_t$ dependence, is given in \eq{Ftop}. At NNLL+NNLO we also need to include the NNLO terms that depend logarithmically on the hard scale $\mu_H$, which follow from the two-loop RGE of the Wilson coefficient (see \eq{C_RGE}), and are given by \begin{align} C^{(2)}(x_H, z) &= \frac{1}{2} C_A^2 \ln^4 x_H + \frac{1}{3} C_A \beta_0 \ln^3 x_H + C_A\Bigl[\Bigl(-\frac{4}{3} + \frac{\pi^2}{6} \Bigr) C_A - \frac{5}{3}\beta_0 - F^{(1)}(z) \Bigr] \ln^2 x_H \nonumber \\ & \quad + \Bigl[\Bigl(\frac{59}{9} - 2\zeta_3 \Bigr) C_A^2 + \Bigl(\frac{19}{9}-\frac{\pi^2}{3}\Bigr) C_A \beta_0 - F^{(1)}(z) \beta_0 \Bigr] \ln x_H \,.\end{align} The remaining $\mu_H$-independent NNLO terms are contained in $F^{(2)}(z)$. Although these are known in an expansion in $z$, we do not include them, since the corresponding $\mu$-independent NNLO terms are not known for the beam and soft functions. To minimize the large logarithms in $C_{ggH}$ we should evaluate \eq{CggH} at the hard scale $\mu_H$ with $\abs{\mu_H^2} \sim q^2 \sim m_t^2$. For the simplest choice $\mu_H^2 = q^2$ the double logarithms of $-q^2/\mu_H^2$ are not minimized since they give rise to additional $\pi^2$ terms from the analytic continuation of the form factor from spacelike to timelike argument, $\ln^2(-1-\img0) = -\pi^2$, which causes rather large perturbative corrections. These $\pi^2$ terms can be summed along with the double logarithms by taking $\mu_H = -\mathrm{i} \sqrt{q^2}$ or in our case $\mu_H = -\mathrm{i} m_H$~\cite{Parisi:1979xd, Sterman:1986aj, Magnea:1990zb, Eynck:2003fn}. For Higgs production this method was applied in refs.~\cite{Ahrens:2008qu, Ahrens:2008nc}, where it was shown to improve the perturbative convergence of the hard matching coefficient. Starting at NNLO, the expansion of $C_{ggH}$ contains single logarithms $\ln(m_t^2/\mu_H^2)$, which in \eq{CggH} are contained as $\ln x_H$ in $C^{(2)}$ with a compensating $-\ln(-4z-\mathrm{i} 0)$ in $F^{(2)}(z)$, which are not large since $m_H/m_t \simeq 1$. In \eq{CggH}, $\alpha_s(\mu_H)$ is defined for $n_f = 5$ flavors. When written in terms of $\alpha_s(\mu_H)$ with $n_f = 6$ flavors similar $\ln(m_t^2/mu_H^2)$ terms would already appear at NLO. The additional terms that are induced by using an imaginary scale in these logarithms are small, because the imaginary part of $\alpha_s(-\mathrm{i} m_H)$ is much smaller than its real part. The alternative two-step matching is briefly discussed in \app{hardapp}, where we compare results with the literature. In this case, one first integrates out the top quark at the scale $m_t$ and then matches QCD onto SCET at the slightly lower scale $\mu_H \simeq m_H$. This allows one to sum the logarithms of $m_H/m_t$ at the expense of neglecting $m_H^2/m_t^2$ corrections. Since parametrically $m_H/m_t \simeq 1$, we use the one-step matching above. Note that we do not include electroweak corrections whose predominant effect (of order $5\%$) is on the normalization of the cross section through the hard function~\cite{Aglietti:2004nj, Actis:2008ug, Anastasiou:2008tj, Chiu:2008vv, Chiu:2009mg, Ahrens:2010rs}. Given the hard matching coefficient at the scale $\mu_H$ we use its renormalization group evolution to obtain it at any other scale $\mu$, \begin{equation} \label{eq:Hrun} H_{gg}(m_t, q^2, \mu) = H_{gg}(m_t, q^2, \mu_H)\, U_H(q^2, \mu_H, \mu) \,,\end{equation} where the evolution factor is given by \begin{align} U_H(q^2, \mu_H, \mu) &= \Bigl\lvert e^{K_H(\mu_H, \mu)} \Bigl(\frac{- q^2 - \mathrm{i} 0}{\mu_H^2}\Bigr)^{\eta_H(\mu_H, \mu)} \Bigr\rvert^2 \,,\nonumber \\ K_H(\mu_H,\mu) &= -2 K^g_\Gamma(\mu_H,\mu) + K_{\gamma_H^g}(\mu_H,\mu) \,, \qquad \eta_H(\mu_H,\mu) = \eta_\Gamma^g(\mu_H,\mu) \,,\end{align} and the functions $K_\Gamma^g(\mu_H, \mu)$, $\eta_\Gamma^g(\mu_H, \mu)$, and $K_\gamma(\mu_H, \mu)$ are given in \app{rgeapp}. They vanish for $\mu = \mu_H$ and therefore $U_H(q^2, \mu_H, \mu_H) = 1$, consistent with \eq{Hrun}. \subsection{Gluon Beam Function} \label{subsec:beam} The gluon beam function can be computed in an operator product expansion (OPE) in terms of standard gluon and quark PDFs (see \app{beamdef} for more details), \begin{equation} \label{eq:Bg_OPE2} B_g(t,x,\mu_B) = \sum_{j = \{g,q,\bar{q}\}} \int_x^1 \! \frac{\mathrm{d} \xi}{\xi}\, {\mathcal I}_{gj}\Bigl(t,\frac{x}{\xi},\mu_B \Bigr) f_j(\xi, \mu_B) \biggl[1 + \ORd{\frac{\Lambda_\mathrm{QCD}^2}{t}}\biggr] \,.\end{equation} In ref.~\cite{Fleming:2006cd} the ${\mathcal I}_{gg}$ matching coefficient was computed at one loop in moment space. The ${\mathcal I}_{gq}$ and ${\mathcal I}_{g\bar q}$ coefficients in the sum over $j$ in \eq{Bg_OPE2} describe the case where a quark or antiquark is taken out of the proton, it radiates a gluon which participates in the hard collision, and the quark or antiquark then continues into the final state. These mixing contributions start at one loop. Our one-loop calculation of ${\mathcal I}_{gj}$ for $j = \{g, q, \bar{q}\}$, which are needed for the gluon beam function in the $0$-jet Higgs cross section at NNLL, is given in some detail in \app{calculation} and follows the analogous computation of the quark beam function in ref.~\cite{Stewart:2010qs}. We write the matching coefficients for the gluon beam function as \begin{align} \label{eq:Ig} {\mathcal I}_{gg}(t,z,\mu_B) &= \delta(t)\,\delta(1-z) + \frac{\alpha_s(\mu_B)}{4\pi}\, {\mathcal I}_{gg}^{(1)}(t,z,\mu_B) + \frac{\alpha_s^2(\mu_B)}{(4\pi)^2}\, {\mathcal I}_{gg}^{(2)}(t,z,\mu_B) \,, \nonumber \\ {\mathcal I}_{gq}(t,z,\mu_B) &= \frac{\alpha_s(\mu_B)}{4\pi}\, {\mathcal I}_{gq}^{(1)}(t,z,\mu_B) + \frac{\alpha_s^2(\mu_B)}{(4\pi)^2}\, {\mathcal I}_{gq}^{(2)}(t,z,\mu_B) \,.\end{align} Our calculation in \app{calculation} yields the one-loop coefficients \begin{align} \label{eq:Ig_results} {\mathcal I}^{(1)}_{gg}(t,z,\mu_B) &= 2C_A\, \theta(z) \biggl\{ \frac{2}{\mu_B^2} {\mathcal L}_1\Bigl(\frac{t}{\mu_B^2}\Bigr) \delta(1-z) + \frac{1}{\mu_B^2} {\mathcal L}_0\Bigl(\frac{t}{\mu_B^2}\Bigr) P_{gg}(z) + \delta(t)\, {\mathcal I}_{gg}^{(1,\delta)}(z) \biggr\} \,, \nonumber \\ {\mathcal I}^{(1)}_{gq}(t,z,\mu_B) &= 2C_F\, \theta(z) \biggl\{\frac{1}{\mu_B^2} {\mathcal L}_0\Bigl(\frac{t}{\mu_B^2}\Bigr) P_{gq}(z) + \delta(t)\, {\mathcal I}_{gq}^{(1,\delta)}(z) \biggr\} \,,\end{align} where \begin{align} \label{eq:Igdel_results} {\mathcal I}_{gg}^{(1,\delta)}(z) &= {\mathcal L}_1(1-z) \frac{2(1-z + z^2)^2}{z} - P_{gg}(z) \ln z - \frac{\pi^2}{6} \delta(1-z) \,, \nonumber \\ {\mathcal I}_{gq}^{(1,\delta)}(z) &= P_{gq}(z)\ln \frac{1-z}{z} + \theta(1-z) z \,.\end{align} Here $P_{gg}(z)$ and $P_{gq}(z)$ are the $g\to gg$ and $q\to gq$ splitting functions given in \eq{split}, and the ${\mathcal L}_n(x)$ denote the standard plus distributions, \begin{align} \label{eq:cL} {\mathcal L}_n(x) = \bigg[\frac{\theta(x)\ln^n\!x}{x}\bigg]_+ \,,\end{align} defined in \eq{plusdef}. From \eq{Ig_results} we see that the proper scale to evaluate \eq{Bg_OPE2} is $\mu_B^2 \simeq t \simeq \mathcal{T}_\mathrm{cm} m_H$. For our final NNLL+NNLO result we also need the $\mu_B$-dependent terms of the two-loop coefficients, contained in ${\mathcal I}_{gg}^{(2)}$ and ${\mathcal I}_{gq}^{(2)}$. They can be computed from the two-loop RGE of the ${\mathcal I}_{gj}$ (see \eq{Igj_RGE}), which follows from the two-loop RGEs of the beam function and the PDFs. Our results for these coefficients are given in \app{beamapp}. Our result for ${\mathcal I}_{gg}^{(1)}$ is converted to moment space in \eq{hC}, and except for a $\pi^2$ term, agrees with the corresponding moment space result given in eq.~(68) of ref.~\cite{Fleming:2006cd}. Another comparison can be made by considering the correspondence with the $p_T$-dependent gluon beam function from ref.~\cite{Mantry:2009qz}, which is given in impact parameter space $y_T$ as $B_g(t_n,x,y_T,\mu)={\mathcal I}_{gg}(t_n,x/\xi,y_T,\mu)\otimes f_g(\xi,\mu)$. Taking the $y_T \to 0$ limit of their bare result should yield agreement with our bare beam function. In principle the renormalization could change in this limit, but their results indicate that this is not the case. Translating their variable $t_n$ into our variables, $t_n = t/z$, the $\lim_{y_T\to 0}{\mathcal I}_{gg}(t_n,z,y_T,\mu)$ from ref.~\cite{Mantry:2009qz} agrees with our result in \eq{Ig_results}. In ref.~\cite{Mantry:2010mk} the authors changed their variable definition from $t_n$ to our $t = t_n z$.\footnote{The translated result for ${\mathcal I}_{gg}$ quoted in ref.~\cite{Mantry:2010mk} has a typo, it is missing a term $-\delta(t) P_{gg}(z) \ln z$ induced by rescaling $(1/\mu^2) {\mathcal L}_0(t_n/\mu^2) P_{gg}(z)$. We thank S.~Mantry and F.~Petriello for confirming this.} Ref.~\cite{Mantry:2010mk} also calculates the other ${\mathcal I}_{ij}$ coefficients at one loop. Our result for the mixing contribution ${\mathcal I}_{gq}^{(1)}$ disagrees with the $y_T\to 0$ limit of their ${\mathcal I}_{gq}$. In particular, their constant term is ${\mathcal I}_{gq}^{(1,\delta)}(z) = - P_{gq}(z) \ln[(1-z)/z] + 2(1-z)/z$ which disagrees with ours in \eq{Igdel_results}. We have also compared the results of ref.~\cite{Mantry:2010mk} for the quark beam function for $y_T\to 0$ with our earlier results in refs.~\cite{Stewart:2009yx, Stewart:2010qs}. The coefficient ${\mathcal I}_{qq}^{(1)}$ agrees, but the mixing term ${\mathcal I}_{qg}^{(1)}$ also disagrees. The $\lim_{y_T\to 0} {\mathcal I}_{qg}^{(1)}$ result in ref.~\cite{Mantry:2010mk} is missing a term $-\delta(t) P_{qg}(z)$ that is present in refs.~\cite{Stewart:2009yx, Stewart:2010qs}. Given the beam function at the scale $\mu_B$ from \eq{Bg_OPE2}, we can evaluate it at any other scale using its renormalization group evolution~\cite{Stewart:2010qs} \begin{equation} \label{eq:Bgrun} B_g(t, x, \mu) = \int\!\mathrm{d} t'\, B_g(t - t', x, \mu_B)\, U_B(t', \mu_B, \mu) \,,\end{equation} with the evolution kernel \begin{align} \label{eq:U_def} U_B(t, \mu_B, \mu) &= \frac{e^{K_B-\gamma_E\, \eta_B}}{\Gamma(1+\eta_B)} \, \biggl[\frac{\eta_B}{\mu_B^2} {\mathcal L}^{\eta_B} \Bigl(\frac{t}{\mu_B^2}\Bigr) + \delta(t) \biggr] \,,\nonumber\\ K_B(\mu_B,\mu) &= 4 K^g_\Gamma(\mu_B,\mu) + K_{\gamma_B^g}(\mu_B,\mu) \,, \qquad \eta_B(\mu_B,\mu) = -2\eta^g_{\Gamma}(\mu_B,\mu) \,.\end{align} The plus distribution ${\mathcal L}^\eta(x) = [\theta(x)/x^{1-\eta}]_+$ is defined in \eq{plusdef}, and the functions $K_\Gamma^g(\mu_B, \mu)$, $\eta_\Gamma^g(\mu_B, \mu)$, and $K_\gamma(\mu_B, \mu)$ are given in \app{rgeapp}. Note that for $\mu = \mu_B$ we have $U_B(t, \mu_B, \mu_B) = \delta(t)$, which is consistent with \eq{Bgrun}. \begin{figure}[t] \includegraphics[width=0.485\textwidth]{plots/Bg_muB_NLO_lxl}% \hfill% \includegraphics[width=0.495\textwidth]{plots/Bg_muB_NLO_rel_lxl}% \vspace{-0.5ex} \caption{The gluon beam function integrated up to $t_\max = 0.1 (x\, 7 \,\mathrm{TeV})^2$. The left plot shows $x\widetilde{B}_g(t_\max, x, \mu_B)$. The right plot shows all results relative to the LO result. The solid lines show the LO and NLO results with the perturbative uncertainties shown by the bands. The dashed, dotted, and dot-dashed lines show the NLO result without quark contribution, in the large $x$ limit, and the small $x$ limit, respectively. See the text for further details.} \label{fig:Bg_muB} \end{figure} To illustrate our results for the gluon beam function we define its integral over $t \leq t_\max$, \begin{equation} \label{eq:tB_def} \widetilde{B}_g(t_\max, x, \mu_B) = \int\! \mathrm{d} t\, B_g(t, x, \mu_B)\,\theta(t_\max - t) \,.\end{equation} In \fig{Bg_muB} we plot $\widetilde{B}_g(t_\max, x, \mu_B)$ for a representative fixed value of $t_\max = 0.1 (x\, 7\,\mathrm{TeV})^2$. (Similar plots for the quark and antiquark beam functions can be found in refs.~\cite{Stewart:2009yx, Stewart:2010qs}.) The left panel shows $x \widetilde{B}_g(t_\max, x, \mu_B)$. The right panel shows the relative corrections to the LO result $\widetilde{B}_g^\mathrm{LO}(t_\max, x, \mu_B) = f_g(x, \mu_B)$. We use MSTW2008 NLO PDFs~\cite{Martin:2009iq} with their $\alpha_s(m_Z) = 0.12018$ and two-loop, five-flavor running for $\alpha_s$. The bands show the perturbative uncertainties from varying the matching scale $\mu_B$. Since at the scale $\mu_B$ there are no large logarithms in the beam function, the $\mu_B$ variation can be used as an indicator of higher-order perturbative uncertainties. At LO the only scale variation is that of the PDF and the minimum and maximum scale variation are obtained by $\mu_B = \{\sqrt{t_\max}/2,2\sqrt{t_\max}\}$ with $\mu_B = \sqrt{t_\max}$ the central value. At NLO the maximum variation does not occur at the endpoints of the range $\sqrt{t_\max}/2 \leq \mu_B \leq 2\sqrt{t_\max}$, but rather for approximately $\mu_B = \{0.8\sqrt{t_\max},2.0\sqrt{t_\max}\}$ with the central value at $\mu_B = 1.5\sqrt{t_\max}$. The $\alpha_s$ corrections to the gluon beam function are quite large, between $20\%$ to $40\%$, which is significantly larger than the $\sim 10\%$ corrections to the quark beam function. The main reason for this is the larger color factor for gluons than quarks. The size of the various perturbative contributions to the beam function is illustrated in~\fig{Bg_muB}. The dashed line shows the result obtained from ${\mathcal I}_{gg}$, without adding the mixing contribution ${\mathcal I}_{gq}$, using the same central value $\mu_B = 1.5\sqrt{t_\max}$. The mixing contributions are only relevant above $x \gtrsim 0.2$, and are suppressed at small $x$, because of their smaller color factor compared to ${\mathcal I}_{gg}$ and the dominance of the gluon PDF at small $x$. This means they will be numerically small for a light Higgs. The dotted line in~\fig{Bg_muB} shows the result in the threshold limit (again for $\mu_B = 1.5\sqrt{t_\max}$), where we drop ${\mathcal I}_{gq}$ and in addition only keep the terms in ${\mathcal I}_{gg}$ that are singular as $z \to 1$, and which are expected to become dominant as $x\to 1$ in \eq{Bg_OPE2}, \begin{align} \label{eq:I_thres} {\mathcal I}^{z\to 1}_{gg}(t,z,\mu_B) & = \delta(t)\delta(1-z) + \frac{\alpha_s(\mu_B)}{2\pi}\, C_A\, \theta(z) \biggl\{ \frac{2}{\mu_B^2} {\mathcal L}_1\Bigl(\frac{t}{\mu_B^2}\Bigr) \delta(1-z) \nonumber \\ &\qquad + \frac{2}{\mu_B^2} {\mathcal L}_0\Bigl(\frac{t}{\mu_B^2}\Bigr) {\mathcal L}_0(1-z) + \delta(t)\Bigl[ 2 {\mathcal L}_1(1-z) - \frac{\pi^2}{6} \delta(1-z) \Bigr] \biggr\} \,.\end{align} The dotted line indeed approaches the dashed line for $x \gtrsim 0.2$. However, since in this region the mixing contributions become important, the threshold result does not provide a good approximation to the full result (solid line) anywhere. Finally, the dot-dashed line in~\fig{Bg_muB} shows the result only keeping the terms singular as $z\to 0$ (but including tree level), \begin{align} \label{eq:I_small_x} {\mathcal I}^{z\to 0}_{gg}(t,z,\mu_B) &= \delta(t)\delta(1-z) + \frac{\alpha_s(\mu_B)}{\pi}\, C_A\, \theta(z) \theta(1-z) \biggl[ \frac{1}{\mu_B^2} {\mathcal L}_0\Bigl(\frac{t}{\mu_B^2}\Bigr) \frac{1}{z} - \delta(t) \frac{\ln z}{z} \biggr] \,, \nonumber \\ {\mathcal I}^{z\to 0}_{gq}(t,z,\mu_B) &= \frac{\alpha_s(\mu_B)}{\pi}\, C_F\, \theta(z) \theta(1-z) \biggl[ \frac{1}{\mu_B^2} {\mathcal L}_0\Bigl(\frac{t}{\mu_B^2}\Bigr) \frac{1}{z} - \delta(t) \frac{\ln z}{z} \biggr] \,,\end{align} which one might expect to dominate for $x\to 0$. Since these only have single-logarithmic $\mu$ dependence, their central value is obtained for $\mu_B = \sqrt{t_\max}$. This contribution indeed grows towards smaller $x$, and makes up more than half of the total contribution at $x=0.01$, but does not yet dominate. \subsection{Soft Function} \label{subsec:soft} The soft function $S_B^{gg}(k, \mu_S)$ appearing in \eq{TauBcm_run} is defined by the vacuum matrix element of a product of Wilson lines. For $k \simeq \mathcal{T}_\mathrm{cm} \gg \Lambda_\mathrm{QCD}$, it can be computed in perturbation theory. We write the perturbative soft function as \begin{equation} S^{gg}_{\rm pert}(k,\mu_S) = \delta(k) + \frac{\alpha_s(\mu_S)}{\pi}\, C_A\, S^{(1)}_{gg}(k, \mu_S) + \frac{\alpha_s^2(\mu_S)}{\pi^2}\, C_A\, S_{gg}^{{(2)}}(k, \mu_S) + \ORd{\frac{\alpha_s^3}{k}} \,,\end{equation} where the one- and two-loop coefficients are \begin{align} \label{eq:Scoeff12} S_{gg}^{(1)}(k, \mu_S) &= -\frac{4}{\mu_S} {\mathcal L}_1\Bigl(\frac{k}{\mu_S}\Bigr) + \frac{\pi^2}{12}\, \delta(k) \,,\nonumber\\ S_{gg}^{(2)}(k, \mu_S) &= 8 C_A\, \frac{1}{\mu_S} {\mathcal L}_3 \Bigl(\frac{k}{\mu_S}\Bigr) + \beta_0\, \frac{1}{\mu_S} {\mathcal L}_2 \Bigl(\frac{k}{\mu_S}\Bigr) - \Bigl[\Bigl( \frac{4}{3} + \frac{8\pi^2}{3} \Bigr) C_A + \frac{5}{3} \beta_0 \Bigr] \frac{1}{\mu_S} {\mathcal L}_1 \Bigl(\frac{k}{\mu_S}\Bigr) \nonumber \\ & \quad + \Bigl[\Bigl(\frac{8}{9} + \frac{25}{2} \zeta_3\Bigr)C_A + \Bigl(\frac{7}{9} - \frac{\pi^2}{12}\Bigr) \beta_0\Bigr] \frac{1}{\mu_S} {\mathcal L}_0 \Bigl(\frac{k}{\mu_S}\Bigr) + S^{(2,\delta)}_{gg} \delta(k) \,,\end{align} In ref.~\cite{Stewart:2009yx} the quark beam-thrust soft function was obtained from the one-loop hemisphere soft function for outgoing jets~\cite{Schwartz:2007ib, Fleming:2007xt}. The gluon beam-thrust soft function has Wilson lines in the adjoint rather than fundamental representation and at one loop $S^{gg}_{{\rm pert}}$ is obtained from the quark result by simply replacing $C_F$ by $C_A$. The $\mu_S$-dependent terms needed at NNLL+NNLO are shown in \eq{Scoeff12} and were obtained by perturbatively solving the two-loop RGE of the soft function (see \eq{S_RGE}). The determination of the $\mu_S$-independent constant term, $S^{(2,\delta)}_{gg} \delta(k)$, requires the two-loop calculation of the soft function. The RG evolution of the soft function has the same structure as that of the beam function, \begin{equation} \label{eq:SBrun} S_B^{gg}(k,\mu) = \int\! \mathrm{d} k'\, S_B^{gg}(k - k',\mu_S)\, U_S(k',\mu_S,\mu) \,,\end{equation} with the evolution kernel \begin{align} U_S(k, \mu_S, \mu) & = \frac{e^{K_S -\gamma_E\, \eta_S}}{\Gamma(1 + \eta_S)}\, \biggl[\frac{\eta_S}{\mu_S} {\mathcal L}^{\eta_S} \Bigl( \frac{k}{\mu_S} \Bigr) + \delta(k) \biggr] \,, \nonumber \\ K_S(\mu_S,\mu) &= -4K_\Gamma^g(\mu_S,\mu) + K_{\gamma_S^g}(\mu_S,\mu) \,, \qquad \eta_S(\mu_S,\mu) = 4\eta_\Gamma^g(\mu_S,\mu) \,,\end{align} where ${\mathcal L}^\eta(x) = [\theta(x)/x^{1-\eta}]_+$ is defined in \eq{plusdef}, and $K_\Gamma^g(\mu_S, \mu)$, $\eta_\Gamma^g(\mu_S, \mu)$, and $K_\gamma(\mu_S, \mu)$ are given in \app{rgeapp}. The nonperturbative corrections can be modeled and included using the methods of refs.~\cite{Hoang:2007vb, Ligeti:2008ac}. The perturbative component $S_\mathrm{pert}^{gg}$ and nonperturbative component $F^{gg}$ of the soft function can be factorized as \begin{equation} \label{eq:SF} S_{B}^{gg}(k,\mu_S) = \int\! \mathrm{d} k'\, S_\mathrm{pert}^{gg}(k-k',\mu_S)\, F^{gg}(k') \,. \end{equation} At small $\mathcal{T}_\mathrm{cm} \sim \Lambda_\mathrm{QCD}$ the nonperturbative corrections to the soft function are important. When the spectrum is dominated by perturbative momenta with $\mathcal{T}_\mathrm{cm} \gg\Lambda_\mathrm{QCD}$ \eq{SF} can be expanded in an OPE as \begin{align}\label{eq:OPE} S_B^{gg}(k,\mu_S) = S^{gg}_\mathrm{pert}(k,\mu_S) - 2\Omega_1^{gg}\, \frac{\mathrm{d} S^{gg}_\mathrm{pert}(k,\mu_S)}{\mathrm{d} k} +\ORd{\frac{\Lambda_\mathrm{QCD}^2}{k^3}} \,,\end{align} where the leading power correction is determined by the dimension-one nonperturbative parameter $\Omega_1^{gg}= \int\!\mathrm{d} k'\, (k'/2) F^{gg}(k')$ which is parametrically $\ord{\Lambda_\mathrm{QCD}}$. The positivity of $F^{gg}(k)$ implies that $\Omega_1^{gg}>0$, so the factorization in \eq{SF} predicts the sign of the correction caused by the nonperturbative effects. We will see that this simple OPE result with one nonperturbative parameter $\Omega_1^{gg}$ gives an accurate description of the nonperturbative effects in the $\mathcal{T}_\mathrm{cm}$ spectra for the entire region we are interested in, which includes the peak in the distribution. The OPE in \eq{OPE} implies that the leading nonperturbative effects can be computed as an additive correction to the spectrum \begin{equation} \label{eq:OPEdsdT} \frac{\mathrm{d}\sigma^\sing}{\mathrm{d}\mathcal{T}_\mathrm{cm}} = \frac{\mathrm{d}\sigma^\sing_\mathrm{pert}}{\mathrm{d}\mathcal{T}_\mathrm{cm}} - 2\Omega_1^{gg} \frac{\mathrm{d}^2\sigma^\sing_\mathrm{pert}}{\mathrm{d}\mathcal{T}_\mathrm{cm}^2} \,, \end{equation} and likewise for the cumulant \begin{align} \label{eq:OPEsigc} \sigma^\sing(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) = \sigma^\sing_\mathrm{pert}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) - 2\Omega_1^{gg} \frac{\mathrm{d}}{\mathrm{d}\mathcal{T}_\mathrm{cm}^\mathrm{cut}} \sigma^\sing_\mathrm{pert}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) \,. \end{align} To first order in the OPE expansion this is equivalent to a shift in the variable used to evaluate the perturbative spectrum, $\mathcal{T}_\mathrm{cm} \to \mathcal{T}_\mathrm{cm} - 2 \Omega_1^{gg}$, or cumulant, $\mathcal{T}_\mathrm{cm}^\mathrm{cut}\to \mathcal{T}_\mathrm{cm}^\mathrm{cut}-2\Omega_1^{gg}$. For the cumulant the nonperturbative corrections always reduce the cross section, whereas the distribution is reduced before the peak and increased in the tail region. Since the nonsingular terms in the cross section are an order of magnitude smaller than the singular terms we can also replace $\sigma^\sing$ by $\sigma$, that is include the nonsingular $\mathrm{d}\sigma^\ns/\mathrm{d}\mathcal{T}_\mathrm{cm}$ in \eq{OPEdsdT}. For simplicity, we will use the purely perturbative result in most of our numerical analysis. However, in \subsec{Nonperturbative} we will use \eqs{SF}{OPEdsdT} to analyze the effect of nonperturbative corrections on our predictions. \subsection{Nonsingular Contributions} \label{subsec:nonsingular} In this section we discuss how we incorporate the nonsingular contributions to the cross section using fixed-order perturbation theory. For the two beam thrust cross sections considered in this paper, the full cross section in fixed-order perturbation theory can be written as \begin{align} \label{eq:sigmaFO} \frac{\mathrm{d} \sigma}{\mathrm{d} \tau \mathrm{d} Y} &= \sigma_0\, \alpha_s^2(\mu) \Bigl\lvert F^{(0)}\Bigl(\frac{m_H^2}{4m_t^2}\Bigr)\Bigr\rvert^2 \nonumber\\ & \quad\times \int\! \frac{\mathrm{d} \xi_a}{\xi_a}\, \frac{\mathrm{d} \xi_b}{\xi_b}\, \sum_{i,j} C_{ij}\Bigl(\frac{x_a}{\xi_a},\frac{x_b}{\xi_b},\tau, Y, \mu, m_H, m_t\Bigr)\, f_i(\xi_a,\mu)\, f_j(\xi_b,\mu) \,,\end{align} where $i,j={g,q,\bar q}$ sum over parton types, and $\tau = \mathcal{T}_\mathrm{cm}/m_H$.\footnote{For $H\to\gamma\gamma$ where the boost between the partonic and hadronic center-of-mass frames is accounted for with $\tau_B=\mathcal{T}_B/m_H$, the appropriate replacements in \eq{sigmaFO} are to take $\tau\to \tau_B$, and in the fourth argument of $C_{ij}$ to set $Y=0$.} To simplify the notation in the following we will suppress the $m_H$ and $m_t$ dependence of the coefficients $C_{ij}$. The contributions to the $C_{ij}$ can be separated into singular and nonsingular parts, \begin{equation} \label{eq:Cijdecomp} C_{ij}(z_a,z_b,\tau, Y,\mu) = C_{ij}^\sing(z_a,z_b,\tau, Y,\mu) + C_{ij}^\ns(z_a,z_b,\tau, Y,\mu) \,,\end{equation} where the singular terms scale as $\sim 1/\tau$ modulo logarithms and can be written as \begin{equation} \label{eq:Cijsing} C_{ij}^\sing(z_a,z_b,\tau, Y,\mu) = C_{ij}^{-1} (z_a,z_b,Y,\mu)\,\delta(\tau) + \sum_{k\geq 0} C_{ij}^k (z_a,z_b,Y,\mu)\, {\mathcal L}_k(\tau) \,,\end{equation} where ${\mathcal L}_k(\tau) = [\theta(\tau)(\ln^k\!\tau)/\tau]_+$ is defined in \eq{plusdef}. The resummed result for the cross section in \eq{TauBcm_run} sums the singular contributions at small $\tau$ to all orders, counting $(\alpha_s\ln\tau)\sim 1$. The nonsingular contributions, $C_{ij}^\ns$, are suppressed relative to the singular ones by $\ord{\tau}$ and it suffices to determine them in fixed-order perturbation theory. Hence, we can obtain them by simply subtracting the fixed-order expansion of the singular result from the full fixed-order result, \begin{equation} \frac{\mathrm{d}\sigma^{\ns,\mathrm{FO}}}{\mathrm{d} \tau} = \frac{\mathrm{d}\sigma^\mathrm{FO}}{\mathrm{d} \tau} - \frac{\mathrm{d}\sigma^{\sing,\mathrm{FO}}}{\mathrm{d} \tau} \,,\end{equation} and analogously for the cumulant \begin{align} \sigma^{\ns,\mathrm{FO}}(\tau^\mathrm{cut}) = \sigma^\mathrm{FO}(\tau^\mathrm{cut})-\sigma^{\sing,\mathrm{FO}}(\tau^\mathrm{cut}) \,.\end{align} We must take care to use the same PDFs and renormalization scale $\mu$ for both the $\sigma^\mathrm{FO}$ and $\sigma^{\sing,\mathrm{FO}}$ terms. In our analysis we obtain $\sigma^\mathrm{NLO}(\tau^\mathrm{cut})$ and $\sigma^\mathrm{NNLO}(\tau^\mathrm{cut})$ numerically using the publicly available \textsc{FEHiP}\xspace program~\cite{Anastasiou:2004xq, Anastasiou:2005qj}, which allows one to obtain the fixed-order NNLO Higgs production cross section for generic phase-space cuts. At tree level, the only nonzero coefficient is $C_{ij}^{-1}$, and the nonsingular contribution vanishes, $\sigma^{\ns,\mathrm{LO}}(\tau^\mathrm{cut}) = 0$. At NLO, the $C_{ij}^k$ are nonzero for $k \leq 1$ and are fully contained in the resummed NNLL result for $\mathrm{d}\sigma^\sing/\mathrm{d}\tau$. Hence, we can obtain them by expanding our NNLL singular result to fixed next-to-leading order, \begin{equation} \label{eq:singNLO} \sigma^{\sing,\mathrm{NLO}}(\tau^\mathrm{cut}) = \sigma^{\sing,\mathrm{NNLL}}(\tau^\mathrm{cut}) \big\vert_\mathrm{NLO} \,.\end{equation} The explicit expressions are given in \app{singular}. Subtracting this from the full NLO result we get the nonsingular contribution at NLO, \begin{equation} \label{eq:nsNLO} \sigma^{\ns,\mathrm{NLO}}(\tau^\mathrm{cut}) = \sigma^\mathrm{NLO}(\tau^\mathrm{cut}) - \sigma^{\sing,\mathrm{NNLL}}(\tau^\mathrm{cut}) \big\vert_\mathrm{NLO} \,.\end{equation} In the left panel of \fig{nonsing} we plot the NLO nonsingular cross section determined by this procedure for three different choices of $\mu$, namely $\mu=m_H/2, m_H, 2m_H$. The scaling of the nonsingular distribution implies that it involves only integrable functions, therefore the cumulant $\sigma^{\ns,\mathrm{NLO}}(\tau^\mathrm{cut})$ vanishes for $\tau^\mathrm{cut} \to 0$. The $\mu$ dependence of the NLO nonsingular cross section is sizeable since this is the leading term in this part of the cross section. This $\mu$ dependence is canceled by the nonsingular terms at NNLO which we turn to next. \begin{figure}[t] \includegraphics[width=0.485\textwidth]{plots/ggHsignons_7_165_NLO_lxl}% \hfill% \includegraphics[width=0.495\textwidth]{plots/ggHsigres_7_165_NNLO_lxl}% \vspace{-0.5ex} \caption{The left panel shows the nonsingular contribution to the NLO cross section as a function of $\mathcal{T}_B^{\cm,\mathrm{cut}}$, for the LHC at $7\,\mathrm{TeV}$. The residual NNLO cross section shown in the right panel, is the nonsingular NNLO cross section, $\sigma^{\ns,\mathrm{NNLO}}$, plus a constant term $c^\res$, as explained in the text.} \label{fig:nonsing} \end{figure} At NNLO the singular cross section is determined by a result analogous to \eq{singNLO} \begin{align} \label{eq:singNNLO} \sigma^{\sing,\mathrm{NNLO}}(\tau^\mathrm{cut}) = \sigma^{\sing,\mathrm{NNLL}}(\tau^\mathrm{cut}) \big\vert_{\mathrm{NNLO}, k\geq 0} + \sigma^{\sing,\mathrm{NNLO}}\big\vert_{k = -1} \,.\end{align} The NNLO singular coefficients $C_{ij}^k$ are nonzero for $-1\leq k \leq 3$. Those for $k \ge 0$ can be obtained by expanding the singular NNLL result to fixed NNLO. Their explicit expressions are given in \app{singular}. For $k=-1$ the NNLO contribution to the coefficient $C_{ij}^{-1}$ of the $\delta(\tau)$ is not fully contained in the NNLL result. Therefore, the $\tau^\mathrm{cut}$-independent $k=-1$ contribution is not included in the first term on the right-hand side of \eq{singNNLO}, but in the second term. For this second term we proceed as follows. First, we write the NNLO contribution to $C_{ij}^{-1}$ as \begin{equation} \label{eq:Cm1terms} C_{ij}^{-1} (z_a,z_b, Y,\mu)\big\vert_\mathrm{NNLO} = \frac{\alpha_s^2(\mu)}{(2\pi)^2} \Bigl[ c_{ij}^\pi(z_a,z_b,Y) + c_{ij}^{\mu}(z_a,z_b,Y, \mu) + c_{ij}^{\res}(z_a,z_b,Y) \Bigr] \,.\end{equation} The first term in brackets denotes the $\mu$-independent terms proportional to $\pi^2$ that are part of the $\pi^2$ summation. The second term contains all terms proportional to $\ln(\mu/m_H)$, which cancel the $\mu$ dependence in the NLO result. We can obtain these two contributions analytically, and they are given in \app{singular}. The remaining $\mu$-independent terms, $c_{ij}^{\res}$, are currently not known analytically. They could be obtained when the complete NNLO results for the hard, beam, and soft functions become available. Using \eq{Cm1terms}, the second term on the right-hand side of \eq{singNNLO} is given by \begin{equation} \sigma^{\sing,\mathrm{NNLO}}\big\vert_{k = -1} = c^\pi(\mu) + c^\mu(\mu) + c^\res(\mu) \,,\end{equation} with ($x=\{\pi, \mu,\res\}$) \begin{align} c^x(\mu) &= \sigma_0\,\frac{\alpha_s^4(\mu)}{(2\pi)^2} \Bigl\lvert F^{(0)}\Bigl(\frac{m_H^2}{4m_t^2}\Bigr)\Bigr\rvert^2 \int\!\mathrm{d} Y\!\int\! \frac{\mathrm{d} \xi_a}{\xi_a} \frac{\mathrm{d} \xi_b}{\xi_b}\,\sum_{i,j} c_{ij}^{x} \Bigl(\frac{x_a}{\xi_a},\frac{x_b}{\xi_b}, Y, \mu \Bigr)\, f_i(\xi_a,\mu)\, f_j(\xi_b,\mu) \,.\end{align} The $\mu$ dependence of $c^\mu(\mu)$ cancels that of $\sigma^{\ns,\mathrm{NLO}}(\tau^\mathrm{cut})$ up to terms of $\ord{\alpha_s^5}$, whereas that of $c^\pi(\mu)$ and $c^\res(\mu)$ only starts at $\ord{\alpha_s^5}$. Since $x_{a,b} = (m_H/E_\mathrm{cm})e^{\pm Y}$, the $c^x(\mu)$ have a nontrivial dependence on $m_H$. To determine the constant $c^\res$ numerically, we now consider \begin{equation} \sigma^\res(\tau^\mathrm{cut}) \equiv \sigma^\mathrm{NNLO}(\tau^\mathrm{cut}) - \sigma^{\sing,\mathrm{NNLL}}(\tau^\mathrm{cut}) \Big\vert_{\mathrm{NNLO},k\geq 0} - c^\pi - c^\mu = c^\res + \sigma^{\ns,\mathrm{NNLO}}(\tau^\mathrm{cut}) \,.\end{equation} Since $\sigma^{\ns,\mathrm{NNLO}}(\tau^\mathrm{cut})$ vanishes as $\tau^\mathrm{cut} \to 0$, the coefficient $c^\res$ is determined by $\sigma^\res(\tau^\mathrm{cut})$ as $\tau^\mathrm{cut}\to 0$, while the nonsingular corrections are given by the remainder $\sigma^\res(\tau^\mathrm{cut}) - c^\res$. Hence, we can obtain both by fitting our numerical results for $\sigma^\mathrm{res}(\tau^\mathrm{cut})$ at different values of $\tau^\mathrm{cut}$ to the following function: \begin{equation} \label{eq:fit_ns} \sigma^\mathrm{res}(\tau) = c^\res + a_0\, \tau \ln\tau + a_1\, \tau + a_2\, \tau^2 \ln \tau + a_3\, \tau^2 \,.\end{equation} The $a_0$ through $a_3$ terms are sufficient to describe the $\tau^\mathrm{cut}$ dependence of $\sigma^{\ns,\mathrm{NNLO}}(\tau^\mathrm{cut})$ over the whole range of $\tau^\mathrm{cut}$. The results of the fit for $pp$ collisions at $7\,\mathrm{TeV}$ and $m_H=165\,\mathrm{GeV}$ for $\mu=m_H$, $\mu=m_H/2$, and $\mu=2 m_H$ are shown in the right panel of \fig{nonsing}. At $\mu=m_H$ this fit gives \begin{align} c^\res &= 0.86\pm 0.02\,, &a_0 &= 7.6 \pm 0.6 \,, &a_1 &= 9.3 \pm 1.5 \,, &a_2 &= 3.9 \pm 1.1 \,, &a_3 &= -9.9 \pm 1.6 \,. \end{align} Similarly, for $p\bar p$ collisions at $1.96\,\mathrm{TeV}$, $m_H = 165\,\mathrm{GeV}$, and $\mu = m_H$, we obtain \begin{align} c^\res &= 0.028\pm 0.001\,, &a_0 &= 0.28 \pm 0.03 \,, &a_1 &= 0.44 \pm 0.08 \,, &a_2 &= 0.18 \pm 0.06 \,, \nonumber\\ &&a_3 &= -0.42 \pm 0.08 \,. \end{align} We have checked that we obtain the same values for $c^\res$ within the uncertainties when the fit range is restricted to the region of small $\tau^\mathrm{cut}$. (In this case the fit is not sensitive to $a_2$ and $a_3$, so either one or both of them must be set to zero.) Note that $\sigma^\mathrm{NNLO}(\tau^\mathrm{cut})$ and $\sigma^{\sing,\mathrm{NNLL}}\vert_\mathrm{NNLO}$ both diverge as $\tau^\mathrm{cut} \to 0$. The fact that the difference in $\sigma^\res(\tau^\mathrm{cut})$ does not diverge for $\tau^\mathrm{cut} \to 0$ provides an important cross check between our analytic results for the NNLO singular terms and the full numerical NNLO result. The numerical uncertainty from fitting $c^\res(\mu)$ is much smaller than its $\mu$ dependence, and so can be ignored for our final error analysis. The $\mu$ dependence of $c^\res(\mu)$ comes from the PDFs and the overall $\alpha_s^4(\mu)$, where the latter is by far the dominant effect. To see this we can rescale $c^\res(m_H/2) = 1.35$ and $c^\res(2m_H) = 0.58$ as $c^\res(m_H/2) \alpha_s^4(m_H)/\alpha_s^4(m_H/2) = 0.91$ and $c^\res(2m_H) \alpha_s^4(m_H)/\alpha_s^4(2m_H) = 0.83$, giving values that are very close to the central value $c^\res(m_H) = 0.86$. Having determined the nonsingular contributions to the cross section we can compare their size to the dominant singular terms. In \fig{singcomp} we plot the singular, nonsingular, and full cross sections at NNLO for $\mu = m_H$. The left panel shows the absolute value of these components of the differential cross sections (obtained by taking the derivative of $\sigma(\tau^\mathrm{cut})$ with respect to $\tau^\mathrm{cut}$). For $\mathcal{T}_\mathrm{cm} \ll m_H$ the nonsingular terms are an order of magnitude smaller than the singular ones. On the other hand for $\mathcal{T}_\mathrm{cm} \gtrsim m_H/2$ the singular and nonsingular terms become equally important and there is a large cancellation between the two contributions. These features of the fixed-order cross section will have implications on our choice of running scales discussed in \subsec{profiles}. \begin{figure}[t] \includegraphics[width=0.505\textwidth]{plots/ggHdsdTauBcm_7_165_compsing_lxl}% \hfill% \includegraphics[width=0.475\textwidth]{plots/ggHsigTauBcmcut_7_165_compsing_lxl}% \vspace{-0.5ex} \caption{Comparison of the singular, nonsingular, and full cross sections at NNLO for $\mu = m_H$. The left panel shows the magnitude of the differential cross sections on a logarithmic scale. The right panel shows the corresponding cumulant cross sections.} \label{fig:singcomp} \end{figure} To determine the singular NNLO contributions in \eq{singNNLO} for the above analysis we only considered the $k\geq 0$ terms contained in $\sigma^{\sing,\mathrm{NNLL}}$. Of course $\sigma^{\sing,\mathrm{NNLL}}$ also contains some $k = -1$ terms at NNLO, in particular the $c^\pi(\mu)$ and $c^\mu(\mu)$ contributions, but also parts of the $c^\res(\mu)$ contribution from cross terms between the NLO matching corrections. Since we know $c^\res(\mu)$ numerically, we are able to determine the missing $k = -1$ contribution at NNLO numerically, which corresponds to the sum of the unknown $\mu$-independent NNLO matching corrections to the hard, beam, and soft functions. It is given by the difference \begin{equation} \label{eq:cdelta} c^\delta(\mu) = \sigma^{\sing,\mathrm{NNLO}} - \sigma^{\sing,\mathrm{NNLL}} \big\vert_\mathrm{NNLO} = c^\pi(\mu) + c^\mu(\mu) + c^\res(\mu) - \sigma^{\sing,\mathrm{NNLL}} \big\vert_{\mathrm{NNLO},k = -1} \,.\end{equation} Since we include the $\mu$-dependent NNLO matching corrections in $\sigma^{\sing,\mathrm{NNLL}}$, its NNLO expansion is obtained by setting $\mu_S = \mu_B = \mu_H = \mu$. Thus, we can easily evaluate \eq{cdelta} numerically. For $m_H=165 \,\mathrm{GeV}$, we find for the LHC at $7\,\mathrm{TeV}$, \begin{equation} c^\delta(m_H/2) = 0.002 \,,\qquad c^\delta(m_H) = -0.035 \,,\qquad c^\delta(2m_H) = -0.028 \,,\end{equation} and for the Tevatron, \begin{equation} c^\delta(m_H/2) = -0.0043 \,,\qquad c^\delta(m_H) = -0.0026 \,,\qquad c^\delta(2m_H) = -0.0027 \,.\end{equation} Comparing this to $c^\res(m_H) = 0.86$ (LHC) and $c^\res(m_H) = 0.028$ (Tevatron), we see that these coefficients are almost fully accounted for by cross terms between the NLO hard, beam, and soft functions. The remaining NNLO terms in $c^\delta$ are in fact very small, and our NNLL+NNLO results are therefore numerically very close to the complete NNLL$^\prime$+NNLO result. \subsection{Cross Section at NNLL+NNLO} \label{subsec:NNLLNNLO} Using the results of sections~\ref{subsec:hard} to \ref{subsec:nonsingular} our final result at NNLL+NNLO for the distribution and cumulant is obtained as \begin{align} \label{eq:NNLL_NNLO} \frac{\mathrm{d}\sigma^\mathrm{NNLL+NNLO}}{\mathrm{d} \mathcal{T}_\mathrm{cm}} &= \frac{\mathrm{d}\sigma^{\sing,\mathrm{NNLL}}}{\mathrm{d} \mathcal{T}_\mathrm{cm}} + \frac{\mathrm{d}\sigma^\mathrm{\delta}}{\mathrm{d} \mathcal{T}_\mathrm{cm}} + \frac{\mathrm{d}\sigma^{\ns,\mathrm{NNLO}+\pi^2}}{\mathrm{d} \mathcal{T}_\mathrm{cm}} \,,\nonumber\\[1ex] \sigma^\mathrm{NNLL+NNLO}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) &= \sigma^{\rm s,NNLL}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) + \sigma^\delta(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) + \sigma^{\ns,\mathrm{NNLO}+\pi^2}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) \,.\end{align} The first term in each equation contains the resummed singular result obtained from \eq{TauBcm_run} to NNLL order, including the $\mu$-dependent NNLO matching corrections. The last term contains the NNLO nonsingular corrections determined in the previous subsection, but including $\pi^2$ summation by using \begin{align} \label{eq:NNLOnspi2} \frac{\mathrm{d}\sigma^{\ns,\mathrm{NNLO}+\pi^2}}{\mathrm{d} \mathcal{T}_\mathrm{cm}} &= U_H(m_H^2,-\mathrm{i}\mu_\ns,\mu_\ns) \biggl[ \frac{\mathrm{d}\sigma^{\ns,\mathrm{NNLO}}}{\mathrm{d} \mathcal{T}_\mathrm{cm}} - \frac{\alpha_s(\mu_\ns) C_A}{2\pi}\, \pi^2\, \frac{\mathrm{d}\sigma^{\ns,\mathrm{NLO}}}{\mathrm{d} \mathcal{T}_\mathrm{cm}} \biggr] \,, \nonumber\\ \sigma^{\ns,\mathrm{NNLO}+\pi^2}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) &= U_H(m_H^2,-\mathrm{i}\mu_\ns,\mu_\ns) \Bigl[ \sigma^{\ns,\mathrm{NNLO}}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) - \frac{\alpha_s(\mu_\ns) C_A}{2\pi}\, \pi^2 \sigma^{\ns,\mathrm{NLO}}(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) \Bigr] \,. \end{align} Here $U_H(m_H^2,-\mathrm{i}\mu_\ns,\mu_\ns) = \exp[\alpha_s(\mu_\ns) C_A\pi/2 + \ldots]$ contains the $\pi^2$ summation. There are two reasons we include the $\pi^2$ summation for the nonsingular terms. First, using SCET one can derive factorization theorems for the nonsingular terms when $\mathcal{T}_\mathrm{cm} \ll m_H$, and the results will involve a combination of leading and subleading hard, jet, and soft functions. Many of these terms will have the same LL evolution for their hard functions, and hence they predominantly require the same $\pi^2$ summation. As a second reason we observe from \fig{singcomp} that there are important cancellations between the singular and nonsingular cross sections for $\mathcal{T}_\mathrm{cm} \gtrsim m_H/2$. Since the $\pi^2$ summation modifies the cross section for all $\mathcal{T}_\mathrm{cm}$ and $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ values it is important to include it also in the nonsingular terms to not spoil these cancellations. The middle terms in \eq{NNLL_NNLO} incorporate the singular NNLO terms that are not reproduced by our resummed NNLL result. At fixed order, they are given by \begin{equation} \frac{\mathrm{d}\sigma^\mathrm{\delta}}{\mathrm{d} \mathcal{T}_\mathrm{cm}}\bigg\vert_\mathrm{NNLO} = c^\delta(\mu)\,\delta(\mathcal{T}_\mathrm{cm}) \,,\qquad \sigma^\delta(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) \bigg\vert_\mathrm{NNLO} = c^\delta(\mu) \,.\end{equation} As we saw in \subsec{nonsingular}, $c^\delta(\mu)$ turns out to be very small numerically, which means we might as well neglect the $\sigma^\delta$ term entirely. We include it for completeness in \eq{NNLL_NNLO}, in order to formally reproduce the complete NNLO cross section. In fact, at this level other contributions that we neglect here, such as the bottom-quark contributions or electroweak corrections, are likely more relevant numerically. Formally, $c^\delta(\mu)$ is reproduced by the complete two-loop matching required at NNLL$'$ or N$^3$LL. When it is properly incorporated into the predictions at that order it is multiplied by a Sudakov exponent that ensures that the total $\sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})\to 0$ as $\mathcal{T}_\mathrm{cm}^\mathrm{cut}\to 0$. Hence, we can include it in the resummed result by multiplying it with the NNLL evolution factors, \begin{align} \label{eq:de_sm} \frac{\mathrm{d}\sigma^\delta}{\mathrm{d} \mathcal{T}_\mathrm{cm}} &= c^\delta(\mu_{\ns})\, U_H(m_H^2,\mu_H,\mu) \int\! \mathrm{d} t_a \mathrm{d} t_b\, U_B^g(t_a,\mu_B,\mu)\, U_B^g(t_b,\mu_B,\mu)\, U_S\Bigl(\mathcal{T}_\mathrm{cm} - \frac{t_a + t_b}{m_H},\mu_S,\mu\Bigr) \,.\end{align} The scale where we evaluate $c^\delta(\mu)$ here is an N$^3$LL effect, and so beyond the order we are working. We choose $\mu = \mu_\ns$, which is the scale at which we evaluate $\sigma^\ns(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$. \subsection{Choice of Running Scales} \label{subsec:profiles} The factorization theorem in \eq{TauBcm_run} resums the singular cross section by evaluating the hard, beam, and soft function at their natural scales $\mu_H \simeq -\mathrm{i} m_H$, $\mu_B \simeq \sqrt{\tau} m_H$, $\mu_S \simeq \tau m_H$ where they have no large logarithms in fixed-order perturbation theory. Their renormalization group evolution is then used to connect these functions at a common scale. This resums logarithms in the ratios of $\mu_H$, $\mu_B$, and $\mu_S$, which are logarithms of $\tau$. The $\tau$-spectrum has three distinct kinematic regions where this resummation must be handled differently and we will do so using $\tau$-dependent scales given by profile functions $\mu_S(\tau)$ and $\mu_B(\tau)$. Profile functions of this type have been previously used to analyze the $B\to X_s\gamma$ spectrum~\cite{Ligeti:2008ac} and the thrust event shape in $e^+e^-\to $ jets~\cite{Abbate:2010xh}. For $\Lambda_\mathrm{QCD}/m_H \ll \tau \ll 1$ the scales $\mu_H$, $\mu_B$, $\mu_S$, and $\Lambda_\mathrm{QCD}$ are all widely separated and the situation is as described above. We define this region to be $\tau_1< \tau< \tau_2$. In the $\tau < \tau_1$ region the scale $\mu_S$ drops below $1 \,\mathrm{GeV}$, we have $\Lambda_\mathrm{QCD}/m_H \sim \tau$, and nonperturbative corrections to the soft function become important. In this case the scales are $\mu_H \simeq -\mathrm{i} m_H$, $\mu_B \simeq \sqrt{\Lambda_\mathrm{QCD} m_H}$, and $\mu_S \simeq \Lambda_\mathrm{QCD}$. Finally for $\tau > \tau_2$ we have $\tau \sim 1$, the resummation is not important, and the nonsingular corrections, which are evaluated at a fixed scale $\mu_\ns$, become just as important as the singular corrections. In this region there is only one scale $\mathrm{i} \mu_H = \mu_B = \mu_S = \mu_\ns \simeq m_H$. Furthermore it is known from $B\to X_s\gamma$ and thrust~\cite{Ligeti:2008ac,Abbate:2010xh} that there can be important cancellations between the singular and nonsingular terms in this limit, and that to ensure these cancellations take place the scales $\mu_B(\tau)$ and $\mu_S(\tau)$ must converge to $\abs{\mu_H} = \mu_\ns$ in a region, rather than at a single point. To ensure this we make the approach to $\mu_H$ quadratic for $\tau_2< \tau< \tau_3$ and set $\mu_B(\tau)=\mu_S(\tau)=\mathrm{i} \mu_H$ for $\tau\ge \tau_3$ (recall that $\mathrm{i} \mu_H>0$). As we saw in \subsec{nonsingular}, the singular and nonsingular contributions in our case become equally important for $\tau \gtrsim 1/2$. Accordingly, we choose the profile functions such that the scales converge around this value and stay equal for larger $\tau$. A transition between these three regions is given by the following running scales \begin{align} \mu_H &= -\mathrm{i}\, \mu \,, \nonumber \\ \mu_B(\tau) & = \Big[1 + e_B\, \theta(\tau_3-\tau) \Big(1 - \frac{\tau}{\tau_3}\Big)^2\,\Big] \sqrt{\mu\, \mu_\mathrm{run}(\tau,\mu)} \,, \nonumber \\ \mu_S(\tau) & = \Big[1 + e_S\, \theta(\tau_3-\tau) \Big(1 - \frac{\tau}{\tau_3}\Big)^2\,\Big] \mu_\mathrm{run}(\tau,\mu) \,, \nonumber \\ \mu_\ns & = \mu \,. \end{align} For the profile $\mu_\mathrm{run}(\tau,\mu)$ we use a combination of two quadratic functions and a linear function as in ref.~\cite{Abbate:2010xh}. For $\tau> \tau_3$ our choice for $\mu_\mathrm{run}(\tau,\mu)$ ensures that our cross section formula becomes precisely the fixed-order result. \begin{align} & \mu_\mathrm{run}(\tau,\mu) = \begin{cases} \mu_0 + a\tau^2/\tau_1 & \tau \leq \tau_1 \,,\\ 2a\, \tau + b & \tau_1 \leq \tau \leq \tau_2 \,,\\ \mu - a (\tau-\tau_3)^2/(\tau_3 - \tau_2) & \tau_2 \leq \tau \leq \tau_3 \,,\\ \mu & \tau > \tau_3 \,,\end{cases} \nonumber \\ & a= \frac{\mu_0-\mu}{\tau_1-\tau_2-\tau_3} \,, \qquad b = \frac{\mu \tau_1 - \mu_0 (\tau_2 + \tau_3)}{\tau_1-\tau_2-\tau_3} \,.\end{align} The expressions for $a$ and $b$ follow from demanding that $\mu_\mathrm{run}(\tau)$ is continuous and has a continuous derivative. The value of $\mu_0$ determines the scales at $\tau = 0$, while $\tau_{1,2,3}$ determine the transition between the regions discussed above. For our central value we use the following choice of parameters \begin{equation} \mu = m_H \,,\quad e_B = e_S = 0 \,,\quad \mu_0 = 2 \,\mathrm{GeV} \,,\quad \tau_1 = \frac{5 \,\mathrm{GeV}}{m_H} \,,\quad \tau_2 = 0.4 \,,\quad \tau_3 = 0.6 \,.\end{equation} The corresponding running scales are shown in \fig{scales}. \begin{figure}[t] \centering \includegraphics[width=0.5\textwidth]{plots/scales_lxl}% \vspace{-0.5ex} \caption{Profiles for the running scales $\mu_H$, $\mu_B$, and $\mu_S$. The central lines for $\mu_B$ and $\mu_S$ show our central scale choices. The upper and lower curves for $\mu_B$ and $\mu_S$ correspond to their respective variations b) and c) in \eq{scales}.} \label{fig:scales} \end{figure} Since the factorization theorem is not affected by $\ord{1}$ changes of the renormalization scales, we should vary them to determine the perturbative uncertainty. For a reasonable variation of the above parameters, the cross section is most sensitive to $\mu$, $e_B$ and $e_S$. We therefore estimate our uncertainties from higher order terms in perturbation theory by taking the envelope of the following three separate variations, \begin{align} \label{eq:scales} &\text{a)}& \mu &= 2^{\pm 1} m_H\,,& e_B &= 0\,,& e_S &= 0 \,,\nonumber\\ &\text{b)}& \mu &= m_H\,,& e_B &= \pm 0.5\,,& e_S &= 0 \,,\nonumber\\ &\text{c)}& \mu &= m_H\,,& e_B &= 0\,,& e_S &= \pm 0.5 \,.\end{align} The effect of variations b) and c) are shown in \fig{scales} by the upper and lower curves for $\mu_B$ and $\mu_S$, respectively. The effect of variation a) is to change the overall vertical scale of the $\|\mu_H\|$, $\mu_B$, and $\mu_S$ curves in \fig{scales} by a factor of $1/2$ or $2$ as indicated by the arrows. In predictions based on fixed-order perturbation theory only a scale variation analogous to a) can be considered. \subsection{PDFs and $\pi^2$ Summation} \label{subsec:pi2pdf} In this subsection we briefly discuss the choice of the order of the PDFs for our resummed results and the effect of the $\pi^2$ summation. As shown in table~\ref{tab:counting}, by default we use the PDFs that correspond to the order of the matching corrections, namely LO PDFs at NLL and NLO PDFs at NNLL. Since the MSTW2008 PDFs \cite{Martin:2009bu} are extracted simultaneously with the value of $\alpha_s(m_Z)$, by using PDFs at different orders we are forced to also use different values of $\alpha_s(m_Z)$. Our NNLL+NNLO results contain two-loop corrections and so at this order our default is to use the MSTW2008 PDFs~\cite{Martin:2009bu} at NNLO with their $\alpha_s(m_Z) = 0.11707$ and with three-loop, five-flavor running for $\alpha_s(\mu)$. For our NLL$'$+NLO and NNLL results, which include one-loop matching, we use the corresponding NLO PDFs with their $\alpha_s(m_Z) = 0.12018$ and two-loop, five-flavor running for $\alpha_s(\mu)$. At LL and NLL, which only includes tree-level matching, we use the LO PDFs with $\alpha_s(m_Z) = 0.13939$ and one-loop, five-flavor running for $\alpha_s(\mu)$. Note that at NLL (NNLL) there is a slight mismatch in the required running of $\alpha_s(\mu)$. The resummation at this order requires two-loop (three-loop) $\alpha_s(\mu)$ running, whereas the used LO (NLO) PDFs employ one-loop (two-loop) running of $\alpha_s(\mu)$. In this case, we use the following compromise. We use the appropriate $\alpha_s(m_Z)$ and running consistent with the PDF set to obtain the numerical value of $\alpha_s$ at some required scale. At the same time, in the NLL (NNLL) RGE solutions we use the QCD $\beta$ function at the appropriate one higher loop order to be consistent with the RGE. There is no such mismatch in the $\alpha_s$ running at LL, NLL$'$+NLO, and NNLL+NNLO, and hence no mismatch for our highest order predictions. \begin{figure}[t] \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_196_165_comppi2_lxl}% \hfill% \includegraphics[width=0.485\textwidth]{plots/ggHsigTauBcmcut_7_165_comppi2_lxl}% \vspace{-0.5ex} \caption{The effect of $\pi^2$ summation and using different orders for the PDFs on the cumulant beam thrust cross section for $m_H=165 \,\mathrm{GeV}$ at the Tevatron (left panel) and the LHC with $7\,\mathrm{TeV}$ (right panel). Shown are the NLL result with/without $\pi^2$ summation and with LO/NLO PDFs, as well as the NNLL cross section with/without $\pi^2$ summation and NLO PDFs. See the text for further explanations.} \label{fig:dsdTauB_nopi2_LHC} \end{figure} Various results which test the effect of $\pi^2$ resummation and the treatment of PDFs are shown for the cumulant cross section, $\sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$, for $m_H=165 \,\mathrm{GeV}$ in \fig{dsdTauB_nopi2_LHC}. The left panel shows the Tevatron case and right panel shows the LHC with $E_\mathrm{cm} = 7 \,\mathrm{TeV}$. The lower four blue curves show the NLL and the upper two orange curves the NNLL results. (The nonsingular corrections do not affect this discussion much, so for simplicity we do not include them in this figure.) The solid lines correspond to our default results, while the dashed, dot-dashed, and dotted are variations with other choices for the PDFs or $\pi^2$ summation. As discussed in \subsec{hard}, the hard function contains large $\alpha_s^n \pi^{2m}$ terms, with $m \leq n$, which in our default results are summed by evaluating the hard function at $\mu_H = -\mathrm{i} m_H$. The $\pi^2$ summation is switched off by taking $\mu_H = m_H$ instead, which is shown by the dotted lines in \fig{dsdTauB_nopi2_LHC}. The effect of $\pi^2$ summation is very large. It almost doubles the NLL cross section and increases the NNLL cross section by about $30\%$. From the fact that the NLL results with $\pi^2$ summation (blue solid line) is very close to the NNLL result without $\pi^2$ summation (orange dotted line) we can conclude that the large corrections from NLL to NNLL are caused by the large $\pi^2$ terms in the virtual hard-matching contributions. This result for the cross section with a cut on $\mathcal{T}_\mathrm{cm}$ agrees with the observations made in ref.~\cite{Ahrens:2008qu} for the total cross section. Similarly, we have checked that the $\pi^2$ summation in the NNLL result brings it much closer to the total NNLO result. As a result, the convergence of the perturbative series is significantly improved by including the $\pi^2$ summation, and we will make use of this for our main results. We can also explore the effect of using the NLO PDFs already at NLL, which amounts to including some higher-order terms in the NLL result but allows us to use the same value for $\alpha_s(m_Z)$ at NLL and NNLL. The corresponding results are shown by the blue dashed (with $\pi^2$ summation) and dot-dashed (without $\pi^2$ summation) in \fig{dsdTauB_nopi2_LHC}. For our default (solid blue curve) we use the LO PDFs in the NLL cross section, which increases it by about $10\%$ compared to using NLO PDFs at NLL. Thus it moves the NLL result in the direction of the NNLL result. The main reason for the upward shift is the higher value of $\alpha_s$ associated with the lower-order PDFs, since the Higgs cross section has an overall $\alpha_s^2$. \subsection{Nonperturbative Corrections} \label{subsec:Nonperturbative} \begin{figure}[t] \includegraphics[width=0.495\textwidth]{plots/ggHdsdTauBcm_7_165_nonpert_lxl}% \hfill\includegraphics[width=0.485\textwidth]{plots/ggHsigTauBcmcut_7_165_nonpert_lxl}% \vspace{-0.5ex} \caption{Shift to the NNLL+NNLO perturbative cross section, shown by solid curves with $\Omega_1^{gg}=0$, caused by the leading nonperturbative hadronization corrections, shown by the dashed and dotted curves for $\Omega_1^{gg}=0.35\,\mathrm{GeV}$ and $\Omega_1^{gg}=1.0\,\mathrm{GeV}$, respectively. } \label{fig:sigNP} \end{figure} As discussed in \subsec{soft}, for $\mathcal{T}_\mathrm{cm}\ll m_H$ the leading nonperturbative hadronization corrections to the beam thrust spectrum are given by a nonperturbative soft function $F^{gg}$. For $\mathcal{T}_\mathrm{cm} \gg \Lambda_\mathrm{QCD}$ the dominant nonperturbative effect can be described by an OPE that yields a single nonperturbative parameter $\Omega_1^{gg} \sim \Lambda_\mathrm{QCD}$, leading to a shift in the beam thrust spectrum by $\mathcal{T}_\mathrm{cm} \to \mathcal{T}_\mathrm{cm} - 2 \Omega_1^{gg}$, and in the cumulant by $\mathcal{T}_\mathrm{cm}^\mathrm{cut}\to \mathcal{T}_\mathrm{cm}^\mathrm{cut}-2\Omega_1^{gg}$. To illustrate the size of this nonperturbative effect we consider two values for $\Omega_1^{gg}$. First, $\Omega_1^{gg}=0.35\,\mathrm{GeV}$ is motivated by the fit result for an analogous parameter for dijet quark production in $e^+e^-$ collisions~\cite{Abbate:2010xh}. Second, $\Omega_1^{gg}=1.0\,\mathrm{GeV}$ is motivated by a potential enhancement by $C_A/C_F=9/4$ from Casimir scaling for adjoint Wilson lines. This choice also reproduces roughly the size of hadronization effects for Higgs production in \textsc{Pythia}\xspace. Using \eqs{OPEdsdT}{OPEsigc}, results from the OPE for the LHC with $E_\mathrm{cm}=7\,\mathrm{TeV}$ are shown in \fig{sigNP}. Comparing the OPE results for the distribution, shown in the left panel, to the full convolution with a model soft function using \eq{SF}, we find that the OPE works well for the entire displayed spectrum when $\Omega_1^{gg}=0.35\,\mathrm{GeV}$ and for $\mathcal{T}_\mathrm{cm} > 10\,\mathrm{GeV}$ when $\Omega_1^{gg}=1.0\,\mathrm{GeV}$. (Thus, for $\Omega_1^{gg}=0.35\,\mathrm{GeV}$ the peak is perturbative.) Examining the right panel of \fig{sigNP}, we see that at $\mathcal{T}_\mathrm{cm}^\mathrm{cut}=20\,\mathrm{GeV}$ a power correction of $\Omega_1^{gg}=0.35\,\mathrm{GeV}$ reduces $\sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$ by $3\%$, while for $\Omega_1^{gg}=1.0\,\mathrm{GeV}$ the reduction is by $7\%$. (The results for the Tevatron are very similar, giving reductions by $2\%$ and $6\%$, respectively, for $\mathcal{T}_\mathrm{cm}^\mathrm{cut}=20\,\mathrm{GeV}$.) The sign of this nonperturbative shift is predicted by the factorization theorem, while its magnitude is determined by $\Omega_1^{gg}$. Examining the $\mathcal{T}_\mathrm{cm}$ spectra from \textsc{Pythia}\xspace before and after hadronization, we find that the hadronization correction in \textsc{Pythia}\xspace is consistent with the nonperturbative shift discussed here with a value $\Omega_1^{gg}=1.0\,\mathrm{GeV}$ for both the Tevatron and LHC. \section{Numerical Results} \label{sec:results} In this section we present our numerical results for the Higgs production cross section for both the differential beam thrust spectrum, $\mathrm{d}\sigma/\mathrm{d}\mathcal{T}_\mathrm{cm}$, and the cumulant, $\sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$, which gives the integrated cross section with a cut on beam thrust, $\mathcal{T}_\mathrm{cm} \leq \mathcal{T}_\mathrm{cm}^\mathrm{cut}$. We are mostly interested in the region of small $\mathcal{T}_\mathrm{cm}$ or $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$, which corresponds to the $0$-jet region. We will show resummed results up to NNLL+NNLO order and also compare with the results obtained in fixed-order perturbation theory at NNLO using \textsc{FEHiP}\xspace~\cite{Anastasiou:2004xq, Anastasiou:2005qj}. An explanation of the various orders is given at the beginning of \sec{calc} and in table~\ref{tab:counting}. Since our focus in this section is on the perturbative results and their uncertainties, we will not include the nonperturbative hadronic correction discussed in \subsec{Nonperturbative} (i.e. we take $\Omega_1^{gg}=0$). The perturbative uncertainties in the resummed predictions are estimated as explained in detail in \subsec{profiles}. For the fixed-order results we use $\mu = m_H/2$ as the default choice for the central value, which tends to give a better convergence for the total cross section, mimicking the effect of the $\pi^2$ summation. The perturbative scale uncertainties at fixed order are then evaluated using $\mu = m_H$ and $\mu = m_H/4$. (We follow ref.~\cite{Anastasiou:2009bt} and do not vary the renormalization and factorization scales independently.) Since our focus here is on the perturbative uncertainties, we do not add PDF and $\alpha_s(m_Z)$ uncertainties in our plots. We have checked that they are essentially independent of the cut on beam thrust and the same as for the total inclusive cross section. We show results for both the Tevatron and the LHC. For the LHC we always use $E_\mathrm{cm} = 7\,\mathrm{TeV}$. The results for higher $E_\mathrm{cm}$ are qualitatively similar, except for the overall increased cross section. For most of our plots we use $m_H = 165\,\mathrm{GeV}$, which is near the $WW$ threshold and where the current Tevatron limits are most sensitive. We also show some plots that illustrate the dependence of our results on $m_H$. \subsection{Convergence of Resummed Predictions} \label{subsec:convergence} \begin{figure}[p] \includegraphics[width=0.5\textwidth]{plots/ggHdsdTauBcm_196_165_lxl}% \hfill% \includegraphics[width=0.48\textwidth]{plots/ggHdsdTauBcm_7_165_lxl}% \vspace{-0.75ex} \caption{The beam thrust spectrum for Higgs production for $m_H = 165\,\mathrm{GeV}$ at the Tevatron (left) and the LHC for $E_\mathrm{cm} = 7\,\mathrm{TeV}$ (right). The bands show the perturbative scale uncertainties as explained in \subsec{profiles}.} \label{fig:dsdTauBcm} \end{figure} \begin{figure}[p] \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_196_165_lxl}% \hfill% \includegraphics[width=0.485\textwidth]{plots/ggHsigTauBcmcut_7_165_lxl}% \vspace{-0.75ex} \caption{Higgs production cross section as a function of $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ for $m_H = 165\,\mathrm{GeV}$ at the Tevatron (left) and the LHC with $E_\mathrm{cm} = 7\,\mathrm{TeV}$ (right). The bands show the perturbative scale uncertainties as explained in \subsec{profiles}.} \label{fig:sigTauBcmcut} \end{figure} \begin{figure}[p] \includegraphics[width=0.495\textwidth]{plots/ggHsigmHcm_196_10_lxl}% \hfill% \includegraphics[width=0.485\textwidth]{plots/ggHsigmHcm_7_20_lxl}% \vspace{-0.75ex} \caption{Higgs production cross section with a cut on beam thrust as function of $m_H$ at the Tevatron for $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = 10\,\mathrm{GeV}$ (left) and the LHC with $E_\mathrm{cm} = 7\,\mathrm{TeV}$ and $\mathcal{T}_\mathrm{cm}^\mathrm{cut}=20\,\mathrm{GeV}$ (right). The bands show the perturbative scale uncertainties as explained in \subsec{profiles}.} \label{fig:sigmHcm} \end{figure} \afterpage{\clearpage} To study the convergence of the resummed results, we consider results at three different orders: NLL, NLL$'$+NLO, and NNLL+NNLO, which contain the matching and nonsingular corrections at LO, NLO, and NNLO, respectively. We choose NLL instead of LL as our lowest order to compare to, since NLL is the lowest order where we get a useful approximation with appropriately large scale uncertainties. (The LL results are lower than the NLL ones and also have a smaller scale uncertainty, which means that they do not contain enough information yet to provide a reasonable lowest-order approximation.) In \figs{dsdTauBcm}{sigTauBcmcut} we show the convergence for the differential spectrum and cumulant, respectively for the Tevatron (left panels) and the LHC (right panels). In \fig{sigmHcm} we show the cumulant for fixed $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ as a function of the Higgs mass. We see that the perturbative corrections are rather large, as is typical for Higgs production. The convergence within our perturbative uncertainty bands is reasonable for the differential spectrum and quite good for the cumulant, both for different $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ and different $m_H$. The large step from NLL to NLL$'$+NLO is mostly due to the NLO matching corrections. As we saw in \fig{singcomp}, for $\mathcal{T}_\mathrm{cm}^\mathrm{cut} \ll m_H$ the nonsingular terms are much smaller than the singular corrections that we have computed analytically. One can also see this by comparing the size of the NLO nonsingular terms in the left panel of \fig{nonsing} with the full cross section in the right panel of \fig{sigTauBcmcut}. The beam thrust spectrum in \fig{dsdTauBcm} is peaked in the $0$-jet region at small beam thrust $\mathcal{T}_\mathrm{cm} \simeq 5 \,\mathrm{GeV}$ with a large tail towards higher values. The peak in the spectrum is a perturbative prediction; hadronization effects only have a mild effect on the peak structure as shown above in \subsec{Nonperturbative}. For the beam thrust spectrum of Drell-Yan, which is Fig.~3 of ref.~\cite{Stewart:2010pd}, the peak occurs at smaller values, around $\mathcal{T}_\mathrm{cm} \sim 2 \,\mathrm{GeV}$, and the tail of the spectrum falls off much faster. The reason for the shifted peak and higher tail for Higgs production compared to Drell-Yan is that the incoming gluons emit much more initial-state radiation than quarks. This is also the main reason why the perturbative uncertainties are still rather large even at the highest order, NNLL+NNLO. One can also see that at the LHC the tail is somewhat higher and the peak less pronounced than at the Tevatron. Correspondingly, the cumulant at the Tevatron starts to level out earlier than at the LHC. The reason is that due to the higher center-of-mass energy at the LHC, more phase space is available for initial-state radiation. \begin{figure}[t] \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_196_165_nopi2_lxl}% \hfill% \includegraphics[width=0.485\textwidth]{plots/ggHsigTauBcmcut_7_165_nopi2_lxl}% \vspace{-0.5ex} \caption{Same as \fig{sigTauBcmcut} but without the $\pi^2$ summation.} \label{fig:sigTauBcmcut_nopi2} \end{figure} In \fig{sigTauBcmcut_nopi2} we illustrate what happens if we turn off the $\pi^2$ summation. Comparing with \fig{sigTauBcmcut}, we see again that the $\pi^2$ summation significantly improves the convergence, and by reducing the overall size of the fixed-order corrections it also reduces the size of the perturbative uncertainties. \subsection{Comparison of Resummed and Fixed-Order Predictions} \label{subsec:fixedorder} In this subsection, we compare our best resummed result at NNLL+NNLO to the NNLO fixed-order prediction without any resummation. In \fig{dsdTauBcm_compNNLO} we compare both predictions for the differential beam-thrust spectrum for the Tevatron (left panel) and the LHC (right panel). For small $\mathcal{T}_\mathrm{cm}$ the large logarithms of $\mathcal{T}_\mathrm{cm}/m_H$ dominate the cross section. The NNLO cross section contains terms up to $\alpha_s^2 \ln^3(\mathcal{T}_\mathrm{cm}/m_H)/\mathcal{T}_\mathrm{cm}$ and diverges as $\mathcal{T}_\mathrm{cm} \to 0$, so we do not expect it to provide a good description of the spectrum at small $\mathcal{T}_\mathrm{cm}$. In the NNLL+NNLO calculation, the series of logarithms is summed to all orders, which regulates the divergences and yields a reliable prediction for the cross section. The resummation also enhances the radiative tail in the spectrum, because it essentially sums up the effects of multiple emissions from ISR. \begin{figure}[t] \includegraphics[width=0.5\textwidth]{plots/ggHdsdTauBcm_196_165_compNNLO_lxl}% \hfill% \includegraphics[width=0.48\textwidth]{plots/ggHdsdTauBcm_7_165_compNNLO_lxl}% \vspace{-0.5ex} \caption{Comparison of the beam thrust spectrum at NNLL+NNLO to the fixed NNLO result at the Tevatron (left) and the LHC with $E_\mathrm{cm} = 7\,\mathrm{TeV}$ (right). The bands show the perturbative scale uncertainties.} \label{fig:dsdTauBcm_compNNLO} \end{figure} \begin{figure}[t] \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_196_165_matchNNLO_lxl}% \hfill% \includegraphics[width=0.485\textwidth]{plots/ggHsigTauBcmcut_7_165_matchNNLO_lxl}% \vspace{-0.5ex} \caption{Illustration that the NNLL+NNLO resummed result reproduces the fixed NNLO result at large beam thrust for the Tevatron (left) and the LHC (right). The resummed result has the $\pi^2$ summation switched off and $\mu = m_H$ is used for the central value of the fixed-order result. The bands show the perturbative uncertainties. See text for further explanations. } \label{fig:matchNNLO} \end{figure} In \fig{matchNNLO} we illustrate that the NNLL+NNLO result correctly reproduces the NNLO result for large $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$. To see this we need to switch off the $\pi^2$ summation, because the NNLL+NNLO result would otherwise contain higher order $\pi^2$ terms at large $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ that are absent at fixed NNLO. Furthermore, we use $\mu = m_H$ for the NNLO central value, and $\mu = 2m_H$ and $\mu = m_H/2$ for the NNLO scale uncertainties. In this way, the NNLL+NNLO and the NNLO are evaluated at the same scales for large $\mathcal{T}_\mathrm{cm} \geq 0.6\,m_H$, where the logarithmic resummation is switched off and our running scales satisfy [see \subsec{profiles}] $\mu_S(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) = \mu_B(\mathcal{T}_\mathrm{cm}^\mathrm{cut}) = \mu_\ns = \mu_H = m_H$. In \fig{matchNNLO} we see that with these choices the NNLL+NNLO indeed smoothly merges into the NNLO result, including the scale uncertainties, at large $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$, as it should. Examining \fig{matchNNLO} for smaller $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ values we see that the resummed result starts to deviate from the fixed-order one for $\mathcal{T}_\mathrm{cm}^\mathrm{cut} \lesssim 40\,\mathrm{GeV}$ at the Tevatron and $\mathcal{T}_\mathrm{cm}^\mathrm{cut} \lesssim 50\,\mathrm{GeV}$ at the LHC. \begin{figure}[p] \includegraphics[width=0.49\textwidth]{plots/ggHsigTauBcmcut_196_165_compNNLO_lxl}% \hfill% \includegraphics[width=0.49\textwidth]{plots/ggHsigTauBcmcut_196_165_compNNLO_rel_lxl}% \vspace{-0.5ex} \caption{Comparison of the NNLL+NNLO result for the Higgs production cross section as a function of $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ to the fixed NNLO result for the Tevatron. The bands show the perturbative scale uncertainties. The left plot shows the cumulant cross section. The right plot shows the same information as percent difference relative to the NNLL+NNLO central value.} \label{fig:compNNLO_Tev} \end{figure} \begin{figure}[p] \includegraphics[width=0.485\textwidth]{plots/ggHsigTauBcmcut_7_165_compNNLO_lxl}% \hfill% \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_7_165_compNNLO_rel_lxl}% \vspace{-0.5ex} \caption{Same as \fig{compNNLO_Tev} but for the LHC with $E_\mathrm{cm} = 7\,\mathrm{TeV}$.} \label{fig:compNNLO_LHC} \end{figure} \begin{figure}[p] \includegraphics[width=0.485\textwidth]{plots/ggHsigTauBcmcut_196_165_compNNLO_full_lxl}% \hfill% \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_7_165_compNNLO_full_lxl}% \vspace{-0.5ex} \caption{Same as the left panels of \figs{compNNLO_Tev}{compNNLO_LHC} but plotted up to $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = m_H$.} \label{fig:compNNLO_full} \end{figure} \afterpage{\clearpage} In \figs{compNNLO_Tev}{compNNLO_LHC} we compare the full NNLL+NNLO including $\pi^2$ summation to the NNLO (using again the default $\mu = m_H/2$ as the central value) for the Tevatron and LHC, respectively. The left panels show the cumulant cross section, and the right panels show the same results as the relative difference in percent to the central NNLL+NNLO curve, which makes it easy to read off uncertainties. The relative plots are cut off below $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = 5\,\mathrm{GeV}$ because the resummed cross section goes to zero there. The central value of the NNLL+NNLO leaves the fixed-order uncertainty band at $\mathcal{T}_\mathrm{cm}^\mathrm{cut} \simeq 25\,\mathrm{GeV}$ at the Tevatron and at $\mathcal{T}_\mathrm{cm}^\mathrm{cut} \simeq 35\,\mathrm{GeV}$ at the LHC. Hence, for any lower values the resummation should be taken into account. At $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = 20\,\mathrm{GeV}$ the central values of the NNLL+NNLO and the NNLO already differ by $20\%$ at the Tevatron and over $25\%$ at the LHC, which both quickly grows beyond $50\%$ towards $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = 10\,\mathrm{GeV}$.% \footnote{One might expect that a better agreement could be achieved by using a dynamical $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$-dependent scale in the fixed-order prediction. We have checked that using the intermediate scale $\mu = \mu_B$ in the fixed-order result however does not improve its behavior relative to the resummed result, but in fact makes it a bit worse.} This clearly shows that it is important to resum the higher-order logarithms that are missing in the fixed-order prediction in order to obtain reliable predictions in the $0$-jet region. This also means that one cannot expect the scale variation in the fixed-order result to give a realistic estimate of the size of the missing higher-order terms, and hence it should not be used to estimate the perturbative scale uncertainty either. In contrast, since the resummation takes into account the presence of large logarithms for small $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$, we are able to obtain reliable estimates of the perturbative uncertainties. The perturbative uncertainties at small $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ are larger than those in the fixed-order result, namely $15-20\%$ at NNLL+NNLO for $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = 15-20\,\mathrm{GeV}$. Implications of this for the Higgs search are taken up in \sec{conclusions}. \begin{figure}[t] \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_196_165_scales_rel_lxl}% \hfill% \includegraphics[width=0.495\textwidth]{plots/ggHsigTauBcmcut_7_165_scales_rel_lxl}% \vspace{-0.5ex} \caption{Contribution to the relative uncertainties in the NNLL+NNLO results shown in \figs{compNNLO_Tev}{compNNLO_LHC} from the individual scale variations. Here, the $\mu_H$, $\mu_B$, and $\mu_S$ variations correspond to cases a), b), and c) in \eq{scales} and are shown in \fig{scales}.} \label{fig:compscales} \end{figure} \begin{figure}[t] \includegraphics[width=0.495\textwidth]{plots/ggHsigmHcm_196_10_compNNLO_lxl}% \hfill% \includegraphics[width=0.485\textwidth]{plots/ggHsigmHcm_7_20_compNNLO_lxl}% \vspace{-0.5ex} \caption{Comparison of the NNLL+NNLO result to the fixed NNLO result for the Higgs production cross section with a cut on beam thrust as function of $m_H$ at the Tevatron for $\mathcal{T}_\mathrm{cm}^\mathrm{cut}=10\,\mathrm{GeV}$ (left) and the LHC with $E_\mathrm{cm} = 7\,\mathrm{TeV}$ and $\mathcal{T}_\mathrm{cm}^\mathrm{cut}=20\,\mathrm{GeV}$ (right). The bands show the perturbative scale uncertainties.} \label{fig:sigmHcm_compNNLO} \end{figure} In \fig{compNNLO_full} we show the same comparison of NNLL+NNLO and NNLO cumulants, but plotted up to $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = m_H$. We can see that the central value of the resummed result including the $\pi^2$ resummation almost exactly reproduces the NNLO result which uses $\mu = m_H/2$. However, the $\pi^2$ summation included in the NNLL+NNLO results leads to a reduction of the scale uncertainties in the inclusive cross section compared to those at NNLO. For the uncertainties at the LHC we find $+3\%$ and $-5\%$ and for the Tevatron $+5\%$ and $-9\%$. In \fig{compscales} we show the relative uncertainties at NNLL+NNLO from the individual scale variations in \eq{scales}. We can see that at small $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ the uncertainties are dominated by $\mu_B$ and $\mu_S$, i.e.\ variations b) and c) in \eq{scales}. By construction, those variations go to zero at large $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$, where the uncertainties are now completely determined by variation a) in \eq{scales} and denoted $\mu_H$ in the figure, which is equivalent to the fixed-order scale variation. Finally, \fig{sigmHcm_compNNLO} shows a comparison of NNLL+NNLO to NNLO as a function of the Higgs mass for a fixed value of $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$. For smaller Higgs masses, the logarithms get smaller and the cross section increases, which reduces the relative differences. This does not change however our overall conclusions for the importance of the resummation for both the central value and determining the perturbative uncertainties. \subsection{Discussion of $K$-Factors} \label{subsec:Kfactor} Using our results we can also address the origin of the large NLO and NNLO $K$-factors that are typically observed for Higgs production. It is sometimes argued that the origin of these large $K$-factors for the inclusive cross section are large perturbative corrections due to hard emissions. This is based on the observation that by vetoing hard jets, the fixed-order $K$-factors are reduced. As a result the fixed-order perturbative series in the presence of a jet veto actually appears to be better behaved than for the inclusive cross section. Our results show that once the large jet-veto logarithms are properly summed, the $K$-factor is mostly independent of the jet veto. Hence, it is not caused by hard real emissions, but rather mostly by hard virtual corrections and to a lesser extent by collinear and soft virtual and real radiation. In our analysis this can be examined directly by comparing the convergence of perturbation theory for the hard, jet, and soft functions. As we have seen in \subsec{pi2pdf}, and was already observed in ref.~\cite{Ahrens:2008qu}, by summing the large $\pi^2$ terms in the hard virtual corrections, the $K$-factors are substantially reduced. There is a simple reason why the large $K$-factors in fixed-order perturbation theory are reduced: At NLO, the jet veto reduces the cross section just because it cuts out available phase space, and since at LO the cross section is not yet affected by the jet veto, the NLO $K$-factor is reduced accordingly. Essentially, the large negative phase-space logarithms resulting from the jet veto happen to cancel the large positive corrections from hard virtual corrections. A similar effect appears at NNLO where the jet veto reduces the available phase space for the second jet. Since the hard virtual corrections are independent of the jet veto one can always choose a particular value for the jet-veto cut such that they are exactly canceled by the large phase-space logarithms at a given order in perturbation theory. However, to conclude that the jet veto in general renders the fixed-order perturbative series better behaved one would have to know that the same level of cancellation will happen at each order in perturbation theory. Since these two types of corrections are a priori not related there is no reason to believe this will be the case. Instead, to obtain reliable predictions both types of large corrections should be rendered as convergent as possible. For the hard virtual corrections the $\pi^2$ summation improves the convergence, and for the large phase-space logarithms this is achieved by the resummation carried out here. With the resulting cross section we can then obtain more realistic estimates for higher-order theoretical uncertainties as discussed in \subsec{fixedorder}. \section{Conclusions} \label{sec:conclusions} A major effort at the LHC and Tevatron is devoted to the search for the Higgs boson. In the current Tevatron exclusion limits and the early LHC searches the $H \to WW \to \ell^+ \nu \ell^- \bar \nu$ channel plays an important role, since it is the dominant decay channel for Higgs masses above $\sim 130\,\mathrm{GeV}$. A large background for this channel are $t\bar t \to W W b \bar b$ events, which must be eliminated by a veto on central jets, so the resulting measurement is $pp\to H + 0$ jets. Such a jet veto causes large double logarithms in the cross section, which need to be summed to all orders in perturbation theory in order to obtain reliable theoretical predictions and uncertainties for the $H + 0$ jet production cross section. In this paper we have studied Higgs production from gluon fusion, using the inclusive event shape beam thrust, $\mathcal{T}_\cm$, to implement a central jet veto. As beam thrust characterizes the event as a whole and does not require a jet algorithm, it is well suited to analytic calculations. This allows us to resum the jet-veto logarithms to NNLL order, based on the factorization theorem for the beam thrust cross section derived in ref.~\cite{Stewart:2009yx}. In our analysis we also include the full set of NNLO corrections, such that our final result at NNLL+NNLO provides the best possible theoretical prediction at any value of beam thrust. Our main results are presented in figs.~\ref{fig:compNNLO_Tev} and \ref{fig:compNNLO_LHC}. We find that in the $0$-jet region at small beam thrust, the resummation of jet-veto logarithms is crucial, and our central value at NNLL+NNLO for the cross section with a cut on beam thrust, $\mathcal{T}_\mathrm{cm} \leq \mathcal{T}_\mathrm{cm}^\mathrm{cut}$, differs significantly from the fixed-order prediction at NNLO. We also find substantially larger perturbative scale uncertainties arising from the jet veto compared to those at NNLO. Since fixed-order perturbation theory is not reliable in the presence of large jet-veto logarithms, one also cannot expect its scale variation to yield a reliable estimate of perturbative uncertainties due to neglecting higher-order corrections. At present, the jet-veto logarithms are taken into account in the experimental analyses using the leading-logarithmic resummation provided by parton-shower Monte Carlo programs, usually supplemented with some reweighting procedure to reproduce the total NNLO cross section. While this might yield a reasonable central value that takes into account the dominant effect of the logarithmic resummation, the uncertainties in the so predicted $0$-jet cross section cannot be taken as those of the inclusive NNLO cross section. They could at best be equal to those in our NNLL+NNLO results. In fact, they are probably larger than that, since we include the resummation at two orders higher than the LL resummation of parton showers. As we have seen in \subsec{convergence}, already one order lower, at NLL$'$+NLO, the perturbative uncertainties are much larger than those in our NNLL+NNLO results. The conventional method to veto central jets is to use a jet algorithm and require $p_T^\mathrm{jet} < p_T^\mathrm{cut}$ for all jets in the event. As we saw in \sec{Intro}, $p_T^\mathrm{cut}$ can be related to $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ by associating $\mathcal{T}_\mathrm{cm}^\mathrm{cut}/m_H \simeq (p_T^\mathrm{cut}/m_H)^{\sqrt{2}}$ or $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = p_T^\mathrm{cut}$, where the former works well at NNLO while the latter is favored by \textsc{Pythia}\xspace 8. Hence, we can use the perturbative uncertainties in our results based on $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ as a benchmark for the perturbative uncertainties from large logarithms relevant for $p_T^\mathrm{cut}$, on which the current experimental analyses are based. For example, the perturbative uncertainties for typical values $p_T^\mathrm{cut} \simeq 20-30\,\mathrm{GeV}$ at the LHC can be as large as those for $\mathcal{T}_\mathrm{cm}^\mathrm{cut}\simeq 10-15\,\mathrm{GeV}$, which are $15-20\%$ at NNLL+NNLO (for $m_H = 165\,\mathrm{GeV}$ and $E_\mathrm{cm} = 7\,\mathrm{TeV}$). In the current Tevatron analyses, the perturbative scale uncertainty in the $0$-jet cross section for $p_T^\mathrm{cut} = 15\,\mathrm{GeV}$ is taken as $7\%$ from the fixed-order analysis in ref.~\cite{Anastasiou:2009bt}. In contrast, the perturbative uncertainties at NNLL+NNLO for values $\mathcal{T}_\mathrm{cm}^\mathrm{cut} \leq 15 \,\mathrm{GeV}$ are significantly larger, e.g. about $20\%$ for $\mathcal{T}_\mathrm{cm}^\mathrm{cut} = 10\,\mathrm{GeV}$ and $m_H = 165\,\mathrm{GeV}$. In light of this, the current Tevatron exclusion limits should be reconsidered with an increased theory uncertainty for the $0$-jet Higgs cross section. We note that our conclusions about theoretical uncertainties are based on a systematic study of the jet veto, and are therefore independent of the arguments in ref.~\cite{Baglio:2010um} proposing the use of a larger range for the fixed-order scale variation. To implement our NNLL+NNLO results in the experimental searches, one could reweight the partonic beam-thrust spectrum obtained from Monte Carlo to our results. This would incorporate both the higher-order resummation of large logarithms in the $0$-jet region and at the same time the full NNLO result for the total cross section. This also allows a consistent treatment of the perturbative uncertainties in both regions. To illustrate this, consider dividing the total inclusive cross section, $\sigma_\mathrm{total}$, into a $0$-jet bin, $\sigma_0 \equiv \sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$, and a remaining $(\geq 1)$-jet bin, $\sigma_{\geq 1} \equiv \sigma_\mathrm{total} - \sigma(\mathcal{T}_\mathrm{cm}^\mathrm{cut})$. This separation into exclusive jet bins causes large logarithms of $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ in both $\sigma_0$ and $\sigma_{\geq 1}$ which cancel in their sum. This implies that the uncertainties due to $\mathcal{T}_\mathrm{cm}^\mathrm{cut}$ in both bins are anti-correlated. In particular, if one wants to consider the theory uncertainties in $\sigma_0$ and $\sigma_{\geq 1}$ simultaneously in a Gaussian fashion, one has to consider the full theory covariance matrix \begin{equation} \label{eq:C} C = \begin{pmatrix} \Delta_0^2 & \Delta_0\, \Delta_{\geq 1}\,\rho_{0,\geq 1} \\ \Delta_0\, \Delta_{\geq 1}\,\rho_{0,\geq 1} & \Delta_{\geq1}^2 \end{pmatrix} \approx \begin{pmatrix} \Delta_0^2 & -\Delta_0^2 \\ -\Delta_0^2 & \quad\Delta_0^2 + \Delta_\mathrm{total}^2 \end{pmatrix} \,,\end{equation} where $\Delta_0$ and $\Delta_{\geq 1}$ are the theory uncertainties of $\sigma_0$ and $\sigma_{\geq 1}$, and $\rho_{0,\geq1}$ is their correlation coefficient. In the second step above we used the approximation that the uncertainties in $\sigma_0$ and $\sigma_\mathrm{total}$ are uncorrelated and are added in quadrature in $\sigma_{\geq 1}$. From our results in \fig{compscales} we can see that this is reasonable since the uncertainties in $\sigma_0$ are dominated by the lower scales $\mu_B$ and $\mu_S$, while those in $\sigma_\mathrm{total}$ are determined by $\mu_H$ (which in this case is combined in quadrature with the others rather then by taking the envelope). The uncertainty squared for the total cross section, $\sigma_0 + \sigma_{\ge 1}$, is given by the sum of all entries in the matrix $C$ in \eq{C}. Due to the anti-correlation the uncertainties for the individual jet bins can be larger than that for the total cross section. Our numerical results for the theory uncertainties for the 0-jet bin directly give $\Delta_0$. The full correlation can be taken into account by reweighting the Monte Carlo to both the central value curve as well as the results obtained from the individual scale variations. It would also be useful to have a benchmark theoretical uncertainty that is desired for the experimental searches, since with further effort our NNLL+NNLO results can be extended to NNLL$^\prime$+NNLO or N$^3$LL+NNLO, which has the potential to reduce the uncertainty in the resummed perturbation theory. Given that the theoretical predictions and their uncertainties are very sensitive to the jet veto, it would also be useful to implement the jet veto in the experimental analyses directly in terms of beam thrust, for which resummed theory predictions are available. In addition, a benchmark experimental study of the beam-thrust spectrum can be made with Drell-Yan pairs, as advocated in ref.~\cite{Stewart:2010pd}. In this paper, we have restricted ourselves to studying the case of $gg\to H + 0$ jets. The same methods can be used to calculate the dominant irreducible background from direct $WW$ production, i.e., the process $pp\to WW + 0$ jets using beam thrust for the jet veto. The generalization of beam thrust to processes with $N$ signal jets is provided by the event shape $N$-jettiness introduced in ref.~\cite{Stewart:2010tn}. It can be used in an analogous fashion to study the exclusive $H+1$ jet and $H+2$ jet cross sections, the latter being relevant for Higgs production from vector-boson fusion. \begin{acknowledgments} We thank the ATLAS group at the University of Pennsylvania and Elliot Lipeles for stimulating discussions. We also thank Lance Dixon, Steve Ellis, Zoltan Ligeti, Aneesh Manohar, Kerstin Tackmann, Jesse Thaler, Joey Huston, Fabian St\"ockli and especially Kirill Melnikov and Frank Petriello for useful discussions and comments on the manuscript. For hospitality during part of this work we thank the Max-Planck Institute for Physics (Werner-Heisenberg Institute), the CERN theory group, and the Center for the Fundamental Laws of Nature at Harvard. This work was supported in part by the Office of Nuclear Physics of the U.S.\ Department of Energy under the Contract DE-FG02-94ER40818, by the Department of Energy under the grant DE-SC003916, and by a Friedrich Wilhelm Bessel award from the Alexander von Humboldt foundation. \end{acknowledgments}	{ "redpajama_set_name": "RedPajamaArXiv" }	1,175
Osterdorf (mundartlich: Oaschtərdoarf, Oschtrdoərv) ist ein Gemeindeteil der Marktgemeinde Oberstaufen im Landkreis Oberallgäu und ein Ortsteil von Thalkirchdorf. Geographie Das Dorf liegt circa sechs Kilometer östlich des Hauptorts Oberstaufen im Konstanzer Tal gelegen. Nördlich der Ortschaft verläuft die Bahnstrecke Buchloe–Lindau sowie südlich die Queralpenstraße B 308 und die Konstanzer Ach. Südlich befindet sich das Himmelseck sowie das Geotop Steigbach-Schichten am Osterdorfer Wasserfall. Ortsname Der Ortsname bedeutet im Osten (von Thalkirchdorf) gelegenes Dorf. Vermuten lässt sich auch eine staatsfränkische Siedlung. Geschichte Osterdorf wurde erstmals urkundlich im Jahr 1421 als zum Osterdorf erwähnt. 1808 wurde im Ort 15 Wohnhäuser gezählt sowie elf Güter mit Weiderecht auf der Alpe Egg. Zu dieser Zeit waren alle Güter in Osterdorf nach Eglofs lehnenbar. Osterdorf gehörte bis zur bayerischen Gebietsreform 1972 der Gemeinde Thalkirchdorf an. Baudenkmäler Der Ort Osterdorf steht unter Ensembleschutz Siehe: Liste der Baudenkmäler in Osterdorf Weblinks Einzelnachweise Ort im Landkreis Oberallgäu Geographie (Oberstaufen)	{ "redpajama_set_name": "RedPajamaWikipedia" }	9,475
\section{Introduction and summary} Categorical structures have numerous applications outside of category theory proper as they occur naturally in many branches of mathematics, physics and computer science. In particular, higher-dimensional categories provide a suitable tool for the treatment of an extensive list of issues with recognized mathematical interest in algebraic topology, algebraic geometry, algebraic $K$-theory, string field theory, conformal field theory and statistical mechanics, as well as in the study of geometric structures on low-dimensional manifolds. See the recent book {\em Towards Higher Categories} \cite{Baez-May}, which provides a useful background for this subject. Like small categories \cite{quillen}, small B\'{e}nabou's bicategories and, in particular, Mac Lane's monoidal categories, are closely related to topological spaces through the classifying space construction. This assigns to each bicategory $\B$ a CW-complex $\BB \B$ whose cells give a natural geometric meaning to the cells of the bicategory \cite{ccg-1}. By this correspondence, for example, bigroupoids correspond to homotopy 2-types (CW-complexes whose $n^{\mathrm{th}}$ homotopy group at any base point vanishes for $n\geq 3$), and arbitrary monoidal categories to delooping of the classifying spaces of the underlying categories (up to group completion). The process of taking classifying spaces of bicategories reveals a way to transport categorical coherence to homotopical coherence since the construction $\B\mapsto \BB\B$ preserves products, any lax or oplax functor between bicategories, $F:\A\to\B$, induces a continuous map on classifying spaces $\BB F:\BB\A\to \BB\B$, any lax or oplax transformation between these, $\alpha:F\Rightarrow F'$, induces a homotopy between the corresponding induced maps $\BB\alpha:\BB F\Rightarrow \BB F'$, and any modification between these, $\varphi:\alpha\Rrightarrow\beta$, a homotopy $\BB\varphi:\BB\alpha\Rrightarrow\BB\beta$ between them. Thus, if $\A$ and $\B$ are biequivalent bicategories or if a homomorphism $\A\to \B$ has a biadjoint, then their associated classifying spaces are homotopy equivalent. In this paper we show the subtlety of this theory by analyzing the homotopy fibers of the map $\BB F:\BB \A\to \BB\B$, which is induced by a lax functor between small bicategories $F:\A\to \B$, such as Quillen did in \cite{quillen} where he stated his celebrated Theorems A and B for the classifying spaces of small categories. Every object $b\in \Ob\B$ has an associated {\em homotopy fiber bicategory} $F\!\!\downarrow{\!_b}$ whose objects are the 1-cells $f:Fa\to b$ in $\B$, with $a$ an object of $\A$; the 1-cells consist of all triangles $$ \xymatrix@C=10pt@R=14pt{ Fa\ar[rr]^{Fu}\ar[rd]_{f}& \ar@{}[d]\|(.27){\beta}\|(.48){\Rightarrow}&Fa'\ar[ld]^{f'}\\ &b& } $$ with $u:a\to a'$ a 1-cell in $\A$ and $\beta:f\Rightarrow f'\circ Fu$ a 2-cell in $\B$, and the 2-cells of this bicategory are commutative diagrams of 2-cells in $\B$ of the form $$ \xymatrix@C=18pt@R=18pt{ &f\ar@2[ld]_{\beta}\ar@2[rd]^{\beta'}& \\ f'\circ Fu\ar@2[rr]^{1_{f'}\circ F\alpha}&&f'\circ Fu' } $$ with $\alpha:u\Rightarrow u'$ a 2-cell in $\A$. Compositions, identities, and the structure associativity and unit constraints in $F\!\!\downarrow{\!_b}$ are canonically provided by those of the involved bicategories and the structure 2-cells of the lax functor (see Section \ref{theB} for details). For the case $F=1_\B$, we have the {\em comma} bicategory $\B\!\!\downarrow{\!_b}$. Then, we prove (see Theorem \ref{B}): \begin{quote}{\em ``For every object $b$ of the bicategory $\B$, the induced square $$ \begin{array}{c} \xymatrix{\BB(F\!\!\downarrow{\!_b})\ar[d]\ar[r]&\BB(\B\!\!\downarrow{\!_b}) \ar[d] \\ \BB\A\ar[r]^{\BB F}&\BB\B}\end{array} $$ is homotopy cartesian if and only if all the maps $\BB p:\BB(F\!\!\downarrow{\!_b})\to \BB(F\!\!\downarrow{\!_{b'}})$, induced by the 1-cells $p:b\to b'$ of $\B$, are homotopy equivalences.\!"} \end{quote} Since the spaces $\BB(\B\!\!\downarrow{\!_b})$ are contractible (Lemma \ref{cont}), the result above tells us that, under the minimum necessary conditions, the classifying space of the homotopy fiber bicategory $F\!\!\downarrow{\!_b}$ is homotopy equivalent to the homotopy fiber of $\BB F:\BB\A\to\BB \B$ at its 0-cell $\BB b\in \BB\B$. Thus, the name `homotopy fiber bicategory' is well chosen. Furthermore, as a corollary, we obtain (see Theorem \ref{A}): \begin{quote}{\em ``\! If all the spaces $\BB(F\!\!\downarrow{\!_b})$ are contractible, then the map $\BB F:\BB\A\to\BB \B$ is a homotopy equivalence.\!"} \end{quote} When the bicategories $\A$ and $\B$ involved in the results above are actually categories, then they are reduced to the well-known Theorems A and B by Quillen \cite{quillen}. Indeed, the methods used in the proof of Theorem \ref{B} we give follow similar lines to those used by Quillen in his proof of Theorem B. However, the situation with bicategories is more complicated than with categories. Let us stress the two main differences between both situations: On one hand, every 2-cell $\sigma:p\Rightarrow q:b\to b'$ in $\B$ gives rise to a homotopy $$\BB\sigma:\BB p\simeq\BB q:\BB(F\!\!\downarrow{\!_b})\to\BB(F\!\!\downarrow{\!_{b'}})$$ that must be taken into account. On the other hand, for $p:b\to b'$ and $p':b'\to b''$ any two composable 1-cells in $\B$, we have a homotopy $$\BB p'\circ \BB p \simeq \BB(p'\circ p):\BB(F\!\!\downarrow{\!_b})\to \BB(F\!\!\downarrow{\!_{b''}}),$$ rather than the identity $\BB p'\circ \BB p = \BB(p'\circ p)$, as it happens in the category case. This unfortunate behavior is due to the fact that neither is the horizontal composition of 1-cells in the bicategories involved (strictly) associative nor does the lax functor preserve (strictly) that composition. Therefore, in the process of taking homotopy fiber bicategories, $F\!\!\downarrow: b\mapsto F\!\!\downarrow{\!_b}$, we are forced to deal with {\em lax bidiagrams of bicategories} $$ \F:\B\to\Bicat,\hspace{0.3cm} b\mapsto \F_b, $$ which are a type of lax functors in the sense of Gordon, Power and Street \cite{g-p-s} from the bicategory $\B$ to the tricategory of small bicategories, rather than ordinary diagrams of small categories, that is, functors $\F:\B\to \Cat$, as it happens when both $\A$ and $\B$ are categories. After this introductory Section 1, the paper is organized in four sections. Section 2 is an attempt to make the paper as self-contained as possible; hence, at the same time as we set notations and terminology, we define and describe in detail the kind of lax functors $\F:\B\to \Bicat$ we are going to work with. Section 3 is very technical but crucial to our discussions. It is mainly dedicated to describing in detail a {\em bicategorical Grothendieck construction}, which assembles any lax bidiagram of bicategories $\F:\B \to\Bicat$ into a bicategory $\int_\B\F$. This is similar to what the ordinary construction, due to Grothendieck \cite{groth,grothendieck}, Giraud \cite{giraud-2,giraud}, and Thomason \cite{thomason} on lax diagrams of categories with the shape of any given category. By means of this higher Grothendieck construction, in Section 4 we establish the third relevant result of the paper, namely (see Theorem \ref{th1}): \begin{quote}{\em ``\!If $\F:\B\to \Bicat$ is a lax bidiagram of bicategories such that each 1-cell $p:b\to b'$ in the bicategory $\B$ induces a homotopy equivalence $\BB\F_b\simeq \BB\F_{b'}$, then, for every object $b\in \Ob\B$, there is an induced homotopy cartesian square $$ \xymatrix{ \BB\F_b\ar[r]\ar[d]&\BB\!\int_\B\F\ar[d]\\ pt \ar[r]^{\BB b}&\BB\B.} $$ That is, the classifying space $\BB\F_b$ is homotopy equivalent to the homotopy fiber of the map induced on classifying spaces by the projection homomorphism $\int_\B\F \to \B$ at the 0-cell corresponding to the object $b$.\!"} \end{quote} Thanks to Thomason's Homotopy Colimit Theorem \cite{thomason}, when $\B$ is a small category and $\F$ values in $\Cat$, the result above is equivalent to the relevant lemma used by Quillen in his proof of Theorem B. Similarly here, the proof of the bicategorical Theorem B, given in the last Section 5, essentially consists of two steps: First, to apply that key result above to the lax bidiagram of homotopy fiber bicategories, $F\!\!\downarrow\,:\B\to\Bicat$, of a lax functor $F:\A\to \B$. Second, to prove that there is a homomorphism $\int_\B F\!\!\downarrow\,\, \to \A$ inducing a homotopy equivalence $\BB(\int_\B F\!\!\downarrow)\simeq \BB\A$, so that the bicategory $\int_\B F\!\!\downarrow$ may be thought of as the ``total bicategory" of the lax functor $F$. Section 5 also includes some applications to classifying spaces of monoidal categories. For instance, we find a new proof of the well-known result by Mac Lane \cite{mac} and Stasheff \cite{sta}: \begin{quote}{\em ``\! Let $(\mathcal{M},\otimes)=(\mathcal{M},\otimes,I,\aso,\bl,\br)$ be a monoidal category. If multiplication for each object $x\in \Ob\mathcal{M}$, $y\mapsto y\otimes x$, induces a homotopy autoequivalence on $\BB\mathcal{M}$, then there is a homotopy equivalence $$\BB \mathcal{M}\simeq \Omega\BB(\mathcal{M},\otimes),$$ between the classifying space of the underlying category and the loop space of the classifying space of the monoidal category.\!"} \end{quote} \section{Bicategorical preliminaries: Lax bidiagrams of bicategories}\label{laxbid} In this paper we shall work with small bicategories, and we refer the reader to the papers by B\'{e}nabou \cite{benabou}, Street \cite{street}, Gordon-Power-Street \cite{g-p-s}, Gurski \cite{gurski}, and Leinster \cite{leinster}, for the background. The bicategorical conventions and the notations that we use along the paper are the same as in \cite[\S 2.1]{ccg-2} and \cite[\S2.4]{ccg-1}. Thus, given any bicategory $\B$, the composition in each hom-category $\B(a,b)$, that is, the vertical composition of 2-cells, is denoted by $\beta\cdot \alpha$, while the symbol $\circ$ is used to denote the horizontal composition functors $\B(b,c)\times \B(a,b) \overset{\circ}\to \B(a,c)$. Identities are denoted as $1_f:f\Rightarrow f$, for any 1-cell $f:a\to b$, and $1_a:a\to a$, for any object $a\in\Ob\B$. The associativity, right unit, and left unit constraints of the bicategory are respectively denoted by the letters $\boldsymbol{a}$, $\boldsymbol{r}$, and $\boldsymbol{l}$. We will use that, in any bicategory, the commutativity of the two triangles \begin{equation}\label{tri2}\begin{array}{cc}\xymatrix@C=3pt@R=15pt{(1\circ g)\circ f \ar@2{->}[rr]^{\aso}\ar@2{->}[rd]_{\bl\circ 1}& &1\circ (g\circ f) \ar@2{->}[ld]^{\bl}\\&g\circ f&} &\xymatrix@C=5pt@R=20pt{(g\circ f)\circ 1 \ar@2{->}[rr]^{\aso}\ar@2{->}[rd]_{\br}& &g\circ(f\circ 1) \ar@2{->}[ld]^{1\circ \,\br}\\&g\circ f&} \end{array} \end{equation} and the equality \begin{equation}\label{equide} \br_1=\bl_1: 1\circ 1\cong 1 \end{equation} are consequence of the other axioms (this is not obvious, but a proof can be done paralleling the given, for monoidal categories, by Joyal and Street in \cite[Proposition 1.1]{joyal-street}). A {\em lax functor} is usually written as a pair $ F=(F,\widehat{F}):\A \to \B$ since we will generally denote its structure constraints by $$\widehat{F}_{g,f}:Fg\circ Ff\Rightarrow F(g\circ f),\hspace{0.4cm} \widehat{F}_a:1_{Fa}\Rightarrow F1_a.$$ The lax functor is termed a {\em pseudo functor} or {\em homomorphism} whenever all the structure constraints $\widehat{F}$ are invertible. If the unit constraints $\widehat{F}_a$ are all identities, then the lax functor is qualified as (strictly) {\em unitary} or {\em normal} and if, moreover, the constraints $\widehat{F}_{g,f}$ are also identities, then $F$ is called a $2$-{\em functor}. If $F,G:\A \to \B$ are lax functors, then we follow the convention of \cite{g-p-s} in what is meant by a {\em lax transformation} ${\alpha=(\alpha,\widehat{\alpha}):F\Rightarrow G}$. Thus, $\alpha$ consists of morphisms ${\alpha a:Fa\to Ga}$, $a\in \Ob\A$, and of 2-cells $\widehat{\alpha}_f:\alpha b\circ Ff\Rightarrow Gf\circ \alpha a$, subject to the usual axioms. When the 2-cells $\widehat{\alpha}$ are all invertible, we say that $\alpha:F\Rightarrow G$ is a {\em pseudo transformation}. In accordance with the orientation of the naturality 2-cells chosen, if ${\alpha,\beta:F\Rightarrow G}$ are two lax transformations, then a {\em modification} $\sigma:\alpha\Rrightarrow\beta$ will consist of 2-cells $\sigma a:\alpha a\Rightarrow\beta a$, $a\in \mbox{Ob}\A$, subject to the commutativity condition, for any morphism $f:a\to b$ of $\A$: $$ \begin{array}{c} \xymatrix@R=35pt@C=35pt{ Fa \ar@/_1.2pc/[r] \ar@{}@<-19pt>[r]\|(.4){\alpha a} \ar@{}[r]_{\Uparrow\sigma} \ar[r]^{\beta a}\ar[d]_{F\!f} &Ga\ar[d]^{G\!f} \ar@{}@<30pt>[d]\|{ =} \\ Fb \ar@{}@<-25pt>[u]_(.26){\Rightarrow}_(.4){\widehat{\alpha}} \ar[r]_{\alpha b}&Gb }\hspace{0.3cm} \xymatrix@R=35pt@C=35pt{ Fa\ar[r]^{\beta a}\ar[d]_{F\!f} \ar@{}@<35pt>[d]_(.4){\Rightarrow}_(.25){\widehat{\beta}} &Ga\ar[d]^{G\!f} \\ Fb\ar@/^1.2pc/[r] \ar@{}@<19pt>[r]\|(.6){\beta b} \ar@{}[r]^{\Uparrow\sigma} \ar[r]_{\alpha b}&Gb. } \end{array} $$ $\Bicat$ denotes the tricategory of of bicategories, homomorphisms, pseudo natural transformations, and modifications. In the structure of $\Bicat$ we use, the composition of pseudo transformations is taken to be $$\big(\xymatrix@C=8pt{\B \ar@/^0.6pc/[rr]^{G}\ar@{}[rr]\|{\Downarrow \beta} \ar@/_0.6pc/[rr]_{ G'} & &\C}\big)\big(\xymatrix@C=8pt{\A \ar@/^0.6pc/[rr]^{F}\ar@{}[rr]\|{\Downarrow \alpha} \ar@/_0.6pc/[rr]_{ F'} & &\B}\big)=\big(\xymatrix@C=9pt{\A \ar@/^0.7pc/[rr]^{GF}\ar@{}[rr]\|{\Downarrow \beta\alpha} \ar@/_0.7pc/[rr]_{ G'\!F'} & &\C}\big), $$ where $\beta \alpha=\beta F'\circ G\alpha:\big(\xymatrix{GF\ar@2[r]^{G\alpha}& GF'\ar@2[r]^{\beta F'}& G'\!F'}\big)$, but note the existence of the useful invertible modification \begin{equation}\label{4}\begin{array}{l}\xymatrix@R=15pt@C=20pt{GF\ar@{}@<-4pt>[rd]^(.5){\Rrightarrow} \ar@{=>}[r]^{\beta F}\ar@{=>}[d]_{G\alpha}&G'F \ar@{=>}[d]^{G'\alpha}\\ GF'\ar@{=>}[r]^{\beta F'}&G'F'}\end{array}\end{equation} whose component at an object $a$ of $\A$, is $\widehat{\beta}_{\alpha a}$, the component of $\beta$ at the morphism $\alpha a$. \subsection{Lax bidiagrams of bicategories} The next concept of fibered bicategory in bicategories is the basis of most of our subsequent discussions. Let $\B$ be a bicategory. Regarding $\B$ as a tricategory in which the $3$-cells are all identities, we define a \emph{lax bidiagram of bicategories} \begin{equation}\label{lb}\F=(\F, \chi,\xi,\omega,\gamma,\delta):\B^{\mathrm{op}}\to\Bicat\end{equation} to be a contravariant lax functor of tricategories from $\B$ to $\Bicat$, all of whose coherence modifications are invertible. More explicitly, a lax bidiagram of bicategories $\F$ as above consists of the following data: \vspace{0.2cm} $(\mathbf{D1})$ for each object $b$ in $\mathcal{B}$, a bicategory \hspace{0.1cm}$\F_b$; \vspace{0.2cm} $(\mathbf{D2})$ for each 1-cell $f:a\to b$ of $\B$, a homomorphism \hspace{0.1cm}$f^:\F_b\to \F_a$; \vspace{0.2cm} $(\mathbf{D3})$ for each 2-cell $\xymatrix@C=0.5pc{a \ar@/^0.7pc/[rr]^{ f} \ar@/_0.7pc/[rr]_{ g}\ar@{}[rr]\|{\Downarrow\alpha} & &b }$ of $\B$, a pseudo transformation \hspace{0.1cm}$\alpha^:f^\Rightarrow g^$; \vspace{-0.5cm}$$\xymatrix@C=0.6pc{\F_b \ar@/^0.8pc/[rr]^{ f^} \ar@/_0.8pc/[rr]_{ g^}\ar@{}[rr]\|(.55){\Downarrow\alpha^} & &\F_a}$$ $(\mathbf{D4})$ for each two composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^{g}&c}$ in the bicategory $\B$, a pseudo transformation \hspace{0.1cm}$\chi_{_{g,f}}:f^g^\Rightarrow (g\circ f)^$; $$\xymatrix@C=10pt@R=15pt{ &\F_c\ar[ld]_{g^}\ar[rd]^{(g\,\circ f)^}\ar@{}[d]\|(.65){\Rightarrow} \ar@{}[d]\|(.4){\chi}& \\ \F_b\ar[rr]_{f^}&&\F_a}$$ $(\mathbf{D5})$ for each object $b$ of $\B$, a pseudo transformation \hspace{0.1cm}$\chi_{_b}: 1_{\F_b}\Rightarrow 1_b^$; $$\xymatrix@C=0.6pc{\F_b \ar@/^0.8pc/[rr]^{ 1_{\F_b}} \ar@/_0.8pc/[rr]_{ 1_b^}\ar@{}[rr]\|{\Downarrow\chi} & &\F_b}$$ \vspace{-0.6cm}$(\mathbf{D6})$ for any two vertically composable 2-cells $\xymatrix{a\ar@/^/[r]^f \ar@/_/[r]_g \ar@{}[r]\|{\Downarrow\alpha} & b}$ and $\xymatrix{a\ar@/^/[r]^g \ar@/_/[r]_h \ar@{}[r]\|{\Downarrow\beta} & b}$ in $\B$, an invertible modification $\xi_{_{\beta,\alpha}}\!:\beta^\circ \alpha^* \Rrightarrow (\beta\cdot \alpha)^$; $$\xymatrix@C=15pt@R=15pt{ &{f}^\ar@2[ld]_{\alpha^}\ar@2[rd]^{(\beta\cdot\alpha)^}\ar@{}[d]\|(.65){\Rrightarrow} \ar@{}[d]\|(.4){\xi}& \\ g^\ar@2[rr]_{\beta^}&&{h}^}$$ $(\mathbf{D7})$ for each 1-cell $f:a\to b$ of $\B$, an invertible modification \hspace{0.1cm}$\xi_{_f}:1_{f^}\Rrightarrow 1_f^$; $$ \xymatrix{f^ \ar@{=>}@/_1.2pc/[d]_{1_{f^} } \ar@{=>}@/^1.2pc/[d]^{1_{f}^ }\ar@{}[d]\|{\Rrightarrow}\ar@{}[d]\|(.32){\xi}\\f^} $$ $(\mathbf{D8})$ for every two horizontally composable 2-cells $\xymatrix@C=0.5pc{a\ar@/^0.7pc/[rr]^f \ar@/_0.7pc/[rr]_{h} \ar@{}[rr]\|{\Downarrow\alpha}& & b\ar@/^0.7pc/[rr]^{g}\ar@/_0.7pc/[rr]_{k} \ar@{}[rr]\|{\Downarrow\beta}& & c}$, an invertible modification \hspace{0.1cm}$\chi_{_{\beta,\alpha}}:(\beta\circ\alpha)^\!\circ \chi_{_{g,f}}\Rrightarrow\chi_{_{k,h}}\circ (\alpha^\beta^)$; $$ \xymatrix{f^\,g^\ar@2[r]^{\alpha^\beta^}\ar@2[d]_\chi\ar@{}@<-22pt>[r]\|{\Rrightarrow} \ar@{}@<-15pt>[r]\|{\chi} & h^\,k^\ar@2[d]^{\chi}\\ (g\circ f)^\ar@2[r]_{(\beta\circ \alpha)^} & (k\circ h)^} $$ $(\mathbf{D9})$ for every three composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^g&c\ar[r]^h&d}$ in $\B$, an invertible modification \hspace{0.1cm}$\omega_{_{h,g,f}}:\, \aso^\!\circ (\chi_{_{h\circ g,f}}\circ f^\chi_{_{h,g}} )\Rrightarrow \chi_{_{h,g\circ f}}\circ \chi_{_{g,f}}h^$; $$ \xymatrix@C=20pt{f^g^h^\ar@2[rr]^{\chi h^}\ar@2[d]_{f^\chi} \ar@{}@<-22pt>[rr]\|{\Rrightarrow}\ar@{}@<-16pt>[rr]\|{\omega}&&(g\circ f)^h^\ar@2[d]^{\chi}\\ f^(h\circ g)^\ar@2[r]^-{\chi}&((h\circ g)\circ f)^ \ar@2[r]^{\aso^}&(h\circ (g\circ f))^ } $$ $(\mathbf{D10})$ for any 1-cell $f:a\to b$ of $\B$, two invertible modifications $${\gamma_f:\bl^_f\circ (\chi_{_{1_b,f}}\circ f^\chi_{_b})\Rrightarrow 1_{f^}},\ \ \delta_f: \br^_f\circ (\chi_{_{f,1_a}}\circ \chi_{_a}f^)\Rrightarrow 1_{f^}.$$ $$ \xymatrix{ f^1_b^\ar@2[d]_{\chi}\ar@{}@<-22pt>[r]\|{\Rrightarrow} \ar@{}@<-15pt>[r]\|{\gamma}& f^\ar@2[d]^{1_{f^}}\ar@2[l]_-{f^\!\chi} \ar@2[r]^-{\chi f^} \ar@{}@<-22pt>[r]\|{\Lleftarrow} \ar@{}@<-15pt>[r]\|{\delta} &1_a^f^ \ar@2[d]^{\chi}\\ (1_b\circ f)^\ar@2[r]_{\bl^}&f^&(f\circ 1_a)^\ar@2[l]^{\br^} } $$ \vspace{0.2cm} These data must satisfy the following coherence conditions: \vspace{0.2cm} $(\mathbf{C1})$ for any three composable $2$-cells $\xymatrix@C=13pt{f\ar@2[r]^{\alpha}&g\ar@2[r]^{\beta}&h\ar@2[r]^(.3){\zeta}&k:a\to b}$ in $\B$, the equation on modifications below holds; $$ \xymatrix@C=35pt@R=35pt{g^\ar@2[d]_{\beta^} \ar@{}@<-15pt>[r]\|(0.35){\Rrightarrow}\ar@{}@<-9pt>[r]\|(0.3){\xi}& f^\ar@2[l]_{\alpha^}\ar@2[d]^{(\zeta\cdot\beta\cdot\alpha)^}\ar@2[ld]\|{_{(\beta\cdot\alpha)^}}& \ar@{}@<5pt>[d]\|{\textstyle =} & g^\ar@2[d]_{\beta^}\ar@2[rd]\|{_{(\zeta\cdot\beta)^}} \ar@{}@<-15pt>[r]\|(0.7){\Rrightarrow} \ar@{}@<-9pt>[r]\|(0.65){\xi}&k^\ar@2[l]_{\alpha^}\ar@2[d]^{(\zeta\cdot\beta\cdot\alpha)^}\\ h^\ar@2[r]_{\zeta^}\ar@{}@<10pt>[r]\|(0.75){\Rrightarrow}\ar@{}@<17pt>[r]\|(0.7){\xi}& k^& & h^\ar@{}@<10pt>[r]\|(0.3){\Rrightarrow}\ar@{}@<17pt>[r]\|(0.25){\xi}\ar@2[r]_{\zeta^}&k^} $$ $(\mathbf{C2})$ for any 2-cell $\xymatrix@C=13pt{f\ar@2[r]^(.26){\alpha}&g:a\to b}$ of $\B$, $$ \xymatrix{\ar@{}@<-9pt>[rr]^(.2){1_{f^}}\ar@{}@<20pt>[d]\|(.53){\Rrightarrow}\|(.36){\xi}& f^\ar@2@/_0.9pc/[ld]\ar@2[rd]^{\alpha^} \ar@{=>}@/^0.7pc/[ld] \ar@{}@<8pt>[d]\|(.7){\Rrightarrow}\|(.5){\xi} & \ar@{}@<10pt>[d]\|(.5){\textstyle =}\ar@{}@<28pt>[d]\|(.5){\textstyle \br_{\alpha^}\,,} \\f^\ar@2[rr]_{\alpha^}\ar@{}@<0.5pt>[rr]^(.37){1_f^}&&g^} \hspace{0.4cm} \xymatrix{\ar@{}@<-9pt>[rr]^(.2){1_{g^}}\ar@{}@<20pt>[d]\|(.53){\Rrightarrow}\|(.36){\xi}& g^\ar@{}@<8pt>[d]\|(.7){\Rrightarrow}\|(.5){\xi} & \ar@{}@<10pt>[d]\|(.5){\textstyle =}\ar@{}@<28pt>[d]\|(.5){\textstyle \bl_{\alpha^};} \\g^\ar@2@/_0.7pc/[ru] \ar@{=>}@/^0.9pc/[ru] \ar@{}@<0.7pt>[rr]^(.42){1_{\!{g}}^}&&{f}^\ar@2[lu]_{\alpha^} \ar@2[ll]^{\alpha^}} $$ \begin{notation}{\em Thanks to conditions $(\mathbf{C1})$ and $(\mathbf{C2})$, for each objects $a,b\in \Ob\B$, we have a homomorphism $\B(a,b)\to \Bicat(\F_b,\F_a)$ such that \vspace{-0.3cm}$$\xymatrix@C=0.5pc{a \ar@/^0.7pc/[rr]^{ f} \ar@/_0.7pc/[rr]_{g}\ar@{}[rr]\|{\Downarrow\alpha} & &b } \mapsto \xymatrix@C=0.6pc{\F_b \ar@/^0.8pc/[rr]^{ f^} \ar@/_0.8pc/[rr]_{ g^}\ar@{}[rr]\|(.55){\Downarrow\,\alpha^} & &\F_a,}$$ and whose structure constraints are the deformations $\xi$ in $(\mathbf{D6})$ and $(\mathbf{D7})$. Then, whenever it is given a commutative diagram in the category $\B(a,b)$ of the form \begin{equation}\label{dn1}\xymatrix@C=15pt@R=20pt{f\ar@2[d]_{ \alpha_0}\ar@2[r]^{\beta_0}&g_1\ar@2[r]^{\beta_1}&\cdots\ar@2[r]& g_n\ar@2[d]^{\beta_n}\\ f_1\ar@2[r]^{\alpha_1}&\cdots\ar@2[r]&f_m\ar@2[r]^{ \alpha_m}&g,}\end{equation} we will denote by \begin{equation}\label{dn2}\xymatrix@C=15pt@R=20pt{f^\ar@2[d]_{\alpha^_0} \ar@{}@<-20pt>[rrr]\|{\cong} \ar@{}@<-13pt>[rrr]\|{\xi} \ar@2[r]^{\beta_0^}&{g}^_1\ar@2[r]^{{\beta}^_1} &\cdots\ar@2[r]& {g}^_n\ar@2[d]^{{\beta}^_n}\\ f_1^\ar@2[r]^{\alpha_1^}&\cdots\ar@2[r]&f_m^\ar@2[r]^{ \alpha_m^}&g^}\end{equation} the invertible modification obtained by an (any) appropriate composition of the modifications $\xi$ and their inverses $\xi^{-1}$, once any particular bracketing in the strings $\alpha_0^,\ldots,\alpha_m^$ and $\beta_0^,\ldots,\beta_n^$ has been chosen. That diagram (\ref{dn2}) is well defined from diagram (\ref{dn1}) is a consequence of the coherence theorem for homomorphisms of bicategories \cite[Theorem 1.6]{g-p-s}. Furthermore, for any diagram $\xymatrix@C=0.8pc{a\ar[r]^{f}&b \ar@/^0.7pc/[rr]^{g} \ar@/_0.7pc/[rr]_{g'}\ar@{}[rr]\|{\Downarrow\alpha} & &c\ar[r]^h&d }$ in $\B$, we shall denote by $$\chi_{_{\alpha,f}}: (\alpha\circ 1_f)^\circ \chi_{_{g,f}}\Rrightarrow \chi_{_{g',f}}\circ f^\alpha^,\hspace{0.4cm} \chi_{_{h,\alpha}}: (1_h\circ \alpha)^\circ \chi_{_{h,g}}\Rrightarrow \chi_{_{h,g'}}\circ \alpha^h^ , $$ $$ \xymatrix@C=30pt{f^g^\ar@{}@<34pt>[d]\|(.55){\Rrightarrow} \ar@{}@<34pt>[d]\|(.4){\chi} \ar@2[r]^{f^\!\alpha^}\ar@2[d]_{\chi}&f^g'^\ar@2[d]^{\chi}\\ (g\circ f)^\ar@2[r]_{(\alpha\circ 1_f)^}&(g'\circ f)^* } \hspace{0.4cm} \xymatrix@C=30pt{g^h^\ar@{}@<34pt>[d]\|(.55){\Rrightarrow} \ar@{}@<34pt>[d]\|(.4){\chi} \ar@2[r]^{\alpha^\!h^}\ar@2[d]_{\chi}&g'^h^\ar@2[d]^{\chi}\\ (h\circ g)^\ar@2[r]_{(1_h\circ \alpha)^}&(h\circ g')^* } $$ the modifications obtained, respectively, by pasting the diagrams in $\Bicat$ below. $$ \xymatrix@C=45pt{f^g^ \ar@{}@<38pt>[dd]\|(.75){\Rrightarrow} \ar@{}@<38pt>[dd]\|(.65){\chi} \ar@2[dd]_{\chi} \ar@2[r]\|(.6){\ 1_{\!f^}\!\alpha^}\ar@{}@<3pt>[r]^(.45){\cong} \ar@{}@<-16pt>[r]\|{\cong}\ar@{}@<-9pt>[r]\|{\xi 1_{\alpha^{\!}}} \ar@2@/^1.5pc/[r]^{f^\!\alpha^}\ar@2@/_2pc/[r]_{1_f^\alpha^}&f^g'^* \ar@2[dd]^{\chi} \\ &\\ (g\circ f)^\ar@2[r]_{(\alpha\circ 1_f)^}&(g'\circ f)^& }\hspace{0.4cm} \xymatrix@C=45pt{g^h^* \ar@{}@<38pt>[dd]\|(.75){\Rrightarrow} \ar@{}@<38pt>[dd]\|(.65){\chi} \ar@2[dd]_{\chi} \ar@2[r]\|(.6){\ \!\alpha^\!1_{\!h^}} \ar@{}@<3pt>[r]^(.45){\cong} \ar@{}@<-17pt>[r]\|{\cong}\ar@{}@<-9pt>[r]\|{1_{\alpha^{\!}}\xi } \ar@2@/^1.5pc/[r]^{\alpha^\!h^}\ar@2@/_2pc/[r]_{\alpha^1_h^}&g'^h^* \ar@2[dd]^{\chi} \\ &\\ (h\circ g)^\ar@2[r]_{(1_h\circ \alpha)^}&(h\circ g')^& } $$ } \end{notation} $(\mathbf{C3})$ for every diagram of $2$-cells $ \xymatrix@C=2.5pc{a\ar@/^1.2pc/[r]^(.4){f}_{}="a"\ar[r]\|(.38){\,f'}^{}="b"_{}="e" \ar@/_1.2pc/[r]_(.4){f''}^{}="c" & b \ar@/^1.3pc/[r]^(.4){g}_{}="d"\ar[r]\|(.38){\,g'}^{}="f"_{}="g"\ar@/_1.2pc/[r]_(.4){g''}^{}="h" & c \ar@2"a";"b"^\alpha \ar@2"e";"c"^{\alpha'} \ar@2"d";"f"^{\beta} \ar@2"g";"h"^{\beta'}} $ in $\mathcal{B}$, $$ \xymatrix@C=15pt@R=30pt{ (g\circ f)^ \ar@{}@<-20pt>[rr]\|(.36){\Rrightarrow} \ar@{}@<-13pt>[rr]\|(.36){\chi} \ar@2[d]_{(\beta\circ\alpha)^}&&&f^g^* \ar@{}@<45pt>[dd]\|{\textstyle =} \ar@2[lll]_{\chi} \ar@2@/^1.5pc/[dd]\|(.3){\ \ (\alpha'\cdot\alpha)^(\beta'\cdot\beta)^} \ar@2[lld]_{\alpha^\beta^} \ar@2@/_1.5pc/[dd]\|(.6){(\!\alpha'^\!\circ \alpha^\!)(\!\beta'^\!\circ\beta^\!)} \\ (g'\circ f')^\ar@2[d]_{(\beta'\circ\alpha')^} \ar@{}@<-20pt>[rr]\|(.36){\Rrightarrow} \ar@{}@<-13pt>[rr]\|(.36){\chi} &f'^g'^\ar@2[rrd]_{\alpha'^\beta'^}\ar@2[l]_-{\chi} \ar@{}@<4pt>[r]\|(1.0){\overset{(\ref{4})}\cong} \ar@{}[rr]\|(1.0){\cong} \ar@{}@<7pt>[rr]\|(1.0){\xi\xi}&&\\ (g''\circ f'')^&&&f''^g''^\ar@2[lll]^{\chi}} \xymatrix@C=15pt@R=30pt{ (g\circ f)^ \ar@2@/^2.3pc/[dd]^(.3){((\beta'\cdot\beta)\circ (\alpha'\cdot\alpha))^} \ar@2[d]_{(\beta\circ\alpha)^}&&&f^g^\ar@2[lll]_{\chi} \ar@2@/^1.5pc/[dd]\|(.3){\ \ (\alpha'\cdot\alpha)^(\beta'\cdot\beta)^} \\ (g'\circ f')^\ar@2[d]_{(\beta'\circ\alpha')^} \ar@{}@<10pt>[r]\|(.4){\cong} \ar@{}@<10pt>[r]^(.4){\xi} \ar@{}@<-5pt>[rr]\|(.9){\Rrightarrow} \ar@{}@<-4pt>[rr]^(.9){\chi} &&\\ (g''\circ f'')^&&&f''^g''^\ar@2[lll]^{\chi}} $$ $(\mathbf{C4})$ for every pair of composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^g&c}$, $$ \xymatrix{(g\!\circ\!f)^ \ar@{=>}@/_1.2pc/[d] \ar@{}@<-27pt>[d]\|{1_{(g\circ f)^}} \ar@{=>}@/^1.2pc/[d]^{1_{g\circ f}^} \ar@{}[d]\|{\Rrightarrow}\ar@{}[d]\|(.32){\xi}& f^g^\ar@2[l]_{\chi} \ar@{}@<-10pt>[d]\|{\Rrightarrow}\ar@{}@<-10pt>[d]\|(.32){\chi} \ar@{=>}@/^1.2pc/[d]^{1_{f}^1_{g}^} \ar@{}@<50pt>[d]\|{\textstyle =} \\(g\!\circ\!f)^* & f^g^\ar@2[l]_{\chi} } \hspace{0.4cm} \xymatrix{(g\!\circ\!f)^* \ar@{=>}@/_1.2pc/[d] \ar@{}@<-27pt>[d]\|{1_{(g\circ f)^}} \ar@{}@<8pt>[d]\|(.45){\cong}& f^g^* \ar@{}[d]\|{\Rrightarrow}\ar@{}[d]\|(.32){\xi\xi} \ar@2[l]_{\chi} \ar@{=>}@/^1.2pc/[d]^{1_{f}^1_{g}^} \ar@{=>}@/_1.2pc/[d] \ar@{}@<-26pt>[d]\|(.6){1_{f^}\!1_{g^}} \\(g\!\circ\!f)^* & f^g^\ar@2[l]_{\chi} } $$ $(\mathbf{C5})$ for every $2$-cells $\xymatrix@C=0.5pc{a\ar@/^0.7pc/[rr]^f \ar@/_0.7pc/[rr]_{f'} \ar@{}[rr]\|{\Downarrow\alpha}& & b\ar@/^0.7pc/[rr]^{g}\ar@/_0.7pc/[rr]_{g'} \ar@{}[rr]\|{\Downarrow\beta}& & c\ar@/^0.7pc/[rr]^h \ar@/_0.7pc/[rr]_{h'} \ar@{}[rr]\|{\Downarrow\zeta}& & d}$, the equation $A=A'$ holds, where $$ \xymatrix@C=13pt@R=35pt{\ar@{}[dd]\|{\textstyle A=}& &((h\circ g)\circ f)^* \ar@2[d]\|{((\zeta\circ\beta)\circ\alpha)^} \ar@{}@<-22pt>[r]\|{\Rrightarrow}\ar@{}@<-16pt>[r]\|{\chi} \ar@2[ld]_{\aso^}&f^(h\circ g)^ \ar@{}@<-20pt>[r]\|(.75){\Rrightarrow} \ar@{}@<-13pt>[r]\|(.75){\alpha^\!\chi} \ar@2[l]_(.45){\chi} \ar@2[d]\|{\alpha^(\zeta\circ \beta)^}& &f^g^h^\ar@2[ll]_{f^\chi}\ar@2[d]^{(\alpha^\beta^)\zeta^} \ar@{}@<-6pt>[d]\|(.5){\cong}\ar@2@/_1.2pc/[d]\ar@{}@<-6pt>[d]_(.64){\alpha^(\beta^\zeta^)\ \ }\\ & (h\circ(g\circ f))^\ar@{}[r]\|{\cong}\ar@{}@<7pt>[r]\|{\xi} \ar@2[rd]\ar@{}@<-4pt>[rd]_{(\zeta\circ (\beta\circ \alpha))^} & ((h'\circ g')\circ f')^ \ar@{}@<-22pt>[rr]\|(.4){\Rrightarrow}\ar@{}@<-16pt>[rr]\|(.4){\omega} \ar@2[d]^{\aso^}&f'^(h'\circ g')^\ar@2[l]^(.4){\chi}& & f'^g'^h'^\ar@2[lld]^{\chi h'^* }\ar@2[ll]^{f'^\chi}\\ & & (h'\circ (g'\circ f'))^& (g'\circ f')^h'^\ar@2[l]_-{\chi}& & } $$ $$ \xymatrix@C=13pt@R=35pt{\ar@{}[dd]\|{\textstyle A'=} & &((h\circ g)\circ f)^* \ar@2[ld]_{\aso^} \ar@{}@<-22pt>[rr]\|(.3){\Rrightarrow}\ar@{}@<-16pt>[rr]\|(.3){\omega}&f^(h\circ g)^\ar@2[l]_-{\chi} & & f^g^h^\ar@2[ll]_{f^\chi}\ar@2[d]^{(\alpha^\beta^)\zeta^} \ar@2[lld]_{\chi h^}\\ & (h\circ(g\circ f))^ \ar@{}@<-22pt>[rr]\|(.6){\Rrightarrow}\ar@{}@<-16pt>[rr]\|(.6){\chi} \ar@2[rd]\ar@{}@<-4pt>[rd]_{(\zeta\circ (\beta\circ \alpha))^} &&(g\circ f)^h^* \ar@2[d]^(.35){(\beta\circ \alpha)^\zeta^} \ar@2[ll]_{\chi} \ar@{}@<-4pt>[rr]\|(.5){\Rrightarrow}\ar@{}@<3pt>[rr]\|(.5){\chi\zeta^}&& f'^g'^h'^\ar@2[lld]^{\chi h'^}\\ & & (h'\circ (g'\circ f'))^& (g'\circ f')^h'^\ar@2[l]_-{\chi}&& } $$ $(\mathbf{C6})$ for every four composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^g&c\ar[r]^h&d\ar[r]^k&e}$, the equation $B=B'$ holds, where $$ \xymatrix@C=5pt{B=&&f^g^h^k^\ar@2[rrd]\ar@{}@<3pt>[rrd]^{\chi h^* k^} \ar@2[lld] \ar@{}@<-3pt>[lld]_{f^g^\chi} &&\\ f^g^(k\!\circ\!h)^\ar@2[d]_{f^\chi} \ar@{}@<18pt>[rrrr]\|{\overset{(\ref{4})}\cong} \ar@2[rr]^{\chi(k\circ h)^} && (g\!\circ\!f)^(k\!\circ\!h)^\ar@2[dd]^{\chi} &&(g\!\circ\!f)^h^k^\ar@2[d]^{\chi k^} \ar@2[ll]_{(g\circ f)^\chi}\\ f^((k\!\circ\!h)\!\circ\!g)^* \ar@{}@<-2pt>[rr]\|{\Rrightarrow}\ar@{}@<4pt>[rr]\|{\omega} \ar@2[d]_{\chi} & &\ar@{}@<-2pt>[rr]\|{\Rrightarrow}\ar@{}@<4pt>[rr]\|{\omega}& &(h\!\circ\!(g\!\circ\!f))^k^\ar@2[d]^{\chi} \\ (((k\!\circ\!h)\!\circ\!g)\!\circ\!f)^\ar@2[rr]^{\aso^} \ar@2[rd]\ar@{}@<-3pt>[rd]_{(\aso\circ 1_f)^} & &((k\!\circ\!h)\!\circ\!(g\!\circ\!f))^\ar@2[rr]^{\aso^}\ar@{}[d]\|(.55){\cong}\|(.4){\xi}& &(k\!\circ\!(h\!\circ\!(g\!\circ\!f)))^ \\ &((k\!\circ\!(h\!\circ\!g))\!\circ\!f)^\ar@2[rr]_{\aso^}&&(k\!\circ\!((h\!\circ\!g)\!\circ\!f))^* \ar@2[ru]_{\ (1_k\circ \aso)^}& } $$ $$ \xymatrix@C=10pt{B'=&&f^g^h^k^\ar@2[rrd]\ar@{}@<3pt>[rrd]^{\chi h^ k^} \ar@2[lld] \ar@{}@<-3pt>[lld]_{f^g^\chi} \ar@2[d]^{f^\!\chi \,k^} &&\\ f^g^(k\!\circ\!h)^\ar@2[d]_{f^\chi} \ar@{}[rr]\|{\Rrightarrow} \ar@{}@<7pt>[rr]\|{f^\!\omega} &&f^(h\!\circ\!g)^k^\ar@2[rd]^{\chi k^} \ar@2[ld]_{f^\chi } \ar@{}[rr]\|{\Rrightarrow}\ar@{}@<7pt>[rr]\|{\omega k^} &&(g\!\circ\!f)^h^k^\ar@2[d]^{\chi k^}\\ f^((k\!\circ\!h)\!\circ\!g)^ \ar@2[r]^{f^\!\aso^} \ar@2[d]_{\chi} \ar@{}@<44pt>[dd]\|(.45){\Rrightarrow} \ar@{}@<44pt>[dd]\|(.36){\chi} &f^(k\!\circ\!(h\!\circ\!g))^ \ar@2[dd]^{\chi} \ar@{}@<-20pt>[rr]_{\Rrightarrow} \ar@{}@<-15pt>[rr]_{\omega} &&((h\!\circ\!g)\!\circ\!f)^k^ \ar@2[r]^{\aso^k^} \ar@2[dd]_{\chi} \ar@{}@<30pt>[dd]\|(.5){\Rrightarrow} \ar@{}@<30pt>[dd]\|(.42){\chi} &(h\!\circ\!(g\!\circ\!f))^k^\ar@2[d]^{\chi} \\ (((k\!\circ\!h)\!\circ\!g)\!\circ\!f)^* \ar@2[rd]\ar@{}@<-3pt>[rd]_{(\aso\circ 1_f)^} & && &(k\!\circ\!(h\!\circ\!(g\!\circ\!f)))^ \\ &((k\!\circ\!(h\!\circ\!g))\!\circ\!f)^\ar@2[rr]_{\aso^}&& (k\!\circ\!((h\!\circ\!g)\!\circ\!f))^\ar@2[ru]_{\ (1_k\circ \aso)^}& } $$ $(\mathbf{C7})$ for every 2-cell $\xymatrix@C=13pt{f\ar@2[r]^(.26){\alpha}&g:a\to b}$, the following two equations on modifications hold: $$ \xymatrix{&1_a^f^\ar@2[ld]_{\chi} \ar@{}@<-3pt>[d]\|(.55){\Rrightarrow}\|(.38){\delta} & f^\ar@2[l]_(.4){\chi\!f^}\ar@2[ld]\ar@{}@<3pt>[ld]_(.4){1_{f^}} \ar@2[d]^(.4){\alpha^}\\ (f\!\circ\!1_a)^* \ar@{}@<30pt>[d]\|(.56){\cong}\|(.38){\xi} \ar@2[r]^{\br^}\ar@2[d]_{(\alpha\circ 1)^}&f^\ar@{}[r]\|{\cong}\ar@2[d]^(.4){\alpha^}& g^\ar@2[ld]^{1_{g^}}\\(g\!\circ 1_a)^\ar@2[r]^-{\br^}&g^& } \xymatrix{\\ =\\} \xymatrix{&1_a^f^\ar@2[ld]_{\chi}\ar@2[d]\ar@{}@<-1pt>[d]^{1_a^\alpha^} \ar@{}@<-3pt>[rd]^(.5){\overset{(\ref{4})}\cong} & f^\ar@2[l]_(.4){\chi\! f^} \ar@2[d]^(.4){\alpha^}\\ (f\!\circ\!1_a)^\ar@{}@<-2pt>[r]\|(.55){\Rrightarrow}\ar@{}@<4pt>[r]\|(.55){\chi}\ar@2[d]_{(\alpha\circ 1)^}&1_a^g^ \ar@2[ld]_{\chi} \ar@{}@<-3pt>[d]\|(.55){\Rrightarrow}\|(.38){\delta} & g^\ar@2[l]_(.4){\chi\! g^} \ar@2[ld]^{1_{g^}}\\(g\!\circ 1_a)^\ar@2[r]^-{\br^}&g^& } $$ $$ \xymatrix{&f^1_b^\ar@2[ld]_{\chi} \ar@{}@<-3pt>[d]\|(.55){\Rrightarrow}\|(.38){\gamma} & f^\ar@2[l]_(.4){f^\!\chi}\ar@2[ld]\ar@{}@<3pt>[ld]_(.4){1_{f^}} \ar@2[d]^(.4){\alpha^}\\ (1_b\!\circ\!f)^* \ar@{}@<30pt>[d]\|(.56){\cong}\|(.38){\xi} \ar@2[r]^{\bl^}\ar@2[d]_{(1\circ\alpha)^}&f^\ar@{}[r]\|{\cong}\ar@2[d]^(.4){\alpha^}& g^\ar@2[ld]^{1_{g^}}\\(1_b\!\circ g)^\ar@2[r]^{\bl^}&g^& } \xymatrix{\\ =\\} \xymatrix{&f^1_b^\ar@2[ld]_{\chi}\ar@2[d]\ar@{}@<-1pt>[d]^{\alpha^1_b^} \ar@{}@<-3pt>[rd]^(.5){\overset{(\ref{4})}\cong} & f^\ar@2[l]_(.4){f^\!\chi} \ar@2[d]^(.4){\alpha^}\\ (1_b\!\circ\!f)^\ar@{}@<-2pt>[r]\|(.55){\Rrightarrow}\ar@{}@<4pt>[r]\|(.55){\chi}\ar@2[d]_{(1\circ \alpha)^} &g^1_b^ \ar@2[ld]_{\chi} \ar@{}@<-3pt>[d]\|(.55){\Rrightarrow}\|(.38){\gamma} & g^\ar@2[l]_(.4){g^\!\chi} \ar@2[ld]^{1_{g^}}\\(1_b\!\circ g)^\ar@2[r]^{\bl^}&g^& } $$ $(\mathbf{C8})$ for every pair of composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^g&c}$, the following equation holds: $$ \xymatrix@R=18pt@C=10pt{ f^1_b^g^* \ar@{}@<-17pt>[rr]\|(.55){\Rrightarrow} \ar@{}@<-10pt>[rr]\|(.55){\gamma g^} \ar@2[d]_{f^\chi}\ar@2[rd]^{\chi g^}&& f^g^* \ar@2[d]^{1_{f^g^}}\ar@2[ll]_{f^\chi g^} \ar@{}@<28pt>[ddd]\|{\textstyle =} \\ f^(g\!\circ\!1_b)^ \ar@{}@<-15pt>[r]\|{\Rrightarrow} \ar@{}@<-9pt>[r]\|{\omega} \ar@2[dd]_{\chi} & (1_b\!\circ\!f)^g^ \ar@2[d]^{\chi} \ar@2[r]^(.6){\bl^\!g^}& f^g^\ar@2[dd]^{\chi} \\ &(g\!\circ\!(1_b\!\circ\! f))^* \ar@{}@<4pt>[r]\|(.7){\Rrightarrow} \ar@{}@<10pt>[r]\|(.7){\chi} \ar@2[rd]^{(1_g\circ\, \bl)^}& \\ ((g\!\circ\!1_b)\!\circ\! f)^ \ar@{}@<8pt>[rr]\|{\cong} \ar@{}@<15pt>[rr]\|{\xi} \ar@2[ru]^{\aso^} \ar@2[rr]_{(\br\circ 1_f)^}& &(g\!\circ\!f)^* } \xymatrix@R=14pt@C=28pt{ f^1_b^g^\ar@2[dd]_{f^\chi}& & f^g^\ar@2[ll]_{f^\chi g^} \ar@2@<3pt>[dd]^{1_{f^g^}} \ar@/_1pc/@2[dd]\ar@{}@<-22pt>[dd]\|{f^\!1_{g^}} \ar@{}@<-5pt>[dd]\|{\cong} \\ & &\\ f^(g\!\circ\!1_b)^ \ar@{}@<-28pt>[rr]\|(.5){\Rrightarrow} \ar@{}@<-22pt>[rr]\|(.5){\chi} \ar@{}@<26pt>[r]\|(.7){\Rrightarrow} \ar@{}@<33pt>[r]\|(.7){f^\!\delta} \ar@2[rr]^(.6){f^\!\br^} \ar@2[dd]_{\chi}&&f^g^\ar@2[dd]^{\chi} \\ & ~ & \\ ((g\!\circ\!1_b)\!\circ\! f)^ \ar@2[rr]_{(\br\circ 1_f)^}& &(g\!\circ\!f)^ } $$ A lax bidiagram of bicategories $\F:\B^{\mathrm{op}}\to\Bicat$ is called a {\em pseudo bidiagram of bicategories} whenever each of the pseudo transformations $\chi$, in $(\mathbf{D4})$ and $(\mathbf{D5})$, is a pseudo equivalence; that is, regarding $\B$ as a tricategory whose 3-cells are all identities, a trihomomorphism $\F:\B^{^{\mathrm{op}}}\to \Bicat$ in the sense of Gordon-Power-Street \cite[Definition 3.1]{g-p-s}. \begin{example}\label{1213}{\em If $\C$ is any small category viewed as a bicategory, then a lax bidiagram of bicategories over $\C$, as above, in which the deformations $\xi$ in $(\mathbf{D6})$ and $(\mathbf{D7})$, and $\chi$ in $(\mathbf{D8})$, are all identities is the same thing as a {\em lax diagram of bicategories} $\F:\C^{\mathrm{op}}\to \Bicat$ as in \cite[\S 2.2]{ccg-2}. For instance, let $X$ be any topological space and let $\C(X)$ denote its poset of open subsets, regarded as a category. Then a {\em fibered bicategory in bigroupoids above $X$} is a lax diagram of bicategories $$ \F:\C(X)^{\mathrm{op}}\to \Bicat, $$ such that all the bicategories $\F_U$ are bigroupoids, that is, bicategories whose 1-cells are invertible up to a 2-cell, and whose 2-cells are strictly invertible. In particular, when all the bigroupoids $\F_U$ are strict, that is, 2-categories, and all the homomorphisms $f^:\F_U\to \F_V$ associated to the inclusions of open sets $f:V\hookrightarrow U$ are 2-functors, we have the notion of {\em fibered $2$-category in $2$-groupoids above the space} $X$. Thus, 2-stacks and 2-gerbes on spaces are relevant examples of lax diagrams of bicategories (see e.g. Breen \cite[Definitions 6.1, 6.2, and 6.3]{Breen}). For another example, if $\mathcal{T}$ is any small tricategory, as in \cite[Definition 2.2]{g-p-s}, then its {\em Grothendieck nerve} $$ \mathrm{N}\mathcal{T}:\Delta^{\!{\mathrm{op}}}\to \Bicat, $$ whose bicategory of $p$-simplices is $$ \mathrm{N}_p\mathcal{T} = \bigsqcup_{(x_0,\ldots,x_p)\in \mbox{\scriptsize Ob}\mathcal{T}^{p+1}}\hspace{-0.3cm} \mathcal{T}(x_1,x_0)\times\mathcal{T}(x_2,x_1)\times\cdots\times\mathcal{T}(x_p,x_{p-1}), $$ \cite[Theorem 3.3.1]{c-h} gives a striking example of a pseudo diagram of bicategories.} \end{example} \begin{example}\label{exap2} {\em For any bicategory $\B$, a {\em lax bidiagram of categories} over $\B$, that is, a lax bidiagram $\F:\B^{\mathrm{op}}\to \Bicat$ in which every bicategory $\F_a$, $a\in \Ob\B$, is a category (i.e., a bicategory where all the 2-cells are identities) is the same thing as a contravariant lax functor $\F:\B^{\mathrm{op}}\to \Cat$ to the 2-category $\Cat$ of small categories, functors, and natural transformations, since the condition of all $\F_a$ being categories forces all the modifications in $(\mathbf{D6})- (\mathbf{D10})$ to be identities. For example, any object $b$ of a bicategory $\B$ defines a pseudo bidiagram of categories \cite[Example 10]{street} $$ \B(-,b):\B^{\mathrm{op}}\to \mathbf{Cat}, $$ which carries an object $x\in\Ob\B$ to the hom-category $\B(x,b)$, a 1-cell $g:x\to y$ to the functor $g^:\B(y,b)\to\B(x,b)$ defined by $$ \xymatrix@C=25pt {y \ar@/^0.8pc/[r]^{f} \ar@/_0.8pc/[r]_-{f'} \ar@{}[r]\|{\Downarrow\beta} & b} \xymatrix{\ar@{\|->}[r]^{g^}&} \xymatrix @C=40pt{x \ar@/^0.8pc/[r]^{f\circ g} \ar@/_0.8pc/[r]_-{f'\!\circ g} \ar@{}[r]\|{\Downarrow \beta\circ 1_{\!g}} & b}, $$ and a 2-cell $\alpha:g\Rightarrow g'$ is carried to the natural transformation $\alpha^:g^\Rightarrow g'^$ that assigns to each 1-cell $f:y\to b$ in $\B$ the 2-cell $1_{\!f}\circ \alpha:f\circ g\Rightarrow f\circ g'$. For $x\overset{g}\to y\overset{h}\to z$ any two composable 1-cells of $\B$, the structure natural equivalence $\chi: g^h^\cong (h\circ g)^$, at any $f:z\to b$, is provided by the associativity constraint $\boldsymbol{a}:(f\circ h)\circ g\cong f\circ (h\circ g)$, whereas for any $x\in\Ob \B$, the structure natural equivalence $\chi:1_{\B(x,b)}\cong 1_{x}^$, at any $f:x\to b$, is the right unit isomorphism $\br^{-1}:f\cong f\circ 1_{x}$.} \end{example} \section{The Grothendieck construction on lax bidiagrams of bicategories}\label{gt} The well-known `Grothendieck construction', due to Grothendieck \cite{groth, grothendieck} and Giraud \cite{giraud-2,giraud}, on pseudo diagrams $(\F,\chi):\B^{\mathrm{op}}\to \Cat$ of small categories with the shape of any given small category, was implicitly used in the proof given by Quillen of his famous Theorems A and B for the classifying spaces of small categories \cite{quillen}. Subsequently, since Thomason established his celebrate Homotopy Colimit Theorem \cite{thomason}, the Grothendieck construction has become an essential tool in homotopy theory of classifying spaces. In this section, our work is dedicated to extending the Grothendieck construction to lax bidiagrams of bicategories $\F=(\F, \chi,\xi,\omega,\gamma,\delta):\B^{\mathrm{op}}\to\Bicat$, where $\B$ is any bicategory, since its use is a key for proving our main results in the paper. But we are not claiming here much originality, since extensions of the ubiquitous Grothendieck construction have been developed in many general frameworks. In particular, we should mention here three recent approaches to our construction: In \cite{ccg-2}, Carrasco, Cegarra, and Garz\'on study the bicategorical Grothendieck construction on lax diagrams of bicategories, as in Example \ref{1213}. In \cite{bakovic-2,bakovic}, Bakovi\'c performs the Grothendieck construction on normal pseudo bidiagrams of bicategories, that is, lax bidiagrams $\F$ whose modifications $\chi_b$ in $(\mathbf{D5})$ and $\xi_f$ in $(\mathbf{D7})$ are identities, and whose pseudo transformations $\chi_{g,f}$ in $(\mathbf{D4})$ are pseudo equivalences. Buckley, in \cite{buckley}, presents the more general case of pseudo bidiagrams, that is, when all the pseudo transformations $\chi_{g,f}$ and $\chi_b$ in $(\mathbf{D4})$ and $(\mathbf{D5})$ are pseudo equivalences. The {\em Grothendieck construction on a lax bidiagram of bicategories} $\F:\B^{\mathrm{op}} \to\Bicat$, as in $(\ref{lb})$, assembles it into a bicategory, denoted by $$\xymatrix{\int_{\B}\!\F},$$ which is defined as follows: \underline{The objects}\, are pairs $(x,a)$, where $a\in\Ob\B$ and $x\in \Ob\F_a$. \underline{The 1-cells}\, are pairs $(u,f):(x,a)\to (y,b)$, where $f:a\to b$ is a 1-cell in $\B$ and $u:x\to f^y$ is a 1-cell in $\F_a$. \underline{The 2-cells}\, are pairs $\xymatrix@C=25pt{(x,a)\ar@{}[r]\|{\Downarrow (\phi,\alpha)}\ar@/^0.8pc/[r]^{(u,f)}\ar@/_0.8pc/[r]_{(v,g)}&(y,b),} $ consisting of a 2-cell $\xymatrix@C=0.5pc{a \ar@/^0.6pc/[rr]^{ f} \ar@/_0.6pc/[rr]_{ g}\ar@{}[rr]\|{\Downarrow\alpha} & &b }$ of $\B$ together with a 2-cell $\phi:\alpha^y\circ u\Rightarrow v$ in $\F_a$, $$ \xymatrix@C=15pt@R=10pt{&f^y\ar@{}@<3pt>[d]\|(.6){\Downarrow\phi}\ar[rd]^{\alpha^\!y}&\\ x\ar[ru]^{u}\ar[rr]_{v}&&g^y.} $$ \underline{The vertical composition} of 2-cells in $\int_\B\F$, $\xymatrix@C=25pt{(x,a) \ar@/^0.8pc/[r]^{(u,f)} \ar@/_0.8pc/[r]_{(v,g)} \ar@{}[r]\|{\Downarrow(\phi,\alpha)} & (y,b)}$ and $\xymatrix@C=25pt{(x,a) \ar@/^0.8pc/[r]^{(v,g)} \ar@/_0.8pc/[r]_{(w,h)} \ar@{}[r]\|{\Downarrow(\psi,\beta)} & (y,b)}$, is the 2-cell $$ \xymatrix@C=50pt{(x,a)\ar@/^1pc/[r]^{(u,f)}\ar@{}[r]\|(.5){\Downarrow (\psi\odot \phi,\beta\cdot \alpha)} \ar@/_1pc/[r]_{(w,h)}&(y,b),} $$ where $\beta\cdot\alpha$ is the vertical composition of $\beta$ with $\alpha$ in $\B$, and $\psi\odot \phi:(\beta\cdot\alpha)^\!y\circ u\Rightarrow w$ is the 2-cell of $\F_a$ obtained by pasting the diagram below. $$ \xymatrix@C=40pt{\ar@{}@<-45pt>[dd]\|(.3){\textstyle \psi\odot \phi:} & f^y \ar[rdd]^{(\beta\cdot\alpha)^y} \ar[d]\|{\alpha^y} & \\ & g^y \ar[rd]_{\beta^y} \ar@{}@<3pt>[r]\|(.3){\overset{\xi}\cong}& \\ x \ar[ruu]^u \ar[ru]_{v} \ar[rr]_w \ar@{}@<40pt>[r]\|(.7){\Downarrow \phi} \ar@{}@<20pt>[rr]\|{\Downarrow \psi}&& h^y } $$ The vertical composition of 2-cells so defined is associative and unitary thanks to the coherence conditions $(\mathbf{C1})$ and $(\mathbf{C2})$. The identity 2-cell, for each 1-cell $(u,f):(x,a)\to (y,b)$, is $$\begin{array}{c} 1_{(u,f)}=(\overset{_\cdot}1_{(u,f)},1_f):(u,f)\Rightarrow (u,f).\\ \overset{\cdot}1_{(u,f)}=\big(\xymatrix{1_f^y\circ u\ar@2[r]^-{\xi^{-1}\circ 1}&1_{f^\!y}\circ u\overset{\bl}\Rightarrow u}\big) \end{array} $$ Hence, we have defined the hom-category $\int_\B\F\big((x,a),(y,b)\big)$, for any two objects $(x,a)$ and $(y,b)$ of $\int_\B\F$. Before continuing the description of this bicategory, we shall do the following useful observation: \begin{lemma}\label{isogro} A $2$-cell $(\phi,\alpha):(u,f)\Rightarrow (v,g)$ in $\int_\B\F\big((x,a),(y,b)\big)$ is an isomorphism if and only if both $\alpha:f\Rightarrow g$, in $\B(a,b)$, and $\phi:\alpha^y\circ u\Rightarrow v$, in $\F_a(x,g^y)$, are isomorphisms. \end{lemma} \begin{proof} It is quite straightforward, and we leave it to the reader. \end{proof} We return now to the description of the bicategory $\int_\B\F$. \underline{The horizontal composition} of two 1-cells $\xymatrix@C=20pt{(x,a)\ar[r]^{(u,f)}&(y,b)\ar[r]^{(u',f')}&(z,c)}$ is the 1-cell $$ (u',f')\circ (u,f)=(u'\circledcirc u,f'\circ f):(x,a)\longrightarrow (z,c), $$ where $f'\circ f:a\to c$ is the composite in $\B$ of the 1-cells $f$ and $f'$, while $$u'\circledcirc u=\chi z\circ (f^u'\circ u):x\longrightarrow (f'\circ f)^z$$ is the composite in $\F_a$ of $ \xymatrix@C=15pt{x\ar[r]^-{u}&f^y\ar[r]^{f^u'}&f^{f'}^z\ar[r]^-{\chi z}&(f'\circ f)^z}$. \underline{The horizontal composition} of 2-cells is defined by $$ \xymatrix@C=30pt{(x,a)\ar@{}[r]\|{\Downarrow (\phi,\alpha)}\ar@/^1pc/[r]^{(u,f)} \ar@/_1pc/[r]_{(v,g)}&(y,b) \ar@{}[r]\|{\Downarrow (\phi',\alpha')}\ar@/^1pc/[r]^{(u',f')}\ar@/_1pc/[r]_{(v',g')}&(z,c)} \ \overset{\circ}\mapsto \xymatrix@C=55pt{(x,a)\ar@{}[r]\|{\Downarrow (\phi'\circledcirc\phi,\alpha'\circ \alpha)} \ar@/^1.2pc/[r]^{(u'\circledcirc u,f'\circ f)}\ar@/_1.2pc/[r]_{(v'\circledcirc v,g'\circ g)}&(z,c), } $$ where $\alpha'\circ\alpha$ is the horizontal composition in $\B$ of $\alpha'$ with $\alpha$, and $\phi'\circledcirc\phi$ is the 2-cell in $\F_a$ canonically obtained by pasting the diagram below. $$ \xymatrix@C=40pt{\ar@{}@<-45pt>[dd]\|(.2){\textstyle \phi'\circledcirc\phi:} & f^y \ar[dd]^{\alpha^\!y} \ar[r]^{f^u'}\ar[rd]\ar@{}@<1pt>[rd]_{f^\!v'} \ar@{}@<52pt>[d]\|(.35){\Downarrow f^\!\phi'}& f^f'^z\ar[r]^{\chi z}\ar[d]^{f^\!\alpha'^\!z}& (f'\circ f)^z\ar[dd]^{(\alpha'\circ \alpha)^z} \\ x\ar@{}[r]\|(.6){\Downarrow \phi}\ar[ru]^{u}\ar[rd]_{v}&& f^g'^z\ar[d]^{\alpha^\!g'^\!z} \ar@{}[r]^{\chi z} \ar@{}@<-1pt>[r]\|{\cong} &\\ &g^y \ar@{}@<28pt>[r]\|{\widehat{\alpha^}} \ar@{}@<21pt>[r]\|{\cong} \ar[r]_{g^\!v'}&g^g'^z\ar[r]_{\chi z}&(g'\circ g)^z} $$ Owing to the coherence conditions $(\mathbf{C3})$ and $(\mathbf{C4})$, the horizontal composition so defined truly gives, for any three objects $(x,a),(y,b),(z,c)$ of $\int_{\B}\!\F$, a functor $$\xymatrix{\int_{\B}\!\F((y,b),(z,c))\times \int_{\B}\!\F((x,a),(y,b))\ar[r]^(.65){\circ}& \int_{\B}\!\F((x,a),(z,c)).}$$ \underline{The structure associativity} isomorphism, for any three composable morphisms $$(x,a)\overset{(u,f)}\longrightarrow (y,b)\overset{(v,g)}\longrightarrow (z,c)\overset{(w,h)}\longrightarrow (t,d),$$ $$(\overset{_\circ}\aso,\aso):\big((w\circledcirc v)\circledcirc u,(h\circ g)\circ f \big) \cong\big(w\circledcirc (v\circledcirc u), h\circ (g\circ f) \big),$$ is provided by the associativity constraint $\aso:(h\circ g)\circ f\cong h\circ (g\circ f)$ of the bicategory $\B$, together with the isomorphism in the bicategory $\F_a$ $$ \overset{_\circ}\aso:\aso^t\circ ((w\circledcirc v)\circledcirc u)\cong w\circledcirc (v\circledcirc u),$$ canonically obtained from the 2-cell pasted of the diagram $$ \xymatrix{ && & f^(h\circ g)^t\ar[r]^{\chi t}& ((h\circ g)\circ f)^ t\ar[dd]^{\aso^* t}\\ x \ar@{}@<-16pt>[rr]\|(.75){\cong } \ar[r]^{u}\ar[rrd]_{v\circledcirc u}& f^y\ar[r]\ar@{}@<-2pt>[r]^{f^\!v}\ar@<4pt>[rru]^{f^\!(w\circledcirc v)}& f^g^z\ar[r]\ar@{}@<-2pt>[r]^{f^\!g^\!w} \ar[d]_{\chi z} \ar@{}@<16pt>[r]\|{\cong }& f^g^h^t\ar[u]_{f^\!\chi t}\ar[d]^{\chi h^\!t} \ar@{}@<-3pt>[r]\|(.55){\cong} \ar@{}@<3pt>[r]\|(.55){\omega t}&\\ &&(g\circ f)^z\ar[r]_{(g\circ f)^\!w} \ar@{}@<24pt>[r]\|{\widehat{\chi}} \ar@{}@<17pt>[r]\|{\cong} &(g\circ f)^h^t\ar[r]_{\chi t}&(h\circ(g\circ f))^t } $$ By Lemma \ref{isogro}, these associativity 2-cells are actually isomorphisms in $\xymatrix{\int_{\B}\!\F}$. Furthermore, they are natural thanks to the coherence condition $(\mathbf{C5})$, while the pentagon axiom for them holds because of condition $(\mathbf{C6})$. \underline{The identity 1-cell}, for each object $(x,a)$ in $\xymatrix{\int_{\B}\!\F}$, is provided by the pseudo transformation $\chi_a:1_{\F_a}\Rightarrow 1_a^$, by $$ 1_{(x,a)}=(\chi x,1_a):(x,a)\to (x,a).$$ \underline{The left and right unit constraints}, for each morphism $(u,f):(x,a)\to (y,b)$ in $\xymatrix{\int_{\B}\!\F}$, $$\begin{array}{cc} (\overset{_\circ}\bl,\bl):1_{(y,b)}\circ (u,f)\cong (u,f),& (\overset{_\circ}\br,\br):(u,f)\circ 1_{(x,a)}\cong (u,f), \end{array} $$ are respectively given by the 2-cells $\bl:1_b\circ f\Rightarrow f$ and $\br:f\circ 1_a\Rightarrow f$ of $\B$, together with the 2-cells in $\F_a$ obtained by pasting the diagrams below. $$ \xymatrix@R=62pt{f^y \ar@{}@<-25pt>[d]\|(0.35){\textstyle \overset{_\circ}\bl: }\ar[rrd]\ar@{}@<3pt>[rrd]_{1_{f^\!y}}\ar[r]^{f^\!\chi\!y}& f^1_b^y\ar[r]^{\chi y}&(1_b\circ f)^y\ar[d]^{\bl^\!y}\\ x\ar@{}@<18pt>[r]\|{\cong} \ar@{}@<24pt>[r]\|{\bl } \ar@{}@<52pt>[rr]\|(.70){\cong} \ar@{}@<58pt>[rr]\|(.70){\gamma y} \ar[u]^{u}\ar[rr]_{u}&&f^y }\hspace{0.5cm} \xymatrix{ 1_a^x \ar@{}@<-25pt>[dd]\|(0.35){\textstyle \overset{_\circ}\br: } \ar[r]^{1_a^\!u}&1_a^f^y\ar[r]^{\chi y}&(f\circ 1_a)^y\ar[dd]^{\br^y}\\ \ar@{}@<12pt>[r]\|{\cong} \ar@{}@<20pt>[r]\|{\widehat{\chi}} & f^y \ar[u]\ar@{}@<2pt>[u]_{\chi f^\!y} \ar[rd]\ar@{}@<-3pt>[rd]^{1_{f^\!y}} \ar@{}@<12pt>[r]\|{\cong} \ar@{}@<20pt>[r]\|{\delta y}& \\ x \ar@{}@<16pt>[rr]\|{\cong} \ar@{}@<22pt>[rr]\|{\bl} \ar[ru]\ar@{}@<-2pt>[ru]^{u}\ar[uu]^{\chi x}\ar[rr]_{u}&&f^y} $$ These unit constraints in $\xymatrix{\int_{\B}\!\F}$ are isomorphisms by Lemma \ref{isogro}, natural due to coherence condition $(\mathbf{C7})$, and the coherence triangle for them follows from condition $(\mathbf{C8})$. Hence, $\xymatrix{\int_{\B}\!\F}$ is actually a bicategory. As a consequence of the above construction we obtain the following equalities on lax bidiagram of bicategories, which is used many times along the paper for several proofs: \begin{lemma} \label{gdo}Let $\F=(\F, \chi,\xi,\omega,\gamma,\delta):\B^{\mathrm{op}}\to\Bicat$ be a lax bidiagram of bicategories. The equations on modifications below hold. $(i)$ For any object $a$ of $\B$, $$ \xymatrix@C=45pt{1_a^\ar@2[rdd]^{1}\ar@2[d]_{1_a^\chi }\ar@{}@<-24pt>[r]\|(.7){\cong} &1_{\mathcal{F}_a}\ar@2[dd]^{\chi }\ar@2[l]_{\chi } \ar@{}@<30pt>\|{\textstyle =}[dd] \\ 1_a^1_a^\ar@2[d]_{\chi} \ar@{}@<-15pt>[r]\|(.35){\Rrightarrow} \ar@{}@<-9pt>[r]\|(.35){\gamma} & \\ (1_a\circ 1_a)^\ar@2[r]_{\bl_1^=\br_1^} &1_a^}\hspace{0.3cm} \xymatrix@C=15pt{1_a^\ar@2[d]_{1_a^\chi } &\ar@{}@<-11pt>[d]\|(.4){\overset{(\ref{4})}\cong}&1_{\mathcal{F}_a}\ar@2[dd]^{\chi }\ar@2[ll]_{\chi } \ar@2[ld]\ar@{}@<-6pt>[ld]\|{\chi }\\ 1_a^1_a^\ar@2[d]_{\chi} & 1^_a\ar@2[l]_(.4){\chi 1_a^} \ar@2[rd]_{1} \ar@{}[r]\|(.6){\cong} \ar@{}@<20pt>[l]\|(.35){\Rrightarrow} \ar@{}@<14pt>[l]\|(.35){\delta}&\\ (1_a\circ 1_a)^\ar@2[rr]_{\bl_1^=\br_1^} &&1_a^} $$ $(ii)$ for every pair of composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^g&c}$ in $\B$, $$ \xymatrix@R=18pt@C=16pt{ f^g^1_c^* \ar@{}@<-15pt>[rr]\|(.6){\overset{(\ref{4})}\cong} \ar@2[d]_{f^\chi}\ar@2[rd]^{\chi 1_c^}&& f^g^ \ar@2[d]^{\chi} \ar@2[ll]_{f^g^\chi } \ar@{}@<30pt>[ddd]\|{\textstyle =} \\ f^(1_c\!\circ\!g)^ \ar@{}@<-18pt>[r]\|{\Rrightarrow} \ar@{}@<-12pt>[r]\|{\omega} \ar@2[dd]_{\chi}& (g\!\circ\!f)^1_c^ \ar@2[d]^{\chi} & (g\!\circ\! f)^* \ar@2[dd]^{1_{(g\circ f)^}} \ar@2[l]_(.45){(g\circ f)^\chi } \\ &(1_c\!\circ\!(g\!\circ\! f))^* \ar@{}@<12pt>[r]\|{\Rrightarrow} \ar@{}@<18pt>[r]\|{\gamma} \ar@2[rd]^{\bl^}& \\ ((1_c\!\circ\!g)\!\circ\!f)^ \ar@{}@<8pt>[rr]\|{\cong} \ar@{}@<15pt>[rr]\|{\xi} \ar@2[ru]^{\aso^} \ar@2[rr]_{(\bl\circ 1_f)^}& &(g\!\circ\!f)^* } \xymatrix@R=14pt@C=22pt{ f^g^1_c^\ar@2[dd]_{f^\chi}& & f^g^\ar@2[ll]_{f^g^\chi } \ar@2[dd]^{\chi} \ar@2[ldd]\ar@{}@<4pt>[ldd]_{f^1_{g^}} \\ & &\\ f^(1_c\!\circ\!g)^ \ar@2[r]^{f^\bl^} \ar@{}@<28pt>[r]\|(.75){\Rrightarrow} \ar@{}@<34pt>[r]\|(.75){f^\gamma} \ar@2[dd]_{\chi}&f^g^* \ar@2[rdd]^{\chi}\ar@{}[r]\|{\cong} &(g\!\circ\!f)^\ar@2[dd]^{1_{(g\circ f)^}}\\ & ~ & \\ ((1_c\!\circ\!g)\!\circ\!f)^* \ar@{}@<18pt>[r]\|(.8){\Rrightarrow} \ar@{}@<24pt>[r]\|(.8){\chi} \ar@2[rr]_{(\bl\circ 1_f)^}& &(g\!\circ\!f)^ } $$ $$ \xymatrix@R=18pt@C=10pt{ 1_a^f^g^* \ar@{}@<-15pt>[rr]\|(.5){\Rrightarrow} \ar@{}@<-8pt>[rr]\|(.5){\delta g^} \ar@2[d]_{1_a^\chi}\ar@2[rd]_{\chi g^}&& f^g^* \ar@2[d]^{1_{f^\!g^}} \ar@2[ll]_{\chi f^g^} \ar@{}@<26pt>[ddd]\|{\textstyle =} \\ 1_a^(g\circ\!f)^ \ar@{}@<-16pt>[r]\|{\Rrightarrow} \ar@{}@<-11pt>[r]\|{\omega} \ar@2[dd]_{\chi} & (f\!\circ\!1_a)^\!g^ \ar@2[r]^{\br^\!g^} \ar@2[d]^{\chi} &f^g^ \ar@2[dd]^{\chi}\\ &(g\!\circ\!(f\!\circ\! 1_a))^* \ar@{}@<4pt>[r]\|(.75){\Rrightarrow} \ar@{}@<10pt>[r]\|(.75){\chi} \ar@2[rd]^(.6){(1_g\circ \br)^}& \\ ((g\!\circ\!f)\!\circ\!1_a)^ \ar@{}@<8pt>[rr]\|{\cong} \ar@{}@<15pt>[rr]\|{\xi} \ar@2[ru]^{\aso^} \ar@2[rr]_{\br^}& &(g\!\circ\!f)^* } \xymatrix@R=14pt@C=20pt{ 1_a^f^g^\ar@2[dd]_{1_a^\chi}& & f^g^\ar@2[ll]_{\chi f^g^} \ar@2[dd]^{1_{f^\!g^}} \ar@2[ldd]_{\chi} \\ & &\\ 1_a^(g\!\circ\!f)^ \ar@{}@<26pt>[r]\|(.7){\overset{(\ref{4})}\cong} \ar@2[dd]_{\chi}&(g\!\circ\!f)^* \ar@2[l]_(.45){\chi (g\circ f)^} \ar@2[rdd]\ar@{}@<2pt>[rdd]_(.4){1_{(g\circ f)^}} \ar@{}[r]\|{\cong} &f^g^ \ar@2[dd]^{\chi}\\ & ~ & \\ ((g\!\circ\!f)\!\circ\!1_a)^* \ar@{}@<23pt>[r]\|(.7){\Rrightarrow} \ar@{}@<30pt>[r]\|(.7){\delta} \ar@2[rr]_{\br^}& &(g\!\circ\!f)^ } $$ \end{lemma} \begin{proof} $(i)$ follows from the equality $(\ref{equide})$ in the bicategory $\int_\B\F$, that is, $\br_{1_{(x,a)}}=\bl_{1_{(x,a)}}$, for any $x\in \F_a$. Similarly, $(ii)$ is consequence of the commutativity of triangles $(\ref{tri2})$ in $\int_\B\F$, for any pair of composable 1-cells of the form $$\xymatrix{(f^g^x,a)\ar[r]^{(1,f)}&(g^x,b)\ar[r]^{(1,g)}&(x,c)},$$ for any $x\in\Ob\F_c$. \end{proof} \subsection{A cartesian square} Let $\F=(\F, \chi,\xi,\omega,\gamma,\delta):\B^{\mathrm{op}}\to\Bicat$ be any given lax bidiagram of bicategories. For any bicategory $\A$ and any lax functor $F:\A\to \B$, we shall denote by \begin{equation}\label{f} \F F=(\F F, \chi_F,\xi_F,\omega_F,\gamma_F,\delta_F):\A^{\mathrm{op}}\to\Bicat \end{equation} the lax bidiagram of bicategories obtained by composing, in the natural sense, $\F$ with $F$; that is, the lax bidiagram consisting of the following data: \vspace{0.2cm} $(\mathbf{D1})$ for each object $a$ in $\A$, the bicategory \hspace{0.1cm}$\F_{Fa}$; \vspace{0.2cm} $(\mathbf{D2})$ for each 1-cell $f:a\to b$ of $\A$, the homomorphism \hspace{0.1cm}$(Ff)^:\F_{Fb}\to \F_{Fa}$; \vspace{0.2cm} $(\mathbf{D3})$ for each 2-cell $\xymatrix@C=0.5pc{a \ar@/^0.7pc/[rr]^{ f} \ar@/_0.7pc/[rr]_{ g}\ar@{}[rr]\|{\Downarrow\alpha} & &b }$ of $\A$, the pseudo transformation \hspace{0.1cm}$(F\alpha)^:(Ff)^\Rightarrow (Fg)^$; $(\mathbf{D4})$ for each two composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^{g}&c}$ in the bicategory $\A$, the pseudo transformation \hspace{0.1cm} ${\chi_F} _{_{\!g,f}}:(Ff)^(Fg)^\Rightarrow F(g\circ f)^$ obtained by pasting $$\xymatrix@R=30pt{ &\F_{Fc}\ar[ld]\ar@{}@<4pt>[ld]_{(Fg)^} \ar@/_1.4pc/[rrd]\|(.3){(Fg\circ Ff)^}\ar[rrd]^{F(g\circ f)^}& &\\ \F_{Fb}\ar@<-3pt>[rrr]_(.3){(Ff)^}\ar@{}@<10pt>[r]\|(.7){\Rightarrow} \ar@{}@<17pt>[r]\|(.7){\chi}&&\ar@{}@<8pt>[r]\|(.1){\Rightarrow} \ar@{}@<14pt>[r]\|(.1){\widehat{F}^}&\F_{Fa};}$$ $(\mathbf{D5})$ for each object $a$ of $\A$, the pseudo transformation $ {\chi_{\!F}}_{\!_a}\!=\!\xymatrix@C=15pt{\big(1_{\F_{Fa}}\ar@2[r]^{\chi_{\!_{Fa}}}& 1_{Fa}^* \ar@2[r]^-{\widehat{F}^_{a}}&F(1_a)^\big);}$ $(\mathbf{D6})$ for any two vertically composable 2-cells $\xymatrix@C=12pt{f\ar@2[r]^{\alpha}&g\ar@2[r]^{\beta}&h }$ in $\A$, the invertible modification \hspace{0.1cm}${\xi_F}_{_{\beta,\alpha}}=\xi_{_{F\beta,F\alpha}}\!:F(\beta)^\circ F(\alpha)^ \Rrightarrow F(\beta\cdot \alpha)^$; \vspace{0.2cm}$(\mathbf{D7})$ for each 1-cell $f:a\to b$ of $\A$, the invertible modification \hspace{0.1cm}${\xi_F}_{_f}=\xi_{_{Ff}}\!:1_{{F(f)}^}\Rrightarrow 1_{Ff}^$; $(\mathbf{D8})$ for every two horizontally composable 2-cells $\xymatrix@C=0.5pc{a\ar@/^0.7pc/[rr]^f \ar@/_0.7pc/[rr]_{h} \ar@{}[rr]\|{\Downarrow\alpha}& & b\ar@/^0.7pc/[rr]^{g}\ar@/_0.7pc/[rr]_{k} \ar@{}[rr]\|{\Downarrow\beta}& & c}$ in $\A$, $${\chi_F}_{_{\beta,\alpha}}:F(\beta\circ \alpha)^\!\circ {\chi_F}{_{_{g,f}}}\Rrightarrow{\chi_F}{_{_{k,h}}}\circ (F(\alpha)^F(\beta)^)$$ is the invertible modification obtained by pasting the diagram below; $$\xymatrix@R=40pt@C=40pt{F(f)^F(g)^\ar@2[r]^{F(\alpha)^F(\beta)^} \ar@2[d]_{\chi} \ar@{}@<-22pt>[r]\|{\Rrightarrow} \ar@{}@<-16pt>[r]\|{\chi} & F(h)^F(k)^\ar@2[r]^{\chi} & (Fk\circ Fh)^\ar@2[d]^{\widehat{F}^}\\ (Fg\circ Ff)^\ar@2[r]_{\widehat{F}^} \ar@2[rru]\ar@{}@<-10pt>[rru]\|(.6){(F\beta\circ F\alpha)^} &F(g\circ f)^ \ar@{}@<16pt>[r]\|(.6){\Rrightarrow} \ar@{}@<23pt>[r]\|(.6){\xi} \ar@2[r]_{F(\beta\circ \alpha)^}&F(k\circ h)^ } $$ $(\mathbf{D9})$ for every three composable 1-cells $\xymatrix@C=13pt{a\ar[r]^{f}&b\ar[r]^g&c\ar[r]^h&d}$ in $\A$, the invertible modification $${\omega_F}_{_{h,g,f}}:\, F(\aso)^\!\circ ({\chi_F}_{_{h\circ g,f}}\circ F(f)^{\chi_F}_{_{h,g}} )\Rrightarrow {\chi_F}_{_{h,g\circ f}}\circ {\chi_F}_{_{g,f}}F(h)^$$ is obtained from the modification pasted of the diagram below; $$ \xymatrix@C=20pt{F(f)^F(g)^F(h)^\ar@2[d]_{F(f)^\chi}\ar@2[r]^{\chi\, F(h)^} \ar@{}@<-22pt>[rr]\|(.3){\cong} \ar@{}@<-16pt>[rr]\|(.3){\omega} &(Fg\circ Ff)^F(h)^ \ar@{}@<-22pt>[rr]\|(.5){\cong} \ar@{}@<-16pt>[rr]\|(.5){\chi} \ar@2[rd]^{\chi}\ar@2[r]^{\widehat{F}^F(h)^}& F(g\circ f)^F(h)^\ar@2[rd]^{\chi}&\\ F(f)^(Fh\circ Fg)^ \ar@{}@<-22pt>[r]\|(.45){\cong} \ar@{}@<-16pt>[r]\|(.45){\chi} \ar@2[r]^{\chi} \ar@2[d]_{F(f)^\widehat{F}^} &((Fh\circ Fg)\circ Ff)^* \ar@{}@<-22pt>[rr]\|(.5){\cong} \ar@{}@<-15pt>[rr]\|(.5){\xi} \ar@2[r]^{\aso^} \ar@2[d]_{(\widehat{F}\circ 1)^} & (Fh\circ(Fg\circ Ff))^* \ar@2[r]^{(1\circ \widehat{F})^}&(Fh\circ F(g\circ f))^ \ar@2[d]^{\widehat{F}^} \\ F(f)^F(h\circ g)^\ar@2[r]_{\chi}&(F(h\circ g)\circ Ff)^ \ar@2[r]_{\widehat{F}^}&F((h\circ g)\circ f)^ \ar@2[r]_{F(\aso)^}&F(h\circ (g\circ f))^ } $$ $(\mathbf{D10})$ for any 1-cell $f:a\to b$ of $\A$, the invertible modifications $$\begin{array}{l}{\gamma_F}_{_{\!f}}:F(\bl_f)^\circ ({\chi_F}_{_{1,f}}\circ F(f)^{\chi_F}_{_{\!b}})\Rrightarrow 1_{F(f)^},\\[4pt] {\delta_F}_{_{\!f}}: F(\br_f)^\circ ({\chi_F}_{_{f,1}}\circ {\chi_F}_{_{\!a}}F(f)^)\Rrightarrow 1_{F(f)^},\end{array}$$ are, respectively, canonically obtained from the modification pasted of the diagrams below. $$ \xymatrix@C=40pt{\ar@{}@<-55pt>[dd]\|{\textstyle \gamma_F:} F(f)^F(1_b)^\ar@2[d]_{\chi} \ar@{}@<-23pt>[r]\|{\cong} \ar@{}@<-16pt>[r]\|{\chi} & F(f)^1_{Fb}^\ar@2[d]^{\chi}\ar@2[l]_-{F(f)^\widehat{F}^} \ar@{}@<-33pt>[r]\|(.55){\cong} \ar@{}@<-26pt>[r]\|(.55){\gamma} &F(f)^* \ar@2[dd]^{1}\ar@2[l]_-{F(f)^\chi}\\ (F1_b\circ Ff)^\ar@2[d]_{\widehat{F}^} \ar@{}@<-23pt>[rr]\|(.4){\cong} \ar@{}@<-16pt>[rr]\|(.4){\xi} &(1_{Fb}\circ Ff)^\ar@2[l]^-{(\widehat{F}\circ 1)^} \ar@2[rd]^{\bl^}& \\ F(1_b\circ f)^\ar@2[rr]_{F(\bl)^}&&F(f)^* } $$ $$ \xymatrix@C=40pt{\ar@{}@<-55pt>[dd]\|{\textstyle \delta_F:} F(1_a)^F(f)^\ar@2[d]_{\chi} \ar@{}@<-23pt>[r]\|{\cong} \ar@{}@<-16pt>[r]\|{\chi} & 1_{Fa}^F(f)^\ar@2[d]^{\chi}\ar@2[l]_-{\widehat{F}^F(f)^} \ar@{}@<-33pt>[r]\|(.55){\cong} \ar@{}@<-26pt>[r]\|(.55){\delta} &F(f)^* \ar@2[dd]^{1}\ar@2[l]_-{\chi F(f)^}\\ (Ff\circ F1_a)^\ar@2[d]_{\widehat{F}^} \ar@{}@<-23pt>[rr]\|(.4){\cong} \ar@{}@<-16pt>[rr]\|(.4){\xi} &(Ff\circ 1_{Fa})^\ar@2[l]^-{(1\circ \widehat{F})^} \ar@2[rd]^{\br^}& \\ F(f\circ 1_a)^\ar@2[rr]_{F(\br)^}&&F(f)^* } $$ There is an induced lax funtor \begin{equation}\label{ilf} \xymatrix{\bar{F}:\int_\A\F F \to \int_\B\F} \end{equation} given on cells by $$ \xymatrix@C=30pt{(x,a)\ar@{}[r]\|{\Downarrow (\phi,\alpha)}\ar@/^1pc/[r]^{(u,f)}\ar@/_1pc/[r]_{(v,g)}&(y,b) }\ \overset{\bar{F}}\mapsto\ \xymatrix@C=30pt{(x,Fa) \ar@{}[r]\|{\Downarrow (\phi,F\alpha)}\ar@/^1pc/[r]^{(u,Ff)}\ar@/_1pc/[r]_{(v,Fg)}&(y,Fb), } $$ and whose structure constraints are canonically given by those of $F$, namely: For every two composable 1-cells $\xymatrix@C=20pt{(x,a)\ar[r]^{(u,f)}&(y,b)\ar[r]^{(v,g)}&(z,c)}$ in $\int_\A\F F$, the corresponding structure 2-cell of $\bar{F}$ for their composition is $$ (\aso^{-1},\widehat{F}):\bar{F}(v,g)\circ \bar{F}(u,f)\cong \bar{F}((v,g)\circ (u,f)), $$ where $\widehat{F}=\widehat{F}_{g,f}:Fg\circ Ff\Rightarrow F(g\circ f)$ is the structure 2-cell of $F$, and $$ \aso^{-1}: \widehat{F}_{g,f}^\circ(\chi_{_{Fg,F\!f}}z\circ ( F(f)^(v)\circ u))\cong (\widehat{F}_{g,f}^\circ \chi_{_{Fg,F\!f}}z)\circ (F(f)^(v)\circ u) $$ is the associativity isomorphism in the bicategory $\F_{Fa}$. For $(x,a)$ any object of the bicategory $\int_\A\F F$, the corresponding structure 2-cell of $\bar{F}$ for its identity is $$ (1,\widehat{F}):1_{\bar{F}(x,a)}\Rightarrow \bar{F}1_{(x,a)}, $$ where $\widehat{F}=\widehat{F}_a:1_{Fa}\Rightarrow F(1_a)$ is the structure 2-cell of $F$, and $1$ is the is the identity 2-cell of the 1-cell $\widehat{F}_a^x\circ \chi_{Fa}x:x\to F(1_a)^x$ in the bicategory $\F_{Fa}$. Then, although the category of bicategories and lax functors has no pullbacks in general, if, for any lax bidiagram of bicategories $\F:\B^{\mathrm{op}}\to\Bicat$ as above, we denote by \begin{equation}\label{proj2fun} \xymatrix{P:\int_\B\F \to \B} \end{equation} the canonical projection 2-functor, which is defined by $$\xymatrix@C=30pt{(x,a)\ar@{}[r]\|{\Downarrow (\phi,\alpha)}\ar@/^1pc/[r]^{(u,f)}\ar@/_1pc/[r]_{(v,g)}&(y,b)}\ \overset{P}\mapsto\ \xymatrix@C=25pt{a\ar@{}[r]\|{\Downarrow \alpha}\ar@/^0.7pc/[r]^{f} \ar@/_0.7pc/[r]_{g}&b,} $$ the following fact holds: \begin{lemma}\label{lesqld} Let $\F:\B^{\mathrm{op}}\to \Bicat$ be a lax bidiagram of bicategories. For any lax functor $F:\A\to \B$, the induced square $$ \xymatrix{ \int_\A\F F\ar[d]_{P}\ar[r]^{\bar{F}}&\int_\B\F\ar[d]^{P}\\ \A\ar[r]^{F}&\B } $$ is cartesian in the category of bicategories and lax functors. \end{lemma} \begin{proof} Any pair of lax functors, say $L:\C\to \A$ and $M:\C\to \int_\B\F$, such that $FL=PM$ determines a unique lax functor $N:\C\to \int_\A\F F$ $$ \xymatrix{ \C\ar[rdd]_L\ar[rrd]^M\ar@{.>}[rd]\|N&&\\ &\int_\A\F F\ar[d]^(.4){P}\ar[r]^(.4){\bar{F}}&\int_\B\F\ar[d]^{P}\\ &\A\ar[r]^{F}&\B } $$ such that $PN=L$ and $\bar{F}N=M$, which is defined as follows: Observe that the lax functor $M$ carries any object $a\in \Ob\C$ to an object of $\int_\B\F$ which is necessarily written in the form $Ma=(Da, FLa)$ for some object $Da$ of the bicategory $\F_{FLa}$. Similarly, for any 1-cell $f:a\to b$ in $\C$, we have $Mf=(Df,FLf)$, for some 1-cell $Df:Da\to FL(f)^Db$ in $\F_{FLa}$, and, for any 2-cell $\alpha:f\Rightarrow g\in \C(a,b)$, we have $M\alpha=(D\alpha,FL\alpha)$, for $D\alpha:FL(\alpha)^Db\circ Df\Rightarrow Dg$ a 2-cell in $\F_{FLa}$. Also, for any pair of composable 1-cells $a\overset{f}\to b\overset{g}\to c$ and any object $a$ in $\C$, the structure 2-cells of $M$ can be respectively written in a similar form as $$\widehat{M}_{g,f}=(\widehat{D}_{g,f},F\widehat{L}_{_{g,f}}\circ \widehat{F}_{_{Lg,Lf}}): (Dg,FLg)\circ (Df,FLf)\Rightarrow (D(g\circ f),FL(g\circ f))$$ $$\widehat{M}_{a}=(\widehat{D}_{a},F(\widehat{L}_a)\circ \widehat{F}_{La}):1_{(Da,FLa)}\Rightarrow (D1_a,FL1_a),$$ for some 2-cells $\widehat{D}_{g,f}$ and $\widehat{D}_{a}$ of the bicategory $\F_{FLa}$. Then, the claimed $N:\C\to \int_\A\F F$ is the lax functor which acts on cells by $$ \xymatrix@C=0.5pc{a \ar@/^0.7pc/[rr]^{ f} \ar@/_0.7pc/[rr]_{ g}\ar@{}[rr]\|{\Downarrow\alpha} & &b} \ \overset{N}\mapsto \xymatrix@C=1.4pc{(Da,La) \ar@/^1.2pc/[rr]^{ (Df,Lf)} \ar@/_1.2pc/[rr]_{(Dg,Lg)}\ar@{}[rr]\|{\Downarrow(D\alpha,L\alpha)} & &(Db,Lb)} $$ and whose respective structure 2-cells, for any pair of composable 1-cells $a\overset{f}\to b\overset{g}\to c$ and any object $a$ in $\C$, are $$\widehat{N}_{g,f}=(\widehat{D}_{g,f},\widehat{L}_{_{g,f}}): (Dg,Lg)\circ (Df,Lf)\Rightarrow (D(g\circ f),L(g\circ f)),$$ $$\widehat{N}_{a}=(\widehat{D}_{a},\widehat{L}_a):1_{(Da,La)}\Rightarrow (D1_a,L1_a).$$ \end{proof} \begin{remark}{\em \label{reotlax} There exist different other `dual' notions of \emph{bidiagrams of bicategories}, depending on the covariant or contravariant choices for $(\mathbf{D2})$ and $(\mathbf{D3})$, and the direction of the pseudo transformations $\chi$ in $(\mathbf{D4})$ and $(\mathbf{D5})$, but the results we present about lax bidiagrams are similarly proved for the different cases. For example, in a \emph{covariant oplax bidiagram of bicategories} $\F:\B\to \Bicat$ the data in $(\mathbf{D2})$ are specified with homomorphisms $f_:\F_a\to \F_b$ for the $1$-cells $f:a\to b$ of $\B$, while in $(\mathbf{D4})$, the pseudo transformations are of the form $\chi_{g,f}:(g\circ f)_\Rightarrow g_f_$. The corresponding data in $(\mathbf{D5}), (\mathbf{D8}), (\mathbf{D9})$ and $(\mathbf{D10})$ change in a natural way. The Grothendieck construction on such a bidiagram, has now $1$-cells $(u,f):(x,a)\to (y,b)$ given by $f:a\to b$ a $1$-cell in $\B$ and $u:f_x\to y$ a $1$-cell in $\F_b$. The $2$-cells $(\phi,\alpha):(u,f)\Rightarrow (v,g)$ are now given by a $2$-cell $\alpha: f\Rightarrow g$ in $\B$ and a $2$-cell $\phi:u\Rightarrow v\circ \alpha_x$. The compositions and constraints of this bicategory are defined in the same way as in the contravariant lax case.}\end{remark} \section{The homotopy cartesian square induced by a lax bidiagram} For the general background on simplicial sets we mainly refer to the book by Goerss and Jardine \cite{g-j}. In particular, we will use the following result, which can be easily proved from the discussion made in \cite[IV, 5.1]{g-j} and Quillen's Lemma \cite[Lemma in page 14]{quillen} (or \cite[IV, Lemma 5.7]{g-j}): \begin{lemma}\label{simlem} Let $p: E\to B$ be an arbitrary simplicial map. For any $n$-simplex $x\in B_n$, let $p^{-1}(x)$ be the simplicial set defined by the pullback diagram $$ \xymatrix{ p^{-1}(x)\ar[r]\ar[d]&E\ar[d]^{p}\\ \Delta[n]\ar[r]^{\Delta x}&B, } $$ where $\Delta[n]=\Delta(-,[n])$ is the standard simplicial $n$-simplex, whose $m$-simplices are the maps $[m]\to [n]$ in the simplicial category $\Delta$, and $\Delta x: \Delta[n]\to B$ denotes the simplicial map such that $\Delta x(1_{[n]})=x$. Suppose that, for every $n$-simplex $x\in B_n$, and for any map $\sigma:[m]\to [n]$ in the simplicial category, the induced simplicial map $p^{-1}(\sigma^x)\to p^{-1}(x)$ $$ \xymatrix@R=10pt{p^{-1}(\sigma^x)\ar@{.>}[rd]\ar[rrd]\ar[dd]&& \\ &p^{-1}(x)\ar[dd]\ar[r] &E \ar[dd]^{p}\\ \Delta[m]\ar[rd]_{\Delta\sigma}\ar[rrd]^(.35){\Delta(\sigma^x)}\|!{[ru];[rd]}\hole&& \\ & \Delta[n]\ar[r]_{\Delta x}&B} $$ gives a homotopy equivalence on geometric realizations $\|p^{-1}(\sigma^x)\|\simeq \|p^{-1}(x)\|$. Then, for each vertex $v\in B_0$, the induced square of spaces $$ \xymatrix{ \|p^{-1}(v)\|\ar[r]\ar[d]&\|E\|\ar[d]^{\|p\|}\\ pt\ar[r]^{\|v\|}&\|B\| } $$ is homotopy cartesian, that is, $\|p^{-1}(v)\|$ is homotopy equivalent to the homotopy fiber of the map $\|p\|:\|E\|\to \|B\|$ over the 0-cell $\|v\|$ of $\|B\|$. \end{lemma} Like categories, bicategories are closely related to spaces through the classifying space construction. We shall recall briefly from \cite[Theorem 6.1]{ccg-1} that the {\em classifying space} of a (small) bicategory can be defined by means of several, but always homotopy equivalent, simplicial and pseudo simplicial objects that have been characteristically associated to it. For instance, the classifying space $\BB \B$ of the bicategory $\B$ may be thought of as $$ \BB\B=\|\Delta\B\|, $$ the geometric realization of its (non-unitary) {\em geometric nerve} \cite[Definition 4.3]{ccg-1}; that is, the simplicial set $$ \Delta\B:\Delta^{\!^{\mathrm{op}}} \to \ \Set,\hspace{0.6cm} [n]\mapsto \lfunc([n],\B), $$ whose $n$-simplices are all lax functors ${\z:[n]\to \B}$. Here, the ordered sets ${[n]=\{0,1,\dots,n\}}$ are considered as categories with only one morphism $(i,j):i\to j$ when $0\leq i\leq j\leq n$, so that a non-decreasing map $[m]\rightarrow [n]$ is the same as a functor. Hence, a geometric $n$-simplex of $\B$ is a list of cells of the bicategory $$ \z=(\z_i,\z_{i,j},\widehat{\z}_{i,j,k},\widehat{\z}_i) $$ which is geometrically represented by a diagram in $\B$ with the shape of the 2-skeleton of an oriented standard $n$-simplex, whose faces are triangles $$ \xymatrix{ & \z_j \ar@{}@<3pt>[d]\|(.6){\Downarrow\,\widehat{\z}_{i,j,k}}\ar[dr]^{\z_{j,k}} \\ \z_i \ar[ur]^{\z_{i,j}} \ar[rr]_{\z_{i,k}} && \z_k} $$ with objects $\z_i$ placed on the vertices, 1-cells $\z_{i,j}:\z_i\rightarrow \z_j$ on the edges, and 2-cells $\widehat{\z}_{i,j,k}:\z_{j,k}\circ \z_{i,j}\Rightarrow \z_{i,k}$ in the inner, together with 2-cells $\widehat{\z}_i: 1_{\z_i}\Rightarrow \z_{i,i}$. These data are required to satisfy the condition that, for $0\leq i\leq j\leq k\leq l\leq n$, each tetrahedron is commutative in the sense that $$ \xymatrix@C=30pt@R=30pt{\z_i\ar[r]^{\z_{i,l}}\ar[rd]\|{\z_{i,k}}\ar[d]_{\z_{i,j}}& \z_l \ar@{}[ld]\|(0.3){\widehat{\z}\Uparrow} \ar@{}@<7ex>[d]\|{\textstyle =}\\ \z_j \ar@{}@<8pt>[r]\|(.3){\Rightarrow} \ar@{}@<14pt>[r]\|(.3){\widehat{\z}} \ar[r]_{\z_{j,k}}&\z_k\ar[u]_{\z_{k,l}}}\hspace{0.4cm} \xymatrix@C=30pt@R=30pt{\z_i\ar[r]^{\z_{i,l}}\ar[d]_{\z_{i,j}} \ar@{}[rd]\|(0.3){\widehat{\z}\Uparrow}& \z_l \\ \z_j \ar@{}@<8pt>[r]\|(.7){\Leftarrow} \ar@{}@<14pt>[r]\|(.7){\widehat{\z}} \ar[r]_{\z_{j,k}}\ar[ru]\|{\z_{j,l}}&\z_k\ar[u]_{\z_{k,l}} } $$ and, moreover, $$ \xymatrix{\ar@{}@<-7pt>[rr]^(.2){1_{\z_i}}\ar@{}@<20pt>[d]\|(.53){\Rightarrow}\|(.36){\widehat{\z}}& \z_i\ar@/_0.9pc/[ld]\ar[rd]^{\z_{i,j}} \ar@/^0.7pc/[ld] \ar@{}@<8pt>[d]\|(.7){\Rightarrow}\|(.5){\widehat{\z}} & \ar@{}@<10pt>[d]\|(.5){\textstyle =}\ar@{}@<28pt>[d]\|(.5){\textstyle \br_{\z_{i,j}}\,,} \\\z_i\ar[rr]_{\z_{i,j}}\ar@{}@<0.5pt>[rr]^(.37){\z_{i,i}}&&\z_j} \hspace{0.4cm} \xymatrix{\ar@{}@<-7pt>[rr]^(.2){1_{\z_j}}\ar@{}@<20pt>[d]\|(.53){\Rightarrow}\|(.36){\widehat{\z}}& \z_j\ar@{}@<8pt>[d]\|(.7){\Rightarrow}\|(.5){\widehat{\z}} & \ar@{}@<10pt>[d]\|(.5){\textstyle =}\ar@{}@<28pt>[d]\|(.5){\textstyle \bl_{\z_{i,j}}.} \\ \z_j\ar@/_0.7pc/[ru] \ar@/^0.9pc/[ru] \ar@{}@<0.7pt>[rr]^(.42){\z_{j,j}}&&\z_i\ar[lu]_{\z_{i,j}} \ar[ll]^{\z_{i,j}}} $$ If $\sigma:[m]\to [n]$ is any map in $\Delta$, that is, a functor, the induced $\sigma^:\Delta\B_n\to\Delta\B_m$ carries any ${\z:[n]\to \B}$ to ${\sigma^\z=\z \sigma:[m]\to \B}$, the composite lax functor of $\z$ with $\sigma$. On a small category $\C$, viewed as a bicategory in which all 2-cells are identities, the geometric nerve construction $\Delta\C$ gives the usual Grothendieck nerve of the category \cite{groth}, since, for any integer $n\geq 0$, we have $\lfunc([n],\C)=\mathrm{Func}([n],\C)$. Hence, the space $\BB\C=\|\Delta\C\|$ of a category $\C$, is the usual classifying space of the category, as considered by Quillen in \cite{quillen}. In particular, the geometric nerve of the category $[n]$ is precisely $\Delta[n]$, the standard simplicial $n$-simplex, so the notation is not confusing. Furthermore, for any bicategory $\B$, the simplicial map $\Delta\z:\Delta[n]\to\Delta\B$ defined by a $n$-simplex $\z\in \Delta\B_n$, that is, such that $\Delta\z(1_{[n]})=\z$, is precisely the simplicial map obtained by taking geometric nerves on the lax functor $\z:[n]\to\B$. Thus, if $\sigma:[m]\to [n]$ is any map in $\Delta$, then $$\Delta(\sigma^\z)=\Delta(\z\sigma)=\Delta\z\Delta\sigma:\Delta[m]\to \Delta\B.$$ The following fact, which is proved in \cite[Proposition 7.1]{ccg-1}, will be used repeatedly in our subsequent discussions: \begin{lemma}\label{trans} If $F,G:\A\to\B$ are two lax functors between bicategories, then any lax or oplax transformation, $\varepsilon:F\Rightarrow G$, canonically defines a homotopy $\BB\varepsilon:\BB F\simeq \BB G$ between the induced maps on classifying spaces $\BB F,\BB G:\BB\A\to\BB\B$. \end{lemma} Suppose that $\F:\B^{\mathrm{op}}\to \Cat$ is a functor, where $\B$ is any small category, such that for every morphism $f:b\to c$ of $\B$ the induced map $\BB f^:\BB\F_c\to\BB\F_b$ is a homotopy equivalence. Then, by Quillen's Lemma \cite[Lemma in page 14]{quillen}, the induced commutative square of spaces $$ \xymatrix{\BB\F_a\ar[r] \ar[d]&\mathrm{hocolim}_\B\BB\F\ar[d]\\ pt\ar[r]&\BB\B} $$ is homotopy cartesian. By Thomason's Homotopy Colimit Theorem \cite{thomason}, there is a natural homotopy equivalence $\mathrm{hocolim}_\B\BB\F \simeq \BB\int_\B\F$. Therefore, there is a homotopy cartesian square $$ \xymatrix{\BB\F_a\ar[r] \ar[d]&\BB\int_\B\F\ar[d]\\ pt\ar[r]&\BB\B.} $$ We are now ready to state and prove the following important result in this paper, which generalizes the result above, as well as the results in \cite[Theorem 7.4]{ccg-1} and \cite[Theorem 4.3]{cegarra}: \begin{theorem}\label{th1} Let $\F=(\F, \chi,\xi,\omega,\gamma,\delta):\B^{\mathrm{op}}\to\Bicat$ be any given lax bidiagram of bicategories. For any object $a\in \Ob\B$, there is a commutative square in $\Bicat$ \begin{equation}\label{squbicat1} \begin{array}{c}\xymatrix{ \F_a\ar[r]^{J}\ar[d]&\int_\B\F\ar[d]^{P}\\ [0]\ar[r]^{a}&\B } \end{array} \end{equation} where $P$ is the projection $2$-functor $(\ref{proj2fun})$, $a$ denotes the normal lax functor carrying $0$ to $a$, and $J$ is the natural embedding homomorphism $(\ref{emj})$ described below, such that, whenever each 1-cell $f:b\to c$ in $\B$ induces a homotopy equivalence $\BB f^\!:\BB\F_c\simeq \BB\F_b$, then the square of spaces induced on classifying spaces below is homotopy cartesian. \begin{equation}\label{hsqubicat} \begin{array}{c}\xymatrix{ \BB\F_a\ar[r]^-{\BB J}\ar[d]&\BB\!\int_\B\F\ar[d]^{\BB P}\\ pt \ar[r]^{\BB a}&\BB\B } \end{array} \end{equation} \end{theorem} \begin{proof} This is divided into four parts. \vspace{0.2cm} \noindent{\em Part 1.} Here we exhibit the embedding homomorphism in the square $(\ref{squbicat1})$ \begin{equation}\label{emj}\xymatrix@C=18pt{J\!=\!J(\F,a):\ \F_a\ar[r]& \int_\B\F .}\end{equation} It is defined on cells of $\F_a$ by $$ \xymatrix{x\ar@/^0.8pc/[r]^u\ar@/_0.8pc/[r]_v \ar@{}[r]\|{\Downarrow \phi} &y}\ \overset{J}\mapsto \ \xymatrix@C=30pt{(x,a)\ar@/^1pc/[r]^{(\chi y\circ u,1_a)} \ar@/_1pc/[r]_{(\chi y\circ v,1_a)} \ar@{}[r]\|{\Downarrow (\tilde{\phi},1)} & (y,a)} $$ where $\chi y\circ u$ is the 1-cell of $\F_a$ composite of $\xymatrix@C=16pt{x\ar[r]^{u}&y\ar[r]^-{\chi_{_a}y}&1_a^y}$, $1=1_{1_a}$, the identity 2-cell in $\B$ of the identity 1-cell of $a$, and $\tilde{\phi}$ is the $2$-cell given by the pasting in the diagram below. $$ \xymatrix{\ar@{}@<-22pt>[dd]\|{\textstyle \tilde{\phi}:}&&1_a^y\ar@/^0.7pc/[dd]^{1_{1_a}^y}\ar@/_0.7pc/[dd]\ar@{}@<-10pt>[dd]\|(.4){1} \ar@{}[dd]\|(.41){\xi}\|(.5){\cong}\\ x\ar@/^0.7pc/[r]^u\ar@/_0.7pc/[r]_v\ar@{}[r]\|{\Downarrow\phi} &y\ar[ru]^{\chi y}\ar[rd]_{\chi y} \ar@{}[r]\|{\cong} \ar@{}@<7pt>[r]\|{\bl} &\\ &&1_a^y} $$ For $\xymatrix@C=12pt{x\ar[r]^u&y\ar[r]^v&z}$, two composable 1-cells in $\F_a$, the corresponding constraint $2$-cell for their composition is $(\widehat{J},\bl):Jv\circ Ju\cong J(v\circ u)$, where $\bl=\bl_{1_a}:1_a\circ 1_a\cong 1_a$, while $\widehat{J}=\widehat{J}_{v,u}$ is the $2$-cell of $\F_a$ given by pasting the diagram $$ \xymatrix@C=20pt@R=16pt{x \ar@{}@<-28pt>[dd]\|{\textstyle\widehat{J}_{v,u}:} \ar[r]^u\ar[rdd]_{v\circ u}\ar@{}@<22pt>[d]\|(.7){=} & y\ar[r]^{\chi y}\ar[dd]^v &1^_ay\ar[rr]^{1^_a(\chi z\circ v)}\ar[rd] \ar@{}@<-7pt>[rd]\|{1^_av} \ar@{}@<-10pt>[dd]\|(.45){\cong}\|(.35){\widehat{\chi}} &\ar@{}[d]\|{\cong} &1^_a1^_az\ar[r]^{\chi z} \ar@{}@<20pt>[dd]\|(.45){\cong}\|(.35){\gamma} &(1_a\circ 1_a)^z\ar[dd]^{\bl^z}\\ &&&1^_az\ar[rrd]^1\ar[ru]_{1_a^\chi z} \ar@{}[d]\|(.4){\bl}\|(.6){\cong}&&\\ &z\ar[rru]^{\chi z}\ar[rrrr]^{\chi z}& &&&1^_az,} $$ and, for any object $x$ of $\F_a$, the structure isomorphism for its identity is $(\widehat{J},1):1_{Jx}\cong J(1_x)$, where $1=1_{1_a}$, and $\widehat{J}=\widehat{J}_{x}$ is provided by pasting the diagram in $\F_a$ below. $$ \xymatrix@C=3pc@R=3pc{x \ar@{}@<17pt>[d]\|{\cong} \ar[r]^{\chi x}\ar[d]_{1_x}\ar@{}@<-35pt>[d]\|{\textstyle \widehat{J}_x:} &1_a^x\ar[d]^{1_{1_a}^x}\ar@/_1.5pc/[d]\ar@{}@<-20pt>[d]\|(.4){1}\ar@{}@<-8pt>[d]\|(.37){\xi}\|(.5){\cong} \\ x\ar[r]^{\chi x} &1^_ax} $$ So defined, it is straightforward to verify that $J$ is functorial on vertical composition of 2-cells in $\F_a$. The naturality of the structure 2-cells $Jv\circ Ju\cong J(v\circ u)$ follows from the coherence conditions in (\textbf{C1}) and (\textbf{C2}), whereas the hexagon coherence condition for them is verified thanks to conditions (\textbf{C1}), (\textbf{C2}), and (\textbf{C7}), and the result in Lemma \ref{gdo}$(ii)$ relating $\gamma$ with $\omega$. As for the other two coherence conditions, one amounts to the equality in Lemma \ref{gdo}$(i)$, and the other is easily checked. \vspace{0.2cm} \noindent{\em Part 2.} Let $\z:[n]\to \B$ be any given geometric $n$-simplex of the bicategory, $n\geq 0$. Then, as in $(\ref{f})$, we have a composite lax bidiagram of bicategories $\F\z:[n]\to \Bicat$. In this part of the proof, we prove that the homomorphism $$ \xymatrix@C=18pt{J\!=\!J(\F\z,0)\!:\ \F_{\z_0}\ar[r]& \int_{[n]}\F\z} $$ induces a homotopy equivalence on classifying spaces: \begin{equation}\label{bemjn}\xymatrix{\BB J\!:\BB\F_{\z_0}\,\simeq\, \BB\! \int_{[n]}\F\z.}\end{equation} This is a direct consequence of the following general observation: \begin{lemma}\label{lemK} Suppose $\C$ is a small category with an initial object $0$, and let us regard $\C$ as a bicategory whose $2$-cells are all identities. Then, for any lax bidiagram of bicategories $\LL:\C^{\mathrm{op}}\to\Bicat$, the homomorphism $J\!=\!J(\LL,0):\LL_0\to \int_{\C}\!\LL$ induces a homotopy equivalence on classifying spaces, $\BB J\!:\BB\LL_0\simeq \BB\!\int_{\C}\LL$. \end{lemma} \begin{proof} For any object $a\in\Ob\C$, let $0_a:0\to a$ denote the unique morphism in $\C$ from the initial object to $a$. There is a homomorphism $$\xymatrix{K=K(\LL,0):\int_\C\LL\to \LL_0,}$$ which carries any object $(x,a)$ to $K(x,a)=0_a^x$, the image of $x$ by the homomorphism $0_a^:\LL_a\to \LL_0$, a 1-cell $(u,f):(x,a)\to (y,b)$ to $$K(u,f)=\big(\xymatrix@C=35pt{0_a^x\ar[r]^{0_a^u}&0_a^f^y\ar[r]^-{\chi_{_{f,0_a}}\!y}& (f\circ 0_a)^y=0_b^y}\big), $$ and a 2-cell $ \xymatrix@C=30pt{(x,a)\ar@{}[r]\|{\Downarrow (\phi,1)}\ar@/^0.8pc/[r]^{(u,f)}\ar@/_0.8pc/[r]_{(v,f)}&(y,b)} $ to the 2-cell $K(\phi,1):K(u,f)\Rightarrow K(v,f)$ obtained by pasting the diagram below, where $(A)=\big(\xymatrix@C=18pt{1_{0^_af^y}\ar@2[r]^{\widehat{0^_a}}& 0_a^1_{f^y}\ar@2[r]^{0_a^\xi}&0_a^1_f^y}\big)$. $$ \xymatrix{\ar@{}@<-45pt>[d]\|{\textstyle K(\phi,1):} &0_a^f^y\ar[rrd]^{\chi y}\ar[dd]\|(.3){0_a^1_f^y}\ar@/^20pt/[dd]^{1} \ar@{}@<10pt>[dd]\|(.60){\cong}\|(.52){(A)} &\ar@{}[dd]\|(.55){\cong}\|(.45){\br}\\ 0_a^x\ar[ru]^{0_a^u}\ar[rd]_{0_a^v}\ar@{}[r]\|(.55){\Downarrow\, 0_a^\phi} &&&0_b^y\\ &0_a^f^y\ar[rru]_{\chi y}&} $$ For each object $(x,a)$ of $\int_{\C}\LL$, the structure isomorphism $\widehat{K}:1_{K(x,a)}\cong K1_{(x,a)}$ is $$ \xymatrix{\ar@{}@<-45pt>[d]\|(.3){\textstyle \widehat{K}:} 0_a^x \ar[rr]^{0_a^\chi x}\ar[dd]_1 \ar@{}@<27pt>[dd]\|(.35){\cong}\|(.25){\gamma} & \ar@{}@<20pt>[dd]\|(.65){\cong}\|(.57){\bl} &0_a^1_a^x \ar@/^3pc/[lldd]^{\chi x}\ar[ld]_{\chi x}\\ &0_a^x\ar@/^10pt/[ld]^1\ar@/_10pt/[ld]\|{1_{0_a}^\!x} \ar@{}@<-24pt>[d]\|(.53){\cong\xi} & \\ 0_a^x&& } $$ while the constraint $\widehat{K}:K(v,g)\circ K(u,f)\cong K((v,g)\circ (u,f))$, for each pair of composable 1-cells $\xymatrix@C=1pc{(x,a)\ar[r]^{(u,f)} & (y,b) \ar[r]^{(v,g)} &(z,c)}$ of $\int_{\C}\LL$, is given by pasting in $\LL_0$ the diagram below. $$ \xymatrix{ \ar@{}@<-35pt>[dd]\|(.3){\textstyle \widehat{K}:} 0_a^f^y\ar[r]^{\chi y}\ar@{}@<20pt>[dd]\|{=}&0_b^y\ar@{}@<25pt>[d]\|(.6){\cong}\|(.4){\widehat{\chi}} \ar[r]^{0_b^v}&0_b^g^z\ar[rr]^{\chi z} \ar@{}@<40pt>[dd]\|(.42){\cong}\|(.35){\omega}&\ar@{}@<32pt>[d]\|(.60){\cong}\|(.45){\xi} &0_c^z\\ &0_a^f^y\ar[u]^{\chi y}\ar[r]_{0_a^f^v}\ar@{}@<25pt>[d]\|{\cong}& 0_a^f^g^z\ar[rd]_{0_a^\chi z} \ar[u]^{\chi g^z}&0_c^z\ar@/^10pt/[ru]\|{1_{0_c}^z}\ar@/_10pt/[ru]_1 \ar@{}@<32pt>[d]\|(.40){\cong}\|(.25){\bl}&\\ 0_a^x\ar[uu]^{0_a^u}\ar[ru]_{0_a^u}\ar[rrr]^{0_a^(\chi z\circ (f^v\circ u))} &&&0_a^(g\circ f)^z\ar[u]_{\chi z}\ar@/_2pc/[ruu]_{\chi z}&} $$ There are also two pseudo transformations $$\varepsilon: JK \Rightarrow 1_{\int_{\C}\!\LL},\hspace{0.2cm}\eta: 1_{\LL_0} \Rightarrow KJ,$$ which are defined as follows: The component of $\varepsilon$ at an object $(x,a)$ of $\int_\C\LL$ is $$ \varepsilon(x,a)=(1_{0_a^x},0_a):(0_a^x,0)\to (x,a), $$ and its naturality component at a morphism $(u,f):(x,a)\to (y,b)$ is $$ (\widehat{\varepsilon},1): \varepsilon(y,b)\circ JK(u,f)\cong (u,f)\circ \varepsilon (x,a), $$ where $\widehat{\varepsilon}$ is the 2-cell of $\LL_0$ pasted of the diagram below. $$ \xymatrix{ 0_a^x\ar[dd]_1\ar[r]^{0_a^u}&0_a^f^y\ar[r]^{\chi y}&0_b^y\ar[r]^{\chi 0_b^y} \ar[rrrdd]_{1} &1_0^0_b^y\ar[rr]^{1_0^1_{0_b^y}}_{\cong}\ar@/_12pt/[rr]\|1 \ar@{}@<8pt>[dd]\|(.2){\delta}\|(.3){\cong} \ar[rrd]_{\chi y}&&1_0^0_b^y\ar[d]^{\chi y}\ar@{}@<-12pt>[d]\|(.55){\cong}\|(.40){\br}\\ \ar@{}[rrrrr]\|(.4){\cong}&&&&&0_b^y\ar[d]\ar@{}@<-2pt>[d]^{1_{0_b}^\!y} \\ 0_a^x\ar[r]^{0_a^u}&0_a^f^y\ar[rrrr]^{\chi y}&&&&0_b^y } $$ The pseudo transformation $\eta: 1 \Rightarrow KJ$ assigns to each object $x$ of the bicategory $\LL_0$ the 1-cell $ \eta x=\chi x: x\to 1_0^x$, while its naturality isomorphism at any 1-cell $u:x\to y$, $$\widehat{\eta}: \eta y\circ u \,\cong\,KJ(u)\circ \eta x,$$ is obtained by pasting the diagram below. $$ \xymatrix@C=4pc@R=25pt{x \ar[rrr]^u \ar[ddd]_{\chi x} \ar@{}@<35pt>[d]\|(.43){\widehat{\chi}}\|(.6){\cong} &&& y \ar[ddd]^{\chi y} \ar[lld]_{\chi y} \ar@{}@<-35pt>[d]\|(.76){\bl}\|(.9){\cong} \\ & 1^_0y \ar[dd]^{1^_0 \chi y} \ar@/^3pc/[rrdd]^1 \ar@{}@<35pt>[d]\|(.43){\gamma}\|(.6){\cong} & & \\ && 1^_0y \ar@/^1pc/[rd]^(.3){1^_0y} \ar@/_1pc/[rd]\|1 \ar@{}@<35pt>[d]\|(.43){{\xi}}\|(.6){\cong} \ar@{}[d]\|{\cong}& \\ 1^_0x \ar[r]_{1^_0(\chi y \circ u)} \ar[ruu]^{1^_0u} & 1_0^1_0^y \ar[rr]_{\chi y} \ar[ru]^{\chi y} \ar@<-10pt>@{}[ul]\|(.4){\cong} && 1^_0y } $$ Hence, by Lemma \ref{trans}, there are induced homotopies $\BB\varepsilon:\BB J\,\BB K =\BB(JK)\simeq \BB 1_{\int_{\C}\!\LL}=1_{\BB\!\int_{\C}\LL}$ and $\BB\eta: 1_{\BB\LL_0}= \BB 1_{\LL_0}\simeq \BB(KJ)=\BB K\,\BB J $, and it follows that both maps $\BB J$ and $\BB K$ are actually homotopy equivalences.\end{proof} \vspace{0.2cm} \noindent{\em Part 3.} Let $\sigma:[m]\to [n]$ be a map in the simplicial category. By Lemma \ref{lesqld}, for any geometric $n$-simplex $\z:[n]\to \B$ of the bicategory $\B$, we have the square $$ \xymatrix{ \int_{[m]}\F\z\sigma\ar[r]^{\bar{\sigma}}\ar[d]_{P}&\int_{[n]}\F\z\ar[d]^{P}\\ [m]\ar[r]^{\sigma}&[n] }$$ which is cartesian in the category of bicategories and lax functors. This part has the goal of proving that the lax functor $\bar{\sigma}$ induces a homotopy equivalence on classifying spaces: \begin{equation}\label{bemjnt}\xymatrix{\BB \bar{\sigma}\!:\ \BB\!\int_{[m]}\F\z\sigma\,\simeq\, \BB\! \int_{[n]}\F\z.}\end{equation} To do that, let us consider the square of lax functors $$ \xymatrix@C=55pt{ \F_{\z_{\sigma 0}}\ar[d]_{\z_{0,\sigma 0}^}\ar[r]^-{J=J(\F\z\sigma,0)} &\int_{[m]}\F\z\sigma\ar[d]^{\bar{\sigma}} \\ \F_{\z_0}\ar[r]^-{J=J(\F\z,0)}&\int_{[n]}\F\z, } $$ where $\z_{0,\sigma 0}^$ is the homomorphism attached by the lax diagram $\F:\B^{\mathrm{op}}\to \Bicat$ to the 1-cell $\z_{0,\sigma 0}:\z_0\to\z_{\sigma 0}$ of $\B$, and the homomorphisms $J$ are defined as in $(\ref{emj})$. This square is not commutative, but there is a pseudo transformation $\theta: J\z_{0,\sigma 0}^\Rightarrow \bar{\sigma}J$, whose component at any object $x$ of $\F_{\z\sigma 0}$ is the 1-cell of $\int_{[n]}\F\z$ $$ \theta x=(1_{\z_{0,\sigma 0}^x},(0,\sigma 0)):\,(\z_{0,\sigma 0}^x,0)\to (x,\sigma 0), $$ and whose naturality isomorphism, at any 1-cell $u:x\to y$ in $\F_{\z\sigma 0}$, is $$\widehat{\theta}_u=(\tilde{\theta},1_{\z_{0,\sigma 0}}):\theta y\circ J\z_{0,\sigma 0}^ u\cong \bar{\sigma} J u\circ \theta x,$$ where $\tilde{\theta}$ is given by pasting in $\F_{\z_0}$ the diagram below. $$ \xymatrix@C=50pt{ \z_{0,\sigma 0}^x\ar[r]^{\z_{0,\sigma 0}^u}\ar[ddd]_{1} \ar@{}@<25pt>[dd]\|(.45){\cong} &\z_{0,\sigma 0}^y\ar[r]^{\chi_\z \z_{0,\sigma 0}^y} \ar[ddd]\|{\z_{0,\sigma 0}^\chi_{\z} y} \ar@/^14pt/[rrddd]\|{1} \ar@{}@<-25pt>[ddd]\|(.65){\cong} \ar@{}@<50pt>[ddd]\|(.5){\cong}\|(.45){\gamma} &\z_{0,0}^\z_{0,\sigma 0}^y\ar[r]^{\z_{0,0}^1}_{\cong} \ar@{}@<-13pt>[r]\|(.6){1} \ar@/_10pt/[r] \ar[rd]_{\chi_{\z} y} \ar@{}@<20pt>[dd]\|(.55){\cong}\|(.45){\delta} &\z_{0,0}^\z_{0,\sigma 0}^y\ar[d]^{\chi_{\z} y} \ar@{}@<-15pt>[d]\|{\cong} \\ &&&\z_{0,\sigma 0}^y\ar[dd]^{1^_{\z_{0,\sigma0}}y} \\ &&\z_{0,\sigma0}^y\ar@/^7pt/[rd]\|(.38){1^_{\z_{0,\sigma0}}y} \ar@/_9pt/[rd]\|(.3)1 \ar@{}@<46pt>[d]\|(.6){\cong}\|(.47){\xi} \ar@{}[d]\|{\cong} \\ \z_{0,\sigma 0}^x\ar[r]^-{\z^_{0,\sigma0}(\chi_{\z}y\circ u)} \ar[ruuu]^{\z_{0,\sigma0}^u} & \z_{0,\sigma0}^\z_{\sigma0,\sigma 0}^y\ar[rr]^{\chi_\z y}\ar[ru]^{\chi_\z y} &&\z_{0,\sigma0}^y } $$ Hence, by Lemma \ref{trans}, the induced square on classifying spaces $$ \xymatrix@C=35pt{ \BB\F_{\z_{\sigma 0}}\ar[d]_{\BB\z_{0,\sigma 0}^}\ar[r]^-{\BB J}& \BB\int_{[m]}\F\z\sigma\ar[d]^{\BB\bar{\sigma}}\\ \BB\F_{\z_0}\ar[r]^-{\BB J}&\BB\int_{[n]}\F\z } $$ is homotopy commutative. Moreover, both maps $\BB J$ in the square are homotopy equivalences, as we showed in the proof of Lemma \ref{lemK} above. Since, by hypothesis, the map $\BB \z_{0,\sigma 0}^:\BB\F_{\z_{\sigma0}}\to \BB\F_{\z_{0}}$ is also a homotopy equivalence, it follows that the remaining map in the square have the same property, that is, the map $\BB\bar{\sigma}: \xymatrix@C=15pt{ \BB\int_{[m]}\F\z\sigma\ar[r]& \BB\int_{[n]}\F\z} $ is a homotopy equivalence. \vspace{0.2cm} \noindent{\em Part 4.} Finally, we are ready to complete here the proof of the theorem as follows: Let us consider the induced simplicial map on geometric nerves $\Delta P:\Delta\int_\B\F\to\Delta\B$. This verifies the hypothesis in Lemma \ref{simlem}. In effect, thanks to Lemma \ref{lesqld}, for any geometric $n$-simplex of $\B$, $\z:[n]\to \B$, the square $$ \xymatrix{\int_{[n]}\F\z\ar[r]^{\bar{\z}}\ar[d]_{P} & \int_\B\F \ar[d]^{P}\\ [n]\ar[r]^\z&\B} $$ is a pullback in the category of bicategories and lax functors, whence the square induced by taking geometric nerves $$ \xymatrix{\Delta\int_{[n]}\F \z\ar[r]^{\Delta\bar{\z}}\ar[d]_{\Delta P} & \Delta\int_\B\F \ar[d]^{\Delta P}\\ \Delta[n]\ar[r]^{\Delta\z}&\Delta\B} $$ is a pullback in the category of simplicial sets. Thus, $\xymatrix{\Delta P^{-1}(\Delta\z)=\Delta\int_{[n]}\F\z}$. Furthermore, for any map $\sigma:[m]\to [n]$ in the simplicial category, since the diagram of lax functors $$ \xymatrix@R=10pt{\int_{[m]}\F\z\sigma\ar[rd]_{\bar{\sigma}}\ar[rrd]^{\overline{\z\sigma}}\ar[dd]_{P}&& \\ &\int_{[n]}\F\z\ar[dd]^{P}\ar[r]_{\bar{\z}} &\int_{\B}\F \ar[dd]^{P}\\ [m]\ar[rd]_{\sigma}\ar[rrd]^(.35){\z\sigma}\|!{[ru];[rd]}\hole&& \\ & [n]\ar[r]_{\z}&\B} $$ is commutative, the induced diagram of simplicial maps $$ \xymatrix@R=10pt{\Delta \int_{[m]}\F\z\sigma\ar[rd]_{\Delta\bar{\sigma}}\ar[rrd]^{\Delta\overline{\z\sigma}}\ar[dd]_{\Delta P}&& \\ &\Delta \int_{[n]}\F\z\ar[dd]^{\Delta P}\ar[r]_{\Delta \bar{\z}} &\Delta \int_{\B}\F \ar[dd]^{\Delta P}\\ \Delta [m]\ar[rd]_{\Delta \sigma}\ar[rrd]^(.35){\Delta (\z\sigma)}\|!{[ru];[rd]}\hole&& \\ & \Delta [n]\ar[r]_{\Delta \z}&\Delta \B} $$ is also commutative. Then, as $\sigma^\z=\z\sigma$, the induced simplicial map $\Delta P^{-1}(\sigma^\z)\to \Delta P^{-1}(\z)$ is precisely the map $\Delta\bar{\sigma}:\Delta \int_{[m]}\F\z\sigma\to \Delta \int_{[n]}\F\z$, whose induced map on geometric realizations is the homotopy equivalence $(\ref{bemjnt})$, $\xymatrix@C=15pt{\BB \bar{\sigma}\!:\ \BB\!\int_{[m]}\F\z\sigma\, \simeq \, \BB\! \int_{[n]}\F\z}$. Hence, by Lemma \ref{simlem}, for each object $a\in\Ob\B$, the square $$\xymatrix{\|\Delta P^{-1}(a)\|\ar[r]\ar[d]& \|\Delta\int_\B\F\| \ar[d]^{\|\Delta P\|}\ar@{}@<34pt>[d]\|{\textstyle =}\\ pt\ar[r]^-{\|\Delta a\|}&\|\Delta \B\|} \hspace{0.2cm} \xymatrix{\BB\int_{[0]}\F a\ar[r]^{\BB\bar{a}}\ar[d] & \BB\int_\B\F \ar[d]^{\BB P}\\ \BB\Delta[0]\ar[r]^-{\BB a}&\BB\B} $$ is homotopy cartesian. Furthermore, since the diagram of lax functors $$ \xymatrix@R=10pt@C=30pt{\F a\ar[rd]_{J(\F a,0)}\ar[rrd]^{J(\F,a)}\ar[dd]&& \\ &\int_{[0]}\F a\ar[dd]\ar[r]_{\bar{a}} &\int_{\B}\F \ar[dd]^{P}\\ [0]\ar[rd]\ar[rrd]^(.35){a}\|!{[ru];[rd]}\hole&& \\ & [0]\ar[r]_{a}&\B} $$ commutes, it follows that the square $(\ref{hsqubicat})$ is homotopy cartesian square $(\ref{hsqubicat})$ as it is the composite of the squares $$ \xymatrix@C=40pt{ \BB\F_a\ar[r]^{\BB J(\F a,0)}\ar[d]&\BB\!\int_{[0]}\F a\ar[r]^{\BB \bar{a}}\ar[d]&\BB\!\int_\B\F\ar[d]^{\BB P} \\ pt\ar[r]&pt \ar[r]^{\BB a}&\BB\B } $$ where the map $\BB J(\F a,0)\!:\BB\F_a \simeq\BB\!\int_{[0]}\F a$ in the left square is one of the homotopy equivalences $(\ref{bemjn})$, while the square on the right is homotopy cartesian. \end{proof} \section{The homotopy cartesian square induced by a lax functor}\label{theB} In this section we prove the main theorem of this paper, that is, a generalization to lax functors (monoidal functors, for instance) of the well-known Quillen's Theorem B \cite{quillen}. We shall first extend Gray's construction \cite[Section 3.1]{gray2} of homotopy fiber 2-categories to {\em homotopy fiber bicategories} of an arbitrary lax functor between bicategories, so we can state the corresponding `Theorem B' in terms of them. Let $F:\A\to \B$ be any given lax functor between bicategories. As in Example \ref{exap2}, each object $b$ of $\B$ gives rise to a pseudo bidiagram of categories $$ \B(-,b):\B^{\mathrm{op}}\to \Cat, $$ which carries an object $x\in\Ob\B$ to the hom-category $\B(x,b)$, and then also to the lax bidiagram of categories \begin{equation}\label{cF-} \B(-,b)F\!:\A^{\mathrm{op}}\to \Cat, \end{equation} obtained, as in $(\ref{f})$, by composing $\B(-,b)$ with $F$. The Grothendieck construction on these lax bidiagrams leads to the notions of {\em homotopy fiber} and {\em comma bicategories}: \begin{definition} The {\em homotopy fiber}, $F\!\!\downarrow{\!_b}$, of a lax functor between bicategories $F:\A\to \B$ over an object $b\in\Ob\B$, is the bicategory obtained as the Grothendieck construction on the lax bidiagram $(\ref{cF-})$, that is, $$ \xymatrix{F\!\!\downarrow\!_b=\int_\A\B(-,b)F.} $$ In particular, when $F=1_\B$ is the identity functor on $\B$, $$ \xymatrix{\B\!\!\downarrow{\!_b}=\int_\B\B(-,b)} $$ is the {\em comma bicategory} of objects over $b$ of the bicategory $\B$. \end{definition} It will be useful to develop here the Grothendieck construction, exposed in Section \ref{gt}, in this particular case. Its objects are pairs \begin{equation}\label{obcomf} (f\!:Fa\to b,a)\end{equation} with $a$ a 0-cell of $\A$ and $f$ a 1-cell of $\B$ whose source is $Fa$ and target the fixed object $b$. The 1-cells \begin{equation}\label{1celcomf}(\beta,u):(f,a)\to (f',a')\end{equation} consist of a $1$-cell $u:a\to a'$ in $\A$, together with a $2$-cell $\beta\!:f\Rightarrow f'\circ Fu$ in the bicategory $\B$, $$ \xymatrix@C=10pt@R=14pt{ Fa\ar[rr]^{Fu}\ar[rd]_{f}& \ar@{}[d]\|(.27){\beta}\|(.48){\Rightarrow}&Fa'\ar[ld]^{f'}\\ &b& } $$ A 2-cell in $F\!\!\downarrow{\!_b}$, \begin{equation}\label{2cellcommf}\xymatrix@C=30pt{(f,a)\ar@/^1pc/[r]^{(\beta,u)} \ar@/_1pc/[r]_{(\beta',u')} \ar@{}[r]\|{\Downarrow\alpha}& (f',a')}, \end{equation} is a 2-cell $\alpha:u\Rightarrow u'$ in $\A$, such that the equation below holds in the category $\B(Fa,b)$. \begin{equation}\label{1102} \xymatrix@C=20pt@R=15pt{Fa\ar[rr]^{Fu'}_{\Uparrow F\alpha} \ar[rdd]_{f} \ar@/_1.1pc/[rr] \ar@{}@<-17pt>[rr]\|{Fu}& &Fa'\ar[ldd]^{f'}\ar@{}@<24pt>[dd]\|{\textstyle =}\\ && \\ &b\ar@{}[uu]\|(.35){\Rightarrow}\|(.46){\beta}& } \hspace{0.3cm} \xymatrix@C=20pt@R=15pt{Fa\ar[rr]^{Fu'} \ar[rdd]_{f} & &Fa'\ar[ldd]^{f'}\\ && \\ &b\ar@{}[uu]\|(.55){\Rightarrow}\|(.7){\beta'}& } \end{equation} Compositions, identities, and the structure associativity and unit constraints in $F\!\!\downarrow{\!_b}$ are as follows: For any given objects $(f,a)$ and $(f',a')$ as in (\ref{obcomf}), the vertical composition of 2-cells $$ \xymatrix@C=45pt{(f,a)\ar[r]\|{(\beta',u')} \ar@/^1.5pc/[r]^{(\beta ,u)}\ar@/_1.5pc/[r]_{(\beta'',u'')} \ar@{}[r]<10pt>\|{\Downarrow\alpha} \ar@{}[r]<-10pt>\|{\Downarrow\alpha'}&(f',a')}\overset{\cdot}\mapsto \xymatrix@C=40pt{(f,a)\ar@/^1pc/[r]^{(\beta,u)} \ar@/_1pc/[r]_{(\beta'',u'')}\ar@{}[r]\|{\Downarrow\alpha'\cdot\alpha} &(f',a')} $$ is given by the vertical composition $\alpha'\cdot\alpha$ of 2-cells in $\A$. The horizontal composition of two 1-cells in $F\!\!\downarrow{\!_b}$, $$ \xymatrix{(f,a)\ar[r]^{(\beta,u)}&(f',a')\ar[r]^{(\gamma,v)}&(f'',a'')} $$ is the 1-cell $$ (\gamma,v)\circ (\beta,u)=(\gamma \circledcirc \beta, v\circ u): (f,a)\to (f'',a''), $$ where the second component is the horizontal composition $v \circ u$ in $\A$, while the first one is the 2-cell in $\B$ obtained by pasting the diagram below. \begin{equation}\label{oo}\begin{array}{c} \xymatrix@R=12pt{Fa\ar[r]^{Fu}\ar@/^1.6pc/[rr]^{F(v\circ u)}_{\widehat{F}\Uparrow}\ar[rdd]_{f} \ar@{}@<32pt>[dd]\|(.25){\beta}\|(.38){\Rightarrow} \ar@{}@<-35pt>[dd]\|(.25){\textstyle \gamma \circledcirc \beta:} &Fa'\ar[r]^{Fv}\ar[dd]\|{f'} \ar@{}@<16pt>[dd]\|(.25){\gamma}\|(.38){\Rightarrow} &Fa'' \ar[ldd]^{f''}\\ &&\\ &b& }\end{array}\end{equation} The horizontal composition of 2-cells is simply given by the horizontal composition of 2-cells in $\B$, $$ \xymatrix@C=35pt{ (f,a)\ar@/^1pc/[r]^{(\beta,u)} \ar@/_0.8pc/[r]_{(\beta',u')} \ar@{}[r]\|{\Downarrow\alpha}&(f',a')\ar@/^1pc/[r]^{(\gamma,v)} \ar@/_0.8pc/[r]_{(\gamma',v')}\ar@{}[r]\|{\Downarrow \alpha'}&(f'',a'')} \mapsto \xymatrix@C=40pt{(f,a)\ar@/^1pc/[r]^{(\gamma\circledcirc \beta, v\circ u)} \ar@/_1pc/[r]_{(\gamma'\circledcirc \beta',v'\circ u')} \ar@{}[r]\|{\Downarrow \alpha'\circ \alpha}&(f'',a'')}, $$ and the identity 1-cell of each 0-cell $(f:Fa\to b, a)$ is $$\begin{array}{c} 1_{(f,a)}=(\overset{_\circ}{1}_{(f,a)},1_a ):(f,a)\to (f,a), \\ \overset{_\circ}{1}_{(f,a)}=\big(\xymatrix{f\ar@2[r]^-{\br^{-1}}& f\circ 1_{Fa}\ar@2[r]^-{1_f\circ \widehat{F}}&f\circ F(1_a)}\big) \end{array} $$ Finally, the associativity, left and right unit constraints are obtained from those of $\A$ by the formulas $$ \boldsymbol{a}_{(\beta'',u''),(\beta',u'),(\beta,u)}=\boldsymbol{a}_{u'',u',u},\hspace{0.3cm} \boldsymbol{r}_{(\beta,u)}= \boldsymbol{r}_u,\hspace{0.3cm} \boldsymbol{l}_{(\beta,u)}=\boldsymbol{l}_u. $$ We shall prove below that, under reasonable necessary conditions, the classifying spaces of the homotopy fiber bicategories $\BB (F\!\!\downarrow{\!_b})$, of a lax functor $F:\A\to \B$, realize the homotopy fibers of the induced map on classifying spaces, $\BB F:\BB\A\to \BB \B$. This fact will justify the name of `homotopy fiber bicategories' for them. As a first step to do it, we state the following particular case, when $F=1_\B$ is the identity homomorphism: \begin{lemma}\label{cont} For any object $b$ of a bicategory $\B$, the classifying space of the comma bicategory $\B\!\!\downarrow{\!_b}$ is contractible, that is, $\BB(\B\!\!\downarrow{\!_b})\simeq pt$. \end{lemma} \begin{proof} Let $[0]\to \B\!\!\downarrow{\!_b}$ denote the normal lax functor that carries $0$ to the object $(1_b,b)$, and let $\mathrm{Ct}:\B\!\!\downarrow{\!_b}\to\B\!\!\downarrow{\!_b}$ be the composite of $\B\!\!\downarrow{\!_b}\to [0] \to \B\!\!\downarrow{\!_b}$. Then, the induced map on classifying spaces $$\xymatrix@C=20pt{\BB(\B\!\!\downarrow{\!_b})\ar[r]^{\BB\mathrm{Ct}}&\BB(\B\!\!\downarrow{\!_b})}= \xymatrix@C=15pt{\BB(\B\!\!\downarrow{\!_b})\ar[r]&\BB[0]=pt\ar[r]&\BB(\B\!\!\downarrow{\!_b})}$$ is a constant map. Now, let us observe that there is a canonical oplax transformation $1_{\B\downarrow{_b}}\Rightarrow \mathrm{Ct}$, whose component at any object $(f:a\to b,a)$ is the 1-cell $(\bl^{-1}_f,f):(f,a)\to (1_b,b)$, and whose naturality component at a 1-cell $(\beta,u):(f,a)\to (f',a')$ is $$ \xymatrix{(f,a) \ar[r]^{(\bl^{-1},f)} \ar[d]_{(\beta,u)} \ar@{}@<25pt>[d]\|(.39){\beta\cdot\bl}\|(.51){\Leftarrow} & (1_b,b) \ar[d]^{(\br^{-1},1_b)} \\ (f',a') \ar[r]_{(\bl^{-1},f')} & (1_b,b). } $$ This oplax transformation gives, thanks to Lemma $\ref{trans}$, a homotopy between $\BB(1_{\B\downarrow{_b}})=1_{\BB(\B\downarrow{_b})}$ and the constant map $\BB\mathrm{Ct}$, and so we obtain the result. \end{proof} \begin{example}\label{ombi}{\em Let $\B$ be a bicategory, and suppose $b\in\Ob\B$ is an object such that the induced maps $\BB p^:\BB\B(y,b)\to \BB\B(x,b)$ are homotopy equivalences for the different morphisms $p:x\to y$ in $\B$ (for instance, any object of a bigroupoid). By Theorem \ref{th1}, we have the fiber sequence $$ \BB \B(b,b)\to \BB \B\!\!\downarrow{\!_{b}}\to \BB\B$$ in which the space $\BB \B\!\!\downarrow{\!_{b}}$ is contractible by Lemma \ref{cont}. Hence, we conclude the existence of a homotopy equivalence \begin{equation}\label{loo1} \Omega(\BB \B,\BB b)\simeq \BB(\B(b,b)) \end{equation} between the loop space of the classifying space of the bicategory with base point $\BB b$ and the classifying space of the category of endomorphisms of $b$ in $\B$. The homotopy equivalence above is already known when the bicategory is strict, that is, when $\B$ is a 2-category. It appears as a main result in the paper by Del Hoyo \cite[Theorem 8.5]{Hoyo}, and it was also stated at the same time by Cegarra in \cite[Example 4.4]{cegarra}. Indeed, that homotopy equivalence $(\ref{loo1})$, for the case when $\B$ is a 2-category, can be deduced from a result by Tillman about simplicial categories in \cite[Lemma 3.3]{Tillmann}.} \end{example} Returning to an arbitrary lax functor $F:\A\to\B$, we shall now pay attention to two constructions with fiber homotopy bicategories. First, we have that any 1-cell $p:b\to b'$ in $\B$ determines a 2-functor \begin{equation}\label{h_} p_\!:\,F\!\!\downarrow{\!_b}\to F\!\!\downarrow{\!_{b'}} \end{equation} whose function on objects is defined by $$ p_(Fa\overset{f}\to b,a)=(Fa\overset{p\circ f}\longrightarrow b',a). $$ A 1-cell $(\beta,u):(f,a)\to (f',a')$ of $\B\!\!\downarrow{\!_b}$, as in $(\ref{1celcomf})$ , is carried to the 1-cell of $\B\!\!\downarrow{\!_{b'}}$ $$\begin{array}{c} p_(\beta, u)=(p\circledcirc \beta, u):(p\circ f,a)\to(p\circ f',a'), \\ p\circledcirc \beta=\big( \xymatrix@C=15pt{p\circ f\ar@2[r]^-{1_p\circ \beta} & p\circ(f'\circ Fu) \ar@2[r]^{\aso^{-1}}& (p\circ f')\circ Fu}\big)\end{array}$$ while, for $\alpha:(\beta,u)\Rightarrow (\beta',u')$ any 2-cell in $\B\!\!\downarrow{\!_b}$ as in $(\ref{2cellcommf})$, $$ p_(\alpha)=\alpha:(p\circledcirc \beta, u)\Rightarrow (p\circledcirc \beta', u'). $$ Secondly, by Lemma \ref{lesqld}, we have a pullback square in the category of bicategories and lax functors for any $b\in \Ob \B$ \begin{equation}\label{sq1} \begin{array}{c} \xymatrix{F\!\!\downarrow{\!_b}\ar[d]_{P}\ar[r]^{\bar{F}}&\B\!\!\downarrow{\!_b}\ar[d]^{P} \ar@{}@<25pt>[d]\|{\textstyle =} \\ \A\ar[r]^{F}&\B}\hspace{0.3cm}\xymatrix{ \int_\A \B(-,b)F\ar[d]_{P}\ar[r]^{\bar{F}}&\int_\B\B(-,b)\ar[d]^{P}\\ \A\ar[r]^{F}&\B}\end{array} \end{equation} where, recall, the 2-functors $P$ are the canonical projections $(\ref{proj2fun})$, and $\bar{F}$ is the induced lax functor $(\ref{ilf})$, which acts on cells by $$ \xymatrix@C=30pt{(f,a)\ar@/^0.8pc/[r]^{(\beta,u)} \ar@/_0.8pc/[r]_{(\beta',u')} \ar@{}[r]\|{\Downarrow\alpha}& (f',a')}\ \overset{\bar{F}}\mapsto\ \xymatrix@C=30pt{(f,Fa)\ar@/^0.8pc/[r]^{(\beta,Fu)} \ar@/_0.8pc/[r]_{(\beta',Fu')} \ar@{}[r]\|{\Downarrow F\!\alpha}& (f',Fa'),} $$ and whose structure constraints are canonically given by those of $F$. We are now ready to state and prove the following theorem, which is just the well-known Quillen's Theorem B \cite{quillen} when the lax functor $F$ in the hypothesis is an ordinary functor between categories. The result therein also generalizes a similar result by Cegarra \cite[Theorem 3.2]{cegarra}, which was stated for the case when $F$ is a 2-functor between 2-categories, but the extension to arbitrary lax functors between bicategories is highly nontrivial and the proof we give here uses different tools. \begin{theorem}\label{B} Let $F:\A\to\B$ be a lax functor between bicategories. The following statements are equivalent: $(i)$ For every 1-cell $p:b\to b'$ in $\B$, the induced map $\BB p_:\BB(F\!\!\downarrow{\!_b})\to \BB(F\!\!\downarrow{\!_{b'}})$ is a homotopy equivalence. $(ii)$ For every object $b$ of $\B$, the induced square by $(\ref{sq1})$ on classifying spaces \begin{equation}\label{hs2} \begin{array}{c} \xymatrix{\BB(F\!\!\downarrow{\!_b})\ar[d]_{\BB P}\ar[r]^{\BB\bar{F}}&\BB(\B\!\!\downarrow{\!_b}) \ar[d]^{\BB P} \\ \BB\A\ar[r]^{\BB F}&\BB\B}\end{array} \end{equation} is homotopy cartesian. Therefore, in such a case, for each object $a\in\Ob\A$ such that $Fa=b$, there is a homotopy fibre sequence $$\BB(F\!\!\downarrow{\!_b})\to \BB \A\to\BB\B,$$ relative to the base 0-cells $\BB a$ of $\BB \A$, $\BB b$ of $\BB \B$ and $\BB (1_b,a)$ of $\BB(F\!\!\downarrow{\!_b})$, that induces a long exact sequence on homotopy groups $$\cdots \to \pi_{n+1}\BB\B\to\pi_n\BB(F\!\!\downarrow{\!_b})\to\pi_n\BB\A\to\pi_n\BB\B\to\cdots.$$ \end{theorem} \begin{proof} $(ii)\Rightarrow (i)$ Suppose that $p:b\to b'$ is any 1-cell of $\B$. Then, taking $\z:[1]\to \B$ the normal lax functor such that $\z_{0,1}=p$, we have the path $\BB \z:\BB[1]\!=\!I\to \BB\B$, whose origen is the point $\BB a$ and whose end is $\BB b$ (actually, $\BB\B$ is a CW-complex and $\BB\z$ is one of its 1-cells). Since the homotopy fibers of a continuous map whose over points are connected by a path are homotopy equivalent, the result follows. $(i)\Rightarrow (ii)$ This is divided into three parts. {\em Part 1.} We begin here by noting that the bicategorical homotopy fiber construction is actually the function on objects of a covariant oplax bidiagram of bicategories $$ F\!\!\downarrow \,\,=(F\!\!\downarrow\,, \chi,\xi,\omega,\gamma,\delta):\B\to\Bicat $$ consisting of the following data: \vspace{0.2cm} $(\mathbf{D1})$ for each object $b$ in $\B$, the homotopy fiber bicategory \hspace{0.1cm}$F\!\!\downarrow{\!_b}$; \vspace{0.2cm} $(\mathbf{D2})$ for each 1-cell $p:b\to b'$ of $\B$, the 2-functor $p_\!:\, F\!\!\downarrow{\!_b}\to F\!\!\downarrow{\!_{b'}}$ in $(\ref{h_})$; \vspace{0.2cm} $(\mathbf{D3})$ for each 2-cell $\xymatrix@C=0.5pc{b \ar@/^0.6pc/[rr]^{p} \ar@/_0.6pc/[rr]_{ p'}\ar@{}[rr]\|{\Downarrow\sigma} & &b' }$ of $\B$, the pseudo transformation $\sigma_:p_\Rightarrow p'_$, whose component at an object $(f,a)$ of $F\!\!\downarrow{\!_{b}}$, is the 1-cell $$\begin{array}{c}\sigma_(f,a)=(\sigma \circledcirc f,1_a):(p\circ f,a)\to (p'\circ f,a),\\ \sigma \circledcirc f=\big(\xymatrix@C=15pt{ p\circ f\ar@2[r]^{\sigma\circ 1}& p'\circ f\ar@2[r]^-{\br^{-1}} & (p'\circ f)\circ 1_{Fa}\ar@2[r]^(.45){1\circ \widehat{F}} & (p'\circ f)\circ F1_a}\big) \end{array} $$ and whose naturality component at any 1-cell $(\beta,u):(f,a)\to (f',a')$, as in $(\ref{1celcomf})$, is the canonical isomorphism $\br^{-1}\cdot \bl:1_{a'}\circ u\cong u\circ 1_a$; $$ \xymatrix@C=40pt{(p\circ f,a)\ar[r]^{(\sigma\circledcirc f,1_a)} \ar[d]_{(p \circledcirc \beta,u)}\ar@{}@<45pt>[d]\|(.38){\br^{-1}\cdot\, \bl}\|(.51){\cong} & (p'\circ f,a)\ar[d]^{(p'\circledcirc \beta,u)} \\ (p\circ f',a')\ar[r]_{(\sigma\circledcirc f',1_{a'})}&(p'\circ f',a') } $$ $(\mathbf{D4})$ for each two composable 1-cells $\xymatrix@C=11pt{b\ar[r]^{p}&b'\ar[r]^{p'}&b''}$ in the bicategory $\B$, the pseudo transformation ${\chi} _{_{\!p',p}}:(p'\circ p)_\Rightarrow p'_p_$ has component, at an object $(f,a)$ of $F\!\!\downarrow{\!_{b}}$, the 1-cell $$\begin{array}{c} (\text{{\aa}},1_a):((p'\circ p)\circ f,a)\to (p'\circ(p\circ f),a) ,\\ \text{{\aa}}=\big(\xymatrix@C=15pt{(p'\circ p)\circ f\ar@2[r]^{\aso}& p'\circ (p\circ f)\ar@2[r]^-{\br^{-1}} & (p'\circ (p\circ f))\circ 1_{Fa}\ar@2[r]^(.45){1\circ \widehat{F}} & (p'\circ (p\circ f))\circ F1_a}\big) \end{array} $$ and whose naturality component at a 1-cell $(\beta,u):(f,a)\to (f',a')$, is $$ \xymatrix@C=40pt{((p'\circ p)\circ f,a)\ar[r]^{(\text{{\aa}},1)} \ar[d]_{((p'\circ p) \circledcirc \beta),u)} \ar@{}@<55pt>[d]\|(.38){\br^{-1}\cdot\, \bl}\|(.51){\cong} & (p'\circ (p\circ f),a)\ar[d]^{(p'\circledcirc (p\circledcirc \beta),u)} \\ ((p'\circ p)\circ f',a')\ar[r]_{(\text{{\aa}},1)}&(p'\circ (p\circ f'),a'); } $$ $(\mathbf{D5})$ for each object $b$ of $\B$, \hspace{0.1cm}$\chi_{_b}:{1_{b}}_\Rightarrow 1_{F\downarrow{_{b}}}$ is the pseudo transformation whose component at any object $(f,a)$ is the 1-cell $$\begin{array}{c} (\overset{_\circ}{1}\cdot \bl,1_a):(1_b\circ f,a) \to (f,a),\\ \overset{_\circ}{1}\cdot \bl=\big(\xymatrix@C=15pt{1_b\circ f\ar@2[r]^-{\bl}& f\ar@2[r]^-{\br^{-1}} & f\circ 1_{Fa}\ar@2[r]^(.45){1\circ \widehat{F}} & f\circ F1_a}\big)\end{array} $$ and whose naturality component, at a 1-cell $(\beta,u):(f,a)\to (f',a')$, is $$ \xymatrix@C=40pt{(1_b\circ f,a)\ar[r]^{(\overset{_\circ}{1},1)} \ar[d]_{(1_b\circledcirc \beta,u)}\ar@{} \ar@{}@<45pt>[d]\|(.38){\br^{-1}\cdot\, \bl}\|(.51){\cong} & (f,a)\ar[d]^{(\beta,u)} \\ (1_b\circ f',a')\ar[r]_{(\overset{_\circ}{1},1)}&(f',a'); } $$ $(\mathbf{D6})$ for any two vertically composable 2-cells $\xymatrix@C=12pt{p\ar@2[r]^{\sigma}&p'\ar@2[r]^{\tau}&p'' }$ in $\B$, the invertible modification \hspace{0.1cm}${\xi}_{\tau,\sigma}:\tau_\circ \sigma_ \Rrightarrow (\tau\cdot \sigma)_$ has component, at any object $(f,a)$, the canonical isomorphism $\bl:1_a\circ 1_a\cong 1_a$ $$ \xymatrix@C=4pt@R=14pt{ & (p\circ f,a)\ar[ld]_{(\sigma\circledcirc f,1_a)}\ar[rd]^{\ ((\tau\cdot\sigma)\circledcirc f,1_a)} \ar@{}[d]\|(.6){\cong}\|(.42){\bl}& \\ (p'\circ f,a)\ar[rr]_{(\tau\circledcirc f,1_a)}&&(p''\circ f,a); } $$ \vspace{0.2cm}$(\mathbf{D7})$ for each 1-cell $p:b\to b'$ of $\B$, ${(1_p)}_=1_{p_}$, and \hspace{0.1cm}${\xi}_{_p}$ is the identity modification; $(\mathbf{D8})$ for every two horizontally composable 2-cells $\xymatrix@C=0.5pc{b\ar@/^0.5pc/[rr]^{p} \ar@/_0.5pc/[rr]_{q} \ar@{}[rr]\|{\Downarrow\sigma}& & b'\ar@/^0.5pc/[rr]^{p'}\ar@/_0.5pc/[rr]_{q'} \ar@{}[rr]\|{\Downarrow\tau}& & b''}$ in $\B$, the equality $ (\tau_\sigma_)\!\circ {\chi}{_{p',p}}={\chi}{_{q',q}}\circ (\tau\circ \sigma)_ $ holds and the modification ${\chi}_{_{\tau,\sigma}}$ is the identity; $(\mathbf{D9})$ for every three composable 1-cells $\xymatrix@C=13pt{b\ar[r]^{p}&b'\ar[r]^{p'}&b''\ar[r]^{p''}&b'''}$ in $\B$, the invertible modification ${\omega}_{_{p'',p',p}}$, at any object $(f,a)$, is the canonical isomorphism $\br:(1_a\circ 1_a)\circ 1_a\cong 1_a\circ 1_a$, $$ \xymatrix@C=30pt{ (((p''\circ p')\circ p)\circ f,a)\ar[rr]^{(\aso\circledcirc f,1_a)} \ar[d]_{(\text{{\aa}},1_a)} &\ar@{}[d]\|(.55){\cong}\|(.40){\br}&((p''\circ(p'\circ p))\circ f,a) \ar[d]^{(\text{{\aa}},1_a)}\\ ((p''\circ p')\circ(p\circ f),a)\ar[r]^{(\text{{\aa}},1_a)}& (p''\circ(p'\circ(p\circ f)),a) &(p''\circ((p'\circ p)\circ f),a)\ar[l]_{(p''\circledcirc\, \text{{\aa}},1_a)} ; } $$ $(\mathbf{D10})$ for any $1$-cell $p:b\to b'$ of $\B$, the invertible modifications $\gamma_p$ and $\delta_p$, at any object $(f,a)$ are given by the canonical isomorphism $1_a\circ (1_a\circ 1_a)\cong 1_a$, $$ \xymatrix@C=30pt{(1_{b'}\circ(p\circ f),a)\ar[r]^-{(\overset{_\circ}{1}\cdot \bl,1_a)} \ar@{}@<45pt>[d]\|(.55){\cong}\|(.43){\br\cdot\br} & (p\circ f,a)\ar[d]^-{(\overset{_\circ}{1},1_a)} \\ ((1_{b'}\circ p)\circ f,a)\ar[u]^{(\text{\aa},1_a)}\ar[r]_-{(\bl\circledcirc f,1_a)} & (p\circ f,a) } \xymatrix@C=35pt{(p\circ(1_{b'}\circ f),a)\ar[r]^-{(p\circledcirc(\overset{_\circ}{1}\cdot\bl),1_a)} \ar@{}@<45pt>[d]\|(.55){\cong}\|(.43){\br\cdot\br} & (p\circ f,a)\ar[d]^{(\overset{_\circ}{1},1_a)} \\ ((p\circ 1_{b'})\circ f,a)\ar[u]^{(\text{\aa},1_a)}\ar[r]_-{(\br\circledcirc f,1_a)} & (p\circ f,a).} $$ Observe that all the $2$-cells given above are well defined since all the data is obtained from the constraints of the bicategories involved and the lax functor $F$. Then the coherence conditions of these give us the equality (\ref{1102}) in each case. For the same reason the axioms $(\mathbf{C1})-(\mathbf{C8})$ hold. {\em Part 2.} In this part, we consider the Grothendieck construction on the oplax bidiagram of homotopy fibers $F\!\!\downarrow\,:\B\to \Bicat$, and we shall prove the following: \begin{lemma}\label{lemQ} There is a homomorphism \begin{equation}\label{Q} \xymatrix{Q:\int_{\B}\!F\!\!\downarrow\ \to \A,} \end{equation} inducing a homotopy equivalence on classifying spaces, $\BB Q:\BB\!\int_{\B}\!F\!\!\downarrow\ \simeq \BB\A$. \end{lemma} Before starting the proof of the lemma, we shall briefly describe the bicategory $\int_{\B}\!F\!\!\downarrow$. It has objects the triplets $(f,a,b)$, with $a\in\Ob\A$, $b\in\Ob\B$, and $f:Fa\to b$ a 1-cell of $\B$. Its 1-cells $$(\beta,u,p):(f,a,b)\to (f',a',b'),$$ consist of a $1$-cell $p:b\to b'$ in $\B$, together with a $1$-cell $(\beta,u):p_(f,a)=(p\circ f,a)\to (f',a')$ in $F\!\!\downarrow\!_{b'}$, that is, a 1-cell $u:a\to a'$ in $\A$ and a 2-cell $\beta: p\circ f\Rightarrow f'\circ Fu$ in $\B$ $$ \xymatrix{Fa \ar@{}@<20pt>[d]\|(.4){\beta}\|(.55){\Rightarrow} \ar[r]^{Fu}\ar[d]_{f}&Fa'\ar[d]^{f'}\\ b\ar[r]^{p}&b'.} $$ A 2-cell in $\int_{\B}\!F\!\!\downarrow$, $$\xymatrix@C=30pt{(f,a,b)\ar@/^1pc/[r]^{(\beta,u,p)} \ar@/_1pc/[r]_{(\beta',u',p')} \ar@{}[r]\|{\Downarrow(\alpha,\sigma)}& (f',a',b')}, $$ consists of a 2-cell $\sigma:p\Rightarrow p'$ in $\B$, together with a 2-cell $\alpha:(\beta,u)\Rightarrow (\beta',u')\circ \sigma_(f,a)$ in $F\!\!\downarrow\!_{b'}$, that is, (after some work using coherence equations) a 2-cell $\alpha: u\Rightarrow u'\circ 1_a$ in $\A$, such that the equation below holds. $$ \xymatrix@R=35pt@C=35pt{ Fa\ar[r]^{Fu'}\ar[d]_{f}&Fa'\ar[d]^{f'}\ar@{}@<25pt>[d]\|{\textstyle =}\\ b \ar@/^1pc/[r]_{\Uparrow \sigma}\ar@{}@<16pt>[r]\|(.4){p'}\ar@{}@<-25pt>[u]\|(.6){\Rightarrow}\|(.72){\beta'} \ar[r]_p&b'}\hspace{0.2cm} \xymatrix@R=35pt@C=35pt{ Fa \ar@/_1.3pc/[r]\ar@{}@<-19pt>[r]\|(.4){Fu} \ar[r]^{Fu'}_{\Uparrow F(\br\cdot\alpha)}\ar[d]_{f}&Fa'\ar[d]^{f'}\\ b \ar@{}@<-25pt>[u]\|(.25){\Rightarrow}\|(.37){\beta} \ar[r]_p&b'} $$ We shall look carefully at the vertical composition of $2$-cells and the horizontal composition of $1$-cells in $\int_\B F\!\!\downarrow$ since we will use them later: Given two vertically composable $2$-cells, say $(\alpha,\sigma)$ as above and $(\alpha',\sigma'):(\beta',u',p')\Rightarrow (\beta'',u'',p'')$, their vertical composition is given by the formula $$(\alpha',\sigma')\cdot(\alpha,\sigma)=(\alpha'\cdot \br \cdot \alpha,\sigma'\cdot \sigma): (\beta,u,p)\Rightarrow (\beta'',u'',p'').$$ Given two composable $1$-cells, say $(\beta,u,p)$ as above and $(\beta',u',p'):(f',a',b')\to (f'',a'',b'')$, their horizontal composition is $$(\beta',u',p')\circ (\beta,u,p)=(F\br^{-1}\cdot(\beta'\circledcirc (1_{p'}\circ \beta)),(u'\circ u)\circ 1_a,p'\circ p):(f,a,b)\to (f'',a'',b''),$$ where $\beta'\circledcirc (1_{p'}\circ \beta)$ is as in $(\ref{oo})$, thus $$ \xymatrix{ Fa \ar@{}@<-70pt>[d]\|(.3){\textstyle F\br^{-1}\cdot(\beta'\circledcirc (1_{p'}\circ \beta)):} \ar@/^2pc/[rr]^{F((u'\circ u)\circ 1_a)} \ar@{}@<14pt>[rr]\|{\Uparrow F\br^{-1}\cdot \widehat{F}} \ar@{}@<24pt>[d]\|(.52){\Rightarrow}\|(.38){\beta} \ar[r]^{Fu}\ar[d]_{f}&Fa' \ar@{}@<24pt>[d]\|(.52){\Rightarrow}\|(.38){\beta'} \ar[r]^{Fu'}\ar[d]_{f'}&Fa'' \ar[d]^{f''}\\ b\ar[r]^{p}&b' \ar[r]^{p'}&b''. } $$ The identity $1$-cell at an object $(f,a,b)$ is $$\begin{array}{c}1_{(f,a,b)}=(\overset{_\circ}{1}_{(f,a)}\cdot \bl,1_a,1_b):(f,a,b)\to (f,a,b).\\[5pt] \overset{_\circ}{1}_{(f,a)}\cdot \bl=\Big(\xymatrix@C=10pt{1_b\circ f\ar@{=>}[r]^-{\bl}& f\ar@2[r]^-{\br^{-1}}& f\circ 1_{Fa}\ar@2[r]^-{1_f\circ \widehat{F}}&f\circ F(1_a)}\Big) \end{array} $$ {\em Proof of Lemma \ref{lemQ}.} The homomorphism $Q$ in $(\ref{Q})$ is defined on cells by $$\begin{array}{c}\xymatrix@C=30pt{(f,a,b)\ar@/^1pc/[r]^{(\beta,u,p)} \ar@/_1pc/[r]_{(\beta',u',p')} \ar@{}[r]\|{\Downarrow(\alpha,\sigma)}& (f',a',b')}\ \overset{Q}\mapsto \ \xymatrix{a\ar@/^0.8pc/[r]^{u}\ar@/_0.8pc/[r]_{u'}\ar@{}[r]\|{\Downarrow \br\cdot\alpha}&a',}\\[5pt] \br\cdot\alpha=\big(u\overset{\alpha}\Rightarrow u'\circ 1_a\overset{\br}\Rightarrow u'\big) \end{array} $$ This homomorphism $Q$ is strictly unitary, and its structure isomorphism at any two composable 1-cells, say $(\beta,u,p)$ as above and $(\beta',u',p'): (f',a',b')\to (f'',a'',b'')$, is $$\widehat{Q}=\br_{u'\circ u}:Q((\beta',u',p')\circ (\beta,u,p))\cong Q(\beta',u',p')\circ Q(\beta,u,p). $$ To prove that this homomorphism $Q$ induces a homotopy equivalence on classifying spaces, let us observe that there is also a lax functor $L:\A\to \int_{\B}\!F\!\!\downarrow$, such that $Q\,L=1_{\A}$. This is defined on cells of $\A$ by $$\begin{array}{c} \xymatrix{a\ar@/^0.8pc/[r]^{u}\ar@/_0.8pc/[r]_{u'}\ar@{}[r]\|{\Downarrow\alpha}&a'} \ \overset{L}\mapsto \ \xymatrix@C=55pt{(1_{Fa},a,Fa)\ar@/^1pc/[r]^{(\bl^{-1}\cdot \br,u,Fu)} \ar@/_1pc/[r]_{(\bl^{-1}\cdot \br,u',Fu')} \ar@{}[r]\|{\Downarrow(\br^{-1}\cdot\alpha,F\alpha)}& (1_{Fa'},a',Fa'),}\\[5pt] \br^{-1}\cdot \alpha=\big(u\overset{\alpha}\Rightarrow u'\overset{\br^{-1}}\Rightarrow u'\circ 1_a\big) \end{array} $$ where the first component of $(\bl^{-1}\cdot \br,u,Fu)$ is the canonical isomorphism $Fu\circ 1_{Fa}\cong 1_{Fa'}\circ Fu$. Its structure 2-cells, at any pair of composable 1-cells $a\overset{u}\to a'\overset{u'}\to a''$ and at any object $a$ of $\A$, are respectively defined by $$ \begin{array}{ll} \widehat{L}_{u',u}=(1_{(u'\circ u)\circ 1_a},\widehat{F}_{u',u}):\ Lu'\circ Lu \Rightarrow L(u'\circ u),\\ \widehat{L}_{a}=(\br^{-1}_{1_a},\widehat{F}_{a}): 1_{La} \Rightarrow L1_{a}. \end{array} $$ The equality $QL=1_{\A}$ is easily checked. Furthermore, there is an oplax transformation $\iota:LQ\Rightarrow 1_{\int_{\B}\!F\downarrow}$ assigning to each object $(f,a,b)$ of the bicategory $\int_{\B}\!F\!\!\downarrow$ the 1-cell $$ \iota(f,a,b)=(1_f\circ \widehat{F}_a,1_a,f):(1_{Fa},a,Fa)\to (f,a,b), $$ and whose naturality component at any 1-cell $(\beta,u,p):(f,a,b)\to (f',a',b')$ is the 2-cell $$ \xymatrix@C=45pt{ (1_{Fa},a,Fa) \ar@{}@<60pt>[d]\|(.35){\widehat{\iota}\,=((\bl^{-1}\circ 1)\circ 1,\beta)}\|(.5){\Rightarrow} \ar[r]^{(\bl^{-1}\cdot \br,u,Fu)}\ar[d]_{(1_f\circ \widehat{F}_a,1_a,f)}& (1_{Fa'},a',Fa')\ar[d]^{(1_{f'}\circ \widehat{F}_{a'},1_{a'},f')}\\ (f,a,b)\ar[r]_-{(\beta,u,p)}&(f',a',b'). } $$ Therefore, by taking classifying spaces, we have $\BB Q\,\BB L=1_{\BB\A}$ and, by Lemma \ref{trans}, $\BB L\,\BB Q\simeq 1_{\BB\int_{\B}\!F\downarrow}$, whence $\BB Q$ is actually a homotopy equivalence. \qed {\em Part 3.} We complete here the proof of the theorem as follows: There is a canonical homomorphism \begin{equation}\label{fba2}\xymatrix{\bar{F}:\int_\B F\!\!\downarrow \,\longrightarrow \, \int_\B \B\!\!\downarrow} \end{equation} making commutative, for any object $b\in\Ob\B$, the diagrams $$ \begin{array}{cc} \xymatrix{\ar@{}@<-20pt>[d]\|(.3){\textstyle (A):}&&\\F\!\!\downarrow \!b \ar[r]^{J} \ar[d]_{\bar{F}} \ar@/^1.5pc/[rr]^{P} & \int_\B F\!\!\downarrow \ar[r]^{Q} \ar@{-->}[d]^{\bar{F}} & \A \ar[d]^{F} \\ \B\!\!\downarrow \!b \ar[r]^{J} \ar@/_1.5pc/[rr]_{P} & \int_\B \B\!\!\downarrow \ar[r]^{Q} & \B } \hspace{1cm} \xymatrix{\ar@{}@<-20pt>[d]\|(.3){\textstyle (B):}&&\\ F\!\!\downarrow\! b \ar[r]^{F} \ar[d]_{J} \ar@/^1.5pc/[rr] & \B\!\!\downarrow\! b \ar[r] \ar[d]^J & [0]\ar[d]^{b} \\ \int_\B F\!\!\downarrow \ar@{-->}[r]^{\bar{F}} \ar@/_1.5pc/[rr]_P & \int_\B \B\!\!\downarrow \ar[r]^{P} & \B } \end{array} $$ in which $Q:\int_\B F\!\!\downarrow \to \A$ is the homomorphism in $(\ref{Q})$ and $Q:\int_\B \B\!\!\downarrow \to \B$ is the corresponding one for $F=1_{\B}$, all the 2-functors $P$ are the canonical projections $(\ref{proj2fun})$, and the embedding homomorphisms $J$ are the corresponding ones defined as in $(\ref{emj})$. This homomorphism $(\ref{fba2})$ is defined on cells by $$\begin{array}{c} \xymatrix@C=30pt{(f,a,b)\ar@/^1pc/[r]^{(\beta,u,p)} \ar@/_1pc/[r]_{(\beta',u',p')} \ar@{}[r]\|{\Downarrow(\alpha,\sigma)}& (f',a',b')}\,\overset{\bar{F}}\mapsto\, \xymatrix@C=30pt{(f, F a,b)\ar@/^1pc/[rr]^{(\beta, F u,p)} \ar@/_1pc/[rr]_{(\beta', F u',p')} \ar@{}[rr]\|{\Downarrow( \br^{-1}\cdot F\br\cdot F\alpha,\sigma)}&& (f', F a',b').} \\ \br^{-1}\cdot F\br\cdot F\alpha=\Big(\xymatrix@C=15pt{Fu\ar@2[r]^-{F\alpha}& F(u'\circ 1_a)\ar@2[r]^-{F\br}& Fu'\ar@2[r]^-{\br^{-1}} & Fu'\circ 1_{Fa}}\Big) \end{array} $$ Its composition constraint at a pair of composable $1$-cells, say $(\beta,u,p)$ as above and $(\beta',u',p'):(f',a',b')\to (f'',a'',b'')$, is the 2-cell $$\begin{array}{c} (\widetilde{F},1_{p'\circ p}):\bar{F}(\beta',u',p')\circ \bar{F}(\beta,u,p)\Rightarrow \bar{F}((\beta',u',p')\circ (\beta,u,p)),\\ \widetilde{F}=\Big(\xymatrix@C=30pt{(Fu'\circ Fu)\circ 1_{Fa}\ar@2[r]^{\widehat{F}\circ 1} &F(u'\circ u)\circ 1_{Fa}\ar@2[r]^-{F(\br^{-1})\circ 1} &F((u'\circ u)\circ 1_a)\circ 1_{Fa}}\Big) \end{array} $$ while its unit constraint at an object $(f,a,b)$ is $$\begin{array}{c} (\widetilde{F},1_{1_b}):1_{\bar{F}(f,a,b)}\Rightarrow \bar{F}(1_{(f,a,b)}).\\ \widetilde{F}=\Big(\xymatrix@C=15pt{1_{Fa}\ar@2[r]^-{\br^{-1}}&1_{Fa}\circ 1_{Fa}\ar@2[r]^-{\widehat{F}\circ 1} &F(1_a)\circ 1_{Fa}}\Big)\end{array} $$ Let us now observe that (the covariant and oplax version of) Theorem \ref{th1} applies both to the bidiagram of homotopy fibres $F\!\!\downarrow$, by hypothesis, and to the bidiagram of comma bicategories $\B\!\!\downarrow$, since the spaces $\BB\B\!\!\downarrow{\!_{b}}$ are contractible by Lemma \ref{cont} and therefore any 1-cell $p:b\to b'$ in $\B$ obviously induces a homotopy equivalence $\BB p_:\BB\B\!\!\downarrow{\!_{b}}\simeq \BB\B\!\!\downarrow{\!_{b'}}$. Hence, the squares \begin{equation}\label{squbicat} \begin{array}{c}\xymatrix{ F\!\!\downarrow{\!_{b}}\ar[r]^{J}\ar[d]&\int_\B F\!\!\downarrow\ar[d]^{P}\\ [0]\ar[r]^{b}&\B } \hspace{0.5cm} \xymatrix{ \B\!\!\downarrow{\!_{b}}\ar[r]^{J}\ar[d]&\int_\B \B\!\!\downarrow\ar[d]^{P}\\ [0]\ar[r]^{b}&\B } \end{array} \end{equation} induce homotopy cartesian squares on classifying spaces $$ \begin{array}{c}\xymatrix{ \BB F\!\!\downarrow{\!_{b}}\ar[r]^{\BB J}\ar[d]&\BB\int_\B F\!\!\downarrow\ar[d]^{\BB P}\\ pt\ar[r]^{\BB b}&\BB\B, } \hspace{0.5cm} \xymatrix{ \BB\B\!\!\downarrow{\!_{b}}\ar[r]^{\BB J}\ar[d]&\BB\int_\B \B\!\!\downarrow\ar[d]^{\BB P}\\ pt\ar[r]^{\BB b}&\BB \B. } \end{array} $$ By \cite[II, Lemma 8.22 (2)(b)]{g-j}, it follows from the commutativity of diagram $(B)$ above that the induced square $$ \xymatrix{\BB F\!\!\downarrow{\!_{b}}\ar[r]^{\BB\bar{F}}\ar[d]_{\BB J}&\BB \B\!\!\downarrow{\!_{b}}\ar[d]^{\BB J}\\ \BB\int_\B F\!\!\downarrow\ar[r]^{\BB\bar{F}}&\BB \int_B \B\!\!\downarrow} $$ is homotopy cartesian. Then, by \cite[II, Lemma 8.22 (1), (2)(a)]{g-j}, the theorem follows from the commutativity of diagram $(A)$, since, by Lemma \ref{lemQ}, in the induced square $$ \xymatrix{\BB \int_B F\!\!\downarrow\ar[r]^{\BB\bar{F}}\ar[d]_{\BB Q}&\BB \int_\B\B\!\!\downarrow\ar[d]^{\BB Q}\\ \BB\A\ar[r]^{\BB F}&\BB \B} $$ both maps $\BB Q$ are homotopy equivalences and therefore it is homotopy cartesian. \end{proof} The following corollary generalizes Quillen's Theorem A in \cite{quillen}: \begin{theorem} \label{A} Let $F:\A\to \B$ be a lax functor between bicategories. The induced map on classifying spaces $\BB F:\BB \A\to \BB\B$ is a homotopy equivalence whenever the classifying spaces of the homotopy fiber bicategories $\BB F\!\!\downarrow{\!_{b}}$ are contractible for all objects $b$ of $\B$. \end{theorem} Particular cases of the result above have been also stated in \cite[Theorem 1.2]{b-c}, for the case when $F:\A\to \B$ is any 2-functor between 2-categories, and in \cite[Theorem 6.4]{Hoyo}, for the case when $F$ is a lax functor from a category $\A$ to a 2-category $\B$. In \cite[Th\'{e}or\`{e}me 6.5]{chiche}, it is stated a relative Theorem A for lax functors between 2-categories, which also implies the particular case of Theorem \ref{A} when $F$ is any lax functor between 2-categories. \begin{example}{\em Let $(\mathcal{M},\otimes)=(\mathcal{M},\otimes,I,\aso,\bl,\br)$ be a monoidal category (see e.g. \cite{maclane}), and let $\Sigma (\mathcal{M},\otimes)$ denote its {\em suspension} or {\em delooping} bicategory. That is, $\Sigma (\mathcal{M},\otimes)$ is the bicategory with only one object, say $\star$, whose hom-category is $\mathcal{M}$, and whose horizontal composition is given by the tensor functor $\otimes:\mathcal{M}\times \mathcal{M}\to \mathcal{M}$. The identity 1-cell on the object is the unit object $I$ of the monoidal category, and the constraints $\aso$, $\bl$, and $\br$ for $\Sigma (\mathcal{M},\otimes)$ are just those of the monoidal category. By \cite[Theorem 1]{b-c}, $$ \BB(\mathcal{M},\otimes)=\BB\Sigma (\mathcal{M},\otimes),$$ that is, the classifying space of the monoidal category is the classifying space of its suspension bicategory. Then, Theorem \ref{B} is applicable to monoidal functors between monoidal categories. However, we should stress that the homotopy fiber bicategory of the homomorphism between the suspension bicategories that a monoidal functor $F:(\mathcal{M},\otimes)\to (\mathcal{M}',\otimes)$ defines, $\Sigma F:\Sigma(\mathcal{M},\otimes)\to \Sigma(\mathcal{M}',\otimes)$, at the unique object of $\Sigma(\mathcal{M}',\otimes)$, is not a monoidal category but a genuine bicategory: The 0-cells of $\Sigma F\!\!\downarrow_\star$ are the objects $x'\in\mathcal{M}'$, its 1-cells $(u',x):x'\to y'$ are pairs with $x$ an object in $\mathcal{M}$ and $u':x'\to y'\otimes F(x)$ a morphism in $\mathcal{M}'$, and its 2-cells $$ \xymatrix @C=8pt{x' \ar@/^0.7pc/[rr]^{(u',x)} \ar@/_0.7pc/[rr]_-{ (v',y)}\ar@{}[rr]\|{\Downarrow\! u } & & y'} $$ are those morphisms $u:x\to y$ in $\mathcal{M}$ making commutative the triangle $$ \xymatrix{x'\ar[r]^{u'}\ar[rd]_{v'}&y'\otimes Fx\ar[d]^{y'\otimes Fu}\\ & y'\otimes Fy. } $$ The vertical composition of 2-cells is given by the composition of arrows in $\mathcal{M}$. The horizontal composition of two 1-cells $\xymatrix{x'\ar[r]^{(u',x)}&y'\ar[r]^{(v',y)}&z'}$ is the 1-cell $ (v' \circledcirc u', y\otimes x): x'\to z' $, $$ v' \circledcirc u'=\Big(\xymatrix{x'\ar[r]^-{u'}&y'\otimes Fx\ar[r]^-{v'\otimes Fx}&(z'\otimes Fy)\otimes Fx}\cong z'\otimes (Fy\otimes Fx)\cong z'\otimes F(y\otimes x)\Big) $$ and the horizontal composition of 2-cells is given by tensor product of arrows in $\mathcal{M}$. The identity 1-cell of any 0-cell $x$ is $ (\overset{_\circ}{1}_x,I):x\to x$, where $ \overset{_\circ}{1}_x=(x'\cong x'\otimes I'\cong x'\otimes FI) $. The associativity, left and right constraints are obtained from those of $(\mathcal{M},\otimes)$ by the formulas $$ \boldsymbol{a}_{(w',z),(v',y),(u',x)}=\boldsymbol{a}_{z,y,x},\hspace{0.3cm} \boldsymbol{r}_{(u',x)}= \boldsymbol{r}_x,\hspace{0.3cm} \boldsymbol{l}_{(u',x)}=\boldsymbol{l}_x. $$ Following the terminology of \cite[page 228]{b-c2}, we shall call this bicategory $\Sigma F\!\!\downarrow_\star$ the {\em homotopy fiber bicategory} of the monoidal functor $F:(\mathcal{M},\otimes)\to (\mathcal{M}',\otimes)$, and write it by $\mathcal{K}_F$. Every object $z'$ of $\mathcal{M}'$, determines a 2-endofunctor $z'\otimes -:\mathcal{K}_F\to \mathcal{K}_F$, which is defined on cells by $$ \xymatrix @C=8pt{x' \ar@/^0.7pc/[rr]^{(u',x)} \ar@/_0.7pc/[rr]_-{ (v',y)}\ar@{}[rr]\|{\Downarrow\! u } & & y'} \mapsto \ \xymatrix @C=10pt{z'\otimes x' \ar@/^0.7pc/[rr]^{(z'\circledcirc u',x)} \ar@/_0.7pc/[rr]_-{ (z'\circledcirc v',y)}\ar@{}[rr]\|{\Downarrow\! u } & & z'\otimes y',} $$ where $z'\circledcirc u'=\Big(\xymatrix{z'\otimes x'\ar[r]^-{z'\otimes u'}&z'\otimes (y'\otimes Fx)}\cong (z'\otimes y')\otimes Fx\Big)$, and from Theorems \ref{B} and \ref{A}, we get the following: \begin{theorem} For any monoidal functor $F:(\mathcal{M},\otimes)\to (\mathcal{M}',\otimes)$, the following statements hold: (i) There is an induced homotopy fiber sequence $$\BB\mathcal{K}_F\to\BB(\mathcal{M},\otimes)\overset{\BB F}\longrightarrow \BB(\mathcal{M}',\otimes),$$ whenever the induced maps $\BB(z'\otimes -):\BB\mathcal{K}_F\to\BB\mathcal{K}_F$ are homotopy autoequivalences, for all $z'\in \mbox{Ob}\mathcal{M}'$. (ii) The induced map $\BB F: \BB(\mathcal{M},\otimes)\to \BB(\mathcal{M}',\otimes)$ is a homotopy equivalence if the space $\BB\mathcal{K}_F$ is contractible. \end{theorem} For any monoidal category $(\mathcal{M},\otimes)$, pseudo bidiagrams of categories over its suspension bicategory, $$ \mathcal{N}=(\mathcal{N},\chi):\Sigma(\mathcal{M},\otimes)^{\mathrm{op}}\to \Cat, $$ are interesting to consider, since they can be regarded as a category $\mathcal{N}$ (the one associated to the unique object of the suspension bicategory) endowed with a coherente right pseudo action of the monoidal category $(\mathcal{M},\otimes)$ (see e.g. \cite[\S 1]{jardine}). Namely, by the functor $\otimes:\mathcal{N}\times \mathcal{M}\to \mathcal{N}$, which is defined on objects by $a\otimes x=x^a$ and on morphism by $$ (a\overset{f}\to b)\otimes (x\overset{u}\to y)=\xymatrix@C=20pt{\big(x^a\ar[r]^-{x^\!f}&x^b\ar[r]^-{u^b}&y^b\big)}= \xymatrix@C=20pt{\big(x^a\ar[r]^-{u^\!a}&y^a\ar[r]^-{y^f}&y^b\big),} $$ together with the coherent natural isomorphisms $$\begin{array}{l} \xymatrix{(a\otimes x)\otimes y=y^x^a\ar[r]^-{\chi_{x,y}a}&(x\otimes y)^a=a\otimes (x\otimes y)}\\ \ \xymatrix{a\ar[r]^-{\chi_{_I}a}&I^a=a\otimes I.} \end{array} $$ For each such $(\mathcal{M},\otimes)$-category $\mathcal{N}$, the cells of the bicategory $\int_{\Sigma(\mathcal{M},\otimes)}\mathcal{N}$ has the following easy description: Its objects are the same as the objects of the category $\mathcal{N}$. A 1-cell $(f,x):a\to b$ is a pair with $x$ an object of $\mathcal{M}$ and $f:a\to b\otimes x$ a morphism in $\mathcal{N}$, and a 2-cell $$\xymatrix @C=6pt {a \ar@/^0.7pc/[rr]^{(f,x)} \ar@/_0.7pc/[rr]_{ (g,y)} \ar@{}[rr]\|{\Downarrow\!u}& {} &b }$$ is a morphism $u:x\to y$ in $\mathcal{M}$ such that the triangle $$ \xymatrix@C=12pt@R=16pt{&a\ar[rd]^{g}\ar[ld]_{f}&\\ b\otimes x \ar[rr]^{b \otimes u}&&b\otimes y.} $$ is commutative. Many of the homotopy theoretical properties of the classifying space of the monoidal category, $\BB(\mathcal{M},\otimes)$, can actually be more easily reviewed by using Grothendieck bicategories $\int_{\Sigma(\mathcal{M},\otimes)}\mathcal{N}$, instead of the Borel pseudo simplicial categories $$E_{(\mathcal{M},\otimes)}\mathcal{N}:\Delta^{\!{\mathrm{op}}}\to \Cat, \hspace{0.3cm}[p]\mapsto \mathcal{N}\times \mathcal{M}^p $$as, for example, Jardine did in \cite{jardine} for $(\mathcal{M},\otimes)$-categories $\mathcal{N}$. Thus, one sees, for example, that if the action is such that multiplication by each object $x$ of $\mathcal{M}$, that is, the endofunctor $-\otimes x:\mathcal{N}\to \mathcal{N}$, induces a homotopy equivalence $\BB\mathcal{N}\simeq \BB\mathcal{N}$, then, by Theorem \ref{th1}, one has an induced homotopy fiber sequence (cf. \cite[Proposition 3.5]{jardine}) $$\xymatrix{\BB \mathcal{N}\to\BB \int_{\Sigma(\mathcal{M},\otimes)}\mathcal{N} \,\overset{\BB P}\longrightarrow\, \BB(\mathcal{M},\otimes).}$$ In particular, the right action of $(\mathcal{M},\otimes)$ on the underlying category $\mathcal{M}$ leads to the bicategory $$\xymatrix{\int_{\Sigma(\mathcal{M},\otimes)}\mathcal{M}= \Sigma(\mathcal{M},\otimes)\!\!\downarrow_{\star}}, $$ the comma bicategory of the suspension bicategory over its unique object, whose classifying space is contractible by Lemma \ref{cont} (cf. \cite[Proposition 3.8]{jardine}). Then, it follows the well-known result by Mac Lane \cite{mac} and Stasheff \cite{sta} that there is a homotopy equivalence $$\BB \mathcal{M}\simeq \Omega\BB(\mathcal{M},\otimes),$$ between the classifying space of the underlying category and the loop space of the classifying space of the monoidal category, whenever multiplication by each object $x\in \Ob\mathcal{M}$, $y\mapsto y\otimes x$, induces a homotopy autoequivalence on $\BB\mathcal{M}$ (cf. Example \ref{ombi}). } \end{example}	{ "redpajama_set_name": "RedPajamaArXiv" }	6,887
Q: 'NoneType' object does not support item assignment when trying to update text in a button on Tkinter import tkinter as tk from tkinter import * window = tk.Tk() window.title("Panda Clicker") window.counter = 0 pps = 1 cost_pps = 20 counter = window.counter def clicked(): global pps global counter counter += pps L["text"] = "Pandas: " + str(counter) C = Label(window, text="+" + str(pps)) C.place(relx = 0.7, rely = 0.01, anchor ="ne") C.config(font=('Helvetica bold', 14)) C.pack() C.after(100, lambda: C.destroy()) def upgrade_pps(): global cost_pps global pps global counter pps += 1 counter -= 20 cost_pps += 20 Upg["text"] = "Upgrade CLicker (+1) = " + str(cost_pps) + " Pandas" L = Label(window, text="No clicks yet.") L.config(font=('Helvetica bold', 18)) L.pack() Upg = Button(window, text="Upgrade CLicker (+1) = " + "20" + " Pandas", command=upgrade_pps).pack(pady=80) window.loadimage = tk.PhotoImage(file="panda_pic.gif") window.roundedbutton = tk.Button(window, image=window.loadimage, command=clicked) window.roundedbutton["bg"] = "white" window.roundedbutton["border"] = "0" window.roundedbutton.pack() On the line: Upg["text"] = "Upgrade CLicker (+1) = " + str(cost_pps) + " Pandas", it comes up with an error of "'NoneType' object does not support item assignment". I understand that this means that something has "None" assigned to it but I cannot figure out what or why.	{ "redpajama_set_name": "RedPajamaStackExchange" }	9,969
Квінт Анті́стій А́двент Посту́мій Аквілі́н (; близько 120 — після 176) — політичний та військовий діяч часів Римської імперії, консул-суффект 167 року. Життєпис Походив з роду Антістіїв, що були homo novus. Народився у місті Тібіліс (Нумідія). Батьки були із заможної та впливової родини, про що свідчить прізвище Квінта Антістія. Обрав собі кар'єру військового. Спочатку обіймав посаду військового трибуна Legio I Minervia. Після цього став квестором у римської провінції Македонія та увійшов до сенату. Слідом за цим був народним трибуном, легатом у провінції Африка та претором. Потім був легатом Legio VI Ferrata. У 162—166 роках як легат Legio II Adiutrix відзначився у одній з парфянських війн. З 165 до 167 року як імператорський легат-пропретор керував провінцією Кам'яниста Аравія. У 167 році став консулом-суффектом. У 167 році очолив Legio III Italica. Під час Першої Маркоманської війни, перебуваючи в Альпах, захищав Італію від вторгнення германських військ до 170 року. Того ж року призначений куратором зведення громадських будівель у Римі. Наприкінці цього ж року призначений проконсулом провінції Нижня Германія. У 173—176 роках керував провінцією Британія. Того ж часу увійшов до колегії жерців-феціалів. Про подальшу долю його немає відомостей. Сім'я Дружина — Новія Криспія Діти: Луцій Антістій Бурр, консул 181 року Джерела Andreas Krieckhaus, Senatorische Familien und ihre patriae (1./2. Jh.). Kovač, Hamburg 2006, ISBN 3-8300-1836-3, S. 116—126. Birley, The Fasti of Roman Britain, (Oxford: Clarendon Press, 1981), pp. 129—132 Консули Римської імперії Претори Квестори Антістії	{ "redpajama_set_name": "RedPajamaWikipedia" }	6,053
{"url":"https:\/\/www.zigya.com\/study\/book?class=11&board=bsem&subject=Physics&book=Physics+Part+II&chapter=Mechanical+Properties+of+Fluids&q_type=&q_topic=Pressure&q_category=&question_id=PHEN11039419","text":"\u00ef\u00bb\u00bf Work has to be done to increase the area of free surface of liquid. Explain it in accordance with the law of conservation of energy. from Physics Mechanical Properties of Fluids Class 11 Manipur Board\n\n### Chapter Chosen\n\nMechanical Properties of Fluids\n\n### Book Store\n\nCurrently only available for.\nCBSE Gujarat Board Haryana Board\n\n### Previous Year Papers\n\nDownload the PDF Question Papers Free for off line practice and view the Solutions online.\nCurrently only available for.\nClass 10 Class 12\nWork has to be done to increase the area of free surface of liquid. Explain it in accordance with the law of conservation of energy.\n\nNo net force is experienced by the molecules in the bulk of liquid and are free to move within. But, a net inward pull is experienced by the molecules in the surface film.\u00a0When the area of the free surface is increased, the molecules from bulk move to surface film against the inward pull and hence work has to be done to increase the area of free surface.\nThis work is stored in the form of potential energy. Therefore, work done to increase the area of the surface of liquid is\u00a0in accordance with the law of conservation of energy.\n116 Views\n\nDo intermolecular or inter-atomic forces follow inverse square law?\n\nNo. Intermolecular and inter-atomic forces do not obey the inverse square law.\n1257 Views\n\nWhat is hydrodynamics?\n\nHydrodynamics is the branch of science that studies about the force exerted by the fluids or acting on the fluids.\n799 Views\n\nWhat is fluid?\n\nAny material that can flow is a fluid.\u00a0Liquids and gases are examples of fluid.\n\n976 Views\n\nWhat is hydrostatics?\n\nHydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest. The study of fluids at rest or objects placed at rest in fluids is hydrostatics.\n854 Views\n\nWhy solids have definite shape while liquids do not have definite shape?\n\nSolids:\u00a0Intermolecular forces are very strong and thermal agitations are not sufficiently strong to separate the molecules from their mean position. Solids are rigid and hence they have definite shapes.\nLiquids: In liquids intermolecular forces are not sufficiently strong to hold the molecules at definite sites, as a result they move freely within the bulk of liquid, therefore, do not possess definite shapes. Liquids take the same shape as that of the container.\n977 Views","date":"2018-08-16 14:04:48","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\": 0.3092348277568817, \"perplexity\": 886.5498367933407}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221211000.35\/warc\/CC-MAIN-20180816132758-20180816152758-00256.warc.gz\"}"}	null	null
Trial date appears to be on track for Solano… Trial date appears to be on track for Solano man accused of teen's murder By Richard Bammer \| rbammer@thereporter.com \| Vacaville Reporter PUBLISHED: January 4, 2022 at 4:21 p.m. \| UPDATED: January 5, 2022 at 9:12 a.m. A March trial date appears to be on track for a 25-year-old Fairfield man accused of killing a teenager more than two years ago in Fairfield. Deemed mentally competent to stand trial, Michael Ruben Russo, who on Tuesday appeared in Department 9 for a trial readiness conference, also will return to Judge Carlos R. Gutierrez's courtroom at 1:30 p.m. Feb. 28 for a trial management conference and at 9 a.m. March 9 for the trial in the Justice Center in Fairfield. Gutierrez in July reinstated criminal proceedings against Russo, a development in the case that came after a January preliminary hearing when the judge determined there was evidence to hold Russo for trial. But the defendant's road to a preliminary hearing and trial was uncertain at one point. Court records show that Gutierrez found Russo mentally incompetent to stand trial and, on Jan. 15, 2020, ordered him evaluated by the county's Department of Mental Health. The defendant was eventually placed for a time with MHM Services Inc., which provides behavioral health and medical specialty services, from mental health and medical to dental and forensic, to state and local government agencies. The company has offices, a secured facility, in Vallejo. On April 10, the judge signed an order to compel involuntary treatment with antipsychotic medication for Russo. Since then, Gutierrez has heard a report from a psychologist or psychiatrist to determine if Russo is capable of understanding court proceedings, the charge against him, and can help his attorney with his defense. Under the law, a defendant who is incompetent cannot be tried or convicted while the incompetency lasts. If eventually declared competent after treatment and counseling, a defendant may be scheduled for a jury trial. Court records indicate that, on July 9, Gutierrez reinstated criminal proceedings against Russo, meaning the defendant would face additional proceedings, including a jury trial. Russo is represented by Fairfield criminal defense attorney Thomas Maas. Deputy District Attorney Eric Charm has led the prosecution during previous court appearances. Russo has pleaded not guilty to the murder charge. He is suspected of fatally shooting Florentino Barron, 18, at about 7:30 p.m. Sept. 28, 2019, in the 1600 block of Clay Street. Responding Fairfield police officers found Barron in the front yard of a residence. After receiving medical aid, he was taken by ambulance to a local trauma center, where he died. Witnesses gave officers a detailed description of the suspect, which helped police in finding Russo and taking him into custody. The investigation showed the shooting was not a random act, police officials said in a prepared statement at the time. The victim and suspect knew each other, and there were no other suspects. The firearm used in the shooting was recovered. Court records show the Solano County District Attorney's Office filed the criminal complaint on Oct. 1, 2019. If found guilty of first-degree murder at trial, Russo, who remains in the Stanton Correctional Facility in Fairfield without bail, faces 25 years to life in state prison, plus the likelihood of additional time for enhancements, including the use of a firearm, and any previous felony convictions. Michael R. Russo, accused of fatally shooting Florentino Barron, 18, to face jurors March 9 in Department 9 in the Justice Center in Fairfield Richard Bammer \| Reporter More in Courts Humming in California court backfires on defendant who didn't want to hear fentanyl warning Here's what's changed as California's revised COVID workplace rules go into effect Prelim-setting reshuffled for siblings linked to teen's October killing in Fairfield	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	7,150
\section{Introduction} The deformation theory of Galois representations has been used extensively to realize and study the proposed Langlands correspondence between Galois representations and automorphic representations. Depending on the setting in which the correspondence is being studied, one wants to set up the deformation theory so that it parameterizes exactly the Galois representations with certain properties. A collection of such properties is known as a ``deformation condition'' or ``deformation datum.'' Ramakrishna \cite{ramakrishna1993} proved that for any deformation condition $\cC$ satisfying a basic stability property, the deformation problem for $\cC$ is representable relative to the unrestricted deformation problem. In applications, and especially when working with residually reducible Galois representations, Galois pseudorepresentations are often more accessible than Galois representations. A Galois pseudorepresentation is the data of a characteristic polynomial for each element of the Galois group, satisfying appropriate compatibility conditions. However, it has not been clear how to apply deformation conditions -- which apply to Galois modules -- to pseudorepresentations. The point of this paper is to solve this problem by axiomatically translating deformation conditions for representations into deformation conditions for pseudorepresentations. In particular, we construct universal deformation rings for pseudorepresentations satisfying such a condition. Also, it follows from our techniques that a deformation condition cuts out a closed locus in any family of Galois representations, generalizing the case of local families considered in \cite{ramakrishna1993}. We expect that the various constructions we make, especially the universal Cayley-Hamilton algebra defined by a condition, will find many applications. We were stimulated to develop this theory for application in the companion paper \cite{WWE3} (see also \cite{WWE5}). After this work was complete, the second-named author identified Galois cohomological data that controls the deformation theory of these pseudodeformation rings: see \cite{CarlAInf}, especially [Thm.\ 3.4.1, \textit{loc.\ cit.}]. \subsection{Background} \label{subsec:background} To give a more thorough explanation of our results, we define our scope. Fix a prime $p$ and a profinite group $G$. We assume that $G$ satisfies the finiteness condition $\Phi_p$ (see \S\ref{subsec:conv}). We are interested in studying the categories ${\mathcal R \mathrm{ep}}_G$ of representations of $G$ and ${\mathrm{PsR}}_G$ of pseudorepresentations of $G$, and the related moduli spaces ${\mathrm{Rep}}_G$ and ${\mathrm{PsR}}_G$. Let $\cC$ be a condition on finite-cardinality $\mathbb{Z}_p[G]$-modules such that $\cC$ is closed under isomorphisms, subquotients, and finite direct sums (we will call such a condition \emph{stable}). Ramakrishna \cite{ramakrishna1993} showed that the formal deformation functor of representations with $\cC$ is representable. The goal of this paper is to sensibly extend the condition $\cC$ so that it applies to pseudorepresentations. One difficulty in doing this is that pseudorepresentations are defined as collections of functions; they need not come in any obvious way from a $\mathbb{Z}_p[G]$-module. The main tool we use to bridge the gap between $G$-pseudorepresentations and $G$-modules is the category ${\mathcal{CH}}_G$ of \emph{Cayley-Hamilton representations of $G$}, which was introduced by Chenevier \cite{chen2014}, building on work of Bella\"{i}che--Chenevier \cite{BC2009}, and further developed by second-named author \cite{WE2018}. The data of a Cayley-Hamilton representation of $G$ includes a $\mathbb{Z}_p[G]$-module, so it is possible to relate these objects to condition $\cC$. Moreover, ${\mathcal{CH}}_G$ is closely related to both ${\mathcal R \mathrm{ep}}_G$ and ${\mathrm{PsR}}_G$; the key properties can be summarized as follows: \begin{itemize}[leftmargin=2em] \item ${\mathcal{CH}}_G$ broadens the category of representations of $G$, in the sense that there is an inclusion ${\mathcal R \mathrm{ep}}_G \hookrightarrow {\mathcal{CH}}_G$ and an induced-pseudorepresentation functor $\psi: {\mathcal{CH}}_G \rightarrow {\mathrm{PsR}}_G$ extending the usual functor ${\mathcal R \mathrm{ep}}_G \rightarrow {\mathrm{PsR}}_G$. \item $\psi$ is essentially surjective, which is not true for $\psi\vert_{{\mathcal R \mathrm{ep}}_G}$. \item ${\mathcal{CH}}_G$ has a universal object, the ``universal Cayley-Hamilton representation'' of $G$, which we denote by $E_G$. \item $E_G$ is finitely generated as a module over its base ring. \end{itemize} The first two conditions say that Cayley-Hamilton representations generalize the notation of representations to an extent sufficient to see all pseudorepresentations. The last two conditions mean that ${\mathcal{CH}}_G$ is still manageable in size; in particular, it reduces constructions to finite-cardinality $\mathbb{Z}_p[G]$-modules, so one can apply the condition $\cC$. We review the theory of Cayley-Hamilton representations in \S\ref{subsec:RepG}. \subsection{Results} \label{subsec:results} We use the above properties of ${\mathcal{CH}}_G$ to give the definition of a \emph{Cayley-Hamilton representation of $G$ with condition $\cC$ }(Definition \ref{defn:PforCH}). The Cayley-Hamilton representations of $G$ with condition $\cC$ form a full subcategory ${\mathcal{CH}}_G^\cC \subset {\mathcal{CH}}_G$. We say that a pseudorepresentation has condition $\cC$ if there is a Cayley-Hamilton representation that induces it. More precisely, we define the category of pseudorepresentations satisfying condition $\cC$, denoted by ${\mathrm{PsR}}_G^\cC$, as the essential image ${\mathrm{PsR}}_G^\cC \subset {\mathrm{PsR}}_G$ of $\psi$ restricted to ${\mathcal{CH}}_G^\cC \subset {\mathcal{CH}}_G$ (Definition \ref{defn:PsR-P}). We view these new definitions as being the heart of our paper. Our main results are meant to justify why we think this notion is a useful generalization of Ramakrishna's theory. We show that it has the same geometric and universality properties as Ramakrishna's, and that it specializes to his definition in cases where both definitions apply. Importantly, we also give tools for checking whether certain pseudorepresentations (including many of the ones that come up naturally) have this property. In some cases, we are also able to give tools for computing these objects in terms of natural Selmer-type groups. Our main general results are as follows. \begin{enumerate}[leftmargin=2em] \item The category ${\mathcal{CH}}_G^\cC$ has a universal object $E_G^\cC$, which is a quotient of the universal object $E_G$ in ${\mathcal{CH}}_G$ (Theorem \ref{thm:univ_CH_P}). \item We define the notion of a faithful Cayley-Hamilton $G$-module $M$ (it is a $G$-module with extra structures giving rise to a Cayley-Hamilton representation $E_M$ of $G$). We prove that $M$ satisfies $\cC$ if and only if $E_M$ satisfies $\cC$ (Theorem \ref{thm:ModToCH}). \item Property $\cC$ is a Zariski closed condition on both ${\mathrm{Rep}}_G$ and ${\mathrm{PsR}}_G$ (Corollary \ref{cor:P-closed}). In particular, for a fixed residual pseudorepresentation, there is a pseudodeformation-with-$\cC$ ring that is a quotient of the universal pseudodeformation ring (Theorem \ref{thm:RDP}) \end{enumerate} Many of the pseudorepresentations that arise naturally in the study of modularity-lifting come from faithful Cayley-Hamilton $G$-modules (see Example \ref{eg:Tate module}), and result (2) is very useful for showing that such a pseudorepresentation has $\cC$. Result (3) is a generalization of a theorem of Ramakrishna \cite[Prop.~1.2]{ramakrishna1993} for residually irreducible representations, and, in that case, we prove that our construction agrees with his (Corollary \ref{cor:us=Ram}). In \S\S3-4, we restrict our attention to the subcategories of ${\mathcal{CH}}_G$, ${\mathcal R \mathrm{ep}}_G$, and ${\mathrm{PsR}}_G$ that are \emph{residually multiplicity-free} (see Definition \ref{defn:RMF}). For the sake of this introduction, we abuse notation and use the same letters for the residually multiplicity-free versions. With this restriction, it is known that $E_G$ admits the extra structure of a \emph{generalized matrix algebra}, a theory developed by Bella\"{i}che--Chenevier (see \cite[\S1.5]{BC2009} and \cite[\S4.1]{WWE2}). Using this extra structure, we prove the following results in \S\ref{sec:GMAs} and \S\ref{sec:gma_ext}, respectively: \begin{enumerate}[resume, leftmargin=2em] \item The quotient $E_G^\cC$ of $E_G$ is uniquely characterized in terms of ${\mathcal R \mathrm{ep}}_G^\cC$ (Theorem \ref{thm:GMA-P}). \item The structure of $E_G^\cC$ is related to extension groups with condition $\cC$ (Theorem \ref{thm:BCs and exts}). \end{enumerate} We view this last result as especially important, as it allows one to compute with some of our constructions. In particular, it allows one to give a kind of upper bound on the size of the pseudodeformation ring in terms of Selmer-type groups, and this is useful in proving modularity-lifting theorems (see Remark \ref{rem:use for R=T}). This method is used in our papers \cite{WWE3,WWE5}. \begin{rem} In the body of the paper, we mostly discuss variants of these results for the versions of the categories that have a fixed residual pseudorepresentation ${\bar D}$. To translate into the form stated in this introduction, one can apply the results of \S \ref{subsec:RepG}, particularly Theorem \ref{thm:RDb}. \end{rem} \subsection{Context} We are motivated by the case that $G$ is a quotient of an absolute Galois group of a global field and $\cC = \{\cC_v\}$ is a condition on the restrictions of representations of $G$ to its decomposition groups $G_v$, especially those conditions $\cC_v$ that arise from $p$-adic Hodge theory when $v \mid p$. We discuss the examples we have in mind in \S\ref{sec:ex}. The unavailability of a notion of pseudorepresentations satisfying said conditions has been an obstacle to generalizing the use of the deformation theory of Galois representations to the residually reducible case. In particular, in the papers \cite{WWE3,WWE5}, we require the particular condition $\cC_v$ that a $\mathbb{Z}_p[G_v]$-module arises as the generic fiber of a finite flat group scheme over the ring of integers of a $p$-adic local field. (This is the same condition that especially motivated \cite{ramakrishna1993}.) In our previous work \cite{WWE1, WWE2}, we considered the ordinary condition on 2-dimensional global Galois representations (see also \cite{CS2019} and \cite[\S7]{WE2018}). However, the ordinary condition is of a rather different flavor, as it does not apply readily to a finite cardinality $\mathbb{Z}_p[G]$-module without extra structure. \subsection{Acknowledgements} We thank K\c{e}stutis \v{C}esnavi\v{c}ius for pointing out an unusual discrepancy about the numbering in \cite{chen2014}. We thank James Newton for helpful comments and discussions about applications to modularity lifting and the paper \cite{tenauthors}, which led us to include Corollary \ref{cor:tenauthors}. We thank the anonymous referees for many helpful comments that have led to improvements in the exposition. The intellectual debt owed to the works of Bella\"{i}che--Chenevier \cite{BC2009} and Chenevier \cite{chen2014} will be clear to the reader. We thank Jo\"el Bella\"{i}che for his interest in and encouragement about this work. P.W.\ was supported by the National Science Foundation under the Mathematical Sciences Postdoctoral Research Fellowship No.\ 1606255 and and Grant No. DMS-1638352. C.W.E. was supported by the Engineering and Physical Sciences Research Council grant EP/L025485/1. \subsection{Notation and conventions} \label{subsec:conv} Rings $R$ are commutative and Noetherian, unless otherwise noted. Algebras are associative but not necessarily commutative, and are usually finitely generated. For an algebra $E$, the term ``$E$-module" is used to mean ``left $E$-module,'' unless otherwise stated, and we let $\mathrm{Mod}_E$ denote the category of left $E$-modules. When $A$ is a local ring, to say that $B$ is commutative local $A$-algebra means that $B$ is a local ring equipped with a morphism of local rings $A \to B$. We study integral $p$-adic representations and pseudorepresentations of a profinite group $G$, which is assumed to satisfy the $\Phi_p$ finiteness condition of Mazur \cite{mazur1989}, i.e.\ the maximal pro-$p$ quotient of every finite index subgroup of $G$ is topologically finitely generated. We will work with categories of topological rings $R$ discussed in \S\ref{subsec:RepG}, giving finitely generated $R$-modules a natural topology. Algebras $E$ over $R$ are also understood to have their natural topology, either being finitely generated as $R$-modules or of the form $R[G]$. Actions of $G$ and homomorphisms $G \rightarrow E^\times$ are understood to be continuous. Pseudorepresentations $D : E \rightarrow R$ (resp.\ $D : G \rightarrow R$) are also understood to be continuous, which means that the characteristic polynomial coefficient functions $E \rightarrow R$ (resp.\ $G \rightarrow R$) defined in Definition \ref{defn:PsR}(8) are continuous. \subsubsection{A note on attribution} We wish to make it clear that the notions of pseudorepresentation, Cayley-Hamilton algebra, and generalized matrix algebra, which are used in an essential way in this paper, are due to Chenevier \cite{chen2014} and Bella\"iche--Chenevier \cite[Ch.\ 1]{BC2009}, building on works of others including Wiles \cite{wiles1988}, Procesi \cite{procesi1987}, Taylor \cite{taylor1991}, Nyssen \cite{nyssen1996}, and Rouquier \cite{rouquier1996}. Where possible, we have attempted to cite the original sources directly. In some cases, we also cite \cite{WE2018} for background on these objects because the category of profinite-topological Cayley-Hamilton representations of a profinite group was developed there, adapting the non-topological version of \cite[\S1.22]{chen2014}. In contrast, we note that Chenevier developed the profinite-topological notion of pseudorepresentation \cite[\S3]{chen2014}, which we rely on without making any additional contributions. \section{Deformation conditions for Cayley-Hamilton representations} \label{sec:prop} In this section, we develop the theory of Cayley-Hamilton representations with a deformation condition $\cC$. The first three subsections are mainly review. In \S \ref{subsec:compat}, we begin by recalling from \cite{chen2014} and \cite[\S2]{WE2018} the non-topological theory of pseudorepresentations and Cayley-Hamilton algebras, their compatible representations, and moduli spaces of these (see also \cite{WWE1} for further background and examples). In \S \ref{subsec:RepG}, we review the modifications of this theory for topological groups. In \S \ref{subsec:stability}, we discuss the stable conditions of Ramakrishna \cite{ramakrishna1993}, and show that there is a well-defined ``maximal quotient with $\cC$" of a finite cardinality $\mathbb{Z}_p[G]$-module. The main new constructions are given in \S \ref{subsec:constr}. There, we show that the ``maximal quotient with $\cC$" of the universal Cayley-Hamilton algebra retains its algebra structure. We also show that, for an algebra quotient of a Cayley-Hamilton algebra, there is a maximal further quotient that is Cayley-Hamilton. Combining these constructions, we obtain a ``maximal Cayley-Hamilton quotient with $\cC$" of the universal Cayley-Hamilton algebra. The main results and definitions of the section are stated and proven in \S \ref{subsec:CH-P}. We define the category of Cayley-Hamilton representations with $\cC$, and show that the object constructed in \S \ref{subsec:constr} is a universal object in this category. We define a pseudorepresentation with $\cC$ as a pseudorepresentation that comes from a Cayley-Hamilton representations with $\cC$, and prove that the corresponding deformation functor is pro-representable. In \S \ref{subsec:ModCH-P}, we introduce the notion of Cayley-Hamilton $G$-module. Many pseudorepresentations that arise from modular forms (and more generally, automorphic forms) come from Cayley-Hamilton $G$-modules (see Example \ref{eg:Tate module}). We prove a criterion for when such pseudorepresentations have $\cC$. In \S \ref{subsec:formal moduli with C}, we apply this criterion to prove a generalization of Ramakrishna's theorem from formal families of representations to more general families. \subsection{Pseudorepresentations, compatible representations, and Cayley-Hamilton algebras} \label{subsec:compat} Our starting point is following definition, due to Chenevier \cite{chen2014}. \begin{defns} \label{defn:PsR} Let $R$ be a ring, let $E$ be an $R$-algebra, and let ${\mathcal{G}}$ a group. \smallskip \begin{enumerate}[leftmargin=2em] \item A \emph{pseudorepresentation}, denoted $D: E \rightarrow R$ or $(E,D)$, is a multiplicative polynomial law\footnote{A polynomial law is a natural transformation $D: \underline{E \otimes_R (-)} \to \underline{(-)}$ of set-valued functors on commutative $R$-algebras, where $\underline{(-)}$ is the forgetful functor from $R$-algebras to sets. It is multiplicative if, for each commutative $R$-algebra $A$, the function $D_A: E \otimes_{R} A \to A$ satisfies $D_A(1)=1$, $D_A(xy)=D_A(x)D_A(y)$ for all $x, y \in E \otimes_R A$. It has degree $d$ if, for each commutative $R$-algebra $A$, we have $D_A(bx) = b^d D_A(x)$ for all $x \in E \otimes_R A$ and all $b \in A$.} of degree $d$ from $E$ to $R$, for some $d \ge 1$. We call $d$ the \emph{dimension} of $D$, and $R$ the \emph{scalar ring} of $(E,D)$. The data of $D$ consists of a function $E \otimes_R A \to A$ for each commutative $R$-algebra $A$, which we denote by $D_A$. \item A \emph{pseudorepresentation of ${\mathcal{G}}$ with coefficients in $R$}, denoted $D : {\mathcal{G}} \rightarrow R$, is a pseudorepresentation of $R[{\mathcal{G}}]$. \item If $D: E \to R$ is a pseudorepresentation, and $x \in E$, we define the \emph{characteristic polynomial} $\chi_D(x,t) \in R[t]$ by $\chi_D(x,t)=D_{R[t]}(t-x)$. It is monic of degree equal to the dimension $d$ of $D$. We define the \emph{trace} $\mathrm{Tr}_D(x)$ to be the additive inverse of the coefficient of $t^{d-1}$ in $\chi_D(x,t)$. \end{enumerate} \end{defns} \begin{rem} This notion of pseudorepresentation is called a ``determinant'' by Chenevier. The notion of ``pseudorepresentation'' of a group was first considered by Wiles \cite[Lem.\ 2.2.3]{wiles1988} in the case $d=2$, and by Taylor \cite[\S 1]{taylor1991} in general. Taylor's definition was also considered by Rouquier \cite{rouquier1996}, who considered $R$-algebras (not just groups), and called the resulting objects ``pseudo-caract\`eres" (or ``pseudocharacters", in English). By \cite[Lem.\ 1.12]{chen2014}, $\mathrm{Tr}_D: E \to R$ is a ``pseudocharacter" in the sense of Taylor and Rouquier. By \cite[Prop.\ 1.29]{chen2014}, the map $D \mapsto \mathrm{Tr}_D$ is a bijection if $(2d)! \in R^\times$. \end{rem} \begin{eg} \label{eg:det} There is a $d$-dimensional pseudorepresentation $\det: M_d(R) \to R$ defined by letting $\det_A: M_d(A) \to A$ be the determinant for all commutative $R$-algebras $A$. More generally, if $E$ is an Azumaya $R$-algebra of degree $d$, e.g.\ $V$ is a projective $R$-module of rank $d$ and $E = \mathrm{End}_R(V)$, then there is a $d$-dimensional pseudorepresentation $\det: E \to R$ given by the reduced norm on $E$. \end{eg} \begin{eg} \label{eg:psi} If $V$ is a projective $R$-module of rank $d$ and $\rho: R[\mathcal{G}] \to \mathrm{End}_R(V)$ is a representation of $\mathcal{G}$, then there is an associated pseudorepresentation $\psi(\rho):\mathcal{G}\to R$. It is defined, for a commutative $R$-algebra $A$ and $x \in A[\mathcal{G}]$, by \[ \psi(\rho)_A(x)=\mathrm{det}_A((\rho\otimes_R A)(x)), \] where $\det$ is as in the previous example. \end{eg} Informally, $\psi(\rho)$ is the composition of $\rho$ and $\det$. In the following definition, we make this `composition' construction explicit, and introduce notation for some other standard ways to produce new pseudorepresentations from old ones. \begin{defns} \label{defn:NewPsFromOld} Let $f: R \to R'$ be a ring homomorphism, let $E$ and $E'$ be an $R$-algebra and an $R'$-algebra, respectively, and let $g: E\otimes_R R' \to E'$ be a morphism of $R'$-algebras. Let ${\mathcal{G}}$ be a group. \smallskip \begin{enumerate}[leftmargin=2em] \item Let $D: E \to R$ be a pseudorepresentation. The \emph{base-change of $D$ by $f$}, denoted $f \circ D: E \otimes_R R' \to R'$, is the pseudorepresentation of $E\otimes_R R'$ defined by $(f \circ D )_A(x)=D_{A}(x)$, where $A$ is a commutative $R'$-algebra, and $x \in E\otimes_{R} A = (E \otimes_{R} R') \otimes_{R'} A$. When $f$ is understood, we write $D \otimes_{R}R'$ instead of $f \circ D$. \item Let $D': E' \to R'$ be a pseudorepresentation. The \emph{composition of $D'$ and $g$}, denoted $D' \circ g: E \otimes_R R' \to R'$, is the pseudorepresentation of $E\otimes_R R'$ defined by $(D' \circ g)_A(x)=D'_A((g\otimes_{R'} A)(x))$, where $A$ is a commutative $R'$-algebra and $x \in E\otimes_{R'} A$. \item A \emph{morphism of pseudorepresentations} $\rho:(E,D) \to (E',D')$ is the data of a pair $(f,g)$, such that $f \circ D= D' \circ g$. We define $\psi(\rho)= f \circ D= D' \circ g$. A \emph{morphism of pseudorepresentations of ${\mathcal{G}}$} is a morphism of pseudorepresentations $(R[{\mathcal{G}}],D) \to (R'[{\mathcal{G}}],D \otimes_R R')$ where the homomorphism $R'[{\mathcal{G}}] \to R'[{\mathcal{G}}]$ is the identity. \item If $D: E \to R$ is a pseudorepresentation and $\rho: {\mathcal{G}} \to E^\times$ is a homomorphism, there is an induced homomorphism $\tilde{\rho}: R[{\mathcal{G}}] \to E$, which defines a morphism of pseudorepresentations $(R[{\mathcal{G}}],D \circ \tilde{\rho}) \to (E,D)$. We abuse notation and denote this morphism by $\rho$, and write $\psi(\rho):{\mathcal{G}} \to R$ for the resulting pseudorepresentation of ${\mathcal{G}}$. \end{enumerate} \end{defns} It is not, in general, true that any pseudorepresentation $D:\mathcal{G} \to R$ is of the form $\psi(\rho)$ for some representation $\rho$ of $\mathcal{G}$ as in Example \ref{eg:psi}. It is, of course, true that $D=\psi(\rho)$ for some morphism $\rho$ of pseudorepresentations $(R[\mathcal{G}],D \circ \tilde{\rho}) \to (E,D)$ as in Definition \ref{defn:NewPsFromOld}(4) (we could take $E=R[\mathcal{G}]$ and $\tilde{\rho}$ to be the identity). However, it is a surprising fact that the analogous statement is true if we restrict $(E,D)$ to be in the more restrictive class of \emph{Cayley-Hamilton} pseudorepresentations {\cite[\S1.17]{chen2014}}: see Corollary \ref{cor: CH to PsR essentially surjective}. \begin{defn} \label{defn:CH} We call a pseudorepresentation $D : E \rightarrow R$ \emph{Cayley-Hamilton} when $E$ is finitely generated as an $R$-algebra, and, for every commutative $R$-algebra $A$ and every $x \in E \otimes_R A$, the element $x$ satisfies the characteristic polynomial $\chi_D(x,t)\in A[t]$ associated to it by $D$. That is, $D$ is Cayley-Hamilton when $\chi_D(x,x) = 0$ for all $x \in E \otimes_R A$. When $D : E \rightarrow R$ is Cayley-Hamilton, we call the pair $(E,D)$ a \emph{Cayley-Hamilton $R$-algebra}. \end{defn} An important property of Cayley-Hamilton algebras is the the following finiteness result. \begin{prop} \label{prop:C-H alg is module-finite} If $(E,D)$ is a Cayley-Hamilton $R$-algebra, then $E$ is finitely generated as an $R$-module. \end{prop} \begin{proof} This follows from \cite[Prop.\ 2.13]{WE2018}, which uses the theory of PI-algebras. Indeed, $E$ is finitely generated as an $R$-algebra (by definition of Cayley-Hamilton algebra) and $R$ is Noetherian by convention. \end{proof} Of course, the pseudorepresentation $(M_d(R),\det)$ of Example \ref{eg:det} is a Cayley-Hamilton algebra (by the Cayley-Hamilton Theorem), and we think of Cayley-Hamilton representations as generalizations of this example. Just as algebra homomorphisms $E \to M_d(R)$ are given the special name \emph{representation}, we follow Chenevier and give certain morphisms of pseudorepresentations the special name \emph{Cayley-Hamilton representation}. \begin{defns} Let $\mathcal{G}$ be a group. \smallskip \begin{enumerate}[leftmargin=2em] \item A \emph{Cayley-Hamilton representation} of a pseudorepresentation $(E,D)$ is a morphism of pseudorepresentations $\rho:(E,D) \to (E',D')$ with $(E',D')$ a Cayley-Hamilton algebra. \item A \emph{Cayley-Hamilton representation of ${\mathcal{G}}$}, denoted $(E',\rho,D')$, is a morphism of pseudorepresentations $(R[\mathcal{G}],D) \to (E',D')$ such that the map $R[\mathcal{G}] \to E'$ comes from a homomorphism $\rho:{\mathcal{G}} \to E'^\times$. \item If $(E,D)$ is also Cayley-Hamilton, we also refer to a Cayley-Hamilton representation $(E,D) \to (E',D')$ as a \emph{morphism of Cayley-Hamilton algebras}. \item A \emph{morphism of Cayley-Hamilton representations of ${\mathcal{G}}$}, written $(E,\rho,D) \to (E',\rho',D')$, is a morphism of Cayley-Hamilton algebras $(E,D) \to (E',D')$ such that $\rho'=(E \to E') \circ \rho$. \end{enumerate} We let ${\mathcal{CH}}_{\mathcal{G}}$ denote the category of Cayley-Hamilton representations of ${\mathcal{G}}$, with morphisms as defined in (3). \end{defns} \begin{eg} \label{eg:compat_is_CH} Let $(E,D)$ be a pseudorepresentation over $R$, and let $V$ be a projective $A$-module of rank $d$ for a commutative $R$-algebra $A$. Let $(\mathrm{End}_A(V), \det)$ be the pseudorepresentation defined in Example \ref{eg:det}, which is a Cayley-Hamilton algebra. A Cayley-Hamilton representation $(E,D) \to (\mathrm{End}_A(V),\det)$ is the data of a representation $\rho: E \to \mathrm{End}_A(V)$ of $E$ that is compatible with the pseudorepresentation, i.e.~such that $D=\det \circ \rho$. \end{eg} This example motivates the following (perhaps more familiar) definition. \begin{defn} \label{defn:compat_functor} Let $D : E \rightarrow R$ be a a $d$-dimensional pseudorepresentation. Let $A$ be a commutative $R$-algebra. \begin{enumerate}[leftmargin=2em] \item A \emph{framed compatible representation of $(E,D)$ over $A$} is an $R$-algebra homomorphism $\rho_A: E \rightarrow M_d(A)$ such that $D \otimes_R A = \det \circ \rho_A$. We denote by ${\mathrm{Rep}}^\square_{E,D}$ the set-valued functor which assigns to a commutative $R$-algebra $A$ the set of all framed compatible representations of $(E,D)$ over $A$. \item A \emph{compatible representation of $(E,D)$ over $A$} is a pair $(V_A, \rho_A)$, where $V_A$ is a projective rank $d$ $A$-module and $\rho_A : E \rightarrow \mathrm{End}_A(V)$ is an $R$-algebra homomorphism such that $D \otimes_R A = \det \circ \rho_A$. We denote by ${\mathrm{Rep}}_{E,D}$ the $\Spec R$-groupoid (i.e.\ stack) which assigns to a commutative $R$-algebra $A$ a category whose objects are the compatible representations of $(E,D)$ over $A$ and whose morphisms are isomorphisms of such data. \end{enumerate} \end{defn} \begin{rem} The functor ${\mathrm{Rep}}^\square_{E,D}$ and the groupoid ${\mathrm{Rep}}_{E,D}$ are representable over $\Spec R$; see e.g.\ Theorem \ref{thm:ED-compat}. In particular, the natural adjoint action of $\GL_d$ on the affine $\Spec R$-scheme ${\mathrm{Rep}}^\square_{E,D}$ provides a smooth presentation for ${\mathrm{Rep}}_{E,D}$ as a $\Spec R$-algebraic stack. \end{rem} Note that, by Example \ref{eg:compat_is_CH}, compatible representations are a special kind of Cayley-Hamilton representation. We view Cayley-Hamilton representations as a natural generalization of compatible representations. \subsection{Representations of a profinite group} \label{subsec:RepG} For the remainder of the section, we fix a profinite group $G$ that satisfies the $\Phi_p$ condition. Recall the conventions from \S \ref{subsec:conv} regarding continuity. Let $\mathrm{Adm}_{\mathbb{Z}_p}$ denote the category of admissible topological (not necessarily Noetherian) rings where $p$ is topologically nilpotent (see \cite[Def.~0.7.1.2, pg.~60]{ega1} for the definition of admissible). We let ${\mathrm{Tfg}}_{\mathbb{Z}_p} \subset\mathrm{Adm}_{\mathbb{Z}_p}$ be the full subcategory of topologically finitely generated objects (i.e.~those $A \in \mathrm{Adm}_{\mathbb{Z}_p}$ for which there exists a (non-topological) homomorphism $\mathbb{Z}_p[x_1,\dots,x_n] \to A$ with dense image), which are Noetherian rings. For $R \in {\mathrm{Tfg}}_{\mathbb{Z}_p}$, we define ${\mathrm{Tfg}}_R$ as the slice category of ${\mathrm{Tfg}}_{\mathbb{Z}_p}$ over $R$. We often use the following fact. \begin{lem} \label{lem:tfg algebra} Let $R \in {\mathrm{Tfg}}_{\mathbb{Z}_p}$ and let $A \in {\mathrm{Tfg}}_R$. For any $x \in A$ non-zero, there exists an commutative local $R$-algebra of finite cardinality and an $R$-algebra homomorphism $f:A \to B$ such that $f(x) \ne 0$. In particular, a surjection $g:A \to A'$ in ${\mathrm{Tfg}}_R$ is determined by the natural transformation $\mathrm{Hom}_{R-\mathrm{alg}}(A',-) \to \mathrm{Hom}_{R-\mathrm{alg}}(A,-)$ of functors on commutative local $R$-algebras of finite cardinality. \end{lem} \begin{proof} We leave this as an exercise. The main point is that, for any ideal of definition $I$ for $A$, the ring $A/I^n$ is a finitely generated $\mathbb{Z}$-algebra, and consequently a Noetherian Jacobson ring. \end{proof} \begin{defn} For $A \in {\mathrm{Tfg}}_{\mathbb{Z}_p}$, a \emph{representation of $G$ with coefficients in $A$} is a finitely generated projective $A$-module $V_A$ of constant rank with an $A$-linear $G$-action $\rho_A$. We write ${\mathrm{Rep}}_G$ for the category of representations, fibered in groupoids over ${\mathrm{Tfg}}_{\mathbb{Z}_p}$ via the forgetful functor $(V_A, \rho_A) \mapsto A$. We write ${\mathrm{Rep}}_G^\square$ for the category defined just as ${\mathrm{Rep}}_G$, but with the additional data of an $A$-basis for $V_A$. Write ${\mathrm{PsR}}_G$ for the category of pseudorepresentations of $G$, fibered in groupoids over ${\mathrm{Tfg}}_{\mathbb{Z}_p}$. We write $\psi$ for the functor $\psi: {\mathrm{Rep}}_G \rightarrow {\mathrm{PsR}}_G$ over ${\mathrm{Tfg}}_{\mathbb{Z}_p}$ that sends a representation $\rho_A$ to its induced pseudorepresentation $\psi(\rho_A)$. We decompose each of these categories by dimension $d \geq 1$, writing ${\mathrm{Rep}}_G^d \subset {\mathrm{Rep}}_G$, etc. \end{defn} To understand ${\mathrm{PsR}}_G^d$, we introduce deformations. Given a finite field $\mathbb{F}/\mathbb{F}_p$, we let $\hat\cC_{W(\mathbb{F})} \subset {\mathrm{Tfg}}_{\mathbb{Z}_p}$ be the category of complete Noetherian commutative local $W(\mathbb{F})$-algebras $(A, \mathfrak{m}_A)$ with residue field $\mathbb{F}$. \begin{defn} Let ${\bar D}: G \rightarrow \mathbb{F}$ be a pseudorepresentation. Its deformation functor ${\mathrm{PsDef}}_{{\bar D}} : \hat\cC_{W(\mathbb{F})} \rightarrow \mathrm{Sets}$ is \begin{equation} \label{eq:PsR_functor} A \mapsto \{D : A[G] \rightarrow A \text{ such that } D \otimes_A \mathbb{F} \simeq {\bar D}\} \end{equation} and elements of ${\mathrm{PsDef}}_{{\bar D}}(A)$ are called \emph{deformations of ${\bar D}$} or \emph{pseudodeformations}. \end{defn} Remarkably, when one varies ${\bar D}$ over a certain set of finite field-valued pseudorepresentations known as \emph{residual pseudorepresentations} (see \cite[Def.\ 3.4]{WE2018} for the definition), ${\mathrm{PsDef}}_{\bar D}$ captures all of ${\mathrm{PsR}}_G^d$, in the following sense. \begin{thm}[Chenevier] \label{thm:RDb} Assume $G$ satisfies $\Phi_p$. \begin{enumerate}[leftmargin=2em] \item Given a finite field $\mathbb{F}$ and a pseudorepresentation ${\bar D} : G \rightarrow \mathbb{F}$, the functor ${\mathrm{PsDef}}_{\bar D}$ is represented by an object $(R_{\bar D}, \mathfrak m_{\bar D}) \in \hat{\cC}_{W(\mathbb{F})}$. \item There is an isomorphism \begin{equation} \label{eq:PsR-decomp} {\mathrm{PsR}}_G^d \cong \coprod_{\bar D} \Spf R_{\bar D}, \end{equation} where ${\bar D}$ varies over $d$-dimensional residual pseudorepresentations. \end{enumerate} \end{thm} \begin{proof} See \cite[Prop.\ 3.3, Prop.\ 3.7 and Cor.\ 3.14]{chen2014}. \end{proof} Let ${\mathrm{Rep}}_{\bar D}$ (resp.\ ${\mathrm{Rep}}_{\bar D}^\square$) denote the fiber in ${\mathrm{Rep}}_G$ (resp.\ ${\mathrm{Rep}}_G^\square$) of $\psi$ over $\Spf R_{\bar D}$, where ${\bar D}$ is a residual pseudorepresentation, so that we have \begin{equation} \label{eq:Rep-decomp} {\mathrm{Rep}}_G^d \cong \coprod_{\bar D} {\mathrm{Rep}}_{\bar D}, \qquad {\mathrm{Rep}}_G^{\square,d} \cong \coprod_{\bar D} {\mathrm{Rep}}_{\bar D}^\square \end{equation} where ${\bar D}$ varies over $d$-dimensional residual pseudorepresentations. Now, and for the rest of this section, we fix a finite field $\mathbb{F}/\mathbb{F}_p$ and a pseudorepresentation ${\bar D}:G \to \mathbb{F}$. In light of the decompositions \eqref{eq:PsR-decomp} and \eqref{eq:Rep-decomp}, we lose no scope in our study of $p$-adic families by fixing this choice. \begin{defn} Let $A \in {\mathrm{Tfg}}_{\mathbb{Z}_p}$. We say that a pseudorepresentation $D : G \rightarrow A$ \emph{has residual pseudorepresentation ${\bar D}$} when $\Spf A \rightarrow {\mathrm{PsR}}_G^d$ is concentrated over $\Spf R_{\bar D}$. We write $D^u_{\bar D} : G \rightarrow R_{\bar D}$ for the universal pseudodeformation of ${\bar D}$. A Cayley-Hamilton representation $(E, \rho: G \to E^\times, D: E \rightarrow A)$ of $G$ over $A \in {\mathrm{Tfg}}_{\mathbb{Z}_p}$ \emph{has residual pseudorepresentation ${\bar D}$} if its induced pseudorepresentation $\psi(\rho) : G \rightarrow A$ has residual pseudorepresentation ${\bar D}$. In particular, a representation $(V_A, \rho_A) \in {\mathrm{Rep}}_G^d(A)$ has residual pseudorepresentation ${\bar D}$ when $\psi(\rho_A)$ does. We let ${\mathcal{CH}}_{G,{\bar D}}$ denote the full subcategory of ${\mathcal{CH}}_G$ (introduced in \S\ref{subsec:background}) whose objects have residual pseudorepresentation ${\bar D}$. The natural transformation \begin{equation} \label{eq: Rep to CH-Rep} {\mathrm{Rep}}_{\bar D} \rightarrow {\mathcal{CH}}_{G,{\bar D}} \end{equation} arises from the Cayley-Hamilton representation structure on a representation of Example \ref{eg:psi}. This natural transformation commutes with the induced pseudorepresentation functor $\psi$. \end{defn} We observe that ${\mathrm{Rep}}_{\bar D}$ parameterizes representations of $G$ with residual pseudorepresentation ${\bar D}$. From \cite[Thm.\ 3.8]{WE2018} we know that ${\mathrm{Rep}}_{\bar D}$ and ${\mathrm{Rep}}^\square_{\bar D}$ admit natural algebraic models over $\Spec R_{\bar D}$. By this we mean that there exists a finite type affine scheme (resp.\ algebraic stack) over $\Spec R_{\bar D}$ whose $\mathfrak{m}_{\bar D}$-adic completion is ${\mathrm{Rep}}^\square_{\bar D}$ (resp.\ ${\mathrm{Rep}}_{\bar D}$). This implies that ${\mathrm{Rep}}^\square_{\bar D}$ is an affine formal scheme and ${\mathrm{Rep}}_{\bar D}$ is a formal algebraic stack, both formally of finite type over $\Spf R_{\bar D}$. This algebraic model arises from a canonical universal Cayley-Hamilton representation of $G$ with residual pseudorepresentation ${\bar D}$, which we now define. \begin{thm} \label{thm:univ_CH} The category ${\mathcal{CH}}_{G,{\bar D}}$ has a universal object $(E_{\bar D}, \rho^u: G \rightarrow E_{\bar D}^\times, D^u_{E_{\bar D}} : E_{\bar D} \rightarrow R_{\bar D})$. In particular, $E_{\bar D}$ is a finitely generated $R_{\bar D}$-module. The map $\rho^u : R_{\bar D}[G] \rightarrow E_{\bar D}$ is surjective and $D^u_{E_{\bar D}} : E_{\bar D} \rightarrow R_{\bar D}$ is a factorization of the universal pseudodeformation $D^u_{\bar D} : G \rightarrow R_{\bar D}$ through $E_{\bar D}$. \end{thm} \begin{proof} See \cite[Prop.\ 3.6]{WE2018}. \end{proof} Another way of stating the final sentence of Theorem \ref{thm:univ_CH} is that the induced pseudorepresentation $\psi(\rho^u) := D^u_{E_{\bar D}} \circ \rho^u : G \rightarrow R_{\bar D}$ of the universal Cayley-Hamilton representation $\rho^u$ (with residual pseudorepresentation ${\bar D}$) is equal to the universal deformation $D^u_{\bar D} : G \rightarrow R_{\bar D}$ of ${\bar D}$. The following theorem is proved using \eqref{eq: Rep to CH-Rep} and the universal property of $\rho^u$. \begin{thm}[{\cite[Thm.\ 3.7]{WE2018}}] \label{thm:ED-compat} There is an isomorphism of topologically finite type formal algebraic stacks (resp.\ formal schemes) on ${\mathrm{Tfg}}_{\mathbb{Z}_p}$, \[ {\mathrm{Rep}}_{\bar D} \buildrel\sim\over\lra {\mathrm{Rep}}_{E_{\bar D}, D^u_{E_{\bar D}}}, \qquad {\mathrm{Rep}}^\square_{\bar D} \buildrel\sim\over\lra {\mathrm{Rep}}^\square_{E_{\bar D}, D^u_{E_{\bar D}}}. \] \end{thm} \begin{cor} \label{cor: CH to PsR essentially surjective} Let $A \in {\mathrm{Tfg}}_{\mathbb{Z}_p}$. Let $D : G \rightarrow A$ be a pseudorepresentation. Then there exists a Cayley-Hamilton algebra over $A$ that induces it. \end{cor} \begin{proof} By Theorem \ref{thm:RDb}, we reduce to the case that $D$ has a fixed residual pseudorepresentation and deduce that there exists a homomorphism $R_{\bar D} \rightarrow A$. Then the Cayley-Hamilton representation \[ (E_{\bar D} \otimes_{R_{\bar D}} A, \rho^u \otimes_{R_{\bar D}} A : G \rightarrow E_{\bar D} \otimes_{R_{\bar D}} A, D^u_{E_{\bar D}} \otimes_{R_{\bar D}} A) \] of $G$ over $A$ induces $D$. \end{proof} \subsection{Stability} \label{subsec:stability} Let ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}} \subset \mathrm{Mod}_{\mathbb{Z}_p[G]}$ be the full subcategory whose objects have finite cardinality. \begin{defn} \label{defn:stable} A \emph{condition} on ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ is a full subcategory $\cC \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$. We will say an object of ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ \emph{satisfies condition $\cC$} or \emph{has $\cC$} if the object is in $\cC$. A condition $\cC$ is \emph{stable} if it is preserved under isomorphisms, subquotients, and finite direct sums in ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$. In other words, $\cC$ is stable if \begin{enumerate} \item for every object $A$ in $\cC$ and all isomorphisms $f:A \to B$ in ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$, the object $B$ is also in $\cC$, and \item for every object $A$ in $\cC$ and all morphisms $f:A \to B$ and $g:C \to A$ in ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$, the kernel of $f$ and cokernel of $g$ are in $\cC$, and \item for every finite collection of objects $A_1, \dots, A_n$ of $\cC$, the direct sum $A_1 \oplus \dots \oplus A_n$ in ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ is an object of $\cC$. \end{enumerate} \end{defn} \begin{eg} \label{eg:factors through is stable} Let $H \subset G$ be a normal subgroup. Let $\cC \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ be the full subcategory of objects where the $G$ action factors through the quotient $G \to G/H$. Then $\cC$ is stable (as $\cC \cong \mathrm{Mod}_{\mathbb{Z}_p[G/H]}^\mathrm{fin}$). \end{eg} \begin{eg} \label{eg:fiber product of stable is stable} Let $H_1,\dots,H_n \subset G$ be subgroups, and, for $i=1,\dots,n$, let $\cC_i \subset \mathrm{Mod}_{\mathbb{Z}_p[H_i]}^{\mathrm{fin}}$ be a condition. Then there is a condition $\cC \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ defined by the Cartesian square \[ \xymatrix{ \cC \ar[r] \ar[d] & \prod_{i=1}^n \cC_i \ar[d] \\ {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}} \ar[r] & \prod_{i=1}^n \mathrm{Mod}_{\mathbb{Z}_p[H_i]}^{\mathrm{fin}}. } \] In other words, an object $M \in {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ has $\cC$ if and only if the restriction $M\|_{H_i} \in \mathrm{Mod}_{\mathbb{Z}_p[H_i]}^{\mathrm{fin}}$ has $\cC_i$ for all $i=1,\dots,r$. If all $\cC_i$ are stable, then this $\cC$ is stable. \end{eg} For examples of conditions $\cC$ that are of use in arithmetic, see \S\ref{sec:ex}. For the rest of this section, we fix a stable condition $\cC$. \begin{thm}[Ramakrishna] \label{thm:Ram} Let $A$ be a complete commutative Noetherian local $\mathbb{Z}_p$-algebra and let $V_A$ be a finitely generated free $A$-module with an $A$-linear $G$-action. Then there exists a maximal quotient $A \twoheadrightarrow A^\cC$ such that for an commutative local $A$-algebra $B$ of finite cardinality, the $\mathbb{Z}_p[G]$-module $V_A \otimes_A B$ satisfies $\cC$ if and only if $A \rightarrow B$ factors through $A^\cC$. \end{thm} \begin{proof} This follows immediately from \cite[Thm.\ 1.1]{ramakrishna1993}. \end{proof} \begin{lem} \label{lem:Cstart} The inclusion functor $\cC \to {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ has a left adjoint $(-)^\cC:{\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}} \to \cC$. For $V \in {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$, the functor $V \mapsto V^\cC$ has the following additional properties: \begin{enumerate} \item there is a quotient map $f_V: V \twoheadrightarrow V^\cC$ of $\mathbb{Z}_p[G]$-modules. \item for $W \in \cC$, the adjunction isomorphism \[ \mathrm{Hom}_\cC(V^\cC,W) \cong \mathrm{Hom}_{\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}(V,W) \] is given by $g \mapsto g\circ f_V$. \item if $A$ is a commutative $\mathbb{Z}_p$-algebra and $V \in \mathrm{Mod}_{A[G]}$ as well, then $V^\cC \in \mathrm{Mod}_{A[G]}$. \end{enumerate} \end{lem} \begin{proof} Let $V \in {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$. First we construct the quotient map $f_V: V \to V^\cC$. Since $V$ has finite cardinality, there are a finite number of quotients of $V$. Let $\{V_1, \dots, V_n\}$ be the (possibly empty) set all of quotients of $V$ that have property $\cC$. Define $f_V: V \to V^\cC$ to be the quotient by the kernel of $V \to \oplus_{i=1}^n V_i$. Then $V^\cC$ is isomorphic to a submodule of $\oplus_{i=1}^n V_i$. Since $\cC$ is closed under isomorphisms, subobjects, and finite direct sums, we see that $V^\cC$ satisfies $\cC$. By definition, any of the quotients $V \onto V_i$ factor through $f_V$, and this factoring is unique since $f_V$ is an epimorphism. We now let $W \in \cC$ and show that the map given in (2)\ is an isomorphism. Let $W \in \cC$ and let $f:V \to W$ be a homomorphism of $\mathbb{Z}_p[G]$-modules. Let $W'=V/\ker(f)$, let $f':V \to W'$ be the quotient map, and let $f'':W' \to W$ be the injection induced by $f$. Since $W'$ is isomorphic to a submodule of $W$, we have $W' \in \cC$. Then $f':V \to W'$ must be one of the quotient maps $V \to V_i$, so $f'$ factors uniquely through $f_V$, i.e.~there is a unique morphism $g': V^\cC \to W'$ such that $f'=g' \circ f_V$. Now let $g:V^\cC \to W$ be $g=f'' \circ g'$; a simple computation shows that this assignment $f \mapsto g$ is inverse to the map given in (2). Now we show that $V \mapsto V^\cC$ is a functor. Let $h: V \to V'$ be a $\mathbb{Z}_p[G]$-module homomorphism. Then $f_{V'}\circ h \in \mathrm{Hom}_{\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}(V,(V')^\cC)$, so by (2)\ there is a unique map $h^\cC: V^\cC \to (V')^\cC$ such that $f_{V'}\circ h=h^\cC \circ f_V$. This shows that $V \mapsto V^\cC$ is a functor, and, together with (2),\ this implies that it is left adjoint to the inclusion. Finally, (3)\ follows from the functoriality. Indeed, let $a \mapsto m_a$ denote the structure map $A \to \mathrm{End}(V)$. Then there is a unique map $m^\cC_{\,\cdot}:A \to \mathrm{End}(V^\cC)$ such that $m^\cC_a \circ f_V = f_V \circ m_a$ for all $a \in A$. The fact that $m^\cC_{\,\cdot}$ is a ring homomorphism follows from the fact that $f_V$ is an epimorphism; for example, if $a,a'\in A$, we have \[ m^\cC_{aa'} \circ f_V=f_V \circ m_{aa'} = f_V \circ m_a \circ m_{a'} \] and \[ m^\cC_{a} \circ m^\cC_{a'} \circ f_V = m^\cC_{a} \circ f_V \circ m_{a'} = f_V \circ m_{a} \circ m_{a'}, \] from which we conclude $m^\cC_{aa'}=m^\cC_{a} \circ m^\cC_{a'}$. \end{proof} \subsection{Constructions} \label{subsec:constr} In what follows, we will treat finite cardinality left $E_{\bar D}$-modules as objects of ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ via the map $\mathbb{Z}_p[G] \rightarrow E_{{\bar D}}$. By Proposition \ref{prop:C-H alg is module-finite}, $E_{\bar D}$ is finite as a $R_{\bar D}$-module. In particular, for any $i \ge 1$, the $\mathbb{Z}_p[G]$-module $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$ has finite cardinality, and thus is an object of ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$. \begin{lem} \label{lem:max-cC-quot-E_Db} For any $i\ge 1$, there is a unique $E_{\bar D}$-module quotient $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \onto E^\cC_{\bar D}(i)$ such that $E^\cC_{\bar D}(i)$ has $\cC$ and such that any $E_{\bar D}$-module quotient $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \onto W$ where $W$ has $\cC$ factors uniquely through $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \onto E^\cC_{\bar D}(i)$. \end{lem} \begin{proof} By Theorem \ref{thm:univ_CH}, $R_{\bar D}[G] \rightarrow E_{\bar D}$ is surjective, so the lattice of $E_{\bar D}$-quotients and $R_{\bar D}[G]$-quotients of $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$ are identical. By Lemma \ref{lem:Cstart}, we can take $E^\cC_{\bar D}(i) = (E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D})^\cC$. \end{proof} \begin{lem} \label{lem:E(i)-compat} For any $i \geq 1$, there is a canonical isomorphism $E^\cC_{\bar D}(i+1) \otimes_{R_{\bar D}} R_{\bar D}/\mathfrak m_{\bar D}^i \buildrel\sim\over\ra E^\cC_{\bar D}(i)$. \end{lem} \begin{proof} Let $E'=E^\cC_{\bar D}(i+1) \otimes_{R_{\bar D}} R_{\bar D}/\mathfrak m_{\bar D}^i$. We show that $E'$ satisfies the universal property of Lemma \ref{lem:max-cC-quot-E_Db}. The composite \[ E_{\bar D}/\mathfrak m_{\bar D}^{i+i} E_{\bar D} \onto E^\cC_{\bar D}(i+1) \onto E' \] factors through $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$, so $E'$ is a quotient of $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$. Since $E^\cC_{\bar D}(i+1)$ has $\cC$ and $E'$ is a quotient of it, we see that $E'$ has $\cC$. Now suppose that $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \onto W$ where $W$ has $\cC$. By the universal property of $E^\cC_{{\bar D}}(i+1)$, the composite \[ E_{\bar D}/\mathfrak m_{\bar D}^{i+1} E_{\bar D} \onto E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \onto W \] factors uniquely through a map $E^\cC_{{\bar D}}(i+1) \onto W$. Since $W$ is a $R/\mathfrak m_{\bar D}^i$-module, this factors uniquely through $E' \onto W$. \end{proof} \begin{lem} \label{lem:EP} For any $i \ge 1$, the module quotient $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \onto E^\cC_{\bar D}(i)$ has the following properties. \begin{enumerate}[leftmargin=2em] \item Let $N$ be a left $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$-module that has finite cardinality. Then $N$ satisfies condition $\cC$ as a $\mathbb{Z}_p[G]$-module if and only if every map of left $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$-modules $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \rightarrow N$ factors through $E^\cC_{\bar D}(i)$. \item There is a natural right action of $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$ on $E^\cC_{\bar D}(i)$, making $E^\cC_{\bar D}(i)$ a quotient $R_{\bar D}$-algebra of $E_{\bar D}$. \item An $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$-module $N$ that has finite cardinality satisfies condition $\cC$ if and only if its $E_{\bar D}$-action factors through $E^\cC_{\bar D}(i)$. \end{enumerate} \end{lem} \begin{proof} (1)\ If $N$ satisfies $\cC$, then so does the image of any $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \rightarrow N$, so this arrow factors through $E^\cC_{\bar D}(i)$ by Lemma \ref{lem:max-cC-quot-E_Db}. Conversely, if every such arrow factors through $E^\cC_{\bar D}(i)$, then because of the finiteness assumption on $N$ there exists some $m \in \mathbb{Z}_{\geq 1}$ and a surjective map $(E^\cC_{\bar D}(i))^{\oplus m} \twoheadrightarrow N$. Consequently $N$ satisfies $\cC$. (2)\ Choose $z \in E_{\bar D}$ and consider the composite morphism of left $E_{\bar D}$-modules (they are also in ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$) \[ E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \buildrel{(\ ) \cdot z}\over\longrightarrow E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D} \longrightarrow E^\cC_{\bar D}(i) \] where the leftmost arrow is right multiplication by $z$. The composite must factor through $E^\cC_{\bar D}(i)$ by (1). The resulting map $E^\cC_{\bar D}(i) \rightarrow E^\cC_{\bar D}(i)$ gives the desired right action of $z$ on $E^\cC_{\bar D}(i)$ and shows that $E^\cC_{\bar D}(i)$ is an $R_{\bar D}$-algebra. (3)\ This follows directly from (1) and (2) in light of the following general fact: for an algebra $E$ and a left $E$-module $M$, the $E$-action on $M$ factors through a quotient algebra $E \twoheadrightarrow Q$ if and only if every morphism of left $E$-modules $E \rightarrow M$ factors through $Q$. This follows from the fact that any such $E \rightarrow M$ is of the form $x \mapsto x \cdot m$ for some $m \in M$. \end{proof} By Lemmas \ref{lem:E(i)-compat} and \ref{lem:EP}(2), we have an inverse system $\{E_{\bar D}^\cC(i)\}$ of $R_{\bar D}$-algebra quotients of $E_{\bar D}$. In particular, we have the $R_{\bar D}$-algebra quotient \begin{equation} \label{eq:KP} E_{\bar D} \twoheadrightarrow \varprojlim_i E^\cC_{\bar D}(i). \end{equation} This quotient almost gives the algebra we are looking for. However, we need to produce, from this algebra quotient, a Cayley-Hamilton algebra. The following lemma gives the general procedure for doing this. A special case of this lemma was employed in \cite{WWE1,WWE2}. \begin{lem} \label{lem:max_CH_quot} Let $(E, D : E \rightarrow R)$ be a Cayley-Hamilton $R$-algebra. Let $I \subset E$ be a two-sided ideal. Let $J \subset R$ be the ideal generated by the non-constant coefficients of the polynomials in the set \[ \{D_{R[t]}(1-xt) \ \| \ x \in I\} \subset R[t]. \] Let $E' := E/(I,J)$ and $R' := R/J$, and let $f:R \to R'$ and $g:E \to E'$ be the quotient maps. \begin{enumerate} \item There exists a unique Cayley-Hamilton pseudorepresentation $D' : E' \rightarrow R'$ such that the pair $(f,g)$ gives a morphism $(E,D) \to (E',D')$ of Cayley-Hamilton algebras. \item For any Cayley-Hamilton representation $\rho: (E,D) \rightarrow (E_A, D_A : E_A \rightarrow A)$, the map $E \to E_A$ sends $I$ to $0$ if and only if $\rho$ factors through $(E,D) \rightarrow (E',D')$. \end{enumerate} \end{lem} \begin{proof} (1)\ The uniqueness follows from (2). To show existence, we start with the pseudorepresentation $\tilde{D}:=(f \circ D): E\otimes_R R' \to R'$. To construct $D'$, we use Chenevier's notion of \emph{kernel of a pseudorepresentation} \cite[\S 1.17]{chen2014}. It is defined by the following universal property: there is an $R'$-algebra quotient $h:E \otimes_R R' \to (E \otimes_R R')/\ker(\tilde{D})$ and a pseudorepresentation $\tilde{D}': (E \otimes_R R')/\ker(\tilde{D}) \to R'$ such that $\tilde{D}=\tilde{D}' \circ h$, and $\ker(\tilde{D})$ is the maximal ideal with this property. There is an equality \begin{equation} \ker(\tilde{D}) = \{x \in E\otimes_R R' \ \| \ \tilde{D}_{R'[t]}(1-xyt)=1 \ \forall y \in E \otimes_R R'\}. \end{equation} This is proven in \cite[Lem.\ 2.1.2]{CS2016}\footnote{This lemma was removed from the final version \cite{CS2019}, having become unnecessary.}. Their proof uses Amitsur's formula \cite{amitsur1980} for pseudorepresentations \cite[(1.5)]{chen2014}, together with the fact that, for any $x,y \in E$, $D_{R[t]}(1-xyt)=D_{R[t]}(1-yxt)$, which can be proven in the same way as for usual determinants in linear algebra. By the definition of $J$, we have $\tilde{D}_{R'[t]}(1-(x\otimes 1)t)=1$ for all $x \in I$, and hence $\tilde{D}_{R'[t]}(1-(x \otimes 1)yt)=1$ for all $x \in I$ and $y \in E\otimes_R R'$, since $I$ is a two-sided ideal. This implies that $I\otimes_R R' \subset \ker(\tilde{D})$. By the property of $\ker(\tilde{D})$, we have $D':E' \to R'$ such that $f \circ D = D' \circ g$. (2)\ If $\rho$ factors through $(E,D) \rightarrow (E',D')$, then the map $E \to E_A$ factors through $E \to E' \to E_A$, so $E \to E_A$ sends $I$ to $0$. Conversely, let $\rho:(E,D) \rightarrow (E_A,D_A)$ be given by $f_A:R \to A$ and $g_A:E \to E_A$, and assume that $g_A(I)=0$. To show that $\rho$ factors through $(E,D) \rightarrow (E',D')$, it suffices to show that $f_A(J)=0$. For any $x \in E$, the naturality of $D$ implies that the image of $D_{R[t]}(1-xt)$ in $A[t]$ under $f_A$ is given by $D_{A[t]}(1-xt) = (f_A \circ D)_{A[t]}(1-xt)$. Hence it is enough to show that, for $x \in I$, we have $(f_A \circ D)_{A[t]}(1-xt)=1$. However, since $\rho$ is a morphism of Cayley-Hamilton algebras, we have \[ (f_A \circ D)_{A[t]}(1-xt) = (D_A \circ g_A)_{A[t]}(1-xt) = (D_A)_{A[t]}(g_A(1-xt)) = (D_A)_{A[t]}(1)=1, \] where we use the fact that $g_A(x)=0$. \end{proof} \begin{eg} \label{eg:CH-quot-with-scalar-ideal} Let $(E,D)$ be a Cayley-Hamilton $R$-algebra and let $I=JE$ with $J \subset R$ being an ideal. Then the Cayley-Hamilton quotient $(E',D')$ of $(E,D)$ by $I$ has $E'=E/JE$ and scalar ring $R'=R/J$. In other words, $(E', D') = (E/JE, D \otimes_R R/J : E/JE \rightarrow R/J)$. \end{eg} \begin{defn} \label{defn:ERP} With the notation of the lemma, we call $(E',D')$ the \emph{Cayley-Hamilton quotient of $(E,D)$ by $I$}. \end{defn} \begin{defn} \label{defn:E^C_Db} Let $K^\cC \subset E_{\bar D}$ be the kernel of the algebra homomorphism \eqref{eq:KP}. Let $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}})$ denote the Cayley-Hamilton quotient of $(E_{\bar D}, D^u_{E_{\bar D}})$ by $K^\cC$, and let $R_{\bar D}^\cC$ denote the scalar ring of $E^\cC_{\bar D}$. \end{defn} \subsection{Extending condition $\cC$ to pseudorepresentations and Cayley-Hamilton representations} \label{subsec:CH-P} We extend $\cC$ to $A[G]$-modules that may not have finite cardinality in the following way. \begin{defn} \label{defn:proP} Let $\cC$ be a stable condition on objects of ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$. Let $(A,\mathfrak{m}_A)$ be a complete commutative Noetherian local $\mathbb{Z}_p$-algebra. For an $A[G]$-module $M$ that is finitely generated as an $A$-module, we say that $M$ \emph{satisfies condition $\cC$} when $M/\mathfrak{m}_A^i M$ satisfies $\cC$ for all $i \geq 1$. \end{defn} Note that, for $(A,\mathfrak{m}_A)$ and $M$ as in the definition, the canonical map $M \to \varprojlim_i M/\mathfrak{m}_A^i M$ is an isomorphism. Using this, one can check that this extension of $\cC$ is stable in the same sense as Definition \ref{defn:stable}. We will use this extension of condition $\cC$ without further comment. Now we give the definition of condition $\cC$ for Cayley-Hamilton representations. \begin{defn} \label{defn:PforCH} Let $(A,\mathfrak{m}_A)$ be a complete commutative Noetherian local $\mathbb{Z}_p$-algebra and let $(E, \rho,D)$ be a Cayley-Hamilton representation of $G$ with scalar ring $A$ and residual pseudorepresentation ${\bar D}$. We say that $(E, \rho, D)$ \emph{satisfies condition $\cC$} if $E$ satisfies condition $\cC$ as an $A[G]$-module. (Note that $E$ is finitely generated as an $A$-module by Proposition \ref{prop:C-H alg is module-finite}.) We let ${\mathcal{CH}}_{G,{\bar D}}^\cC$ denote the full subcategory of ${\mathcal{CH}}_{G,{\bar D}}$ whose objects satisfy condition $\cC$. \end{defn} We can be confident that this notion behaves well by finding a universal object. \begin{thm} \label{thm:univ_CH_P} The Cayley-Hamilton representation $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}} : E^\cC_{\bar D} \rightarrow R^\cC_{\bar D})$ of Definition \ref{defn:E^C_Db} is the universal object in ${\mathcal{CH}}_{G,{\bar D}}^\cC$. \end{thm} \begin{proof} By the definition of $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}} : E^\cC_{\bar D} \rightarrow R^\cC_{\bar D})$, we see that the map $E_{\bar D} \to E_{\bar D}^\cC$ sends $K^\cC$ to $0$. By Lemma \ref{lem:EP}, we see that $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}} : E^\cC_{\bar D} \rightarrow R^\cC_{\bar D})$ satisfies condition $\cC$. We now show that it has the universal property. Let $A$ be a complete commutative Noetherian local $\mathbb{Z}_p$-algebra, and let $(E, \rho, D)$ be a Cayley-Hamilton representation with scalar ring $A$ and residual pseudorepresentation ${\bar D}$. We have to show that $(E, \rho, D)$ satisfies condition $\cC$ if and only if the map of Cayley-Hamilton algebras $(E_{\bar D}, D^u_{E_{\bar D}}) \rightarrow (E, D)$ induced by the universal property of $(E_{\bar D}, \rho^u, D^u_{E_{\bar D}})$ factors through $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}})$. For any $i \geq 1$, $E_{\bar D} \rightarrow E/\mathfrak{m}_A^i E$ factors through $E_{\bar D}/\mathfrak m_{\bar D}^i E_{\bar D}$. (Recall that a local homomorphism of scalar rings $R_{\bar D} \rightarrow A$ is implicit in $(E_{\bar D}, D^u_{E_{\bar D}}) \rightarrow (E, D)$.) By Lemma \ref{lem:EP}, $(E, \rho, D)$ satisfies $\cC$ if and only if $E_{\bar D}/\mathfrak m_{\bar D}^{i} E_{\bar D} \rightarrow E/\mathfrak{m}_A^i E$ factors through $E^\cC_{\bar D}(i)$ for every $i \geq 1$. Equivalently, $(E, \rho, D)$ satisfies $\cC$ if and only if $E_{\bar D} \to E$ maps $K^\cC$ to $0$. By Lemma \ref{lem:max_CH_quot}, $K^\cC$ maps to $0$ in $E_{\bar D} \rightarrow E$ if and only if $(E_{\bar D}, D^u_{E_{\bar D}}) \rightarrow (E, D)$ factors through $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}})$. \end{proof} Following the pattern of \cite[Defn.\ 5.9.1]{WWE1}, we define condition $\cC$ on pseudorepresentations. \begin{defn} \label{defn:PsR-P} Let $A$ be an complete commutative Noetherian local $\mathbb{Z}_p$-algebra. Let $D : G \rightarrow A$ be a pseudorepresentation with residual pseudorepresentation ${\bar D}$. Then $D$ \emph{satisfies condition $\cC$} provided that there exists a Cayley-Hamilton representation $(E, \rho, D')$ satisfying condition $\cC$ such that $D = \psi(\rho) := D' \circ \rho$. \end{defn} We define the $\cC$-pseudodeformation functor ${\mathrm{PsDef}}^\cC_{\bar D}: \hat\cC_{W(\mathbb{F})} \to \mathrm{Sets}$ by \[ {\mathrm{PsDef}}^\cC_{\bar D}(A)= \{\text{pseudodeformations } D : G \rightarrow A \text{ of }{\bar D} \text{ satisfying } \cC\}. \] \begin{thm} \label{thm:RDP} The functor ${\mathrm{PsDef}}^\cC_{\bar D}$ is represented by $R^\cC_{\bar D}$. \end{thm} \begin{proof} Let $A \in \hat\cC_{W(\mathbb{F})}$, and let $D \in {\mathrm{PsDef}}_{\bar D}(A)$. By Theorem \ref{thm:RDb}, there is unique $R_{\bar D} \rightarrow A$ such that $D \cong D^u_{\bar D} \otimes_{R_{\bar D}} A$. We have to show that $D \in {\mathrm{PsDef}}^\cC_{\bar D}(A)$ if and only if $R_{\bar D} \rightarrow A$ factors through $R_{\bar D} \onto R_{\bar D}^\cC$. If $R_{\bar D} \rightarrow A$ factors through $R_{\bar D} \twoheadrightarrow R^\cC_{\bar D}$, then the Cayley-Hamilton representation \[ (E^\cC_{\bar D} \otimes_{R^\cC_{\bar D}} A, \rho^\cC \otimes_{R_{\bar D}} A : G \rightarrow (E^\cC_{\bar D} \otimes_{R^\cC_{\bar D}} A)^\times, D_{E^\cC_{\bar D}} \otimes_{R^\cC_{\bar D}} A) \] induces $D$ via $D = (R^\cC_{\bar D} \rightarrow A) \circ D_{E^\cC_{\bar D}}$ and satisfies condition $\cC$ by Theorem \ref{thm:univ_CH_P}. Consequently $D$ satisfies $\cC$. Now assume $D$ satisfies condition $\cC$, i.e.\ there exists a Cayley-Hamilton representation $(E, \rho, D')$ satisfying $\cC$ such that $D = D' \circ \rho$. By Theorem \ref{thm:univ_CH_P}, there exists a morphism of Cayley-Hamilton algebras $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}}) \rightarrow (E, D)$ inducing $\rho$. In particular, the implicit morphism of scalar rings $R^\cC_{\bar D} \rightarrow A$ factors $R_{\bar D} \rightarrow A$. \end{proof} \subsection{Modules with Cayley-Hamilton structure} \label{subsec:ModCH-P} We introduce the notion of Cayley-Hamilton $G$-module. \begin{defn} Let $A \in \hat \cC_{W(\mathbb{F})}$. A \emph{Cayley-Hamilton $G$-module} over $A$ is the data of a Cayley-Hamilton representation $(E,\rho,D)$ of $G$ with scalar ring $A$, and an $E$-module $N$. We consider $N$ as a $A[G]$-module via the map $\rho: A[G] \to E$. We often refer to a to a Cayley-Hamilton $G$-module simply by the letter $N$, and call $(E,D)$ the \emph{Cayley-Hamilton algebra of $N$}. We say $N$ is \emph{faithful} if it is faithful as $E$-module. \end{defn} \begin{eg} If $N$ is an $A[G]$-module and there is a Cayley-Hamilton pseudorepresentation $D: \mathrm{End}_A(N) \to A$, then the canonical action of $\mathrm{End}_A(N)$ on $N$ makes $N$ a faithful Cayley-Hamilton $G$-module with Cayley-Hamilton algebra $(\mathrm{End}_A(N),\rho,D)$, where $\rho:G \to \mathrm{End}_A(N)$ is the action map. \end{eg} \begin{eg} \label{eg:Tate module} As a special case of the previous example, suppose $N=N_1 \oplus N_2$ as $A$-modules such that $\mathrm{End}_A(N_i)=A$ (i.e.~the only endomorphisms are scalars). Note that $N_1$ and $N_2$ need not be free $A$-modules (for example, they could be dualizing $A$-modules with $A$ non-Gorenstein; or, if $A=\mathbb{Z}_p{[\![} T{]\!]}$, they could be the maximal ideal). Then $\mathrm{End}_A(N)$ has the structure of $A$-GMA $(B,C,m)$ as in Example \ref{eg:GMA1,1}, where $B=\mathrm{Hom}_A(N_1,N_2)$ and $C=\mathrm{Hom}_A(N_2,N_1)$, and $m(f,g)=f\circ g$ (see \S\ref{sec:GMAs}, below, for a discussion of GMAs). In particular, there is a natural Cayley-Hamilton pseudorepresentation $D: \mathrm{End}_A(N) \to A$. This example appears frequently in the study of pseudorepresentations associated to ordinary modular forms, where one takes $A$ to be a Hecke algebra, and $N$ to be the $p$-adic Tate module of a modular Jacobian, in which case $N$ is a direct sum of a free $A$-module of rank $1$ and a dualizing $A$-module (c.f.~\cite{mazur1978,WWE1,WWE2,WWE3}). \end{eg} \begin{thm} \label{thm:ModToCH} Let $N$ be a faithful Cayley-Hamilton $G$-module with Cayley-Hamilton $A$-algebra $(E,D)$. Then $N$ satisfies condition $\cC$ as an $A[G]$-module if and only if $(E,\rho,D)$ satisfies condition $\cC$ as a Cayley-Hamilton representation. \end{thm} \begin{proof} By Definition \ref{defn:proP}, it suffices to prove the theorem in the case that $A$ is Artinian and local. Choose $i \geq 1$ such that $\mathfrak m_{\bar D}^i \cdot A = 0$. Let ${\bar D}:G \to \mathbb{F}$ be the residual pseudorepresentation of $D \circ \rho: G \rightarrow A$. By Theorem \ref{thm:univ_CH}, there is a distinguished morphism of Cayley-Hamilton algebras $(E_{\bar D}, D^u_{E_{\bar D}}) \rightarrow (E,D)$. Then the action map $\mathbb{Z}_p[G] \to \mathrm{End}_A(N)$ factors as \begin{equation} \label{eq:factoring-G-action} \mathbb{Z}_p[G] \to E_{\bar D} \to E_{\bar D}/\mathfrak{m}_{\bar D}^iE_{\bar D} \to E \to \mathrm{End}_A(N). \end{equation} We claim that the following are equivalent: \begin{enumerate} \item The $\mathbb{Z}_p[G]$-module $N$ satisfies $\cC$ \item The map $E_{\bar D}/\mathfrak{m}_{\bar D}^iE_{\bar D} \to \mathrm{End}_A(N)$ factors through $E_{\bar D}^\cC(i)$ \item The map $E_{\bar D} \to \mathrm{End}_A(N)$ sends $K^\cC$ to $0$ \item The map $E_{\bar D} \to E$ sends $K^\cC$ to $0$ \item The morphism of Cayley-Hamilton algebras $(E_{\bar D}, D^u_{E_{\bar D}}) \rightarrow (E,D)$ factors through $(E_{\bar D}^\cC,D_{E^\cC_{\bar D}})$ \item The Cayley-Hamilton representation $(E,D)$ satisfies condition $\cC$. \end{enumerate} (Recall that $K^\cC$ was defined in Definition \ref{defn:E^C_Db}.) The equivalences are proven as follows: \begin{itemize}[leftmargin=1.75em] \item[]$(1) \Longleftrightarrow (2)$: Lemma \ref{lem:EP}(3). \item[]$(2) \Longleftrightarrow (3)$: From the definition of $K^\cC$ and \eqref{eq:factoring-G-action}. \item[]$(3) \Longleftrightarrow (4)$: Since $N$ is faithful, the map $E \to \mathrm{End}_A(N)$ is injective. \item[]$(4) \Longleftrightarrow (5)$: Lemma \ref{lem:max_CH_quot}. \item[]$(5) \Longleftrightarrow (6)$: Theorem \ref{thm:univ_CH_P}. \qedhere \end{itemize} \end{proof} \subsection{Formal moduli of representations with $\cC$} \label{subsec:formal moduli with C} We generalize Ramakrishna's result (Theorem \ref{thm:Ram}) to any family of integral $p$-adic representations of $G$. \begin{thm} \label{cor:P-closed} There exists a unique closed formal substack (resp.\ closed formal subscheme) \[ {\mathrm{Rep}}^{\cC,d}_G \subset {\mathrm{Rep}}_G^d , \qquad (\text{resp.\ }{\mathrm{Rep}}^{\square,\cC,d}_G \subset {\mathrm{Rep}}^{\square,d}_G) \] characterized by the following property. For any commutative local $\mathbb{Z}_p$-algebra $B$ of finite cardinality and free rank $d$ $B$-module $V_B$ with a $B$-linear $G$-action (resp.\ and fixed basis), the corresponding $B$-point of ${\mathrm{Rep}}_G^d$ lies in ${\mathrm{Rep}}^{\cC,d}_G$ (resp.\ of ${\mathrm{Rep}}_G^{\square,d}$ lies in ${\mathrm{Rep}}^{\square,\cC,d}_G$) if and only if $V_B$ has $\cC$. \end{thm} \begin{proof} It suffices to consider the case of ${\mathrm{Rep}}^{\square,d}_G$. Indeed, since condition $\cC$ does not depend on the choice of basis, the closed subscheme ${\mathrm{Rep}}^{\square,\cC,d}_G \subset {\mathrm{Rep}}^{\square,d}_G$ descends to a closed locus in ${\mathrm{Rep}}_G^d$. By Theorem \ref{thm:RDb}, we may consider a fixed residual pseudorepresentation ${\bar D}$ and produce ${\mathrm{Rep}}^{\square,\cC}_{\bar D} \subset {\mathrm{Rep}}^\square_{\bar D}$. We define ${\mathrm{Rep}}^{\square,\cC}_{\bar D}$ via the pullback diagram \[\xymatrix{ {\mathrm{Rep}}^{\square,\cC}_{\bar D} \ar[r] \ar[d] & {\mathrm{Rep}}^\square_{\bar D} \ar[d]^-\wr \\ {\mathrm{Rep}}^\square_{E_{\bar D}^\cC,D_{E^\cC_{\bar D}}} \ar[r] & {\mathrm{Rep}}^\square_{E_{\bar D},D^u_{E_{\bar D}}} }\] where the right vertical arrow is the isomorphism in Theorem \ref{thm:ED-compat}. Let $B$ be an local $\mathbb{Z}_p$-algebra of finite cardinality. By definition, a point $V_B \in {\mathrm{Rep}}^\square_{\bar D}(B)$ lies in ${\mathrm{Rep}}^{\square,\cC}_{\bar D}(B)$ if and only if the map $(E_{\bar D},D^u_{E_{\bar D}}) \to (\mathrm{End}_B(V_B),\det)$ factors through $(E_{\bar D}^\cC,D_{E^\cC_{\bar D}})$. By Theorem \ref{thm:univ_CH_P}, this occurs if and only if the Cayley-Hamilton representation $(\mathrm{End}_B(V_B),\det)$ has $\cC$, which, by Theorem \ref{thm:ModToCH}, is if and only if $V_B$ has $\cC$. By Lemma \ref{lem:tfg algebra}, this characterizes ${\mathrm{Rep}}^{\square,\cC}_{\bar D}$. \end{proof} \begin{cor} \label{thm:univ_EP} Let $A\in {\mathrm{Tfg}}_{\mathbb{Z}_p}$ and $(V_A,\rho_A) \in {\mathrm{Rep}}^d_G(A)$. There exists a unique quotient morphism $A \twoheadrightarrow A^\cC$ in ${\mathrm{Tfg}}_{\mathbb{Z}_p}$ such that, for any local $A$-algebra $B$ of finite cardinality, the object $V_A \otimes_A B$ of ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ satisfies $\cC$ if and only if the homomorphism $A \rightarrow B$ factors through $A^\cC$. \end{cor} \begin{proof} The quotient $A \to A^\cC$ is defined by the pullback square \[ \xymatrix{ \Spec(A^\cC) \ar[r] \ar[d] & {\mathrm{Rep}}^{\cC,d}_G \ar[d] \\ \Spec(A) \ar[r] & {\mathrm{Rep}}^{d}_G. } \] The characterizing property follows from the characterizing property of ${\mathrm{Rep}}^{\cC,d}_G$. \end{proof} \section{Deformation conditions for generalized matrix algebras} \label{sec:GMAs} For the remainder of the paper, we assume that the pseudorepresentation ${\bar D}$ is multiplicity-free (see Definition \ref{defn:RMF}). Under this assumption, as was first noticed by Chenevier \cite{chen2014}, the universal Cayley-Hamilton algebra with residual pseudorepresentation ${\bar D}$ admits the additional structure of a generalized matrix algebra (GMA). Generalized matrix algebras, which were first introduced by Bella\"iche and Chenevier \cite{BC2009}, are a particularly concrete and explicit type of Cayley-Hamilton algebra. In this section, we recall in \S \ref{subsec:GMAs} the theory of generalized matrix algebras and in \S \ref{subsec:RMF} why the universal Cayley-Hamilton algebra admits this structure. In \S\ref{subsec:RMF-P}, we exploit this extra structure to prove that the universal Cayley-Hamilton algebra with property $\cC$, constructed in \S \ref{sec:prop}, is characterized by representations. This is the main new result of the section. Explicitly, in the multiplicity-free case, we show that, if every representation with $\cC$ factors through a quotient of universal Cayley-Hamilton algebra, then that quotient must be universal Cayley-Hamilton algebra with property $\cC$. The results of \S \ref{subsec:GMAs} and \S \ref{subsec:RMF} will also be used in \S\ref{sec:gma_ext}, where we will exploit the GMA structure to compute the Cayley-Hamilton algebra with property $\cC$ in terms of group cohomology. \subsection{Generalized matrix algebras and their adapted representations} \label{subsec:GMAs} A generalized matrix algebra is a particular kind of Cayley-Hamilton algebra with extra data. We learned this notion from Bella\"iche--Chenevier \cite{BC2009}. \begin{defn}[{\cite[\S1.3]{BC2009}}] \label{defn:GMA} Let $R$ be a ring. A \emph{generalized matrix algebra over $R$} (or an \emph{$R$-GMA}) is the data of: \begin{enumerate} \item An $R$-algebra $E$ that is finitely generated as an $R$-module, \item A set of orthogonal idempotents $e_1, \dotsc, e_r \in E$ such that $\sum_i e_i =1$, and \item A set of isomorphisms of $R$-algebras $\phi_i : e_i E e_i \buildrel\sim\over\ra M_{d_i}(R)$ for $i=1,\dots,r$. \end{enumerate} We call $\mathcal{E} = (\{e_i\}, \{\phi_i\})$ the \emph{data of idempotents} or \emph{GMA structure} of $E$, and write the $R$-GMA as $(E,\mathcal{E})$. We call the list of integers $(d_1, \dots, d_r)$ the \emph{type} of $(E,\mathcal{E})$. These data are required to satisfy the condition that the function $\Tr_\mathcal{E}:E \rightarrow A$ defined by \[ \Tr_\mathcal{E}(x) := \sum_{i=1}^r {\mathrm{tr}} (\phi_i(e_ixe_i)) \] is a central function, i.e.\ $\Tr_\mathcal{E}(xy) = \Tr_\mathcal{E}(yx)$ for all $x,y \in E$. Given an $R$-GMA $(E,\mathcal{E})$ and an $R'$-GMA $(E', \mathcal{E}')$, a \emph{morphism of GMAs} $\rho:(E,\mathcal{E}) \rightarrow (E', \mathcal{E}')$ is the data of a ring homomorphism $f:R \to R'$ and morphism of $R$-algebras $\rho: E \rightarrow E'$ such that $\mathcal{E}$ and $\mathcal{E}'$ are of the same type $(d_1,\dots,d_r)$, we have $g(e_i)=e_i'$ and $f \circ \phi_i = \phi_i'\circ g$ for $i=1,\dots,r$. \end{defn} \begin{eg} \label{eg:matrix} The matrix algebra $M_d(R)$ comes with a natural $R$-GMA structure $\mathcal{E} = (1, \mathrm{id}: M_d(R) \buildrel\sim\over\ra M_d(R))$ of type $(d)$. More generally, given any ordered partition of $d = d_1 + \dotsc + d_r$ of $d$, there is a natural $R$-GMA structure $\mathcal{E}_{\mathrm{block}}$ on $M_d(R)$ of type $(d_1, \dotsc, d_r)$. Namely, the natural $R$-algebra map with block-diagonal image \[ \nu_1 \times \dotsm \times \nu_r : M_{d_1}(R) \times \dotsm \times M_{d_r}(R) \hookrightarrow M_d(R) \] induces $\mathcal{E}_{\mathrm{block}} = (e_i = \nu_i(1_i), \phi_i)$, where $1_i \in M_{d_i}(R)$ is the identity matrix, and $\phi_i$ is the inverse to $\nu_i$ on its image $e_iM_d(R)e_i$. \end{eg} \begin{lem} \label{lem:DE} Given an $R$-GMA $(E,\mathcal{E})$, there is a canonical Cayley-Hamilton pseudorepresentation $D_\mathcal{E}: E \rightarrow R$, such that $\Tr_{D_\mathcal{E}}=\Tr_\mathcal{E}$. A morphism of $R$-GMAs $(E,\mathcal{E}) \to (E',\mathcal{E}')$ induces a morphism of Cayley-Hamilton algebras $(E,D_\mathcal{E}) \to (E',D_{\mathcal{E}'})$. \end{lem} \begin{proof} See \cite[Prop.\ 2.23]{WE2018}; its statement includes the first claim. The second claim follows from examining the formula for $D_\mathcal{E}$ given in \emph{loc.\ cit.}, noting that a morphism of GMAs preserves the idempotents that are used to specify $D_\mathcal{E}$. \end{proof} This lemma gives a faithful embedding of the category of $R$-GMAs into the category of $R$-Cayley-Hamilton algebras. We will consider this embedding as an inclusion. We extend the definition of Cayley-Hamilton representation to GMAs as follows. \begin{defn} If $(E',\mathcal{E}')$ is a GMA, we refer to a morphism of pseudorepresentations $(E,D) \to (E',D_{\mathcal{E}'})$ as a \emph{GMA representation} of $(E,D)$. If $(E,\mathcal{E})$ is another GMA, we call a GMA representation $(E,D_\mathcal{E}) \to (E',D_{\mathcal{E}'})$ \emph{adapted} if the same data give a morphism of GMAs $(E, \mathcal{E}) \to (E',\mathcal{E}')$. \end{defn} \begin{lem} \label{lem:gma data} Given an $R$-GMA $(E,\mathcal{E})$ of type $(d_1,\dots,d_r)$, we can associate to it the data of \begin{enumerate} \item $R$-modules $\cA_{i,j}$ for $1 \le i,j \le r$, \item canonical isomorphisms $\cA_{i,i} \isoto R$ for $1 \le i \le r$, and \item $R$-module homomorphisms $\varphi_{i,j,k}:\cA_{i,j} \otimes_R \cA_{j,k} \rightarrow \cA_{i,k}$ for $1 \leq i,j,k \leq r$, \end{enumerate} such that $(\cA_{i,j},\varphi_{i,j,k})$ completely determine $(E,\mathcal{E})$ and there is an isomorphism of $R$-modules \begin{equation} \label{eq:GMA_form} E \buildrel\sim\over\lra \begin{pmatrix} M_{d_1}(\cA_{1,1}) & M_{d_1 \times d_2}(\cA_{1,2}) & \dotsm & M_{d_1 \times d_r}(\cA_{1,r}) \\ M_{d_2 \times d_1}(\cA_{2,1}) & M_{d_2}(\cA_{2,2}) & \dotsm & M_{d_2 \times d_r}(\cA_{2,r}) \\ \vdots & \vdots & \vdots & \vdots \\ M_{d_r \times d_1}(\cA_{r,1}) & M_{d_2}(\cA_{r,2}) & \dotsm & M_{d_r}(\cA_{r,r}) \\ \end{pmatrix}, \end{equation} Moreover, the collection of maps $\varphi_{i,j,k}$ satisfies properties (UNIT), (COM), and (ASSO) given in \cite[\S1.3.2]{BC2009}, and there is a bijection between $R$-GMAs of type $(d_1,\dots,d_r)$ and data $(\cA_{i,j},\varphi_{i,j,k})$ satisfying (UNIT), (COM), and (ASSO). \end{lem} \begin{proof} This is explained in \cite[\S1.3.1-\S1.3.6]{BC2009}. The association is given as follows. Let $E_i:= \phi_i^{-1}(\delta^{1,1})$, where $\delta^{1,1}$ denotes the elementary matrix with 1 as $(1,1)$th entry, and 0 otherwise. Define $\cA_{i,j} := E_j E E_i$. The maps $\varphi_{i,j,k}$ are induced by the multiplication in $E$. In particular, note that $\phi_i$ induces a canonical isomorphism $\cA_{i,i} \isoto R$. \end{proof} We will not spell out the bijection or the properties (UNIT), (COM), and (ASSO) in general. Instead, we explain the content of the lemma in the case of type $(1,1)$. \begin{eg} \label{eg:GMA1,1} There is a bijection between $R$-GMAs $(E,\mathcal{E})$ of type $(1,1)$ and triples $(B,C, m)$ where $B,C$ are finitely generated $R$-modules and $m : B \otimes_R C \rightarrow R$ is an $R$-module homomorphism, such that the squares \[ \xymatrix{ B \otimes_R C \otimes_R B \ar[r]^-{1 \otimes (m \circ \iota)} \ar[d]^-{m \otimes 1} & B \otimes_R R \ar[d] & & C \otimes_R B \otimes_R C \ar[r]^-{1 \otimes m} \ar[d]^-{(m \circ \iota) \otimes 1} & C \otimes_R R \ar[d] \\ R \otimes_R B \ar[r] & B & & R \otimes_R C \ar[r] & C } \] commute. Here $\iota: C \otimes_R B \isoto B \otimes_R C$ is the isomorphism given by $b \otimes c \mapsto c \otimes b$, and the unlabeled maps are the $R$-action maps. The $R$-GMA associated to a triple $(B,C,m)$ is \begin{equation} \label{eq:E-form} E =\begin{pmatrix} R & B \\ C & R \end{pmatrix}. \end{equation} This means that $E=R \oplus B \oplus C \oplus R$ as an $R$-module, and the multiplication on $E$ is given by $2\times2$-matrix multiplication, but where the action maps $R \otimes_R B \to B$, $R \otimes_R C \to C$ and the map $m$ are used in place of the scalar multiplication. The idempotents $e_1, e_2$ are given in this notation by $e_1=\sm{1}{0}{0}{0}$ and $e_2=\sm{0}{0}{0}{1}$, and the isomorphisms $\phi_i$ are given by the identifications $\sm{R}{0}{0}{0} \isoto R$ and $\sm{0}{0}{0}{R} \isoto R$. In the notation of the lemma, $\cA_{1,2}=B$ and $\cA_{2,1}=C$, and the maps $\varphi_{i,j,k}$ are the action maps, except $\varphi_{1,2,1}=m$ and $\varphi_{2,1,2}= m \circ \iota$. Indeed, the (UNIT) property requires that the maps $\varphi_{i,j,k}$ equal the action maps, unless $(i,j,k) \in \{(1,2,1),(2,1,2)\}$. The (ASSO) property is expressed by the commutative squares above, and the (COM) property is that $\varphi_{2,1,2}= \varphi_{1,2,1} \circ \iota$. \end{eg} \begin{eg} \label{eg:m0} In the foregoing example, setting $m$ to be the zero map is always a valid choice. \end{eg} \begin{defn}[{\cite[Defn.\ 1.3.6]{BC2009}}] \label{defn:adapted} Let $A$ be a commutative $R$-algebra and let $(E,\mathcal{E})$ be an $R$-GMA of type $(d_1, \dotsc, d_r)$. Let $d = \sum_{i=1}^r d_i$ and let $(M_d(A),\mathcal{E}_\mathrm{block})$ be the $A$-GMA of type $(d_1, \dotsc, d_r)$ constructed in Example \ref{eg:matrix}. An \emph{adapted representation of $(E,\mathcal{E})$}, denoted $\rho:(E,\mathcal{E}) \to M_d(A)$ is a morphism of GMAs $\rho:(E,\mathcal{E}) \to (M_d(A),\mathcal{E}_\mathrm{block})$. A pseudorepresentation $D :E \otimes_R A \rightarrow A$ is \emph{adapted to $\mathcal{E}$} if $D=D_\mathcal{E} \otimes_R A$. When the data of idempotents $\mathcal{E}$ is understood, we say $D$ is \emph{adapted} instead of adapted to $\mathcal{E}$. \end{defn} Now fix $(E,\mathcal{E})=(E,(\{e_i\},\{\phi_i\}))$, an $R$-GMA of type $(d_1, \dotsc, d_r)$, and let $(\cA_{i,j},\varphi_{i,j,k})$ be the data associated to it by Lemma \ref{lem:gma data}. We define a the set-valued functor on commutative $R$-algebras by \[ {\mathrm{Rep}}_{(E,\mathcal{E}),{\mathrm{GMA}}}^\square:A \mapsto \{\text{Adapted representations } \rho:(E,\mathcal{E}) \to M_d(A)\}. \] There is an explicit presentation for ${\mathrm{Rep}}^\square_{(E,\mathcal{E}),{\mathrm{GMA}}}$ as an affine $\Spec R$-scheme as follows. \begin{thm} \label{thm:Ad_main} The functor ${\mathrm{Rep}}_{(E,\mathcal{E}),{\mathrm{GMA}}}^\square$ is represented by $\Spec S$, where $S$ is an $R$-algebra quotient \[ \Sym^*_R \left(\bigoplus_{1 \leq i \neq j \leq r} \cA_{i,j} \right) \twoheadrightarrow S \] by the ideal generated by the set of all $\varphi(b \otimes c)-b \otimes c$, with $b \in \cA_{i,j}, c \in \cA_{j,k}$ and $\varphi=\varphi_{i,j,k}$ for all $1 \le i, j, k \leq r$. In particular, $S$ is a finitely generated $R$-algebra. Moreover, for $1 \leq i, j \leq r$, the natural $R$-module maps $\cA_{i,j} \rightarrow S$ are split injections of $R$-modules, inclusive of the case $R \buildrel\sim\over\ra \cA_{i,i} \hookrightarrow S$, which is the $R$-algebra structure map of $S$. The universal adapted representation $\rho_{\mathrm{GMA}}:(E,\mathcal{E}) \rightarrow M_d(S)$ is given by the isomorphism of \eqref{eq:GMA_form} along with these $R$-module injections. In particular, the $R$-algebra homomorphism $E \to M_d(S)$ is injective. \end{thm} \begin{proof} See \cite[Prop.\ 1.3.9]{BC2009} and its proof, as well as \cite[Prop.\ 1.3.13]{BC2009} for the split injectivity. \end{proof} By \cite[Prop.\ 2.23]{WE2018}, any adapted representation of $(E,\mathcal{E})$ is compatible with $D_\mathcal{E}$. This determines a monomorphism ${\mathrm{Rep}}_{(E,\mathcal{E}),{\mathrm{GMA}}}^\square\hookrightarrow {\mathrm{Rep}}^\square_{E,D_\mathcal{E}}$, which can easily be observed to be a closed immersion. Let $\rho^\square$ denote the universal object of ${\mathrm{Rep}}^\square_{E,D_\mathcal{E}}$. Let $Z(\mathcal{E}) \subset \GL_d$ denote the subgroup that stabilizes $\rho^\square(e_iEe_i)$, for all $i=1,\dots,r$, under the adjoint action of $\GL_d$ on ${\mathrm{Rep}}^\square_{E,D_\mathcal{E}}$. This $Z(\mathcal{E})$ is a torus of rank $r$. \begin{prop}[{\cite[Thm.\ 2.27]{WE2018}}] \label{prop:adapted_GMA} For any $R$-GMA $(E,\mathcal{E})$, the map ${\mathrm{Rep}}_{(E,\mathcal{E}),{\mathrm{GMA}}}^\square\hookrightarrow {\mathrm{Rep}}^\square_{E,D_\mathcal{E}}$ induces an isomorphism \begin{equation} \label{eq:adapted_to_reg} [{\mathrm{Rep}}_{(E,\mathcal{E}),{\mathrm{GMA}}}^\square / Z(\mathcal{E})] \buildrel\sim\over\lra {\mathrm{Rep}}_{E,{D_\mathcal{E}}} \end{equation} of $\Spec R$-algebraic stacks. \end{prop} \subsection{Residually multiplicity-free representations of profinite groups} \label{subsec:RMF} Let $G$ be a profinite group satisfying the $\Phi_p$ finiteness condition, and let $\mathbb{F}$ be a finite field of characteristic $p$ with algebraic closure $\overline{\mathbb{F}}$. By \cite[Cor.\ 2.9(2)]{WE2018}, which is a mild strengthening of \cite[Thm.~2.16]{chen2014}, for any pseudorepresentation ${\bar D}: G \rightarrow \mathbb{F}$, there is a unique semi-simple representation $\rho^{ss}_{\bar D} :G \to \GL_d(\mathbb{F})$ such that ${\bar D} =\det \circ \rho^{ss}_{\bar D}$. \begin{defn} \label{defn:RMF} A residual pseudorepresentation ${\bar D}: G \rightarrow \mathbb{F}$ is \emph{multiplicity-free} if $\rho^{ss}_{\bar D} \otimes_\mathbb{F} \overline{\mathbb{F}}$ has pairwise non-isomorphic simple factors, and each of these factors is defined over $\mathbb{F}$. Equivalently, $\rho^{ss}_{\bar D}$ is a direct sum of pairwise non-isomorphic absolutely irreducible representations. A representation $(V_A,A) \in {\mathrm{Rep}}_G^d(A)$ is \emph{residually multiplicity-free} if $(V_A,A) \in {\mathrm{Rep}}_{\bar D}(A)$ with ${\bar D}$ multiplicity-free. \end{defn} Note that when $\rho^{ss}_{\bar D} \otimes_\mathbb{F} \overline{\mathbb{F}}$ has distinct simple factors, then $\rho^{ss}_{\bar D}$ is multiplicity-free after replacing $\mathbb{F}$ with a finite extension. \begin{thm} \label{thm:CH-GMA} Let ${\bar D}: G \to \mathbb{F}$ be multiplicity-free, and let $(d_1,\dots,d_r)$ be the dimensions of the simple factors of $\rho^{ss}_{\bar D}$. Let $A$ be a Noetherian Henselian local ring with residue field $\mathbb{F}$, and let $(E,\rho,D)$ Cayley-Hamilton representation over $A$ with residual pseudorepresentation ${\bar D}$. Then there is an $A$-GMA structure $\mathcal{E}$ of type $(d_1,\dots,d_r)$ on $E$ such that $D=D_{\mathcal{E}}$. Moreover, given a morphism $(E,\rho,D) \to (E',\rho',D')$ of such objects, the structures $\mathcal{E}$ and $\mathcal{E}'$ may be chosen so that the map $(E,\mathcal{E}) \to (E',\mathcal{E}')$ is a morphism of GMAs. \end{thm} \begin{proof} The structure $\mathcal{E}$ is constructed in \cite[Thm.\ 2.22(ii)]{chen2014}, and it follows from this construction that $D= D_{\mathcal{E}}$ (see \cite[Thm.\ 2.27]{WE2018}). Moreover, the construction only depends on the choice certain lifts of idempotents. If we first choose the structure $\mathcal{E}=(\{e_i\},\{\phi_i\})$ on $E$, then the images of $e_i$ in $E'$ will give a choice of lifts of idempotents in $E'$, and for the resulting GMA structure $\mathcal{E}'$, the map $(E,\mathcal{E}) \to (E',\mathcal{E}')$ is a morphism of GMAs. \end{proof} For the rest of this section, we fix a pseudorepresentation ${\bar D}: G \to \mathbb{F}$ that is multiplicity-free. By Theorem \ref{thm:RDb}, the ring $R_{\bar D}$ is Noetherian and complete (and hence Henselian). By Theorem \ref{thm:CH-GMA}, we can and do fix a choice of $A$-GMA structure $\mathcal{E}_{\bar D}$ of type $(d_1,\dots,d_r)$ on $E_{\bar D}$ such that $D^u_{E_{\bar D}}=D_{\mathcal{E}_{\bar D}}$. \begin{cor} \label{cor:compat=adapt} Assume that ${\bar D}$ is multiplicity-free, and let $\mathcal{E}_{\bar D}$ be choice of $R_{\bar D}$-GMA structure on $E_{\bar D}$ as in Theorem \ref{thm:CH-GMA}. \begin{enumerate} \item There are isomorphisms \[ [{\mathrm{Rep}}^\square_{(E_{\bar D},\mathcal{E}_{\bar D}),{\mathrm{GMA}}} / Z(\mathcal{E}_{\bar D})] \buildrel\sim\over\lra {\mathrm{Rep}}_{E_{\bar D},{D^u_{E_{\bar D}}}} \buildrel\sim\over\lra {\mathrm{Rep}}_{\bar D} \] of stacks on ${\mathrm{Tfg}}_{\mathbb{Z}_p}$. \item Let $B$ be a commutative Noetherian local $\mathbb{Z}_p$-algebra. Given an adapted representation $(E_{\bar D},\mathcal{E}_{\bar D}) \to M_d(B)$, the map $E_{\bar D} \to M_d(B)$ is a compatible representation of $(E_{\bar D},D^u_{E_{\bar D}})$. \item Let $(E, \rho, D)$ be a Cayley-Hamilton representation of $G$ with residual pseudorepresentation ${\bar D}$. Then there is an $A$-GMA structure $\mathcal{E}$ on $E_A$ such that $D_{\mathcal{E}}=D$ and such that the map $(E_{\bar D},D_{\mathcal{E}_{\bar D}}) \to (E,D_{\mathcal{E}})$ is adapted. \end{enumerate} \end{cor} \begin{rem} \label{rem:formal adapted rep space} Let $(E,\mathcal{E})$ be an $R$-GMA where $R \in {\mathrm{Tfg}}_{\mathbb{Z}_p}$ is local. Let $S$ be the $R$-algebra from Theorem \ref{thm:Ad_main} so that ${\mathrm{Rep}}^\square_{(E,\mathcal{E}),{\mathrm{GMA}}}=\Spec(S)$. Restricting the functor ${\mathrm{Rep}}^\square_{(E,\mathcal{E}),{\mathrm{GMA}}}$ to the subcategory ${\mathrm{Tfg}}_{R}$ of the category of $R$-algebras, we obtain an affine formal scheme $\Spf(\hat S)$, where $\hat S$ is the $\mathfrak{m}_R S$-adic completion of $S$. We also denote $\Spf(\hat S)$ by ${\mathrm{Rep}}^\square_{(E,\mathcal{E}),{\mathrm{GMA}}}$, abusing notation. This is how we consider $[{\mathrm{Rep}}^\square_{(E_{\bar D},\mathcal{E}_{\bar D}),{\mathrm{GMA}}} / Z(\mathcal{E}_{\bar D})]$ as a formal stack on ${\mathrm{Tfg}}_{\mathbb{Z}_p}$. \end{rem} \begin{proof} Statement (1)\ follows from Proposition \ref{prop:adapted_GMA} and Theorem \ref{thm:ED-compat}, while (2)\ follows from the statement of Theorem \ref{thm:CH-GMA} that $D^u_{E_{\bar D}} = D_{\mathcal{E}_{\bar D}}$. Statement (3)\ follows from the second part of Theorem \ref{thm:CH-GMA}. \end{proof} \begin{lem} \label{lem:Einj} Assume that ${\bar D}$ is multiplicity-free. Let $(E_{\bar D},D^u_{E_{\bar D}}) \to (E,D)$ be a morphism of Cayley-Hamilton algebras, where $E$ is an $R$-algebra. \begin{enumerate} \item For any non-zero $x \in E$, there is a commutative local $R$-algebra $B$ of finite cardinality and a compatible representation $\rho_B: E \rightarrow M_d(B)$ such that $\rho_B(x) \neq 0$. \item Let $(E_{\bar D},D^u_{E_{\bar D}}) \to (E',D')$ be another morphism of Cayley-Hamilton algebras, and assume that $E_{\bar D} \to E$ and $E_{\bar D} \to E'$ are surjective. If, for all $B$ as in (1) and all compatible representations $\rho_B:E_{\bar D} \to M_d(B)$, the map $\rho_B$ factors through $E$ if and only if it factors through $E'$, then there is a canonical isomorphism $(E, D) \isoto (E', D')$ of Cayley-Hamilton algebras. \end{enumerate} \end{lem} \begin{proof} Since ${\bar D}$ is multiplicity-free, we may fix a GMA structure $\mathcal{E}_{\bar D}$ on $E_{\bar D}$ as in Theorem \ref{thm:CH-GMA}. By Corollary \ref{cor:compat=adapt}(3), this gives a GMA structure $\mathcal{E}$ on $E$ such that $(E_{\bar D},D^u_{E_{\bar D}}) \to (E,D)$ induces a morphisms of GMAs $(E_{\bar D}, \mathcal{E}_{\bar D}) \to (E,\mathcal{E})$, and similarly for $(E',D')$. By Corollary \ref{cor:compat=adapt}(3) we may work with adapted representations of these GMAs in the place of compatible representations of the Cayley-Hamilton algebras. We will do this for the remainder of the proof. (1)\ By Theorem \ref{thm:Ad_main} and Remark \ref{rem:formal adapted rep space}, there is $\hat S \in {\mathrm{Tfg}}_{R}$ such that ${\mathrm{Rep}}^\square_{(E,\mathcal{E}),{\mathrm{GMA}}}=\Spf(\hat S)$ and, moreover, $E \hookrightarrow M_d(S)$ splits as an $R$-module map. Therefore $\rho_{\mathrm{GMA}}: E \to M_d(\hat S)$ remains injective. Let $x \in E$ be a non-zero element, so $\rho_{\mathrm{GMA}}(x) \ne 0$. Let $y \in \hat S$ be a non-zero entry in the matrix $\rho_{\mathrm{GMA}}(x)$. By Lemma \ref{lem:tfg algebra}, there is a commutative local $R$-algebra $B$ of finite cardinality and an $R$-algebra homomorphism $f:\hat S \to B$ such that $f(y) \ne 0$. Then the composite $f \circ \rho_{\mathrm{GMA}}: E \to M_d(B)$ is an adapted representation such that $(f \circ \rho_{\mathrm{GMA}})(x) \neq 0$. (2)\ Since the maps $(E_{\bar D},D^u_{E_{\bar D}}) \to (E,D)$ and $(E_{\bar D},D^u_{E_{\bar D}}) \to (E',D')$ are morphisms of pseudorepresentations and the maps $g:E_{\bar D} \to E$ and $g':E_{\bar D} \to E'$ are surjective, it suffices to show $\ker(g)=\ker(g')$. Assume for a contradiction that there exists $x \in \ker(g)$ with $x \not \in \ker(g')$. Since $g'(x) \ne 0$, part (1)\ implies that there is a commutative local $R$-algebra $B$ of finite cardinality and a compatible representation $\rho'_B: E' \rightarrow M_d(B)$ such that $\rho'_B(g'(x)) \neq 0$. The adapted representation $\tilde{\rho}_B=\rho'_B \circ g': E_{\bar D} \to M_d(B)$ factors through $E'$, so it must factor through $E$ by assumption. This implies that $\tilde{\rho}_B = \rho_B \circ g$ for some adapted representation $\rho_B$ of $E$. But then \[ 0=\rho_B(g(x)) =\tilde{\rho}_B(x)= \rho'_B(g'(x)) \neq 0 \] a contradiction. \end{proof} \subsection{Condition $\cC$ in the residually multiplicity-free case} \label{subsec:RMF-P} Let $G$ be a profinite group satisfying the $\Phi_p$ finiteness condition. Fix a stable condition $\cC \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ as in Definition \ref{defn:stable} and a residual pseudorepresentation ${\bar D}: G \to \mathbb{F}$. By Theorem \ref{thm:univ_CH_P}, there is a universal Cayley-Hamilton algebra with condition $\cC$, denoted $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}})$. In the case that ${\bar D}$ is multiplicity-free, the following theorem gives an alternate characterization of $(E^\cC_{\bar D}, D_{E^\cC_{\bar D}})$. \begin{thm} \label{thm:GMA-P} Assume that ${\bar D}$ is multiplicity-free. \begin{enumerate} \item Let $B$ be a commutative local $R_{\bar D}$-algebra of finite cardinality. Let $\rho_B: (E_{\bar D},D^u_{E_{\bar D}}) \to M_d(B)$ be a compatible representation, and let $V_B \cong B^d$ denote the corresponding object of ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$. Then $V_B$ satisfies $\cC$ if and only if $\rho_B$ factors through $E^\cC_{\bar D}$. \item The property of (1) characterizes the quotient $E_{\bar D} \to E^\cC_{\bar D}$. \end{enumerate} \end{thm} \begin{proof} Part (1) follows from Theorem \ref{thm:ModToCH} and Theorem \ref{thm:univ_CH_P}. Part (2) is immediate from Lemma \ref{lem:Einj}(2). \end{proof} When ${\bar D}$ is not multiplicity-free, we only know how to characterize $E^\cC_{\bar D}$ by Theorem \ref{thm:univ_CH_P}, cf.\ \cite[\S1.3.4]{BC2009}. This theorem furnishes a convenient way to prove that, in certain cases where a pseudorepresentation comes from a representation, property $\cC$ for the pseudorepresentation is related to property $\cC$ for a representation inducing it. \begin{cor} \label{cor:tenauthors} Let ${\bar D}$ be multiplicity-free with ${\bar\rho}_{{\bar D}}^{ss} = {\bar\rho}_1 \oplus \cdots {\bar\rho}_r$ over $\mathbb{F}$, where the ${\bar\rho}_i$ are absolutely irreducible and pairwise non-isomorphic of dimension $d_i$. For $i=1,\dots,r$, let $\rho_i:G \to M_{d_i}(A)$ be a deformation of ${\bar\rho}_i$, and let $D=\psi(\rho_1 \oplus \cdots \oplus \rho_r)$. Then $D$ has $\cC$ if and only if there are deformations $\rho_i'$ of ${\bar\rho}_i$ with $\cC$ for $i=1,\dots,r$ such that $D=\psi(\rho_1' \oplus \cdots \oplus \rho_r')$. \end{cor} \begin{proof} If $D=\psi(\rho_1' \oplus \cdots \oplus \rho_r')$ for such $\rho'_i$, then its clear that $D$ has $\cC$. Now assume $D$ has $\cC$, so, by definition, there is a Cayley-Hamilton representation $(E,\rho,D')$ of $G$ that has $\cC$ such that $D= D'\circ \rho$. Since ${\bar D}$ is multiplicity-free, we may assume that $D'=D_\mathcal{E}$ for some GMA-structure $\mathcal{E}=(\{e_i\},\{\phi_i\})$ on $E$ such that $\rho_i'=e_i\rho e_i: G \to M_{d_i}(A)$ is a deformation of ${\bar\rho}_i$. Replacing $E$ by the image of $A[G]\to E$ if necessary, we may assume that the maps $\rho_{i,j}$ given by the GMA structure are surjective. Now, the fact that $D=\psi(\rho_1 \oplus \cdots \oplus \rho_r)$ implies that $\phi_{i,j,k}=0$ for all triples $i,j,k$ of distinct integers with $1 \le i,j,k \le r$ (see \cite[\S 1.5.1]{BC2009}). Then the sum of projection maps \[ E \to \bigoplus_{i=1}^r e_i E e_i =\bigoplus_{i=1}^r M_{d_i}(A). \] is an $A$-algebra homomorphism. The resulting map \[ G \to E \to M_d(A) \] is $\rho':=\oplus_{i=1}^r \rho_i'$. Since $(E,D')$ has $\cC$, the map $E_{\bar D} \to E$ factors through $E_{\bar D}^\cC$. This implies that $\rho'$ has $\cC$ by Theorem \ref{thm:GMA-P}(1), and hence each $\rho_i'$ has $\cC$ by stability. \end{proof} \begin{rem} The preceding corollary can be useful in proving automorphy lifting theorems. Indeed, it is a general version of an argument used in the proof of \cite[Prop.~4.4.6]{tenauthors}. \end{rem} \begin{warn} When applying these results to the situation of \S\ref{subsec:global conditions} below (deformations of global Galois representations, where $\cC$ is a condition on the local representation), the preceding corollary only applies to deformations $D$ that are \textbf{globally} reducible. For example, take $\cC$ to be the condition ``de Rham at $p$" and ${\bar D}=\psi(\mathbb{F}_p(1)\oplus \mathbb{F}_p):G_\mathbb{Q} \to \mathbb{F}_p$. Suppose $D=\psi(\rho)$ where $\rho:G_\mathbb{Q} \to \GL_2(\mathbb{Z}_p)$ is irreducible, but $\rho\|_{G_p}$ is a non-split extension of $\mathbb{Z}_p(1)$ by $\mathbb{Z}_p$ (and hence not de Rham). Then $D\|_{G_p}=\psi(\rho\|_{G_p})=\psi(\mathbb{Z}_p(1) \oplus \mathbb{Z}_p)$ is a sum of two de Rham characters, and hence de Rham, but $D$ is not (globally) de Rham. \end{warn} If ${\bar\rho}$ is absolutely irreducible, let $R_{\bar\rho}^\cC$ be the deformation-with-$\cC$ ring defined by Ramakrishna (denoted $R_\cC({\bar\rho})$ in \cite[Prop.~1.2]{ramakrishna1993}) and let $\rho^\cC_{\bar\rho}$ be the universal deformation. \begin{cor} \label{cor:us=Ram} Suppose that ${\bar\rho}_{\bar D}^{ss}={\bar\rho}$ is absolutely irreducible. Then there is canonical isomorphism $R_{\bar D}^\cC \to R_{\bar\rho}^\cC$ determined by the pseudodeformation $\psi(\rho_{\bar\rho}^\cC)$ of ${\bar D}$. \end{cor} \begin{proof} It is well-known that $R_{\bar D}=R_{\bar\rho}$ -- that is, every deformation $D$ of ${\bar D}$ is of the form $D=\psi(\rho)$ for a unique deformation $\rho$ of ${\bar\rho}$. (Using pseudo-characters, this is due to Carayol and Mazur. For pseudo-representations, this is proved by Chenevier \cite[Thm.\ 2.22(i)]{chen2014}; it also follows from Theorem \ref{thm:CH-GMA}.) In this situation, by the previous corollary, $D$ has $\cC$ if and only if $\rho$ has $\cC$, so the two quotients $R_{\bar D}^\cC$ and $R_{\bar\rho}^\cC$ of $R_{\bar D}=R_{\bar\rho}$ coincide. \end{proof} This corollary shows that our $R_{\bar D}^\cC$ is really a generalization of Ramakrishna's theory. \section{Deformation conditions and extensions} \label{sec:gma_ext} In \cite{BC2009}, Bella\"{i}che and Chenevier used the explicit nature of the generalized matrix algebras discussed in the previous section to partially compute the universal pseudodeformation ring in terms of extension groups, in the category $\mathrm{Mod}_{\mathbb{Z}_p[G]}$, of the constituent irreducible representations of the residual representation. This calculation is a generalization of the description of the tangent space of the deformation ring of an irreducible representation in terms of adjoint cohomology. In this section, we show that pseudodeformations-with-$\cC$ may be controlled by similar calculations, but where the extension groups are taken in the category $\cC$. Whereas Bella\"{i}che and Chenevier work with arbitrary multiplicity-free ${\bar D}$, we, for reasons of simplicity and clarity, have chosen only to consider the case where ${\bar D}$ is a sum of two characters (but see Remark \ref{rem:ExtP_expectation}). In this situation, where ${\bar D} = \psi(\bar\chi_1 \oplus \bar\chi_2)$ for distinct characters $\bar\chi_1$ and $\bar\chi_2$, we now outline what we can compute about $R_{\bar D}^\cC$. First, we consider a simpler deformation ring $R_{\bar D}^{\mathrm{red}, \cC}$ where we only consider deformations $D$ that remain reducible (i.e.~are of the form $D=\psi(\chi_1 \oplus \chi_2)$). This ring can be computed in terms of deformation rings of the characters $\bar\chi_1$ and $\bar\chi_2$. Next, we wish to compute the kernel $J^\cC=\ker(R_{\bar D}^\cC \to R_{\bar D}^{\mathrm{red}, \cC})$, which is called the reducibility ideal. By the work of Bella\"{i}che and Chenevier, this ideal can be described as a quotient of $B^\cC \otimes_{R_{\bar D}^\cC} C^\cC$, where these modules come from the GMA structure on $E_{\bar D}^\cC$. Finally, $B^\cC/J^\cC B^\cC$ and $C^\cC/J^\cC C^\cC$ can be described in terms of extension groups in the category $\cC$ of the universal deformations $\chi_1$ and $\chi_2$. In total, this gives an upper bound for the size of $J^\cC/(J^\cC)^2$ in terms of extension groups of $\chi_1$ and $\chi_2$ in the category $\cC$, where $R_{\bar D}^\cC/J^\cC$ is a quotient of $R_{\bar D}^\cC$ that is fairly simple. \begin{rem} \label{rem:use for R=T} When proving modularity lifting theorems, this is often the kind of upper bound one needs. In that situation, one has a surjection $R_{\bar D}^\cC \to \mathbb{T}$, where $\mathbb{T}$ is the quotient of `modular deformations.' To prove modularity, one needs to show that $R_{\bar D}^\cC$ is not `too big' -- i.e.\ give an upper bound on the size of $R_{\bar D}^\cC$. In this case, the extension groups in $\cC$ are given as certain Selmer groups in Galois cohomology (see Example \ref{eg:selmer H1}). Often, the size of these Selmer groups can be bounded in terms of modular invariants (e.g.~zeta values), and the resulting upper bound on $J^\cC/(J^\cC)^2$ can be enough to show that $R_{\bar D}^\cC \to \mathbb{T}$ is injective. This is the strategy of \cite{WWE3,WWE5}. \end{rem} \begin{rem} In fact, our results allow a similar computation about $I/I^2$ for any ideal $I$ containing $J^\cC$. In particular, taking $I$ to be the maximal ideal of $R_{\bar D}^\cC$, we get a description of the (co)tangent space of $R_{\bar D}^\cC$. This computation about $J^\cC/(J^\cC)^2$ is giving information about the `relative tangent space' over the reducible deformation ring, and is, in practice, often more useful. \end{rem} We now outline the contents of this section. In \S \ref{subsec:RedExt}, we review the results of Bella\"{i}che--Chenevier. In \S \ref{subsec:PExt}, we prove our generalization to deformations-with-$\cC$. \subsection{Conventions} \label{subsec:ext-conventions} Throughout this section, we fix $G$, a profinite group satisfying the $\Phi_p$ finiteness condition, and two distinct characters $\bar\chi_1, \bar\chi_2: G \rightarrow \mathbb{F}^\times$. We let ${\bar D} = \psi(\bar\chi_1 \oplus \bar\chi_2)$. By Theorem \ref{thm:CH-GMA}, we can and do fix a $R_{\bar D}$-GMA structure $\mathcal{E}_{\bar D} =(\{e_1,e_2\},\{\phi_1,\phi_2\})$ on $E_{\bar D}$. We write $(E_{\bar D},\mathcal{E}_{\bar D})$ and $\rho^u: G \to E_{\bar D}^\times$ as \begin{equation} \label{eq:CH-GMA} E_{\bar D} \cong \begin{pmatrix} R_{\bar D} & B^u \\ C^u & R_{\bar D}\end{pmatrix}, \qquad \rho^u(\sigma) = \ttmat{\rho^u_{1,1}(\sigma)}{\rho^u_{1,2}(\sigma)}{\rho^u_{2,1}(\sigma)}{\rho^u_{2,2}(\sigma)} \end{equation} and write $m: B^u \otimes_{R_{\bar D}} C^u \to R_{\bar D}$ for the induced map, as in Example \ref{eg:GMA1,1}. For $b \in B^u$ and $c \in C^u$, we define $b \cdot c = m(b \otimes c) \in R_{\bar D}$. We can and do assume that the idempotents are ordered so that the image of $\rho_{i,i}(\sigma)$ under $R_{\bar D} \to \mathbb{F}$ is $\bar\chi_i(\sigma)$. By Corollary \ref{cor:compat=adapt}(3), a Cayley-Hamilton representation $(E, \rho, D)$ of $G$ with residual pseudorepresentation ${\bar D}$ inherits a GMA structure from the data above. We will use matrix notation for $E$ and $\rho$ according to this structure, as in \eqref{eq:CH-GMA} (for example, writing the coordinates of $\rho$ as $\rho_{i,j}$). \subsection{Review of reducibility, extensions, and GMAs} \label{subsec:RedExt} In this subsection, we review known results of \cite[\S1.5]{BC2009} relating the structure of GMAs to $\mathrm{Ext}$-groups. \begin{defn} \label{defn:reducible} Let $A \in \hat\cC_{W(\mathbb{F})} $. We call a pseudodeformation $D: G \rightarrow A$ of ${\bar D}$ \emph{reducible} if $D=\psi(\chi_1 \oplus \chi_2)$ for characters $\chi_i: G \rightarrow A^\times$ deforming $\bar\chi_i$. Otherwise, we call $D$ \emph{irreducible}. A GMA representation $(E, \rho: G \rightarrow E^\times, D_\mathcal{E})$ of $G$ with residual pseudorepresentation ${\bar D}$ is called \emph{reducible} (resp.\ \emph{irreducible}) provided that pseudodeformation $D_\mathcal{E} \circ \rho : G \rightarrow A$ is reducible (resp.\ irreducible). \end{defn} \begin{prop} \label{prop:reducible} Let $A$ be a commutative Noetherian local $\mathbb{Z}_p$-algebra. Let $D: G \rightarrow A$ pseudorepresentation with residual pseudorepresentation ${\bar D}$. \begin{enumerate} \item $D$ is reducible if and only if $D=\psi(\rho)$ for some GMA representation $\rho$ with scalar ring $A$ such that $\rho_{1,2}(G) \cdot \rho_{2,1}(G)$ is zero. \item Let $J = B^u \cdot C^u \subset R_{\bar D}$ be the image ideal of $B^u \otimes_{R_{\bar D}} C^u$ in $R_{\bar D}$ under $m : B^u \otimes_{R_{\bar D}} C^u \rightarrow R_{\bar D}$. Then $D$ is reducible if and only if the corresponding local homomorphism $R_{\bar D} \rightarrow A$ kills $J$. \end{enumerate} \end{prop} \begin{proof} See \cite[\S1.5.1]{BC2009}. \end{proof} In light of Proposition \ref{prop:reducible}, we establish the following terminology. \begin{defn} \label{defn:univ_red} The ideal $J \subset R_{\bar D}$ of Proposition \ref{prop:reducible}(2) is called the \emph{reducibility ideal} of $R_{\bar D}$. The image of $J$ under the map $R_{\bar D} \rightarrow A$ corresponding to a pseudodeformation $D : G \rightarrow A$ of ${\bar D}$ is called the \emph{reducibility ideal of $D$}. We define $E^\mathrm{red}_{\bar D}$ to be the Cayley-Hamilton quotient of $E_{\bar D}$ by $J E_{\bar D}$, which as in Example \ref{eg:CH-quot-with-scalar-ideal} is the usual algebra quotient $E^\mathrm{red}_{\bar D} = E_{\bar D}/JE_{\bar D}$. Its scalar ring is $R^\mathrm{red}_{\bar D} = R_{\bar D}/J$ and is called the \emph{universal reducible pseudodeformation ring} for ${\bar D}$. We let $(E^\mathrm{red}_{\bar D}, \rho^\mathrm{red}, \mathcal{E}^\mathrm{red}_{\bar D})$ denote the corresponding GMA representation of $G$, and write the decomposition arising from \eqref{eq:CH-GMA} as \begin{equation} \label{eq:red-CMA} E^\mathrm{red}_{\bar D} = \begin{pmatrix} R^\mathrm{red}_{\bar D} & B^\mathrm{red} \\ C^\mathrm{red} & R^\mathrm{red}_{\bar D}\end{pmatrix}. \end{equation} \end{defn} \begin{warn} Being reducible implies that the map $m: B^\mathrm{red} \otimes C^\mathrm{red} \to R^\mathrm{red}_{\bar D}$ is zero, but it \textbf{does not} imply that either $B^\mathrm{red}$ or $C^\mathrm{red}$ is zero. In fact, for a fixed $g \in G$, it is possible that both $\rho^\mathrm{red}_{1,2}(g) \in B^\mathrm{red}$ and $\rho^\mathrm{red}_{2,1}(g) \in C^\mathrm{red}$ are non-zero. \end{warn} Let for $i=1,2$, let $R_{\bar\chi_i}$ denote the universal deformation ring of $\bar\chi_i$ and let $\chi_i^u: G \to R_{\bar\chi_i}^\times$ denote the universal deformation. \begin{prop} \label{prop:univ_red} \hfill \begin{enumerate} \item If $\rho:G \to (E,D)$ is a GMA representation of $G$ with scalar ring $A$ and residual pseudorepresentation ${\bar D}$, then the resulting GMA map $(E_{\bar D}, \mathcal{E}_{\bar D}) \rightarrow (E,\mathcal{E})$ factors through $(E^\mathrm{red}_{\bar D}, \mathcal{E}^\mathrm{red}_{{\bar D}})$ if and only if $\rho$ is reducible. \item There is a canonical isomorphism $R^\mathrm{red}_{\bar D} \cong R_{\bar\chi_1} \hat\otimes_{W(\mathbb{F})} R_{\bar\chi_2}$. Letting $\chi_i = \chi_i^u \otimes_{R_{\bar\chi_i}} R^\mathrm{red}_{\bar D}$, the universal reducible pseudodeformation of ${\bar D}$ is $\psi(\chi_1\oplus \chi_2)$. \item In terms of the decomposition \eqref{eq:red-CMA}, we have $\rho^\mathrm{red}_{i,i} = \chi_i$. \end{enumerate} \end{prop} \begin{proof} (1)\ By Lemma \ref{lem:max_CH_quot}, the GMA map $(E_{\bar D}, \mathcal{E}_{\bar D}) \rightarrow (E,\mathcal{E})$ factors through $(E^\mathrm{red}_{\bar D}, \mathcal{E}^\mathrm{red}_{{\bar D}})$ if and only if $E_{\bar D} \to E$ sends $JE_{\bar D}$ to zero. Because $R \rightarrow E$ is injective whenever there is a pseudorepresentation $D: E \rightarrow R$ \cite[Lem.\ 5.2.5]{WWE1}, $JE_{\bar D}$ maps to zero in $E$ if and only if $R_{\bar D} \to A$ factors through $R_{\bar D}^\mathrm{red}$, which, by Proposition \ref{prop:reducible}(2), is equivalent to $D \circ \rho$ being reducible. (2)\ By Yoneda's lemma, it suffices to construct a canonical functorial isomorphism \[ \mathrm{Hom}(R^\mathrm{red}_{\bar D},A) \cong \mathrm{Hom}(R_{\bar\chi_1} \hat\otimes_{W(\mathbb{F})} R_{\bar\chi_2},A) \] for $A \in \hat\cC_{W(\mathbb{F})}$. Given $R_{\bar\chi_1} \hat\otimes_{W(\mathbb{F})} R_{\bar\chi_2} \to A$, we define a reducible deformation $D$ of ${\bar D}$ by $D=\psi(\chi_1^u \otimes A \oplus \chi_2^u \otimes A)$, which determines an element of $\mathrm{Hom}(R^\mathrm{red}_{\bar D},A)$ by Proposition \ref{prop:reducible}(2) and the universal property of $R_{\bar D}$. Conversely, given $R^\mathrm{red}_{\bar D} \to A$, consider the GMA representation $\rho^\mathrm{red} \otimes_{R^\mathrm{red}_{\bar D}} A$. Since $B^\mathrm{red} \cdot C^\mathrm{red}=0$, we have that $\rho^\mathrm{red}_{i,i} \otimes_{R^\mathrm{red}_{\bar D}} A: G \to A^\times$ (for $i=1,2$) is a character that, by the conventions of \S \ref{subsec:ext-conventions}, is a deformation of $\bar\chi_i$. This pair of characters determines an element of $\mathrm{Hom}(R_{\bar\chi_1} \hat\otimes_{W(\mathbb{F})} R_{\bar\chi_2},A)$. We have defined maps between $\mathrm{Hom}(R^\mathrm{red}_{\bar D},A)$ and $\mathrm{Hom}(R_{\bar\chi_1} \hat\otimes_{W(\mathbb{F})} R_{\bar\chi_2},A)$. The reader can check these are mutually inverse and functorial in $A$. (3)\ It follows from the construction of the isomorphism in (2). \end{proof} The following key result relates $R^\mathrm{red}_{\bar D}[G]$-module extensions to the structure of $E^\mathrm{red}_{\bar D}$. For ease of notation, we write $\chi_i$ for the base change from $R_{\bar\chi_i}$ to $R^\mathrm{red}_{\bar D}$ of the universal deformation $\chi_i^u$ of $\bar\chi_i$. \begin{prop} \label{prop:GMA-Ext} Let $A \in \hat\cC_{W(\mathbb{F})}$ and let $M$ be a finitely generated $A$-module. For $i=1,2$, let $\chi_{i,A}:G \to A^\times$ be characters deforming $\bar\chi_i$. By Proposition \ref{prop:univ_red}, this induces a unique local homomorphism $R_{\bar D}^\mathrm{red} \to A$. There is a natural isomorphism \[ \mathrm{Hom}_{A}(B^\mathrm{red}\otimes_{R^\mathrm{red}_{\bar D}} A,M) \buildrel\sim\over\lra \mathrm{Ext}^1_{A[G]}(\chi_{2,A}, \chi_{1,A} \otimes_{A} M). \] as well as a similar isomorphism in the $C$-coordinate. \end{prop} \begin{proof} The details may be found in \cite[Lem.\ 4.1.5, proof of (4.1.7)]{WWE2}, cf.~also \cite[Thm.\ 1.5.6]{BC2009}. We reproduce the construction of the map here with notation that will be convenient in \S\ref{subsec:PExt}. Let $E_M=\sm{A}{M}{0}{A}$, with GMA structure as in Example \ref{eg:m0}. Given a homomorphism $f:B^\mathrm{red} \otimes_{R^\mathrm{red}_{\bar D}} A \rightarrow M$, we have morphism of GMAs $E^\mathrm{red}_{\bar D}\otimes_{R^\mathrm{red}_{\bar D}} A \to E_M$ as the composition \begin{equation} \label{eq:EM-factor} E^\mathrm{red}_{\bar D}\otimes_{R^\mathrm{red}_{\bar D}} A \xrightarrow{\sm{\mathrm{id}}{\mathrm{id}}{0}{\mathrm{id}}} \ttmat{A}{B^\mathrm{red} \otimes_{R^\mathrm{red}_{\bar D}} A}{0}{A} \xrightarrow{\sm{\mathrm{id}}{f}{0}{\mathrm{id}}} \ttmat{A}{M}{0}{A}. \end{equation} Using the fact that $B^\mathrm{red} \cdot C^\mathrm{red} =0$, we see that $e_1E_Me_2 = \sm{0}{M}{0}{0}$ is a left $E_{\bar D}$-submodule of $E_M e_2 = \sm{0}{M}{0}{A}$. Noting that $e_1E_Me_2 \cong \chi_{1,A} \otimes_{A} M$ and $E_Me_2 / e_1E_Me_2 \cong \chi_{2,A}$ as $A[G]$-modules, we obtain an exact sequence \begin{equation} \label{eq:GMA-SES} 0 \longrightarrow \chi_{1,A} \otimes_{A} M \rightarrow E_M e_2 \longrightarrow \chi_{2,A} \longrightarrow 0, \end{equation} which determines the corresponding element of $\mathrm{Ext}^1_{A[G]}(\chi_{2,A}, \chi_{1,A} \otimes_{A} M)$. Conversely, any such extension can be realized in the form $E_Me_2$. \end{proof} \subsection{GMA structures corresponding to extensions with an abstract property} \label{subsec:PExt} Let $\cC \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ be a stable property, as in Definition \ref{defn:stable}. The following lemma is a well-known consequence of stability, and we leave the proof to the reader. To state the lemma clearly, we introduce some notation for extension classes. If $E$ is an algebra and $V_1, V_2$ are $E$-modules, given an extension class $c \in \mathrm{Ext}^1_{E}(V_1,V_2)$, and an exact sequence \begin{equation} \label{eq:ses-representing-ext-class} 0 \to V_2 \to V \to V_1 \to 0 \end{equation} representing $c$, we call $V$ an \emph{extension module} for $c$. Let $V_1, V_2 \in \cC$, and assume that $V_1, V_2 \in \mathrm{Mod}_{R[G]}$ as well for commutative $\mathbb{Z}_p$-algebra $R$. Define $\mathrm{Ext}^1_{R[G],\cC}(V_1,V_2)$ as the subset of $\mathrm{Ext}^1_{R[G]}(V_1,V_2)$ consisting of extension classes $c$ such that, for every extension module $V$ for $c$, we have $V \in \cC$. \begin{lem} \label{lem:C-exts} With $V_1, V_2$ as above, we have the following. \begin{enumerate} \item For a class $c \in \mathrm{Ext}^1_{R[G]}(V_1,V_2)$, if some extension module $V$ for $c$ has $V \in \cC$, then $c \in \mathrm{Ext}^1_{R[G],\cC}(V_1,V_2)$. \item The subset $\mathrm{Ext}^1_{R[G],\cC}(V_1,V_2) \subset \mathrm{Ext}^1_{R[G]}(V_1,V_2)$ is a sub-$R$-module. \end{enumerate} \end{lem} \begin{eg} \label{eg:selmer H1} Let $H_1,\dots,H_n \subset G$ be subgroups, and, for $i=1,\dots,n$, let $\cC_i \subset \mathrm{Mod}_{\mathbb{Z}_p[H_i]}^{\mathrm{fin}}$ be a stable condition. Assume that the condition $\cC \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ arises from the $\cC_i$ as in Example \ref{eg:fiber product of stable is stable}. Then $\mathrm{Ext}^1_{R[G],\cC}(V_1,V_2)$ is the kernel of the map \[ \mathrm{Ext}^1_{R[G]}(V_1,V_2) \to \bigoplus_{i=1}^n \frac{\mathrm{Ext}^1_{R[H_i]}(V_1,V_2)}{\mathrm{Ext}^1_{R[H_i],\cC_i}(V_1,V_2)} \] given by the restrictions $\mathrm{Ext}^1_{R[G]}(V_1,V_2) \to \mathrm{Ext}^1_{R[H_i]}(V_1,V_2)$ followed by the quotients. (This is sometimes referred to as a Selmer group.) \end{eg} Let $E^\cC_{\bar D}$ be as in Theorem \ref{thm:GMA-P}, and let $E^{\cC,\mathrm{red}}_{\bar D}=E^\cC_{\bar D} / J E^\cC_{\bar D}$ where $J \subset R_{\bar D}$ is the reducibility ideal. Following the notation of \eqref{eq:CH-GMA}, we write them as \[ E^\cC_{\bar D} =\begin{pmatrix} R^\cC_{\bar D} & B^\cC \\ C^\cC & R^\cC_{\bar D}\end{pmatrix}, \qquad E^{\cC,\mathrm{red}}_{\bar D} = \begin{pmatrix} R^{\cC,\mathrm{red}}_{\bar D} & B^{\cC,\mathrm{red}} \\ C^{\cC,\mathrm{red}} & R^{\cC,\mathrm{red}}_{\bar D}\end{pmatrix}. \] We denote the Cayley-Hamilton representations by $\rho^\cC:G \to (E^\cC_{\bar D})^\times$ and $\rho^{\cC,\mathrm{red}}:G \to (E^{\cC,\mathrm{red}}_{\bar D})^\times$. By Ramakrishna's result \cite[Prop.\ 1.2]{ramakrishna1993}, for $i=1,2$, there is a quotient $R_{\bar\chi_i} \onto R_{\bar\chi_i}^\cC$ that represents the functor of deformations of $\bar\chi_i$ having property $\cC$ and $\chi_i^{u,\cC} = \chi_i^{u} \otimes_{R_{\bar\chi_i}} R_{\bar\chi_i}^\cC$ is the universal deformation with property $\cC$. \begin{prop} \label{prop:univ-red-with-C} There is a canonical commutative diagram \[ \xymatrix{ R^{\cC,\mathrm{red}}_{\bar D} \ar[r]^-\sim & R^\cC_{\bar\chi_1} \hat\otimes_{W(\mathbb{F})} R^\cC_{\bar\chi_2} \\ R^\mathrm{red}_{\bar D} \ar@{->>}[u] \ar[r]^-\sim & R_{\bar\chi_1} \otimes_{W(\mathbb{F})} R_{\bar\chi_2} \ar@{->>}[u] } \] where the vertical maps are the quotients and the lower horizontal map is the isomorphism of Proposition \ref{prop:univ_red}(2). \end{prop} \begin{proof} For simplicity of notation, let $R=R^\cC_{\bar\chi_1} \hat\otimes_{W(\mathbb{F})} R^\cC_{\bar\chi_2}$ and, for $i=1,2$, let $\chi_i^\cC = \chi_i^{u,\cC} \otimes_{R^\cC_{\bar\chi_i}} R$. By Proposition \ref{prop:univ_red}(2), the composite map $R^\mathrm{red}_{\bar D} \to R$ corresponds to the reducible pseudodeformation $D=\psi(\chi_1^\cC \oplus \chi_2^\cC)$. The $G$-module $N=\chi_1^\cC \oplus \chi_2^\cC$ is a faithful Cayley-Hamilton $G$-module with Cayley-Hamilton algebra $(E=\sm{R}{0}{0}{R}, D_\mathcal{E})$. By Theorem \ref{thm:ModToCH}, the map $(E_{\bar D},D^u_{E_{\bar D}}) \to (E,D_\mathcal{E})$ factors through $E_{\bar D}^\cC$. Since $D$ is reducible, Proposition \ref{prop:reducible} implies that it further factors through $E_{\bar D}^{\cC,\mathrm{red}}$. This implies that $R^\mathrm{red}_{\bar D} \to R$ factors through $R^{\cC,\mathrm{red}}_{\bar D}$. On the other hand, the composite map $R_{\bar\chi_1} \otimes_{W(\mathbb{F})} R_{\bar\chi_2} \to R^{\cC,\mathrm{red}}_{\bar D}$ corresponds to the pair of characters $e_1\rho^{\mathrm{red},\cC}e_1, e_2\rho^{\mathrm{red},\cC}e_2: G \to R^{\cC,\mathrm{red}}_{\bar D}$. Since these characters are quotient $R^\mathrm{red}_{\bar D}[G]$-modules of the $R^\mathrm{red}_{\bar D}[G]$-module $E^{\cC,\mathrm{red}}_{\bar D}$, which has $\cC$, we see that they both have $\cC$ as well. This implies that the map $R_{\bar\chi_1} \otimes_{W(\mathbb{F})} R_{\bar\chi_2} \to R^{\cC,\mathrm{red}}_{\bar D}$ factors through $R$, completing the proof. \end{proof} We will write $\chi^\cC_i$ for the base change from $R^\cC_{\bar\chi_i}$ to $R^{\cC,\mathrm{red}}_{\bar D}$ of the universal deformation of $\bar\chi_i$ satisfying $\cC$. \begin{thm} \label{thm:BCs and exts} Let $A \in \hat\cC_{W(\mathbb{F})}$ and let $M$ be a finitely generated $A$-module. For $i=1,2$, let $\chi_{i,A}:G \to A^\times$ be characters deforming $\bar\chi_i$ and having property $\cC$. By Proposition \ref{prop:univ-red-with-C}, this induces a unique local homomorphism $R_{\bar D}^{\mathrm{red},\cC} \to A$. There is a natural isomorphism \[ \mathrm{Hom}_{A}(B^{\cC,\mathrm{red}}\otimes_{R^{\mathrm{red},\cC}_{\bar D}} A,M) \buildrel\sim\over\lra \mathrm{Ext}^1_{A[G],\cC}(\chi_{2,A}, \chi_{1,A} \otimes_{A} M), \] as well as a similar isomorphism in the $C$-coordinate. \end{thm} \begin{rem} \label{rem:ExtP_expectation} Throughout this section, we have restricted our attention to 2-dimensional pseudorepresentations and GMAs of type (1,1). However, in \cite[Thm.\ 1.5.6]{BC2009}, Bella\"{i}che and Chenevier prove a version of Proposition \ref{prop:GMA-Ext} for GMAs of any dimension $d$ and any type $(d_1,\dots,d_r)$. We have made this choice strictly for clarity of exposition. We fully expect a version of Theorem \ref{thm:BCs and exts} to be true for GMAs of any dimension $d$ and any type $(d_1,\dots,d_r)$, and that a proof can be given by applying the methods of this section to the general setup of \cite[Thm.\ 1.5.6]{BC2009}. For example, when $A = \mathbb{F}$, such a statement can be deduced directly from \cite[Thms.\ 11.2.1 and 12.3.1]{CarlAInf}. In fact, a version for types $(d_1,d_2)$ can be deduced from Theorem \ref{thm:BCs and exts} by Morita equivalence, as in \cite[\S1.3.2]{BC2009}. \end{rem} \begin{proof} We set $B=B^{\mathrm{red}}\otimes_{R^{\mathrm{red}}_{{\bar D}}}A$ and $B^\cC = B^{\cC,\mathrm{red}}\otimes_{R^{\cC,\mathrm{red}}_{\bar D}} A$ to simplify notation. We consider the diagram \[\xymatrix{ \mathrm{Hom}_{A}(B^{\cC},M) \ar@{^(->}[d] \ar@{-->}[r]^-\sim & \mathrm{Ext}^1_{A[G],\cC}(\chi_{2,A}, \chi_{1,A} \otimes_{A} M) \ar@{^(->}[d] \\ \mathrm{Hom}_{A}(B ,M) \ar[r]_-\sim^-\Psi & \mathrm{Ext}^1_{A[G]}(\chi_{2,A}, \chi_{1,A} \otimes_{A} M), }\] where $\Psi$ isomorphism of Proposition \ref{prop:GMA-Ext}, and where the dotted arrow is the isomorphism we wish to construct. Since this diagram is canonically isomorphic to the one obtained by replacing $A$ by the image of $A \to \mathrm{End}_A(M)$, we can and do assume that $M$ is a faithful $A$-module. Given a class $c \in \mathrm{Ext}^1_{A[G]}(\chi_{2,A}, \chi_{1,A} \otimes_{A} M)$, we have to show that the map $f_c=\Psi^{-1}(c):B \to M$ factors through $B^{\cC}$ if and only if $c \in \mathrm{Ext}^1_{A[G],\cC}(\chi_{2,A}, \chi_{1,A} \otimes_{A} M)$. Following the proof of Proposition \ref{prop:GMA-Ext}, we see that the class $c$ has an extension module $E_Me_2$ where $E_M =\sm{A}{M}{0}{A}$ is a GMA and $f_c$ induces a morphism of GMAs $E^{\mathrm{red}}_{{\bar D}}\otimes_{R^{\mathrm{red}}_{{\bar D}}}A \to E_M$ by \begin{equation} \label{eq:ext-to-map-of-GMA} E^{\mathrm{red}}_{{\bar D}}\otimes_{R^{\mathrm{red}}_{{\bar D}}}A \xrightarrow{\sm{\mathrm{id}}{\mathrm{id}}{0}{\mathrm{id}}} \ttmat{A}{B}{0}{A} \xrightarrow{\sm{\mathrm{id}}{f_c}{0}{\mathrm{id}}} \ttmat{A}{M}{0}{A}. \end{equation} Since $M$ is a faithful $A$-module, we see that $E_Me_2$ is a faithful Cayley-Hamilton $G$-module with Cayley-Hamilton algebra $(E_M,D_{\mathcal{E}_M})$. By Theorem \ref{thm:ModToCH}, $E_Me_2$ has $\cC$ if and only if $(E_M,D_{\mathcal{E}_M})$ has $\cC$ as a Cayley-Hamilton representation of $G$. By Lemma \ref{lem:C-exts}, Lemma \ref{lem:max_CH_quot}, and Theorem \ref{thm:univ_CH_P}, we are reduced to showing that $f_c$ factors through $B^{\cC}$ if and only if the map $E_{\bar D} \onto E^{\mathrm{red}}_{{\bar D}}\otimes_{R^{\mathrm{red}}_{{\bar D}}}A\to E_M$ given by \eqref{eq:ext-to-map-of-GMA} factors through $E_{\bar D}^\cC$. If $f_c$ factors through $B^{\cC}$, then we see that the map $E^{\mathrm{red}}_{{\bar D}}\otimes_{R^{\mathrm{red}}_{{\bar D}}}A\to E_M$ of \eqref{eq:ext-to-map-of-GMA} agrees with the map \[ E^{\mathrm{red}}_{{\bar D}}\otimes_{R^{\mathrm{red}}_{{\bar D}}}A \onto E^{\cC,\mathrm{red}}_{{\bar D}}\otimes_{R^{\cC,\mathrm{red}}_{{\bar D}}}A \xrightarrow{\sm{\mathrm{id}}{\mathrm{id}}{0}{\mathrm{id}}} \ttmat{A}{B^\cC}{0}{A} \xrightarrow{\sm{\mathrm{id}}{f_c}{0}{\mathrm{id}}} \ttmat{A}{M}{0}{A}. \] Hence the map $E_{\bar D} \to E_M$ factors though $E^{\cC,\mathrm{red}}_{{\bar D}}$, which is a quotient of $E_{\bar D}^\cC$. Conversely, suppose that $(E_{\bar D},D^u_{E_{\bar D}}) \to (E_M,D_{\mathcal{E}_M})$ factors through $(E_{\bar D}^\cC,D_{E^\cC_{\bar D}})$. Since $(E_M,D_{\mathcal{E}_M})$ is reducible, this implies that the map $E_{\bar D} \to E_M$ factors through a map $g:E^{\cC,\mathrm{red}}_{{\bar D}}\otimes_{R^{\cC,\mathrm{red}}_{{\bar D}}}A \to E_M$. By \eqref{eq:ext-to-map-of-GMA}, we see that $f_c$ factors through $e_1ge_2: B^\cC \to M$. \end{proof} \section{Examples} \label{sec:ex} The follow examples of conditions $\cC$ could be useful in arithmetic applications. \subsection{Unramified local condition} Let $\ell$ be a prime number and let $K$ be a finite field extension of $\mathbb{Q}_\ell$. Let $G = \mathrm{Gal}(\overline{K}/K)$, which satisfies the $\Phi_p$ finiteness condition because it is topologically finitely generated. Let $H = I_\ell \subset G$ be the inertia subgroup. The condition $\mathrm{Mod}_{\mathbb{Z}_p[G/H]}^\mathrm{fin} \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$, as in Example \ref{eg:factors through is stable}, is called \emph{unramified}. \subsection{Local conditions at $\ell=p$} \label{subsec:Lex} Retain the same notation as the previous subsection, but now assume $\ell=p$. Let $O_K \subset K$ denote the ring of integers. In this case, there are many conditions on representations of $G$ in $\mathbb{Q}_p$-vector spaces, coming from $p$-adic Hodge theory. Some of these conditions descend to ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ as follows. \begin{defns} Let $V$ be an object of ${\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ and let $a \le b$ be integers. \begin{enumerate} \item We call $V$ \emph{finite-flat} if there is a finite flat group scheme ${\mathcal{G}}$ over $O_K$ such that $V \cong {\mathcal{G}}(\overline{K})$ as $\mathbb{Z}_p[G]$-modules. \item We call an object $V \in {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ \emph{torsion crystalline (resp.~semi-stable) with Hodge-Tate weights in $[a,b]$} if there is a crystalline (resp.~semi-stable) representation $\rho:G \to \GL(W)$ with Hodge-Tate weights in $[a,b]$ and a $G$-stable $\mathbb{Z}_p$-lattice $T \subset W$ such that $V$ is isomorphic to a subquotient of $T$. \end{enumerate} \end{defns} Let $\cC_{{\mathrm{flat}}} \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ denote the full subcategory of finite-flat objects. Ramakrishna has proven that $\cC_{{\mathrm{flat}}}$ is stable \cite[\S2]{ramakrishna1993}. Let $\cC_{\mathrm{crys},[a,b]}, \cC_{\mathrm{st},[a,b]} \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$ denote the full subcategories that are torsion crystalline (resp.\ semi-stable) with Hodge-Tate weights in $[a,b]$. Since the category of crystalline (resp.\ semi-stable) representations with Hodge-Tate weights in $[a,b]$ is closed under finite direct sums, we see that $\cC_{\mathrm{crys},[a,b]}$ and $\cC_{\mathrm{st},[a,b]}$ are closed under finite direct sums. They are also closed under isomorphisms and subquotients by definition, so we see that they are stable. \begin{rem} It is known that, for a $\mathbb{Z}_p[G]$-module $M$ that is finitely generated and free as a $\mathbb{Z}_p$-module, $M \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$ is crystalline (resp.~semi-stable) with Hodge-Tate weights in $[a,b]$ if and only if $M \otimes_{\mathbb{Z}_p} \mathbb{Z}/p^n\mathbb{Z}$ is torsion crystalline (resp.~semi-stable) with Hodge-Tate weights in $[a,b]$ for all $n \ge 1$. This was conjectured by Fontaine, and proven by Liu using results of Kisin, following partial results of Ramakrishna, Berger, and Breuil. It is also known that there is an equivalence of categories $\cC_{\mathrm{flat}} \cong \cC_{\mathrm{crys}, [0,1]}$ for $p > 2$. See \cite{liu2007} and the references given there. \end{rem} \begin{rem} If ${\bar D}$ is residually multiplicity-free and $\cC$ is one of the conditions about, then a Zariski-closed substack $\widetilde{{\mathrm{Rep}}}^\cC_{\bar D} \subset {\mathrm{Rep}}_{\bar D}$ and a quotient $R_{\bar D} \onto \tilde{R}^\cC_{\bar D}$ were constructed in \cite[\S7.1]{WE2018}. The two constructions will have the same generic fiber (over $\mathbb{Z}_p$), as this is where the construction of crystalline loci is done in \textit{loc.\ cit.} That is, $\tilde{R}^\cC_{\bar D}[1/p] \cong R^\cC_{\bar D}[1/p]$, and similarly for $\widetilde{{\mathrm{Rep}}}^\cC_{\bar D}$ and ${\mathrm{Rep}}^\cC_{\bar D}$. In particular, the geometric properties of the generic fibers proved in Prop.\ 6.4.4 and Cor.\ 7.1.5 of \emph{loc.\ cit.}\ apply to the generic fibers of the ${\mathrm{Rep}}^\cC_{\bar D}$ and $R^\cC_{\bar D}$ constructed in this paper. When the $\mathbb{Z}_p$-integral structures differ, we believe that the constructions in this paper will be better behaved: here, we make natively integral constructions, while the integral model of \textit{loc.\ cit.}\ is made by flat closure relative to $\mathbb{Z}_p$. \end{rem} \subsection{Global conditions} \label{subsec:global conditions} Let $F$ be a number field with algebraic closure $\overline{F}$ and let $S$ be a finite set of places of $F$. Let $G=\mathrm{Gal}(F_S/F)$, where $F_S \subset \overline{F}$ is the maximal extension of $F$ unramified outside $S$. Then $G$ satisfies $\Phi_p$ by class field theory. For each $v \in S$, choose a decomposition group $G_v \subset G$ (so $G_v$ is of the type considered in the previous subsections) and a stable condition $\cC_v \subset \mathrm{Mod}_{\mathbb{Z}_p[G_v]}^\mathrm{fin}$. Then there is a corresponding stable condition $\cC \subset {\mathrm{Mod}^\mathrm{fin}_{\bZ_p[G]}}$, as in Example \ref{eg:fiber product of stable is stable}. The Selmer groups of Example \ref{eg:selmer H1} correspond to such a $\cC$. \bibliographystyle{alpha}	{ "redpajama_set_name": "RedPajamaArXiv" }	5,837
Bilingual schooling offers college students with the opportunity to study curriculum content in and through two languages, in a balanced way. It's develop into so accepted that being progressive doesn't require one to assist public schools that even Matt Damon, who helps LIFO, hates testing, complains about racism, and is extraordinarily far left on schooling sends his kids to non-public colleges, even Rahm Emmanuel who continuously accuses others of racism. The rest of this paper focuses on evaluations and overviews of research studies on the effectiveness of bilingual training kinds. The element of data on this web page relates to the 2016 Māori Medium Bilingual Education Study Award. In Might 2014, the California Assembly passed laws decreasing the obligation of Marin County to construct low- and reasonable-income housing. English-language learners usually stay in these programs all through elementary school, or from grades K-5 or 6. Much like these educating in transitional bilingual training, teachers in this program ought to be proficient in English and the native language and state licensed on the respective grade stage and in bilingual training. By 1923, the legislatures of 34 states had dictated English-solely instruction in all personal and public primary schools (Kloss, 1998) Nonetheless, from the sixties to the 1980s, bilingual education within the United States was within the opportunist period. At times, bilingual education was favored, or not less than not slated for elimination. During the last 10 years, the number of bilingual packages inside the district has quadrupled, in keeping with faculty board member Steve Zimmer. University of Houston online college for schooling gives Grasp's of education in administration and supervision for higher training; Master's of schooling in curriculum and instruction (Arithmetic schooling); Master's of education in gifted and proficient training. When students graduate from elementary school to center school, they often are positioned in all-English school rooms and no longer obtain bilingual schooling providers. The segregation of these college students is reflective of both neighborhood segregation and a decision on the a part of some districts to group these college students collectively in order to present them with qualified academics and bilingual programs which might be scarce, stated Richard Fry, a senior analysis affiliate for the Pew Hispanic Center. I say that they were bilingual as a result of though they did not have a big vocabulary they understood each languages and reacted accordingly when spoken in English or Spanish. The university is highly ranked by the U.S News and World report which placed it on fifth place within the class of finest online colleges for training with greatest online education program. One other important situation of this chapter lies on the general public debate of bilingual training in which theoretical views of supporters of bilingual education are in opposition to those of opponents of bilingual training. Next PostNext How To Get Your Dog To Stop Peeing All Over The House?	{ "redpajama_set_name": "RedPajamaC4" }	7,104
Ungkyrkorörelsen, ibland kallad Uppsalateologin, var en nationellt inriktad väckelserörelse inom Svenska kyrkan under början av 1900-talet. Rörelsen utgick från Uppsala och Uppsala Kristliga Studentförbund (grundat 1901) under medverkan av ärkebiskop Nathan Söderblom, biskop Einar Billing, biskop J.A. Eklund och uppsalastudenten Manfred Björkquist. Åren 1909–1911 genomförde Ungkyrkorörelsen så kallade "studentkorståg" i många av Sveriges församlingar och Folkets Hus-lokaler under parollen "Sveriges folk - ett Guds folk". Mottot representerande folkkyrkotanken, det vill säga att kyrkan har en kallelse att förverkliga Kristi evangelium hos hela svenska folket, dvs "förmedla syndernas förlåtelse till Sveriges folk, tron på den levande Guden, historiens levande Gud", som det uttrycktes då. Denna nya typ orientering i kyrkans historia och fokus på personliga tolkningen av kristendom kom att kallas för "Uppsalateologi", på grund av dess direkta relation till Uppsala Kristliga Studentförbund och ungkyrkorörelsen. År 1909 grundades tidskriften Vår Lösen, vars redaktör Manfred Björkquist var 1909–1926, och 1917 skapades Sigtunastiftelsen som en hemvist för rörelsen. Ungkyrkorörelsen fick stor betydelse för teologins och det kyrkliga livets förnyelse i Sverige under den första halvan av 1900-talet. Psalmerna Vår Gud är oss en väldig borg, och särskilt Fädernas kyrka, är psalmer som ofta förknippas med rörelsen. Källor "Den svenska folkkyrkans arkitekt" Svenska Dagbladet 2008-07-14 Se även Sigtunastiftelsen Manfred Björkquist Ungdomsrörelsen Svenska kyrkan Ugglan Kristendom i Uppsala Ungdomsorganisationer i Sverige	{ "redpajama_set_name": "RedPajamaWikipedia" }	971
{"url":"https:\/\/scicomp.stackexchange.com\/questions\/20241\/hcurl-conforming-n%C3%A9d%C3%A9lec-elements-to-satisfy-divb-0","text":"# H(curl) conforming N\u00e9d\u00e9lec-Elements to satisfy div(B)=0\n\nMost authors are very clear that it's very dangerous to just use $\\mathrm{H}(curl)$ conforming edge elements, which are divergence free, to satisfy $\\mathrm{div}(\\mathbf{B})=0$ and implement this condition in no other way. I just don't understand why. Could anyone explain in which case this could fail or point me to some good source explaining in which case $\\mathrm{H}(curl)$ conforming Nedelec Elements would fail to satisfy $\\mathrm{div}(\\mathbf{B})=0$ and why?","date":"2021-12-04 02:31:07","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\": 0.7176876664161682, \"perplexity\": 398.49271847865896}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964362923.11\/warc\/CC-MAIN-20211204003045-20211204033045-00228.warc.gz\"}"}	null	null
<a href='http://github.com/angular/angular.js/edit/master/src/ng/interpolate.js' class='improve-docs btn btn-primary'><i class="glyphicon glyphicon-edit"> </i>Improve this doc</a> <a href='http://github.com/angular/angular.js/tree/master/src/ng/interpolate.js#L86' class='view-source pull-right btn btn-primary'> <i class="glyphicon glyphicon-zoom-in"> </i>View Source </a> <header class="api-profile-header"> <h1 class="api-profile-header-heading">$interpolate</h1> <ol class="api-profile-header-structure naked-list step-list"> <li> <a href="api/ng/provider/$interpolateProvider">- $interpolateProvider</a> </li> <li> - service in module <a href="api/ng">ng</a> </li> </ol> </header> <div class="api-profile-description"> <p>Compiles a string with markup into an interpolation function. This service is used by the HTML <a href="api/ng/service/$compile">$compile</a> service for data binding. See <a href="api/ng/provider/$interpolateProvider">$interpolateProvider</a> for configuring the interpolation markup.</p> <pre><code class="lang-js"> var $interpolate = ...; // injected var exp = $interpolate('Hello {{name \| uppercase}}!'); expect(exp({name:'Angular'}).toEqual('Hello ANGULAR!');</code></pre> </div> <div> <h2 id="dependencies">Dependencies</h2> <ul> <li><a href="api/ng/service/$parse"><code>$parse</code></a></li><li><a href="api/ng/service/$sce"><code>$sce</code></a></li> </ul> <h2 id="usage">Usage</h2> <p><code>$interpolate(text, [mustHaveExpression], [trustedContext]);</code></p> <section class="api-section"> <h3>Arguments</h3> <table class="variables-matrix input-arguments"> <thead> <tr> <th>Param</th> <th>Type</th> <th>Details</th> </tr> </thead> <tbody> <tr> <td> text </td> <td> <a href="" class="label type-hint type-hint-string">string</a> </td> <td> <p>The text with markup to interpolate.</p> </td> </tr> <tr> <td> mustHaveExpression <div><em>(optional)</em></div> </td> <td> <a href="" class="label type-hint type-hint-boolean">boolean</a> </td> <td> <p>if set to true then the interpolation string must have embedded expression in order to return an interpolation function. Strings with no embedded expression will return null for the interpolation function.</p> </td> </tr> <tr> <td> trustedContext <div><em>(optional)</em></div> </td> <td> <a href="" class="label type-hint type-hint-string">string</a> </td> <td> <p>when provided, the returned function passes the interpolated result through <a href="api/ng/service/$sce#getTrusted">$sce.getTrusted(interpolatedResult,</a> trustedContext) before returning it. Refer to the <a href="api/ng/service/$sce">$sce</a> service that provides Strict Contextual Escaping for details.</p> </td> </tr> </tbody> </table> </section> <h3>Returns</h3> <table class="variables-matrix return-arguments"> <tr> <td><a href="" class="label type-hint type-hint-function">function(context)</a></td> <td><p>an interpolation function which is used to compute the interpolated string. The function has these parameters:</p> <ul> <li><code>context</code>: an object against which any expressions embedded in the strings are evaluated against.</li> </ul> </td> </tr> </table> <h2>Methods</h2> <ul class="methods"> <li id="startSymbol"> <h3><p><code>startSymbol();</code></p> </h3> <div><p>Symbol to denote the start of expression in the interpolated string. Defaults to <code>{{</code>.</p> <p>Use <a href="api/ng/provider/$interpolateProvider#startSymbol">$interpolateProvider#startSymbol</a> to change the symbol.</p> </div> <h4>Returns</h4> <table class="variables-matrix return-arguments"> <tr> <td><a href="" class="label type-hint type-hint-string">string</a></td> <td><p>start symbol.</p> </td> </tr> </table> </li> <li id="endSymbol"> <h3><p><code>endSymbol();</code></p> </h3> <div><p>Symbol to denote the end of expression in the interpolated string. Defaults to <code>}}</code>.</p> <p>Use <a href="api/ng/provider/$interpolateProvider#endSymbol">$interpolateProvider#endSymbol</a> to change the symbol.</p> </div> <h4>Returns</h4> <table class="variables-matrix return-arguments"> <tr> <td><a href="" class="label type-hint type-hint-string">string</a></td> <td><p>end symbol.</p> </td> </tr> </table> </li> </ul> </div>	{ "redpajama_set_name": "RedPajamaGithub" }	7,750
Part of respecting your opponents is respecting them as opponents and thus being willing to have them win or lose, to be hurt or happy, to succeed or be crushed or be somewhere in the middle. They are grownups now: they can and should take care of themselves. They are your opponent and that is what they are there for, so respect them for it and give them the roughest, most aggressively unforgiving, unrelenting and challenging game you can along with and as part of your respect.	{ "redpajama_set_name": "RedPajamaC4" }	5,505
6 Steps Necessary to Change Your Life Many people want to change their lives, but there's a difference between wanting to change and taking steps to change. Change is possible. But first, a person must be willing to change. When obstacles arise (and they will), it's tempting to throw in the towel altogether. Instead, by changing one's mindset, it's possible to change one's life for the better. When someone sees the need for change, they have reached illumination. It is, to be precise, a light-bulb moment—the point at which a person finally understands their situation in a whole new way. As Ephesians 5:8 (MSG) says, "You groped your way through that murk once, but no longer. You're out in the open now. The bright light of Christ makes your way plain. So, no more stumbling around. Get on with it." (more…) Controlling Anger We have all been there at one point in our lives. Our spouse, kids, boss, neighbor, or friend does or says something that upsets us, and all of a sudden, we feel the pressure begin to build. You start to feel tight around the collar like the dry cleaner put too much starch on it. Heat builds up around the ears, and you feel your face begin to flush. As you continue to think about the action or what was said, it increases. Your heart pounds, and life begins to go in slow motion. You can almost hear NASA control: "T-minus 10, 9, 8…" What you do or say at this moment may affect the next few minutes, hours, days, or even years. (more…)	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	251
package com.gordonwong.materialsheetfab.sample.adapters; import android.content.Context; import android.support.v7.widget.CardView; import android.support.v7.widget.RecyclerView; import android.text.TextUtils; import android.view.LayoutInflater; import android.view.View; import android.view.ViewGroup; import android.widget.ImageView; import android.widget.LinearLayout; import android.widget.TextView; import com.gordonwong.materialsheetfab.sample.R; import com.gordonwong.materialsheetfab.sample.models.Note; /** * Created by Gordon Wong on 7/18/2015. * * Adapter for the all items screen. */ public class NotesAdapter extends RecyclerView.Adapter<NotesAdapter.ViewHolder> { private Note[] notes; public NotesAdapter(Context context, int numNotes) { notes = generateNotes(context, numNotes); } @Override public NotesAdapter.ViewHolder onCreateViewHolder(ViewGroup parent, int viewType) { View v = LayoutInflater.from(parent.getContext()).inflate(R.layout.list_item_note, parent, false); return new ViewHolder(v); } @Override public void onBindViewHolder(ViewHolder holder, int position) { Note noteModel = notes[position]; String title = noteModel.getTitle(); String note = noteModel.getNote(); String info = noteModel.getInfo(); int infoImage = noteModel.getInfoImage(); int color = noteModel.getColor(); // Set text holder.titleTextView.setText(title); holder.noteTextView.setText(note); holder.infoTextView.setText(info); // Set image holder.infoImageView.setImageResource(infoImage); // Set visibilities holder.titleTextView.setVisibility(TextUtils.isEmpty(title) ? View.GONE : View.VISIBLE); holder.noteTextView.setVisibility(TextUtils.isEmpty(note) ? View.GONE : View.VISIBLE); holder.infoLayout.setVisibility(TextUtils.isEmpty(info) ? View.GONE : View.VISIBLE); // Set padding int paddingTop = (holder.titleTextView.getVisibility() != View.VISIBLE) ? 0 : holder.itemView.getContext().getResources() .getDimensionPixelSize(R.dimen.note_content_spacing); holder.noteTextView.setPadding(holder.noteTextView.getPaddingLeft(), paddingTop, holder.noteTextView.getPaddingRight(), holder.noteTextView.getPaddingBottom()); // Set background color ((CardView) holder.itemView).setCardBackgroundColor(color); } @Override public int getItemCount() { return notes.length; } private Note[] generateNotes(Context context, int numNotes) { Note[] notes = new Note[numNotes]; for (int i = 0; i < notes.length; i++) { notes[i] = Note.randomNote(context); } return notes; } public static class ViewHolder extends RecyclerView.ViewHolder { public TextView titleTextView; public TextView noteTextView; public LinearLayout infoLayout; public TextView infoTextView; public ImageView infoImageView; public ViewHolder(View itemView) { super(itemView); titleTextView = (TextView) itemView.findViewById(R.id.note_title); noteTextView = (TextView) itemView.findViewById(R.id.note_text); infoLayout = (LinearLayout) itemView.findViewById(R.id.note_info_layout); infoTextView = (TextView) itemView.findViewById(R.id.note_info); infoImageView = (ImageView) itemView.findViewById(R.id.note_info_image); } } }	{ "redpajama_set_name": "RedPajamaGithub" }	4,326
int main(int argc, char** argv) { ros::init(argc, argv, "move_square"); ros::NodeHandle nh; ros::Time::waitForValid(); const double squareSize = nh.param("size", 500.0); ROS_INFO_STREAM("move_square node launched for a square of " << squareSize << " mm"); snd_msgs::MoveLinearGoal linearGoal; linearGoal.distance = squareSize; snd_msgs::MoveAngularGoal angularGoal; angularGoal.angle = 90; actionlib::SimpleActionClient<snd_msgs::MoveLinearAction> linearAc( nh, "/polar_control/move_linear", true); ROS_INFO("Waiting for linear action server to start."); linearAc.waitForServer(); actionlib::SimpleActionClient<snd_msgs::MoveAngularAction> angularAc( nh, "/polar_control/move_angular", true); ROS_INFO("Waiting for angular action server to start."); angularAc.waitForServer(); for(size_t i = 0; i < 4; ++i) { ROS_INFO_STREAM("Goal distance: " << linearGoal.distance); linearAc.sendGoal(linearGoal); // wait for the action to return bool finished_before_timeout = linearAc.waitForResult(ros::Duration(40.0)); if(finished_before_timeout) { actionlib::SimpleClientGoalState state = linearAc.getState(); ROS_INFO_STREAM("Action finished: " << state.toString()); ROS_INFO_STREAM("Goal desc: " << state.getText()); if(state != actionlib::SimpleClientGoalState::SUCCEEDED) { ROS_WARN("Abort"); break; } } else { ROS_INFO("Action did not finish before the time out."); } ROS_INFO_STREAM("Goal angular: " << angularGoal.angle); angularAc.sendGoal(angularGoal); // wait for the action to return finished_before_timeout = angularAc.waitForResult(ros::Duration(40.0)); if(finished_before_timeout) { actionlib::SimpleClientGoalState state = angularAc.getState(); ROS_INFO_STREAM("Action finished: " << state.toString()); ROS_INFO_STREAM("Goal desc: " << state.getText()); if(state != actionlib::SimpleClientGoalState::SUCCEEDED) { ROS_WARN("Abort"); break; } } else { ROS_INFO("Action did not finish before the time out."); } } return 0; }	{ "redpajama_set_name": "RedPajamaGithub" }	6,384
\section{Introduction} Motivated by potentially detectable but minuscule signatures from Planck-scale or other new physics, there has been a substantial increase in tests of spacetime symmetry in gravity in recent years.\cite{string,datatables} Some novel hypothetical effects that break local Lorentz symmetry and CPT symmetry in gravitational experiments as well as solar system and astrophysical observations have been studied in recent works.\cite{review} Much of this work uses the effective field theory framework, the Standard-Model Extension (SME), that includes gravitational couplings.\cite{sme1,sme2} In other cases, the parameters in specific hypothetical models of Lorentz violation in gravity have been tested.\cite{vecCS} \section{Framework} The general framework of the SME in the pure-gravity sector can be realized as the Einstein-Hilbert action plus a series of terms formed from indexed coefficients, explicit or dynamical, contracted with increasing powers of curvature and torsion. Each term in this series maintains observer invariance of physics, while breaking ``particle" invariance, with respect to local Lorentz symmetry and diffeomorphism symmetry.\cite{sme2} One interesting and practical subset of the SME is a general description of CPT and Lorentz violation that is provided by an expansion valid for linearized gravity ($g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$). For instance, in this approximation the Lagrange density for General Relativity (GR) plus the mass dimension 4 and 5 operators controlling local Lorentz and CPT violation are given by\cite{bk06,bkx15,km16} \begin{equation} {\cal L}=-\frac {1}{4\kappa} (h^{\mu\nu} G_{\mu\nu} - \overline{s}^{\mu\kappa} h^{\nu\lambda} {\cal G}_{\mu\nu\kappa\lambda} +\frac {1}{4} h_{\mu\nu} (q^{(5)})^{\mu\rho\alpha\nu\beta\sigma\gamma} \partial_\beta R_{\rho\alpha\sigma\gamma}+...), \label{L} \end{equation} where $\kappa=8\pi G_N$, and the double dual curvature ${\cal G}$ and the Riemann curvature $R_{\rho\alpha\sigma\gamma}$ are linearized in $h_{\mu\nu}$. This lagrange density maintains linearized diffeomorphism invariance, though generalizations exist\cite{km18}, and $\overline{s}_{\mu\nu}$ and $(q^{(5)})^{\mu\rho\alpha\nu\beta\sigma\gamma}$ are the coefficients controlling the degree of symmetry breaking (they are zero in GR). \section{Experiment and Observation} The mass dimension 4 Lagrange density, the minimal gravity SME, has now been studied in a plethora of tests. The best controlled and simultaneous parameter-fitting limits come from lunar laser ranging\cite{llr}, and other laboratory experiments such as gravimetry.\cite{gravi} These place limits on the $\overline{s}_{\mu\nu}$ coefficients at the level of approximately $10^{-7}-10^{-8}$ on the 3 $\overline{s}_{TJ}$ and $10^{-10}-10^{-11}$ on 5 of the $\overline{s}_{JK}$ coefficients. Stronger limits can be countenanced from distant cosmic rays\cite{kt15}, and one combination of coefficients is bounded at $10^{-15}$ by the multimessenger neutron star inspiral event in 2017.\cite{GRBGW} Other searches for these coefficients include ones with pulsars.\cite{shao} For the mass dimension 5 coefficients in \rf{L} that break CPT symmetry, the post-Newtonian phenomenology includes a velocity-dependent inverse cubic force. This leads to an extra term in the relative acceleration of two bodies given by\cite{bh17} \begin{eqnarray} \delta a^j &=&\fr {G_N M v^k}{r^3} \big( 15 n^l n^m n^n n_{[ j } K_{k] lmn} \nonumber\\ && + 9 n^l n^m K_{[jk] lm}-9 n_{[j} K_{k] ll m} n^m -3 K_{[jk]ll} \big), \label{de_acc} \end{eqnarray} where $K_{jklm}$ are linear combinations of the coefficients $q$ in the lagrange density \rf{L}, $\vec r$ is the separation between the bodies and ${\hat n}={\vec r}/r$. Measurements of the mass dimension $5$ coefficients in \rf{de_acc} are currently scarce. There is one constraint on a combination of dimension 5 and 6 coefficients from Ref.\ \refcite{km16} in searches for dispersion of gravitational waves from distant sources and analysis with multiple gravitational wave events is underway.\cite{gwfit} Disentangled constraints on the $K_{jklm}$ coefficients from analysis of pulsar observations exists at the level of $10^{6}$ meters.\cite{shao2} This leaves room for potentially large, ``countershaded" symmetry breaking to exist in nature.\cite{kt09} Higher-order terms in the series, at mass dimension $6$ and beyond, have been constrained in short-range gravity tests.\cite{sr} \section{Extension to the nonlinear regime} While the general form for linearized gravity has been explored, only several works have explored the general SME framework beyond linearized gravity.\cite{nl} One approach is to extend the general lagrange density for linearized gravity (which is quadratic order in the metric fluctuations) to include terms of cubic and higher order terms. If we adopt the point of view of spontaneous symmetry breaking (SSB), one must consider the dynamics of the coefficients for Lorentz violation. Considering the case of a symmetric two tensor $s_{\mu\nu}$ being the Lorentz-breaking field, it is expanded in the SSB scenario as $s_{\mu\nu} = \overline{s}_{\mu\nu} + \tilde{s}_{\mu\nu}$, where $\overline{s}_{\mu\nu}$ are the vacuum expectation values and $\tilde{s}_{\mu\nu}$ are the fluctuations. The Lagrange density is a series ${\cal L} = {\cal L}^{(2)} + {\cal L}^{(3)}+...$ where $(2)$ and $(3)$ indicate the order in fluctuations $h_{\mu\nu}$ or $\tilde{s}_{\mu\nu}$. A general conservation law\cite{bluhm15}, contained in equation (9) of Ref.\ \refcite{bluhm16}, can be used to constrain the terms in the series. In the example of $s_{\mu\nu}$ it takes the form \begin{equation} \partial_\beta \left( \fr {\delta {\cal L}}{\delta h_{\gamma\beta}} \right) + \Gamma^\gamma_{\pt{\beta}\alpha\beta} \left(\fr {\delta {\cal L}}{\delta h_{\alpha\beta}} \right) + g^{\delta\gamma} s_{\delta\alpha} \partial_\beta \left( \fr {\delta {\cal L}}{\delta \tilde{s}_{\alpha\beta}} \right) + g^{\delta\gamma} {\tilde \Gamma}_{\delta\alpha\beta} \fr {\delta {\cal L}}{\delta h_{\alpha\beta}}=0, \label{constr} \end{equation} where ${\tilde \Gamma}_{\delta\alpha\beta}= (\partial_\alpha \tilde{s}_{\beta\delta}+\partial_\beta \tilde{s}_{\alpha\delta}-\partial_\delta \tilde{s}_{\alpha\beta})/2$. This equation holds ``off-shell", assuming the action obtained from ${\cal L}$ is diffeomorphism invariant. In the case of the minimal SME with just $s_{\mu\nu}$, the Lagrange density is constructed from all possible contractions of generic terms of the quadratic form $\overline{s}_{\alpha\beta} h_{\gamma\delta} \partial_\epsilon \partial_\zeta h_{\eta\theta}, \tilde{s}_{\alpha\beta} \partial_{\gamma} \partial_\delta \tilde{s}_{\epsilon\zeta}, \tilde{s}_{\alpha\beta} \partial_\gamma \partial_\delta h_{\epsilon\zeta}, ...$, the cubic form $\overline{s}_{\alpha\beta} h_{\gamma\delta} h_{\epsilon\zeta} \partial_\eta \partial_\theta h_{\kappa\lambda}, \overline{s}_{\alpha\beta} h_{\gamma\delta} \partial_\epsilon h_{\zeta \eta} \partial_\theta h_{\kappa\lambda}, h_{\alpha\beta} \tilde{s}_{\gamma\delta} \partial_\epsilon \partial_\zeta \tilde{s}_{\theta\kappa},...$, and potential terms. The sum of all such terms, each with an arbitrary parameter, is inserted into \rf{constr} and the resulting linear equations for the parameters are solved. What remains, up to total derivative terms in the action, are a set of independently diffeomorphism invariant terms. As an example of such a term produced by this expansion, we find to cubic order \begin{eqnarray} {\cal L} &\supset& \overline{s}_{\alpha\beta} \tilde{s}^{\alpha\beta} R^{(1)} + \frac 12 \tilde{s}_{\alpha\beta} \tilde{s}^{\alpha\beta} R^{(1)} -2 h^{\alpha\beta} \overline{s}_\alpha^{\pt{\alpha}\gamma} \tilde{s}_{\beta\gamma} R^{(1)} \nonumber\\ && +\overline{s}_{\alpha\beta} \tilde{s}^{\alpha\beta} (\Gamma_{\gamma\delta\epsilon} \Gamma^{\gamma\delta\epsilon} - \Gamma^{\gamma\delta}_{\pt{\gamma\delta}\delta} \Gamma_{\gamma\epsilon}^{\pt{\gamma\epsilon}\epsilon} + \frac 12 h^\gamma_{\pt{\gamma}\gamma} R^{(1)} -2 h^{\gamma\delta} R^{(1)}_{\gamma\delta}), \label{sample} \end{eqnarray} where the $(1)$ superscript indicates linear order in $h_{\mu\nu}$ and the connection coefficients are at linear order. Note that this construction generally includes dynamical terms for the fluctuations and so does not assume ``decoupling".\cite{ms} The construction including all such terms allows exploration of the regime in gravity where nonlinearities need to be considered.\cite{b19,b16} This includes higher order post-Newtonian gravity in weak-field systems and developing a multipole expansion for gravitational waves affected by Lorentz violation. \section*{Acknowledgments} This work was supported by the National Science Foundation under grant no.\ 1806871 and Embry-Riddle Aeronautical University's FIRST grant program.	{ "redpajama_set_name": "RedPajamaArXiv" }	1,543
"Warning! Do not get fooled by fake messages circulating claiming to be from Royal Hospital UAE. Please take note that this is our official website & our official mail ID is info@royalhospitaluae.com. Personalized Medicare with Love and Compassion" Royal Hospital, UAE, a 160 bed multi specialty, high-tech healthcare facility is the region's biggest private referral hospital offering effective medical care in a professional and caring environment. Top notch Specialists from premier institutions all over the world come together to provide specialty and super specialty consultations and treatment utilizing the most modern technology available ! Royal Hospital, UAE, a 160 bed multi specialty, high-tech healthcare facility is the region's biggest private referral hospital offering effective medical care in a professional and caring environment.. A State of Art Laboratory is functioning in the hospital managed by the renowned Gordon Laboratory Group (ME) Ltd. from the U.K. The Radiology Complex is equipped with state of art Imaging Modalities. Powered by Meridian IT Solutions © 2010 Royal Hospital. All rights reserved.	{ "redpajama_set_name": "RedPajamaC4" }	1,915
Xenotilapia nasus é uma espécie de peixe da família Cichlidae. Pode ser encontrada nos seguintes países: Burundi e República Democrática do Congo. Os seus habitats naturais são: lagos de água doce. Referências Bigirimana, C. 2005. Xenotilapia nasus. 2006 IUCN Red List of Threatened Species. Dados de 5 de Agosto de 2007. Xenotilapia Peixes descritos em 1995	{ "redpajama_set_name": "RedPajamaWikipedia" }	6,888
Q: XCode 5.1 -> Xcode schemes not visible on another machine I created two schemes in a project. Live and staging and it's working fine for my xcode 5.0. But then I updated my xcode to 5.1. Now xcode shows schemes only on my system and when I send code to another machine in zip-format. It opens with only default scheme and Live & Staging schemes are not visible. Although it's showing build configuration that I made to differentiate between schemes. Any ideas for this problem? A: Open scheme editor (select scheme, near stop button -> Scheme Editor) and check Shared for schemes you want to share then commit new files or so	{ "redpajama_set_name": "RedPajamaStackExchange" }	1,056
\section{Introduction} Black holes (BH) are physical entities that provide insight into the foundations of (quantum-) gravity theories. Such objects are characterized by three basic properties: mass, angular momentum, and charge. Among these properties the most fundamental is the mass, which characterizes all BHs. Angular momentum is only relevant for rotating (Kerr) BHs while charge is less relevant for astrophysical BHs. Currently, the best evidence for the existence of BH is astrophysical. In particular, a broad range of BH masses is implied by astronomical observations: from solar mass BHs as the end products of massive stars to supermassive BHs (SMBH) at the centers of galaxies with masses that may easily exceed a million solar masses \citep{sal64,lyn69}. Perhaps the first evidence for the presence of SMBH at the centers of galaxies is associated with quasars and other types of active galactic nuclei (to which we shall henceforth refer to, generally, as quasars). In those objects, gas from the medium surrounding the SMBH, that is able to lose angular momentum and spiral inwards, eventually accretes onto it. As the gas accretes, its gravitational energy is effectively converted to radiation with most of the emitted flux originating from the immediate vicinity of the SMBH, where the gas reaches the highest temperatures \citep{ss73}. The mass of SMBHs in small samples of low redshift non-active galaxies has been measured using, for example, maser- and stellar-kinematics \citep{miy95,gen97,geb00a}. For most low redshift galaxies, SMBH mass estimation is, however, indirect and involves scaling relations that were derived for small samples of objects for which direct mass measurements were possible \citep{mag98}. As one attempts to estimate the mass of black holes at higher redshifts, various complications arise: it is not clear that locally determined scaling laws apply also to higher redshift objects since galaxies and their environments evolve with cosmic time. Furthermore, high redshift objects discovered by flux-limited samples are often much brighter than their low-$z$ counterparts, and it is yet to be established whether extrapolated relations are at all relevant in that regime \citep{net03}. For these reasons, direct SMBH mass determination of intermediate- and high-redshift objects is the most reliable way to further our understanding of SMBH-galaxy formation and co-evolution, and to reach a reliable census of black holes in the universe. Direct mass determination of SMBHs at high redshifts is currently limited to quasars, which are among the brightest objects in the universe and are powered by the SMBH itself. While quasars certainly trace the galaxy population, they also present a particular epoch in a galaxy lifetime, in which the black hole is accreting gas, and is at a stage of rapid growth. Therefore, better understanding of the SMBH demography in low- and high-$z$ quasars has far-reaching implications for galaxy/black-hole formation and co-evolution \citep{fer01,sh03,os04,hop06,kol06,sh06,gh06,woo08,tra10}. A reliable method to directly measure the mass of SMBHs in quasars involves the reverberation mapping of the broad emission line region (BLR) in those objects \citep{bah72,pet93}. This region is known to be photo-ionized by the central continuum source \citep{bal78}, and its emission, in the form of broad permitted emission lines, is seen to react to continuum variations. Studies have shown that the BLR lies at distances, $R_{\rm BLR}$, that are much larger than the SMBH Schwarzschild radius, and also larger than the size of the continuum emitting region, i.e., the accretion disk. That being said, the BLR lies at small enough distances so that its kinematics is primarily set by the SMBH, which has a prominent contribution to the gravitational potential on those scales. That this is the case may be deduced from the velocity dispersion of the emitted gas, as inferred from the width of the broad emission lines, which is typically of order a few\,$\times 10^3\,{\rm km~s^{-1}}$, as well as from the fact that, in luminous quasars, line variations lag behind relatively rapid continuum variations with typical time delays, $t_{\rm lag}\sim R_{\rm BLR}/c\simeq 200$\,days, where $c$ is the speed of light \citep[and references therein]{kas00}. Technically, reverberation mapping involves multi-epoch spectroscopic observations so that continuum and emission line variations may be individually traced \citep{di93,net97,kas00}. Continuum and emission line light curves can then be used to infer the size of the broad emission line region using a cross-correlation analysis \citep[and \S2.2]{bm82}. By measuring $R_{\rm BLR}$ and the velocity dispersion of the gas (via spectral modeling of the emission line profile and by defining a measure of the velocity dispersion of the gas), the SMBH mass may be deduced, up to a geometrical factor of order unity \citep{pw99,kro01}. Mass determinations using this technique have been shown to be consistent with the results of other independent methods \citep{geb00,fer01,pet10}, and are thought to be accurate to within 0.5\,dex \citep{kro01}. Nowadays, reverberation mapping has a central role in determining SMBH masses in active galaxies and for shedding light on the physics of the central engine in these objects\footnote{It is worth noting that, in addition to SMBH mass measurement, the structure and geometry of the BLR may be probed by means of velocity-resolved reverberation mapping of the broad emission lines \citep{kor95,wan95,kol03,ben10}. Such investigations cannot be carried out using broadband photometric reverberation techniques, which do not resolve the emission lines, and will not be discussed here any further.}. Nevertheless, despite several decades of spectroscopic monitoring of quasars, reliable BLR sizes and corresponding SMBH mass measurements have been determined for only a few dozens of low-$z$ objects \citep{gas86,kor91,di93,kas00,pet04,kas05,pet05,ben09,pet10,bar11}. Using the results of reverberation mapping, various scaling relations were found and quantified including, for example, the $R_{\rm BLR}$ (or SMBH mass) vs. quasar luminosity relation \citep{kor91,kas00,pet04,kas07,ben09}, the SMBH mass vs. the quasar's optical spectral properties relation \citep{gh05}, and the black hole-host galaxy bulge mass relation in quasars \citep{wan99}. Despite being derived from small samples of nearby objects, the aforementioned relations are often extrapolated to reflect on the quasar population as a whole \citep{ves02,ves09}, sometimes with little justification \citep{net03,kas07,mcg08,z08}. Potential biases arising from uncertain extrapolations to high redshifts, or to different sub-classes of quasars, may crucially affect our understanding of black hole growth and structure formation over cosmic time \citep{kh00,net03,col04,pel07,tra10}. Clearly, an efficient way to implement the reverberation mapping technique is highly desirable as it would alleviate many of the uncertainties currently plaguing the field. Motivated by upcoming photometric surveys that will cover a broad range of wavelengths and will regularly monitor a fair fraction of the sky with good photometric accuracy (e.g., the {\it Large Synoptic Survey Telescope}, LSST), we aim to provide proof-of-concept that photometric reverberation mapping of the BLR in quasars is feasible. In a nut shell, instead of spectrally separating line and continuum light curves from multi-epoch spectroscopic observations, we take advantage of the different variability properties of continuum and line processes and separate them at the light curve level. In this work we have no intention to address the full scope of issues associated with time-series analysis but rather hope to trigger further observational and theoretical work in broadband reverberation mapping of quasar BLRs. This paper is organized as follows: section 2 presents the photometric approach to reverberation mapping using analytic means and by resorting to simulations of quasar light curves. The reliability of the photometric method for line-to-continuum time lag measurement is investigated by means of simulations in \S3. In \S4 we apply the method to a subset of the Palomar-Green (PG) quasar sample \citep{sg83}, for which previous measurements of the BLR size are available \citep{kas00}. Summary follows in \S5. \begin{figure} \plotone{f1.eps} \caption{A realization of the broadband photometric light-curves of a typical quasar (see text). Flux variations in the $X$ (black) and $Y$ (red) bands show an overall similarity due to the large contribution of highly correlated continuum emission processes to both bands. The response of an emission line to continuum variations in the $X$-band is shown in dashed blue lines ($t_{\rm lag}^l=200$\,days was assumed). The $Y$-band light-curve includes a $13\%$ contribution to the total flux from a broad emission line (solid blue line). A case where the noise level is negligible ($\delta /\left < f \right > =10^{-3}$) was assumed for clarity.} \label{lc} \end{figure} \section{The formalism} Here we present the algorithm for reverberation mapping the broad emission line regions in quasars using broadband photometric means. We first describe our model for quasar light curves (\S2.1) and then analyze them using new statistical methods that are developed in \S2.2. \subsection{A model for the light-curve of quasars} To demonstrate the feasibility of photometric reverberation mapping, we first resort to simulations. Specifically, we rely on current knowledge of the typical spectrum of quasars, and the variability properties of such objects, to construct a model for the light curves in different spectral bands. \subsubsection{Continuum emission} Quasar light-curves are reminiscent of red noise, following a power spectrum $P(\omega) \sim \omega^{\alpha}$ ($\omega$ is the angular frequency) with $\alpha\simeq -2$ \citep[and references therein]{giv99}. Here we model the pure continuum light-curve in some band $X$ as a sum of discrete Fourier components, each with a random phase $\phi_i$: \begin{equation} f_X^c(t)=\sum_{i=1}^\infty A(\omega_i){\rm sin}(\omega_it+\phi_i). \end{equation} The amplitude of each Fourier mode, $A(\omega_i)\propto \omega_i^{\alpha/2}$ and $t$ is time\footnote{We note that, realistically, the power at each frequency is distributed around the above-defined powerlaw, forming an effective envelope with some characteristic width. We ignore this additional complication here, which introduces yet another degree of freedom in the model but does not qualitatively change the results. Real light curves are analyzed in \S4.}. We define the normalized variability measure, $\chi=\sqrt{\sigma_f^2-\delta^2}/\left < f \right >$ [c.f. \citet{kas00} who express this in per-cent], where $ \left < f \right >$ is the mean flux of the light-curve, $\sigma_f$ is its standard deviation, and $\delta$ is the mean measurement uncertainty of the light curve. For the Palomar-Green (PG) sample of quasars \citep{sg83}, the continuum normalized variability measure $\chi_c \sim 0.1-0.2$ \citep{kas00} and, typically, $\delta\gtrsim 0.01$ (see table 1). A light-curve realization, with $\delta\ll \sigma_f$ and $\chi_c\sim0.2$, is shown in Fig. \ref{lc}. Quasar emission occurs over a broad range of photon energies, from radio to hard X-ray wavelengths and originates from spatially distinct regions in the quasar. In the optical bands, there is some evidence for red continua to lag behind blue continua \citep{kro91,col98,ser05,bac09}. Without loss of generality, we associate the $Y$-band with the redder parts of the spectrum so that its instantaneous continuum flux may be expressed as a function of the flux in the $X$-band, \begin{equation} f_Y^c(t)= f_X^c * \psi^c \equiv \int_{-\infty}^\infty d\tau f_X^c(\tau) \psi^c(t-\tau), \label{conVol} \end{equation} where $\psi^c(\delta t)$ is the continuum transfer function\footnote{All transfer functions, $\psi$, satisfy the normalization condition $\int_{-\infty}^{+\infty} d(\delta t) \psi(\delta t)=1$.}. Presently, $\psi^c$ is poorly constrained by observations and only the inter-band continuum time delay, $t_{\rm lag}^c$, is loosely determined \citep[and references therein]{bac09}. Nevertheless, so long as $R_{\rm BLR}/c \gg t_{\rm lag}^c$, the particular form of $\psi^c$ is immaterial to our discussion (see \S3.2.3). For simplicity, we shall therefore take $\psi^c$ to be a gaussian with a mean of $t_{\rm lag}^c$ and a standard deviation $t_{\rm lag}^c/2$. \subsubsection{Line emission} \begin{figure} \plotone{f2.eps} \caption{The emission line transfer functions used in this work to model the broad lines' response to continuum variations (see text). While a gaussian transfer function (thick line) is relatively well localized in the time domain, the transfer function which is reminiscent of those discussed in \citet{mao90} extends to much longer times. The latter transfer function naturally results in long-range correlations between line and continuum processes (whose effects are discussed in \S3.2.5), and roughly characterizes the reaction of a geometrically thick BLR that is isotropically encompassing the central continuum source. We note that both transfer functions are normalized such that the line-to-continuum time-delay is $\simeq t_{\rm lag}^l$.} \label{trans} \end{figure} Emission by photo-ionized (BLR) gas is naturally sensitive to the continuum level of the ionizing source. In particular, for typical BLR densities and distances from the SMBH, the response time of the photoionized gas to continuum variations is much shorter than the light travel time across the region, and so may be considered instantaneous. Therefore, the flux in some line $i$, which is emitted by the BLR, can be written, in the linear approximation, as \begin{equation} f_i^l= f_X^c * \psi_i^l, \end{equation} where the transfer function $\psi_i^l$ depends, essentially, on the geometry of the BLR region as seen from the direction of the observer \citep[and references therein]{hor04}. Our present knowledge of $\psi^l(\delta t)$ for the quasar population as a whole, is, however, rather limited [see \citet{go93} for a few relevant forms for $\psi^l$]. In what follows we shall consider two forms for the transfer function: a) a gaussian kernel with a mean $t_{\rm lag}^l$ and a standard deviation $t_{\rm lag}^l/4$, and b) a transfer function with a flat core in the range $ 0 \leq \delta t \leq 1.36 t_{\rm lag}^l$, with a following linear decline to zero at $\delta t=2.72t_{\rm lag}^l$ (such a transfer function also results in a time-lag being $\simeq t_{\rm lag}^l$). The latter transfer function is motivated by the observations of the Seyfert 1 galaxy NGC\,4151 \citep{mao90} and corresponds to a case where a geometrically thick, spherical shell of broad emission line gas isotropically encompasses the continuum emitting region. The different transfer functions defined above will henceforth be termed (a) the gaussian and (b) the thick-shell BLR models. The transfer functions are depicted in figure \ref{trans}. The main qualitative difference between the two transfer functions, as far as light curve behavior is concerned, is that while the gaussian one is relatively localized in time, the thick shell BLR model is much broader, and results in longer range correlations between the light curves of the two bands. Unless otherwise stated, our results are presented for the gaussian model. Cases for which we find a qualitative difference between the two transfer functions are explicitly treated (e.g., \S3.2.5). \begin{figure} \plottwo{f3a.eps}{f3b.eps} \caption{Statistical estimators for evaluating the line to continuum time delay. {\it Left:} The ACFs for the $X$ and $Y$ bands are shown (same color coding applies as that in Fig. \ref{lc}), as well as the CCF of the two bands (black curve). Note the excess power in the ACF of the $Y$-band, as well as in the CCF of the two bands, relative to the ACF of the $X$-band (gray hatched region). This results from relatively long-range correlations between line and continuum variations, which occur on BLR light travel times. The excess power is the sought after signal (see text). {\it Right:} The statistical estimators, $\xi_{\rm CA},~\xi_{\rm AA}$, which quantify the excess power due to line emission, are consistent with $\xi_{lc}$ (see legend). The position of the peaks at positive time-lags (the only ones that preserve causality) is identified with the line-to-continuum time-delay. Evidently, all methods give consistent predictions for the time-lag, which is also in agreement with the input value, $t_{\rm lag}^l$ (dash-dotted black line).} \label{acfccf} \end{figure} A simulated light-curve of an emission line that is driven by continuum variations (assuming $t_{\rm lag}^l=200$\,days) is shown in figure \ref{lc} (blue curves). Time-delay and smearing effects, with respect to the continuum light-curve, are clearly visible. Here too, we define a normalized variability measure for the line, $\chi_l$, and consider $\chi_l\simeq 0.75 \chi_c$ \citep[see their table 5]{kas00}. The sum of continuum and lines' emission to the total flux in band $Y$ is therefore given by \begin{equation} f_Y(t)= f_Y^c(t)+f_Y^l(t) = f_Y^c(t)+\sum_i f_i^l(t), \label{y} \end{equation} where $f_Y^l$ is the total flux in the emission lines. While a similar expression applies, in principle, also for the total flux in band $X$, unless otherwise stated, we shall assume that $f_X=f_X^c$, i.e., that emission line contribution to band $X$ is either negligible or that the line variability associated with it operates on timescales that are much too short to be resolved by the cadence of the experiment, or too long to be detectable in the time-series. For simplicity, we shall limit our discussion to a single emission line contributing to the $Y$-band (this restriction is relaxed in \S3.2.4). The contribution of the emission line(s) to the total flux in the $Y$-band requires knowledge of the equivalent width of the broad emission line, $W$ [typically $\lesssim 100$\AA\ in the optical band for strong lines \citep{dvb01}], the transmission curve of the broadband filter, as well as the redshift of the quasar. For example, for filters with sharp features in their response function, even small redshift changes could lead to substantial differences in the relative contribution of the emission line to the total flux in the band, $\eta$. For a flat filter response function over a wavelength range $W_b$ (typically, $W_b\sim10^3$\,\AA\ for broadband filters), $\eta=W/W_b$. (A more accurate treatment of the line contribution to the broadband flux for the case of Johnson-Cousins filters is given in \S4.) Example light-curves for the $X$ and $Y$ bands, assuming $\eta=0.133$, are shown in figure \ref{lc}. In the following calculations, we shall work with normalized light-curves that have a zero mean and a unit variance, i.e., $f \to (f- \left < f \right > )/\sigma_f$. \subsection{Measuring the Line to Continuum Time Delay} The spectroscopic approach to measuring the size of the BLR cross-correlates pure line and continuum light-curves so that the position of the peak (or the center of mass) of the cross-correlation function (CCF)\footnote{Evaluation of the cross- and auto- correlation functions is done here via the definition of \citet[see his equation 6]{wel99} for the local cross-correlation function. Unless otherwise stated, we use the interpolated CCF method of \citet{gas86}.}, $\xi_{lc}(\delta t)= f_Y^c * f_Y^l$, marks the line to continuum time-delay \citep[and references therein]{pet88}. However, line and continuum light curves cannot be disentangled using pure broadband photometric means and the data include only their combined signal. In fact, light-curves in different bands are rather similar due to the dominant contribution of highly correlated continuum emission processes (Fig. \ref{lc}). Therefore, a naive analysis in which the CCF between the light curves of different bands is calculated would reflect more on $t_{\rm lag}^c$ than on $t_{\rm lag}^l$ (see Fig. \ref{acfccf}). Seeking to measure the time-delay between line and continuum processes using broadband photometric means, it is important to realize that instead of spectrally distinguishing between line and continuum light curves, it is possible, under certain circumstances, to separate these processes at the light curve level. In particular, the line light curve, $f_Y^l = f_Y-f_Y^c\simeq f_Y-f_X^c$ since $f_Y^c\simeq f_X^c$\footnote{This is expected to be true at least for adjacent bands since the signal is largely due to continuum emission processes, such as black-body radiation, which cover a broad wavelength range. Interestingly, this broad similarity of light curves in different bands seems to be true also for the (hidden) far-UV and optical bands given the success of the spectroscopic reverberation mapping technique in measuring line to continuum time-delays. The effects of inter-band time-delay and time-smearing of the continuum signal may account for the observed structure function differences between adjacent bands \citep{mcl10}, and are further discussed in \S3.2.3.}. This is justified for quasars since the continuum auto-correlation timescale [$\gtrsim 10^2$\,days \citep{giv99}, possibly reflecting on viscous timescales in the accretion flow] is much longer than $t_{\rm lag}^c$ [being of order days \citep{ser05} and resulting from irradiation processes in the accretion disk, which take place on light travel time scales]. Furthermore, as $f_X=f_X^c$, $f_Y^l\simeq f_Y-f_X$, and we obtain, \begin{equation} \xi_{lc}\simeq \xi_{\rm CA}(X,Y) \equiv (f_Y-f_X) f_X=f_Y f_X -f_X * f_X, \label{theta} \end{equation} where the rightmost term is just the difference between the CCF of the light-curves and the auto-correlation function (ACF) of the light curve with the negligible line contribution (see \S3.2 for cases in which emission lines contribute also to $f_X$). Graphically, this difference is simply the shaded region shown in the left panel of figure \ref{acfccf}, which arises due to correlated line and continuum processes adding power on $t_{\rm lag}^l$-scales. Using these statistics, and given a rather non-restrictive set of assumptions (see more in \S3), it seems, in principle, possible to recover the line-to-continuum time-delay using {\it pure} broadband photometric data. In particular, for the example shown in figure \ref{acfccf}, $\xi_{\rm CA}$ peaks at the correct time-lag, and is in agreement with the time-lag estimate as obtained using the spectroscopic method, i.e., using $\xi_{lc}$. The estimator defined by equation \ref{theta} provides a good approximation for $\xi_{lc}$ in cases where the emission line contribution to the band is sub-dominant (see below). For the more general case (e.g., when narrow band filters are concerned), it is possible to define a somewhat more generalized version where $\xi_{\rm CA}'\equiv f_Y * f_X - (1-\eta) f_X * f_X$, which is applicable for cases in which the emission line contribution to the $Y$-band is considerable or even dominant\footnote{In this case, $\lim_{\eta \to 1 }\xi_{\rm CA}'=\xi_{lc}$. For $\eta \ll 1$, as appropriate for broadband photometry, $\xi_{\rm CA}'\simeq \xi_{\rm CA}$.}. Nevertheless, with this definition, $\eta$ needs to be observationally known, which is not always the case, especially when large (photometric) redshift uncertainties are concerned, the filter response function has sharp features \citep{che11}, or the quasar spectrum, if available, is noisy. Focusing on broadband photometric light curves in this work, the contribution of emission lines to the band is, typically, of order a few per cent (see \S4). We therefore restrict the following discussion to $\xi_{\rm CA}$ as defined by equation \ref{theta}. \begin{figure} \epsscale{1.2} \plotone{f4.eps} \caption{Averages of $\xi_{\rm CA}$ (blue) ,$\xi_{\rm AA}$ (red) as well as for $\xi_{lc}$ (black) for ensembles of statistically equivalent quasars (the number of objects in each ensemble is indicated above each column). Upper (lower) panels correspond to a noise-level $\delta / \left < f \right >=0$ (0.03). In all simulations the total duration of the light curve is $4t_{\rm lag}^l$, and visits are $t_{\rm lag}^l/50$ apart. The relative contribution of the emission line to the filter flux, $\eta \simeq 0.13$. Clearly, finite sampling of the light curves, even in the absence of measurement noise, could result in the peaks being offset with respect to $t_{\rm lag}^l$. This is true for individual objects as well as for small samples. Generally, however, upon averaging over many objects, the mean estimators peak at $\delta t \simeq t_{\rm lag}^l$ (right panels). When measurement noise is present, larger ensemble are required for the peak of the $\left < \xi_{\rm CA} \right > ,~ \left < \xi_{\rm AA} \right >$ and $\left < \xi_{lc} \right >$ to coincide with $t_{\rm lag}^l$ to good accuracy (e.g., compare the top and bottom panels in the second column from the left). Overall, the photometric and the spectroscopic estimators give consistent results, which is especially true for large ensembles.} \label{ensembles} \end{figure} It is possible to define an additional statistical estimator to measure line to continuum time-delays using broadband photometry. Specifically, by using the fact that, for broadband filters and to zeroth order approximation, $f_X \simeq f_Y$ (see Fig. \ref{lc}), we may approximate $\xi_{\rm CA}$ by \begin{equation} \xi_{\rm AA}(X,Y) \equiv f_Y * f_Y -f_X * f_X. \label{xiaa} \end{equation} This estimator is just the difference between the ACFs of the light-curves in each band. A more intuitive way to understand the above definition may be gained by looking at figure \ref{acfccf}, where the ACF of filter $Y$ shows more power on $t_{\rm lag}^l$ timescales due to relatively long-term correlations between line and continuum processes. In contrast, the ACF of filter $X$, being devoid of line emission, does not show a corresponding power-excess. We note that both definitions of $\xi$ are quasi anti-symmetric with respect to $X$ and $Y$ exchange. We find that, as far as $t_{\rm lag}^l$ measurements are concerned, both estimators, $\xi_{\rm CA}$ and $\xi_{\rm AA}$, give, in principle, consistent results (see Fig. \ref{acfccf}). While $\xi_{\rm CA}$ has the advantage of being a more reliable estimator of $\xi_{lc}$ (see more in \S3), $\xi_{\rm AA}$ has the advantage of not containing a cross-correlation term of the form $f_Y * f_X$. This property allows one to evaluate $\xi_{\rm AA}$ even in cases where non-contemporaneous light curves are available so long as a) both light curves are adequately sampling the line transfer function, and b) quasar variability results from a stationary process (see \S3.2.5). Moreover, if the variability properties for an ensemble of objects (of some given property) are statistically similar, and the auto-correlation terms, $f_Y * f_Y$ and $f_X * f_X$, are well determined in the statistical sense (see \S3.1.1), then it is not required for the same objects to be observed in both bands. This opens up the possibility of monitoring different parts of the sky in each band, and using the acquired data to statistically measure the mean line to continuum time-delay. That being said, large ensembles may be required to achieve a reliable measurement of $t_{\rm lag}^l$ since $\xi_{\rm AA}$ is, in fact, an approximation to an approximation of $\xi_{lc}$, and its results are expected to be somewhat less reliable than both $\xi_{\rm CA}$ and $\xi_{lc}$ (this is further discussed in \S3). A note is in order concerning the maximal value of $\xi_{\rm AA}(\delta t)$ and $\xi_{\rm CA}(\delta t)$. Unlike the spectroscopic technique where $\xi_{lc}(\delta t)$ is, in fact, the Pearson correlation coefficient and its value measures how well continuum and line processes are correlated at any $\delta t$, for the photometric estimators defined here, the peak value depends also on the relative contribution of the emission line(s) to the broad band flux. The latter depends not only on the properties of the quasar, but also on the transmission curve of the broadband filter, the variability measure of the line, and the redshift of the object. Therefore, the significance of the result is not directly implied by the value of the peak in $\xi_{\rm AA}$ and $\xi_{\rm CA}$, and quantifying it requires a different approach, as we shall discuss in \S\S3,4. \section{Simulations} While the reliability of the spectroscopic reverberation mapping technique has been assessed in numerous works [e.g., \citet{gas87} and references therein], we have yet to demonstrate it for the photometric method. In this section we take a purely theoretical approach and use a suite of Monte Carlo simulations to test the method and quantify its strengths and weaknesses. \subsection{Sampling \& Measurement Noise} All light curves are finite with respect to their duration time, $t_{\rm tot}$, as well as the sampling interval, $t_{\rm sam}$. This fact alone introduces potentially substantial "noise" in the cross-correlation analyses (whether photometric or spectroscopic). The effect of finite sampling is seen in Fig. \ref{ensembles} (left panels) where the light curves of individual objects are analyzed, showing that $\xi_{\rm CA},~\xi_{\rm AA}$ formally peak at values that could be significantly removed from $t_{\rm lag}^l$. This implies that even in the ideal case, where measurement noise is negligible, the deduced time-lag might be significantly offset from the true value. While similar situations occur also with spectroscopic data, our simulations indicate that the photometric estimators are, statistically, more prone to such effects. In addition to finite sampling there is measurement noise, which reflects on the ability of any statistical estimator to recover the true time-lag with good accuracy. On an individual case basis, the effect of noise is seen to give rise to more erratic $\xi_{\rm CA}$ and $\xi_{\rm AA}$ behavior, potentially leading to substantial deviations in the position of the peak from the true time-lag. In fact, for the particular case shown (Fig. \ref{ensembles}), $\xi_{\rm CA}$ has a kink at $t_{\rm lag}^l$ and peaks only at $\sim 3t_{\rm lag}^l$. In this particular case, $\xi_{\rm AA}$ happens to be less affected by noise, and formally peaks at $\sim 1.15t_{\rm lag}^l$ due to a narrow feature being the result of noise. Sampling and measurement noise are manifested as small scale fluctuations in all statistical estimators. In particular, in some cases (see Figs. \ref{ensembles} \& \ref{smooth}), large amplitude fluctuations are observed, which peak at long timescales, $\delta t / t_{\rm lag}^l>1$, and their value could exceed the value of the true peak (i.e., that which is associated with, and peaks at, $\sim t_{\rm lag}^l$). Clearly, an algorithm that would aid to properly identify the relevant signal, would be of advantage. To this end we use a physically-motivated algorithm related to the geometry of the BLR: the fact that the BLR encompasses the continuum emitting region means that the width of its line transfer function should be of order the time-lag. Therefore, one expects the relevant signal in $\xi_{\rm CA},~\xi_{\rm AA}$ to peak at $t_{\rm lag}^l$ with a width, $\Delta (\delta t)\sim t_{\rm lag}^l$. In particular, narrower features with $\Delta (\delta t)/t_{\rm lag}^l \ll 1$ are likely to be merely the result of noise or sampling, and although they could reach large amplitudes, they are of little relevance to the time-lag measurement. The aforementioned properties of the line transfer function motivate us to consider the following algorithm to aid in the identification of the time-lag: we resample $\xi_{\rm CA}$ and $\xi_{\rm AA}$ onto a uniform logarithmic grid, which preserves resolution in $\Delta (\delta t)/\delta t$ units. We then apply a running mean algorithm such that the boxcar size corresponds to a fixed range in logarithmic scale. This procedure smooths out narrow features having $\Delta (\delta t)/\delta t \ll 1$. Examples for smoothed statistical estimators are shown for two levels of noise in figure \ref{smooth}. Clearly, narrow features at large $\delta t$ are significantly suppressed without diminishing the power of similar $\Delta (\delta t)$ features centered at smaller $\delta t$. It should be noted, however, that using this technique does not assure that the correct time-lag is recovered (see above). Furthermore, in some cases, multiple peaks may be found and additional methods are required to assess their significance (see \S4). \begin{figure} \plotone{f5.eps} \caption{Photometric time-lag measurements for individual quasars having noisy data ($\delta/\left < f \right >=0.03/0.01$ in the top/bottom panel). Calculated $\xi_{\rm CA}$'s (blue curves) show multiple peaks, some of which are narrow and peak at $\delta t > t_{\rm lag}^l$, potentially with large amplitudes (note the feature at $\delta t/t_{\rm lag}^l\simeq 3$). Smoothing of the signal using a running mean algorithm over a uniform logarithmic grid suppresses narrow features and highlights the peak associated with the time lag (cyan curves; see text). Red arrows mark the peak at $t_{\rm lag,measured}^l$, which, in this example, deviates only slightly from the input lag (dashed vertical line at $\delta/t_{\rm lag}^l=1$).} \label{smooth} \end{figure} \begin{figure} \plotone{f6.eps} \caption{Measured time-lag distributions for a sample of 100 quasars (having 800 points in each light curve and with $t_{\rm tot}/t_{\rm lag}^l=5$; $\eta\simeq 0.13$ was assumed), and for several noise levels (see legend). Clearly, for all cases, the distributions peak at the input time lag (dashed line), demonstrating the feasibility of the photometric approach for time-lag measurement. As expected, noisier light curves result in broader time lag distributions. The distributions show a tail extending to large $t_{\rm lag,measured}^l/t_{\rm lag}^l$-values and are clearly non-gaussian.} \label{smooth1} \end{figure} Testing for the accuracy of time-lag measurements using noisy data, we have conducted uniformly sampled light curves simulations of a quasar (100 realizations were used having 800 points in each light curve, with a total light curve duration, $t_{\rm tot}$, satisfying $t_{\rm tot}/t_{\rm lag}^l=5$ and $\eta\sim 0.13$) at several levels of measurement noise. For each realization, $\xi_{\rm CA}$ was calculated and $\delta t$ where it peaks was identified with $t_{\rm lag,measured}^l$ and logged. The resulting time-lag statistics are shown in figure \ref{smooth1}. The distribution is clearly non-gaussian and shows a peak at the input time lag with a tail extending to long time-lags. Not unexpectedly, the dispersion in time-lag measurements increases with the noise level. For the particular examples shown, taking $\delta / \left < f \right >=0.01$ [a noise level typical of photometric light curves of quasars \citep{kas00}], $\sim 60$\% of the measurements will result in $t_{\rm lag, measured}^l$ being within $\pm 10$\% of the true lag. For noisier data with $\delta / \left < f \right >=0.1$, only $\sim 30$\% of all realizations result in a time lag within $\pm10$\% of the true time lag. These calculations demonstrate that photometric reverberation mapping in the presence of measurement noise typical of photometrically monitored quasars is indeed feasible, and that the measurement is, in principle, reliable. When noisier data are concerned, the uncertainty on the time-lag is larger. \begin{figure} \plottwo{f7a.eps}{f7b.eps} \caption{Relative measurement uncertainties on $t_{\rm lag,measured}^l$. {\it Left:} One standard deviation of the measured time-lag distribution functions (i.e., the uncertainty on $t_{\rm lag,measured}^l$) as obtained by measuring the time delay for individual realizations in a large ensemble of objects (having the same BLR size and sampling characteristics as those of figure \ref{smooth}). One hundred simulations were used to calculate each data point (6 equally log-spaced points were calculated in each curve and a second degree polynomial fit to the results is shown). Noise level $\delta/\left < f \right >=0.01~(0.1)$ is shown in solid (dashed) lines, and the usual color coding scheme applies ($\xi_{\rm CA}/\xi_{\rm AA}/\xi_{lc}$ results in blue/red/black curves). Clearly, better signal-to-noise is obtained by either reducing the measurement noise or by better sampling the light curve. Generally, $\xi_{\rm CA}$ is the more reliable photometric estimator, while both photometric methods are less accurate than the spectroscopic one ($\xi_{lc}$). {\it Right:} The uncertainty on the {\it mean} time-lag deduced for an ensemble of objects using $\xi_{\rm CA}$, for two noise levels ($\delta/\left < f \right >=0.01,~0.1$ are shown in solid and dashed curves, respectively, and smoothing was applied). The total observing effort (number of points for the entire sample) was held fixed. The time-lag was deduced in two ways: (1) by measuring the time-lag for individual objects with the $\xi_{\rm CA}$ estimator and taking the mean result (thick lines), and by measuring the position of the peak of $\left < \xi_{\rm CA} \right >$ (thin lines). The uncertainty on the mean time-lag reported for the latter quantity was deduced from multiple simulations of statistically equivalent ensembles. The results show that, for the same observing effort, monitoring more objects with lower cadence (so long as the line transfer function is properly sampled) results in more accurate time-lag measurements. Also, measuring the peak in the mean statistical estimator provides more accurate results than measuring the time-lag for individual objects [$\sigma(\left < \xi_{\rm CA} \right >)<\sigma(t_{\rm lag,measured}^l)$].} \label{1sigma} \end{figure} To better quantify the ability of the photometric method to uncover the time-lag under various observing strategies, we repeated the above analysis for two noise levels ($\delta / \left < f \right >=0.01,0.1$) while varying the number of points in the light curves (i.e., observing visits). More specifically, we maintained a fixed $t_{\rm tot}/t_{\rm lag}^l=5$ and varied the sampling period, $t_{\rm sam}$ (we made sure that $t_{\rm lag}^l/t_{\rm sam} \geq 5$ so that the BLR transfer function is never under-sampled). Monte Carlo simulations were carried out to obtain reliable statistics, and we report the one standard deviations of the measured time-lag distributions, $\sigma (t_{\rm lag,measured}^l)$, in figure \ref{1sigma} (also shown, for comparison, are the corresponding $\xi_{lc}$-statistics). As expected, better sampled light curves result in a lower dispersion in $t_{\rm lag,measured}^l$ for all statistical methods. Also, unsurprisingly, $\xi_{\rm CA}$ is the more reliable photometric estimator for the time-lag, and both photometric estimators are worse than $\xi_{lc}$. In particular, quasar light curves having $\sim 100$ points in each filter with photometric errors of order 1\% result in a $\xi_{\rm CA}$-deduced time-lag dispersion of order $\sim 35\%$. Similar quality light curves can be found in the literature \citep{giv99,kas00}, and are analyzed in \S4. Consider the following question: having finite resources, is it better to observe fewer objects with better cadence or more objects having sparser light curves? [We assume that the {\it cadence is adequate} for sampling the line transfer function (see more in \S3.2), and that the same signal-to-noise is reached in both cases.] The answer to this question is given in figure \ref{1sigma} (right panel) where the total number of visits was kept fixed yet the number of objects in the ensemble varied (correspondingly, the number of visits per object changed to maintain their product constant). Interestingly, for the same observing effort, it is advantageous to observe more objects with worse cadence. Using this strategy, one is less sensitive to unfavorable variability patterns of particular quasars. For the case considered here, the time-lag uncertainty drops by a factor $\sim 3$ by increasing the sample size from one to ten objects (the uncertainty for the case of ensembles was determined from Monte Carlo simulations of many statistically similar ensembles). Similar results are obtained also for $\xi_{\rm AA}$ (not shown). \subsubsection{Ensemble averages} In the upcoming era of large surveys, statistical information may be gathered for many objects. For example, the LSST is expected to provide photometric light curves for $>10^6$ quasars having a broad range of luminosities and redshifts. With such large sets of data at one's disposal, sub-samples of quasars may be considered having a prescribed set of properties (e.g., luminosity and redshift), and the line to continuum time-lag measurement may be quantified for the ensembles. In this case, it is possible to measure the characteristic line-to-continuum time-delay in two ways: by measuring the peak of $\xi_{\rm CA} ( \delta t )$ or $\xi_{\rm AA} ( \delta t )$ on a case by case basis and then constructing time-lag distributions to measure their moments, or by defining the ensemble averages of the statistical estimators. Specifically, we define $\left < \xi_{\rm CA} (\delta t) \right >,~ \left < \xi_{\rm AA} (\delta t) \right >$ as the arithmetic averages\footnote{We similarly define, for comparison purposes, $\left < \xi_{lc} (\delta t) \right >$. These definitions were, in fact, previously used to compute the curves shown in figure \ref{acfccf}.} of the corresponding statistical estimators at every $\delta t$. (Alternative definitions for the mean do not seem to be particularly advantageous and are not considered here.) The average statistical estimators for ensembles of varying sizes are shown in figure \ref{ensembles}. As expected, averaging leads to a more robust measurement of the peak, as the various $\xi_{\rm CA} (\delta t)$'s "constructively-interfere" at $\sim t_{\rm lag}^l$. The ensemble size required to reach a time-lag measurement of a given accuracy naturally depends on the noise level. We find that measuring the time-lag from the peak of $\left < \xi_{\rm CA} \right >$ (or $\left < \xi_{\rm AA} \right >$) is more accurate than measuring the time-lags for individual objects in the ensemble, and then quantifying the moments of the distribution (Fig. \ref{1sigma})\footnote{The uncertainty on the time-lag, as deduced from $\left < \xi_{\rm CA} \right >$, was calculated by simulating numerous statistically-equivalent ensembles of objects, calculating $\left < \xi_{\rm CA} \right >$, and measuring the time-lag from its peak. A time-lag distribution is obtained with the uncertainty given by its standard deviation, $\sigma(\left < \xi_{\rm CA} \right >)$.}. This is particularly important when noisy data are concerned, in which case reliable time-lag measurements for individual objects are difficult if not impossible to obtain. To see this, we note the rising uncertainty on the deduced time-lag, $\sigma(t_{\rm lag,measured}^l)$, as the number of visits per object is reduced (and despite the fact that a correspondingly larger sample of objects is considered; see the right panel of Fig . \ref{1sigma} for a case in which $\delta/\left < f \right >=0.1$). In contrast, when working with $\left < \xi_{\rm CA} \right >$-statistics, the uncertainty monotonically declines so long as the sampling of the transfer function is adequate. The above analysis highlights the importance of using large samples of objects (even with less than ideal sampling) in addition to studying individual objects having relatively well-sampled light curve. In particular, by using large enough samples of quasars, the effects of noise, and even sparse sampling, may be overcome. \begin{figure} \plotone{f8.eps} \caption{The effect of scatter in the broad line region properties on (arbitrarily scaled) $\left < \xi_{\rm CA} (\delta t) \right >$. A large ensemble of objects (typically 100-1000) was used, whose time-lag distribution follows a modified Gaussian (see text), which is characterized by the mean time-lag, $t_{\rm lag}$ and its standard deviation, $\sigma(t_{\rm lag})$. $t_{\rm tot}/t_{\rm lag}=4$ was assumed. For $\sigma(t_{\rm lag})/t_{\rm lag}<1$, the distribution is nearly Gaussian, $\left < \xi_{\rm CA} (\delta t) \right >$ is, likewise, symmetric, and its peak is at $t_{\rm lag}$. For $\sigma(t_{\rm lag})/t_{\rm lag}\gtrsim 1$, the time-lag distribution is, by definition, modified (see text), and the effective mean time lag is $>t_{\rm lag}$ (black squares for each curve). This, and the fact that, for some objects, $t_{\rm lag}^l \sim t_{\rm tot}$ (hence the line transfer function is poorly sampled) result in an asymmetric $\left < \xi_{\rm CA} (\delta t) \right >$, which only roughly peaks at the effective mean time lag. } \label{lrs} \end{figure} Current studies suggest that there is a scatter of 30\%-40\% in the BLR size vs. quasar luminosity relation \citep{ben09,kas00}. To assess the effect of such a scatter on $\left < \xi_{\rm CA} \right >$ (similar results apply also for $\left < \xi_{\rm AA} \right >$), we averaged the results for an ensemble of simulated objects whose time-lags follow a Gaussian distribution with a mean $t_{\rm lag}$ and standard deviation $\sigma(t_{\rm lag})$. We modify this Gaussian such that the input lag is always positive definite: for $\sigma(t_{\rm lag})/t_{\rm lag} \ll 1$ the distribution is Gaussian while for $\sigma(t_{\rm lag})/t_{\rm lag} \sim 1$ there is an excess of systems with zero lag. $\left < \xi_{\rm CA} (\delta t) \right >$, is shown in figure \ref{lrs}: evidently, the average statistical estimator peaks at the mean of the distribution so long as $\sigma(t_{\rm lag})/t_{\rm lag} < 0.8$. For larger ratios, the distribution considerably deviates from Gaussian and its mean is $>t_{\rm lag}$. In this case, $\left < \xi_{\rm CA} (\delta t) \right >$ still peaks around the mean but its shape is highly asymmetric. This is partly due to the time-lag distribution, but also so due the fact that the transfer functions with the largest $t_{\rm lag}^l$'s are not adequately sampled by the light curve ($t_{\rm tot}/t_{\rm lag}^l=4$ was assumed). To conclude, photometric reverberation should yield accurate mean BLR sizes for a sample of quasars, given the observed scatter in the BLR properties. \subsection{Systematics} We identified several potentially important biases in the method, which we shall now quantify using a suite of Monte Carlo simulations. We work in the limit of large ensembles so that the effect of sampling and measurement noise are negligible (typically, the results of several hundreds of realizations are averaged over and the line to continuum time delay is identified with the peak of $\left < \xi_{\rm CA} (\delta t) \right >$ and $\left < \xi_{\rm AA} (\delta t) \right >$). \subsubsection{Measurement noise} Biases are present in the time-lag measurement from noisy data. These are negligible for noise levels $\delta / \left < f \right > < \eta$ but may surface otherwise. Specifically, for a (Gaussian) noise level of $\sim 50\%$, and a relative line contribution to the band of 10\%, both photometric estimators over-estimate $t_{\rm lag}^l$ by as much as 20\% with $\xi_{\rm AA}$ being more biased than $\xi_{\rm CA}$ (Fig. \ref{biases}). This bias could therefore be present in situations where large and noisy ensembles are considered, or when weak emission lines are targeted. Quantifying the bias requires good understanding of the noise properties. A related bias exists when the noise level in the two bands differs substantially. More specifically, when $\vert \delta_X/\left < f_X\right>-\delta_Y/\left < f_Y\right> \vert \sim \eta$ ($\delta_X/\delta_Y$ is the noise level in the $X$-/$Y$- bands), our simulations show that $t_{\rm lag,measured}$ may differ from $t_{\rm lag}$ by as much as 25\%. Whether an over-estimate or an under-estimate of $t_{\rm lag}$ is obtained depends on the particular noise properties, as mentioned above. Observationally, one should therefore aim to work with data having comparable noise levels for the different bands to obtain a bias-free result. As noted above, for relatively strong lines or low noise levels, such biases are negligible. \begin{figure} \plotone{f9.eps} \caption{Biases in measuring line to continuum time delays as calculated for the spectroscopic (abscissa) and photometric methods (ordinate). Dashed line is for a 1:1 ratio. Upper-left panel shows the effect of noise with the largest data point [blue (red) reflecting on $\xi_{\rm CA}$-($\xi_{\rm AA}$-) deduced measurements] corresponding to $\delta/\left < f \right >=0.512$ with subsequently smaller size symbols corresponding to values smaller by a factor 2 (the same symbol-size value applies to all panels). Upper-right panel shows the bias as a function of $t_{\rm gap}$ and implies consistency between the spectroscopic and photometric results (largest symbol is for $t_{\rm gap}/t_{\rm lag}^l=2$). Lower-left panel quantifies the bias for the $X$-band leading (circles) or trailing (squares) the $Y$-band. Largest symbol is for $\vert t_{\rm lag}^c \vert /t_{\rm lag}^l\simeq 0.3$. Lower-right panel shows the bias in the presence of a subordinate emission line whose time-lag is $t_{\rm lag}^l/2$, which is contributing to the $Y$ (circles) and to the $X$ (squares) filter. The largest symbol corresponds to $\eta_s/\eta = 0.8$ (see text).} \label{biases} \end{figure} Varying seeing conditions leading to aperture losses, as well as improperly accounting for stellar variability in the field of the quasar, could lead to correlated errors between the bands. This in turn could lead to artificial zero-lag peaks in the correlation function of the light curves \citep{wel99}, and several solutions have been devised to overcome this problem \citep[and references therein]{ale97,zu11}. Nevertheless, the formalism developed here subtracts off, by construction, the contribution of similar processes (e.g., correlated noise) to both light curves. In fact, as far as the algorithm is concerned, the continuum contribution to the $Y$-band is merely an effective correlated noise to be corrected for, which it does. Therefore, photometric reverberation mapping, as proposed here, is considerably less susceptible to the effects of correlated noise than the spectroscopic method. That being said, if the noise between the bands correlates over sufficiently long (e.g., $\gg$\,day) timescales, a secure detection of the line-to-continuum time delay may be more challenging (for an effectively similar case see \S3.2.3). \subsubsection{Sampling} \begin{figure} \plottwo{f10a.eps}{f10b.eps} \caption{{\it Left:} the bias in the time-lag measurement as a function of the observable quantity $t_{\rm tot}/t_{\rm lag,measured}^l$ ($\xi_{\rm CA}/\xi_{\rm AA}/\xi_{lc}$ results are shown in blue/red/black curves). For all statistical estimators there is a minimum $t_{\rm tot}/t_{\rm lag,measured}^l$ below which the sampling is inadequate. Above this threshold, the spectroscopic method converges to the input time lag (e.g., for a gaussian transfer function, $t_{\rm lag,measured}^l\simeq t_{\rm lag}^l$ for $t_{\rm tot}/t_{\rm lag,measured}^l>3$). In contrast, there is a bias using either photometric method wherein $t_{\rm lag,measured}^l/t_{\rm lag}^l$ is a rising function of $t_{\rm tot}/t_{\rm lag,measured}^l$, and is more pronounced for geometrically thick BLR configurations. {\it Right:} a particular example for $\left < \xi_{\rm CA}(\delta t) \right >$, calculated for an ensemble of 100 statistically identical objects with $t_{\rm tot}/t_{\rm lag}^l=50$, resulting in a peak at $\delta t \simeq 2 t_{\rm lag}^l$ (dark-black solid line). When the light curves of both bands are de-trended by a polynomial of degree $d$, the bias is gradually removed. For large enough $d$-values, substantial power on relevant timescales is removed, thereby affecting the significance of the results (see legend). When the quasar power-spectrum is truncated at frequencies $\omega < \omega_{\rm min}$ [here $(2\pi/\omega_{\rm min})/t_{\rm tot}\simeq 0.13$], the light curve appears to be stationary, and the bias disappears (dash-dotted dark blue line).} \label{biases1} \end{figure} Sampling is rarely uniform. In fact, seasonal gaps could be of the order of the time lag in some objects \citep{kas00}. The full treatment of sampling related issues is beyond the scope of this paper, and we restrict our discussion to light curve gaps (real data are treated in \S4). To check for potential systematics, we resampled our simulated light curves with equal "on" and "off" observing periods with durations $t_{\rm gap}$. We identify a bias when $t_{\rm gap}\simeq t_{\rm lag}^l$. Nevertheless, both the spectroscopic and the photometric methods are equally affected by it so that their results remain consistent. As our aim is to focus on cases in which the photometric approach leads to new or different biases, we do not further quantify this bias\footnote{See \S4 for the usage of discrete correlation functions, which are less susceptible to sampling issues.}. \subsubsection{Inter-band continuum time delays} Thus far, $t_{\rm lag}^c \ll t_{\rm lag}^l$ was assumed. While the spectroscopic method is unaffected by inter-band continuum delays (the continuum in the immediate vicinity of the emission line is considered), our definition of the continuum is that which corresponds to the filter having the least contribution of emission lines to its flux, i.e., the $X$-band. For a large set of simulation runs (obtained by varying both $\eta$ and $t_{\rm lag}^c/t_{\rm lag}^l$; not shown), we find that biases become non-negligible for $\eta^{-1}\vert t_{\rm lag}^c \vert /{t_{\rm lag}^l} \gtrsim 1$ ($t_{\rm lag}^c>0$ implies that continuum emission in the $Y$-band lags behind the $X$-band). Here we consider particular cases in which $\vert t_{\rm lag}^c \vert/t_{\rm lag}^l \le 0.3$ ($\eta \simeq 0.13$ is assumed), and find that $\vert t_{\rm lag}^c\vert /t_{\rm lag}^l \gtrsim 0.1$ results in substantial, $>20\%$ bias (Fig. \ref{biases}) for the following reason: although modest values of $\vert t_{\rm lag}^c \vert /t_{\rm lag}^l$ are considered, the contribution of continuum processes to both light curves by far exceeds that of the emission line. Specifically, when the continuum in the $Y$-band lags behind the $X$-band, the deduced time lag reflects more on $t_{\rm lag}^c$ than on $t_{\rm lag}^l$. When the $X$-band lags behind the $Y$-band, and $t_{\rm lag}^c/t_{\rm lag}^l$ is significant, the peak in both statistical estimators is strongly suppressed and time-lag measurements become less reliable. Realistically, inter-band continuum time-delays seem to be of order $10^{-2}t_{\rm lag}^l$ \citep{bac09}, and it is therefore likely that biases of the type discussed here would be negligible for most strong broad emission lines. \subsubsection{Multiple emission lines} Realistic quasar spectra could results in more than one emission line contributing to the $Y$-band, or even different emission lines contributing to both bands. While such complications are irrelevant for the spectroscopic method (note the small statistical scatter around unity in Fig. \ref{biases}), the photometric method is only sensitive to the combined contribution of all emission processes in some waveband. Here we study the biases introduced by including the contribution of a weaker secondary emission line to either of the bands whose time lag, $t_{\rm lag,s}^l \ne t_{\rm lag}^l$. We find potentially considerable biases when the relative contribution of the weaker line to the flux, $\eta_s$ satisfies $\eta_s/\eta \lesssim 1$. However, the magnitude of the bias is also affected by $(t_{\rm lag,s}^l - t_{\rm lag}^l)/t_{\rm lag}^l$. We do not attempt to fully cover the parameter space in this work, and focus on particular cases. Figure \ref{biases} shows an example with $t_{\rm lag,s}^l/t_{\rm lag}^l=0.5$ where we find $\gtrsim20\%$ biases for $\eta_s/\eta \gtrsim 0.5$: when the weaker line contributes to the $Y$-band, $t_{\rm lag,measured}^l < t_{\rm lag}^l$ while the opposite occurs when the weaker line contributes to the $X$-band. Comparing to a case in which $t_{\rm lag,s}^l=t_{\rm lag}^l/10$ and $\eta_s/\eta=0.8$, we find similar trends but an overall smaller ( $<5\%$) bias. Assuming instead $t_{\rm lag,s}^l=2t_{\rm lag}^l$ and $\eta_s/\eta=0.8$, we find a bias of order 20\% (10\%) when the weaker line contributes to the $Y$- ($X$-) band. While the magnitude of the bias is clearly not solely determined by $\eta/\eta_s$, we find that the bias is, generally, small for $\eta_s/\eta \ll 1$. Clearly, if $t_{\rm lag,s}^l \ll t_{\rm sam}$ or $t_{\rm lag,s}^l \gg t_{\rm tot}$ then the time-series is insensitive to the presence of the secondary line, and the bias would be negligible regardless of $\eta_s/\eta$. Further quantifying the bias as a function of the line properties (e.g., transfer functions and time-delays), filter response functions, quasar redshifts, and observing patterns is beyond the scope of this paper, and should be carried out on a case by case basis. \subsubsection{Total light curve duration} It is well known that the ratio of the total duration of the light curve to the time-lag, $t_{\rm tot}/t_{\rm lag}^l$, needs to be greater than some value to be able to adequately sample the transfer function and reliably measure the time lag using spectroscopic means. Our calculations demonstrate that the same holds also for the photometric method. In particular, for the gaussian (thick shell) transfer function, reliable measurements require that $t_{\rm tot}/t_{\rm lag}^l\gtrsim 3~(9)$; see figure \ref{biases1}. Differences between the photometric and the spectroscopic approach arise for $t_{\rm tot}/t_{\rm lag}^l \gg 1$: while the spectroscopic method gradually converges to $t_{\rm lag}^l$ (assuming sufficient data and adequate sampling; see Fig. \ref{biases1}), the photometrically deduced time lag, $t_{\rm lag,measured}^l$, develops a bias with respect to $t_{\rm lag}^l$ with increasing $t_{\rm tot}/t_{\rm lag}^l$. Generally, the bias is comparable (but not identical) for $\left < \xi_{\rm CA} \right >$ and $\left < \xi_{\rm AA} \right >$, and over-estimate the true lag. We quantify the bias for the two line transfer functions, $\psi^l$, defined above (\S2.1.2), and find it to be different in each case: broader $\psi^l$ that extend over larger $\delta t$ intervals, lead to more biased results. For example, while the thick shell BLR model shows a factor $\sim 2$ bias for $t_{\rm tot}/t_{\rm lag,measured}^l=50$, the corresponding bias for the Gaussian BLR is only $\sim 40$\% (see Fig. \ref{biases1}). The bias is seen to grow logarithmically with $t_{\rm tot}/t_{\rm lag,measured}^l$, and so would be most prominent when analyzing long time-series of low-luminosity quasars. Considering, for example, $\sim 10$\,years of LSST data for a low-$z$ quasar with a thick shell BLR whose size is $\sim 100$ light-days. In this case, $t_{\rm tot}/t_{\rm lag}^l\sim 30$, which would result in $t_{\rm lag,measured}^l/t_{\rm lag}^l\sim 2$. Not correcting for this bias would lead to an over-estimation of the black hole mass, by a similar factor, with considerable implications for black hole formation and evolution. Consider the bias in $\left < \xi_{\rm CA} \right> $: as its value is sensitive to the line transfer function, its origin may be traced to the $f_Y* f_X$ term in Eq. \ref{theta}, which is akin to the bias discussed in \citet{wel99}. In a nutshell, with the quasar's power spectrum being red, most of the power is associated with the lowest frequency modes. Nevertheless, those are the same modes that are not adequately sampled by any finite time series. As such, the data {\it appear} to originate in a non-stationary process where, for example, the mean flux changes substantially between the extreme ends of the light curve. This and the fact that emission lines' contribution to the flux is always additive results in a positive bias. More extended line transfer functions, resulting in longer term correlations between the light curves, are therefore more prone to such biases. With $\left < \xi_{\rm AA}(\delta t) \right> \simeq \left < \xi_{\rm CA}(\delta t) \right> $, similar biases are also encountered there, that trace back to the $f_Y * f_Y$ term. \begin{table} { \begin{center} \caption{Photometric properties of the PG sample of quasars$^{(a)}$} \begin{tabular}{lllllllllllllll} \tableline Object & & $\lambda L_\lambda(5100{\rm \AA})$ & & B & & & R & & net & net & H$\alpha$ lag$^{(b)}$ & Predicted lag & Photo. lag\\ Name & $z$ & ($10^{44}\,{\rm erg~s^{-1}}$) & pts & $\chi_B$ & $\left < \delta_B \right>$ & pts & $\chi_{R}$ & $\left < \delta_R \right>$ & wo/Fe & w/Fe & (days) & (days) & (days) \\ (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) & (12) & (13) & (14) \\ \tableline PG\,0026+129 & 0.142 & $7.0\pm1.0$ & 70 & 0.19 & 0.02 & 71 & 0.14 & 0.02 & 0.03 & 0.07 & $132^{+29}_{-31}$ & $147\pm25$ &- \\ PG\,0052+251 & 0.155 & $6.5\pm 1.1$ & 75 & 0.26 &0.03 & 75 & 0.20 & 0.02 & 0.04 & 0.08 & $211^{+66}_{-44}$ & $141\pm27$ & - \\ {\bf PG\,0804+761} & {\bf 0.100} & $ {\bf 6.6\pm 1.2}$ & {\bf 83} & {\bf 0.17} & {\bf 0.01} & {\bf 88} & {\bf 0.14} & {\bf 0.01} & {\bf 0.05} & {\bf 0.07} & ${\bf 193^{+20}_{-17}}$ & ${\bf 136\pm27}$ & ${\bf 200 \pm 100}$\\ {\bf PG\,0844+349} & {\bf 0.064} & ${\bf 1.72\pm 0.17}$ & {\bf 65} & {\bf 0.10} & {\bf 0.02} & {\bf 66} &{\bf 0.10} &{\bf 0.02} &{\bf 0.08} & {\bf 0.06} & ${\bf 39^{+16}_{-16}}$ & ${\bf 51\pm6}$ & ${\bf 500\pm 200 }$ \\ PG\,0953+414 & 0.239 & $11.9\pm1.6$ & 60 & 0.14 & 0.03 & 60 & 0.12 &0.01 &0.06 & 0.11 & $187^{+27}_{-33}$$^{(c)}$ & $231\pm39$ & - \\ PG\,1211+143 & 0.085 & $4.93\pm0.80$ & 25 & 0.16 &0.01 & 24 &0.13 &0.01 & 0.06 & 0.07 & $116^{+38}_{-46}$ & $109\pm20$ & - \\ PG\,1226+414 & 0.158 & $64.4\pm7.7$ & 45 & 0.12 &0.01 & 45 &0.10 &0.01 &0.04 & 0.08 & $514^{+65}_{-64}$ & $703\pm135$ & - \\ {\bf PG\,1229+204} & {\bf 0.064} & ${\bf 0.94\pm0.10}$ & {\bf 44} & {\bf 0.16} & {\bf 0.03} & {\bf 45} & {\bf 0.16} & {\bf 0.03} & {\bf 0.08} & {\bf 0.06} & ${\bf 71^{+39}_{-46}}$ & ${\bf 34\pm4}$ & {\bf N/A} \\ PG\,1307+085 & 0.155 & $5.27\pm0.52$ & 30 &0.14 &0.01 & 30 &0.10 &0.01 &0.04 & 0.08 & $179^{+94}_{-145}$ & $122\pm16$ & - \\ PG\,1351+640 & 0.087 & $4.38\pm0.43$ & 35 &0.11 &0.01 & 35 &0.10 &0.01 &0.06 & 0.07 & $247^{+162}_{-78}$ & $101\pm13$ & - \\ PG\,1411+442 & 0.089 & $3.25\pm0.28$ & 29 &0.10 &0.01 & 29 &0.07 &0.01 &0.06 & 0.07 & $103^{+40}_{-37}$ & $82\pm9$ &- \\ PG\,1426+015 & 0.086 & $4.09\pm0.63$ & 26 &0.16 &0.01 & 25 &0.13 &0.02 &0.06 & 0.07 & $90^{+46}_{-52}$ & $96\pm17$ &- \\ {\bf PG\,1613+658} & {\bf 0.129} & ${\bf 6.96\pm0.87}$ & {\bf 66} &{\bf 0.11} &{\bf 0.01} & {\bf 64} &{\bf 0.10} &{\bf 0.02} &{\bf 0.04} & {\bf 0.07} & ${\bf 43^{+40}_{-22}} $ & ${\bf 144\pm22}$ & ${\bf 300 \pm 130}$ \\ PG\,1617+175 & 0.114 & $2.37\pm0.41$ & 56 &0.19 &0.02 & 56 &0.14 &0.02 &0.04 & 0.07 & $111^{+31}_{-37}$ & $67\pm12$ &- \\ {\bf PG\,1700+518} & {\bf 0.292} & ${\bf 27.1\pm1.9}$ & {\bf 53} &{\bf 0.07} &{\bf 0.02} & {\bf 53} &{\bf 0.07} &{\bf 0.01} &{\bf 0.05} & {\bf 0.11} & ${\bf 114^{+246}_{-235}}$${\bf ^{(c)}}$ & ${\bf 428\pm61}$ & ${\bf 300 \pm 160}$\\ {\bf PG\,1704+608} & {\bf 0.371} & ${\bf 35.6\pm5.2}$ & {\bf 38} &{\bf 0.13} &{\bf 0.01} & {\bf 40} &{\bf 0.11} &{\bf 0.01} &{\bf 0.02} & {\bf 0.05} & ${\bf 437^{+252}_{-391}}$${\bf ^{(c)}}$ & ${\bf 550\pm109}$ & ${\bf 400\pm200}$ \\ {\bf PG\,2130+099} & {\bf 0.061} & ${\bf 2.16\pm0.2}$ & {\bf 78} &{\bf 0.10} &{\bf 0.02} & {\bf 79} &{\bf 0.08} &{\bf 0.02} &{\bf 0.08} & {\bf 0.06} &${\bf 237^{+53}_{-28} }$ & ${\bf 60\pm7}$ & ${\bf 240\pm100 }$ \\ \tableline \end{tabular} \label{tab} \end{center} } { Columns: (1) Object name, (2) redshift, (3) monochromatic luminosity in units of $10^{44}\,{\rm erg~s^{-1}}$, (4)-(6) properties of the photometric light curve of the $B$-band: number of points, the normalized variability measure, and the mean measurement uncertainty. (7)-(9) having the same meaning as (4)-(6) but for the $R$ band. (10) the net contribution of emission lines to the bands excluding the iron blends. (11) like (10) but including the iron emission complexes. (12) the spectroscopically measured time lag for the H$\alpha$ line from \citet{kas00}. (13) The {\it observed} lag for the Balmer emission lines as predicted by the BLR size-luminosity relation of \citet[see their Eq. 6]{kas00}. (14) The time lag measured in this work using broadband light curves.\\ (a) Highlighted table rows indicate objects for which a statistically significant signal is detected (see \S4.3).\\ (b) Values correspond to centroid observed-frame values \citep[see their table 6]{kas00}. \\ (c) Data for H$\alpha$ are not available. Time lag corresponds to the H$\beta$ line.} \end{table} To verify the source of the bias, we repeated the calculations using a truncated form for the power-spectrum where all frequencies below a minimum frequency, $\omega_{\rm min}$, have no power associated with them (we have experimented with $2\pi/t_{\rm lag} \gg \omega_{\rm min} \gg 2\pi/t_{\rm tot}$). The resulting $\left < \xi_{\rm CA}(\delta t) \right> $ is indeed unbiased and peaks at $t_{\rm lag}^l$ (not shown). For intermediate cases where, for example, the power spectrum flattens at low frequencies \citep{cze99}, the results will be less biased but may not be entirely bias-free. Having identified the source of the bias, it is now possible to correct for it using one of several methods which we term: de-trending, sub-sampling, and bootstrapping. De-trending has been shown to be useful to correct for biases in the spectroscopic method \citep{wel99}. Basically, one filters out long-term trends in the light curve by fitting (done here in the least-squares' sense) and subtracting off a polynomial of some degree, $d$, from the data. As a polynomial of degree $d$ has $d$ zeros, subtracting a higher degree polynomial from the data results in a suppressed variance at frequencies $\omega \lesssim d (2\pi/t_{\rm tot})$. Conversely, the highest degree polynomial that can be subtracted from the data, whilst not suppressing the reverberation signal, is $d\lesssim \lfloor t_{\rm tot}/t_{\rm lag} \rfloor $. The effect of de-trending is shown in figure \ref{biases1} for a range of $d$-values, and assuming $t_{\rm tot}/t_{\rm lag}^l=50$. While subtracting a $d=2$ polynomial already decreases the bias substantially, a bias-free result is recovered by de-trending with a $d=20~(=0.4 t_{\rm tot}/t_{\rm lag}^l)$ polynomial. Further increasing $d$ suppresses the sought after signal to insignificant levels by removing modes with frequencies $\sim 2\pi/t_{\rm lag}^l$. The dependence of the remaining bias on the de-trending polynomial degree, $d$, is shown in the inset of figure \ref{biases1}. An alternative method to remove the bias is to analyze continuous sub-samples of the time series, thereby reducing the effective $t_{\rm tot}$ and decreasing the bias. This method can be implemented in cases where one is not data-limited. The third method for removing the bias involves bootstrapping: given an observed continuum light curve (i.e., $f_X$), and an assumed $\psi^l$, the light curve of the line-rich band is simulated. Upon sampling it using the cadence of the real data, an artificial $f_Y$ light curve is obtained. By calculating the statistical estimators for the semi-artificial data, it is possible to compare $t_{\rm lag,measured}^l$ to the input lag, and correct for the bias. Conversely, should the bias be known by other means, inferences concerning the nature of the line transfer function may be obtained. \section{Data Analysis} Having gained some understanding of the advantages and limitations of broadband photometric reverberation mapping, we now wish to test the method using real data. Unlike the models described in \S3, real data are often characterized by uneven sampling, gaps, and a varying signal to noise across the light curve. Here we consider a subset of the Palomar-Green (PG) sample of objects\footnote{We choose to not focus our attention on active galactic nuclei (i.e., low luminosity quasars) since, in this case, great care is needed to make sure that varying seeing conditions do not introduce additional noise whose characteristics are poorly understood. In this case, specific data reduction techniques, such as aperture photometry, may be of advantage.} \citep{sg83}, for which Johnson-Cousins $B$ and $R$ photometry is publicly available\footnote{http://wise-obs.tau.ac.il/PG.html} \citep{mao94,giv99}, and independent BLR size measurements are available through spectroscopic reverberation mapping \citep{kas00}. The properties of the sample are given in table 1. While the PG sample provides a natural testbed for our purpose, it is far from being ideal: the typical number of points per light curve is small, $\sim 50$, and the photometric errors non-negligible, $\lesssim2$\% (c.f. Fig. \ref{1sigma}). Other photometric data sets are, in principle, available \citep{geh03,meu11} but do not yet have BLR size measurements with which our findings may be compared [see \citet{che11}]. \subsection{Preliminaries} \begin{figure} \plottwo{f11a.eps}{f11b.eps} \caption{Emission line and continuum contribution to the Johnson-Cousins broadband filters. {\it Left:} the composite quasar spectrum \citep[assumed to be at $z=0.1$; see table \ref{tab}]{dvb01} is shown together with the $B$- and $R$-filter response functions. Spectral decomposition into prominent emission lines (magenta) and iron line blends (gray curve) is shown. A double powerlaw continuum emission model is also included (dashed and dash-dotted magenta lines), and the "best-fit" model is shown in green. {\it Right:} the net emission line contribution to the broadband light curves as a function of the quasar redshift (dashed black lines; gray hatched patterns indicate the uncertainties assuming a 10\% uncertainty on the flux of individual lines). The net contribution of a few particular lines is shown in colored curves. The sum of all lines excluding iron (light grey) and including iron (dark grey) are shown, indicating that in either case the $R$-filter is relatively line-rich for $z<0.1$ objects with the Balmer emission line contribution being the dominant one. Typical redshift intervals over which line-rich filter identifications change are greater than the typical uncertainty associated with photometric redshifts \citep{wei04,wu10}.} \label{wise} \end{figure} We first identify line-rich and line-poor bands given the typical quasar spectrum \citep{dvb01}, the spectroscopically-determined redshift for the PG sample \citep{kas00}, and the relevant filter throughput curves (Fig. \ref{wise}). In particular, we need to identify the filter with the larger relative contribution of emission lines to the variance of the light curve. This is, however, difficult to estimate, since the variability properties of only a few lines, in only a few objects, are sufficiently well understood. Instead, we adopt a simplified approach and {\it assume} that the contribution of emission lines to the variance is proportional to their relative contribution to the flux. This is a fair approximation if, for example, the relative flux variations in all emission lines are comparable. This is supported by theory \citep{kas99} and by the fact that the fractional variability of the Balmer lines is independent of their flux \citep[see their table 5]{kas00}. To properly account for the contribution of emission lines to the flux in some band, spectral decomposition of all prominent line, line blends, and continuum emission processes is required. To this end, we considered the high signal-to-noise (S/N) composite quasar spectrum of \citet{dvb01}, and identified all prominent emission lines and blends in the relevant wavebands. To fit for continuum emission, we identified line-free regions over the entire spectral range from rest-UV to the optical \citep{kas00,dvb01}, and fitted a double powerlaw model (see Fig. \ref{wise}), which accounts for the changing slope at optical wavelengths. The stronger Balmer emission lines were fitted with a two Gaussian model, which accounts for their relatively narrow line cores and extended wings. Weaker lines were fitted with a single Gaussian model. Iron blend emission templates, as derived for IZw I \citep{ver04,ves01}, were convolved with a Gaussian kernel and fitted to the data. Independently adjusting the width and normalization for each Gaussian kernel (individual lines and blends), a reasonable agreement was sought. We model the Balmer continuum using the analytic expression of \citet{gra82}. We emphasize that our model is by no means physical, nor uniquely-determined, and that its sole purpose is to allow for the qualitative estimate of the emission line contribution to the flux in each band. The decomposed spectrum is shown in figure \ref{wise}. The line-rich band, as a function of the quasar redshift, for the Johnson-Cousins filter scheme, is shown in figure \ref{wise}. Clearly, depending on the object's redshift, either the $B$ or the $R$ filter may be identified with the line-rich band. Typically, a redshift accuracy of 0.2 is required to securely identify the line-rich band, which is within reach of photometric redshift determination techniques \citep{ric01}. We find that, for $z<0.5$ quasars, the $R$ band may be identified with the line-rich band (i.e., with the $Y$-band, using the nomenclature of \S2.1.2). This result is rather insensitive to whether or not the optical iron blends contribute to the variance on the relevant timescales. For $z<0.25$ objects, the flux and, by assumption, the variance, are largely dominated by the Balmer lines. Uncertainties of order 10\% in the flux of individual lines and line blends do not significantly change this result. \subsection{Methods} Aiming to analyze individual objects in the PG sample having a marginally adequate number of points in their light curves (thereby resulting in a large uncertainty on the determined lag; Fig. \ref{1sigma}), we outline the formalism used below to assess the significance of our results. \subsubsection{Cross-correlation analysis} \begin{figure} \epsscale{1.2} \plotone{f12.eps} \caption{Light curve analysis for PG\,0804+761. {\it Left:} The $B$ and $R$ light curves show significant non-monotonic variability over a few hundreds days timescales (see table 1). {\it Right:} the calculated statistical estimators, $\xi_{\rm CA}$ (blue shades) and $\xi_{\rm AA}$ (red shades). Also shown is the renormalized $\xi_{lc}$ (black points) based on the spectroscopic data of \citet{kas00}. Two analysis methods are used to calculate the photometric estimators: ICCF (solid lines and shades) and ZDCF (discrete points), which lead to consistent results. The uncertainty on each estimator was calculated using 100 Monte Carlo simulations: the FR-RSS method was used to estimate the error on the ICCF, and a random independent pair selection was used to estimate the error on the ZDCF (see text). All estimators peak at $\delta t \simeq 200$\,days. The result obtained using $\xi_{\rm CA}$ has a higher statistical significance, and both photometric estimators are less significant than the spectroscopic one.} \label{0804} \end{figure} In analyzing real data for individual objects in the PG sample (table 1), it is important to verify that the detected signal is real and is not some artifact of non-even sampling or the particular light curve observed. There are several methods in the literature for cross-correlating realistic data sets: the interpolated cross-correlation function \citep[ICCF; used in \S3]{gas86} and the discrete correlation function \citep[DCF]{edl88}. While the ICCF may be somewhat more sensitive to low-level signals, it is also more prone to sampling artifacts \citep{pet93}. To verify the significance of the ICCF signal, two algorithms are often used to estimate the uncertainty in the deduced correlation function: the flux randomization (FR) and the random subset selection (RSS). In a nutshell, the first algorithm adds noise according to the measurement errors, and repeats the analysis many times to estimate the uncertainty associated with the results, while the latter method resamples the light curves (while losing $\sim 37$\% of the data), to reach the same goal. Quite often both tests are combined to form the FR-RSS method, which is our method of choice in the present analysis. When considering the DCF approach, the $z$-transformed DCF algorithm (ZDCF) by \citet{ale97} often produces the best results, and is the approach adopted here. While all the aforementioned algorithms have been thoroughly tested with respect to spectroscopic reverberation mapping, their adequacy for the photometric reverberation mapping method is yet to be examined. Our implementation of the ICCF (FR and RSS algorithms alike) is standard \citep{pet98}. Nevertheless, because photometric reverberation mapping requires the subtraction of two cross-correlation functions, and to avoid interpolations of any kind\footnote{Although we do not use this method here, we note that due to the correlated nature of the correlation function, interpolation at the correlation function level is likely to provide results which are less sensitive to sampling artifacts than the ICCF method.}, our implementation of the ZDCF algorithm requires that both the $f_Y * f_X$ and $f_X * f_X$ terms in equation \ref{theta} are evaluated over an identical time grid, and that the bin center is identified with the time tag for that bin, i.e. $\delta t$. Similar considerations apply also when evaluating $\xi_{\rm AA}$. Furthermore, we require that the number of independent pairs contributing to each bin in the correlation functions is $\ge 11$ \citep{ale97}. For simplicity, we use equally spaced bins to evaluate $\xi_{\rm CA}$ and $\xi_{\rm AA}$, and repeat the analysis many times by randomly selecting independent pairs \citep{ale97}. This provides us with a distribution of $\xi_{\rm CA}$ and $\xi_{\rm AA}$ values at each time bin, from which the mean and its uncertainty (associated with one standard deviation) are evaluated. Spectroscopic reverberation mapping often employs two distinct approaches to estimate the time lag: by measuring the position of the peak of the correlation function, or by measuring its centroid. While the results are comparable, they are not identical and many works have discussed the pros and cons associated with each approach \citep[and references therein]{kas00}. Similar algorithms are implemented here. Estimating the uncertainty on the measured time-lag is done in the following way: we use the set of Monte Carlo simulations described above and calculate $\xi_{\rm CA},~\xi_{\rm AA}$ for each realization. The time-lag is identified either with the peak or with the centroid of the statistical estimators\footnote{Here we use the smoothing algorithm defined in \S3.1, to avoid identifying spurious peaks with the physically meaningful peak.}, and a distribution of time-lags is obtained. The uncertainty on the time-lag is associated with one standard deviation of that distribution. \subsubsection{Controlling systematics} In addition to the above algorithms, aimed at testing the robustness of the signal, we provide an additional test to identify systematic effects of the type discussed in \S3.2.5 and to assist in screening against problematic data sets that may lead to spurious signals. To this end we use the bootstrapping method discussed in \S3.2.5, verifying that the mean contribution of the simulated emission line to the flux of the line-rich band is that given in Fig. \ref{wise}, and that the variance of the line light curve is consistent with the \citet[see their table 5]{kas00} results. \subsection{Results} Here we outline the results for individual objects enlisted in table 1: we first discuss quasars for which a significant signal is detected\footnote{The significance criterion used here requires a $\gtrsim 1\sigma$ signal (with $\sigma$ being the one standard deviation uncertainty on the measured value) detected in both statistical estimators, and for which the ICCF and ZDCF analysis yield consistent results.}, and then cases where the analysis does not result in a discernible peak in either estimator. Our treatment of the ensemble as a whole is given in \S4.3.9. \subsubsection{PG\,0804+761} The $B$ and $R$ light curves for PG\,0804+761 (henceforth termed PG\,0804), at $z=0.1$, are shown in figure \ref{0804} where large non-monotonic flux variations are evident. This object has the best sampled light curves in the \citet{kas00} sample, with a mean uncertainty on the flux measurement in both bands being of order 1\% (with maximum uncertainties on individual measurements being $\sim 2$\%). The normalized variability measure is among the largest in the sample $\sim 17\%/14\%$ for the $B/R$ bands (Table 1). As such, it is a promising candidate for photometric reverberation mapping. \begin{figure} \plotone{f13.eps} \caption{The peak (left panels) and centroid (right panels) $t_{\rm lag,measured}^l$ distributions for PG\,0804, obtained using sets of 100 Monte Carlo simulations (standard color scheme is used). ICCF (ZDCF) results are shown in the upper (lower) panels. When using the ZDCF approach, two different bin sizes were assumed (solid and dashed lines; red/blue shades correspond to $\xi_{\rm AA}/\xi_{\rm CA}$), and consistent results are obtained in both cases. The mean lag and its uncertainty (assumed to be one standard deviation) corresponding to each distribution are denoted in each panel. Centroid statistics results in slightly larger time-lags due to the non-symmetric shape of the statistical estimators around the peak (Fig. \ref{acfccf}). While the time-lags inferred by all statistical estimators and methods are statistically consistent, the uncertainties inferred by the ZDCF approach are relatively small, and possibly less reliable (see text).} \label{0804_dist} \end{figure} \begin{figure} \plotone{f14.eps} \caption{Photometric analysis of real $B$ data and simulated $R$ data for PG\,0804. Several artificial time-lags (see legend) are assumed and the line-rich $R$ band light curve is simulated using the line-poor $B$-band light curve (see \S4.2.2). Analysis results for the real data are shown, for comparison, in blue shades. Both ICCF and ZDCF results are shown. Different artificial time-lags result in statistically different signals. As expected, longer time-lags result in a hump extending to longer timescales. The current data set is insensitive to short lags due to inadequate cadence. Qualitatively, an input time lag of 100-200\,days is consistent with the observed signal.} \label{0804_sys} \end{figure} Figure \ref{wise} indicates that, at its redshift, the $R$ band is relatively line-rich (the net contribution of the lines to the bands is $\sim 6$\%), with H$\alpha$ being the dominant emission line. We calculated $\xi_{\rm CA}$ and $\xi_{\rm AA}$ using the ICCF and ZDCF algorithms. Unsurprisingly, the ZDCF method is somewhat more noisy than the ICCF methods \citep[see their Fig. 4]{kas00}. That said, we find the results of all methods of analyses to be consistent given the uncertainties. This means that interpolation is not a major source of uncertainty for this object, which is consistent with the relative regular sampling of its light curves. \begin{figure} \epsscale{1.2} \plotone{f15.eps} \caption{Analysis of 15 PG quasars from the sample of \citet[see our table 1]{kas00}. In each pair of columns, the left panel shows the $B$ (blue points) and $R$ (red points) light curves, and the right panel the calculated $\xi_{\rm CA}$ (blue curves and shades) and $\xi_{\rm AA}$ (red curves and shades) using the ICCF (solid lines) and ZDCF (points with error-bars) numerical schemes. Cases for which a statistically significant signal is detected in both estimators and using both numerical schemes, are highlighted. For each quasar, the results of three $\xi_{\rm CA}$ simulations are shown (in dashed blue-shaded curves) based on the expected time-lag from the BLR size vs. luminosity relation, $t_{\rm lag, ex}^l$ (table 1): light to dark shades correspond simulated time-lags of $t_{\rm lag, ex}^l/2 \to t_{\rm lag, ex}^l \to 2t_{\rm lag, ex}^l$ (cyan curves correspond to $t_{\rm lag, ex}^l$). For PG\,0844, a simulation with a 500\,days lag is also included (dashed magenta line). The ICCF results reported here are used to construct the ensemble averages considered in figure \ref{pg_mean}.} \label{all_pgs} \end{figure} Our analysis shows the expected features: a hump in both $\xi_{\rm CA}$ and $\xi_{\rm AA}$, which peaks at similar ($\delta t \sim 200$\,days) timescales. For comparison, we used a published light curve for the broad H$\alpha$ emission line \citep{kas00}\footnote{see http://wise-obs.tau.ac.il/~shai/PG/}, and calculated $\xi_{lc}$ using the ZDCF algorithm. Clearly, both statistical estimators defined here trace the line-to-continuum cross-correlation function with a varying degree of significance: $\xi_{\rm CA}$ shows a $\gtrsim 2\sigma$ peak ($\sigma$ being the uncertainty around the maximum), while the peak in $\xi_{\rm AA}$ is only $\gtrsim 1\sigma$ significant. Using the Monte Carlo simulations described in \S4.2.1, peak or centroid time distributions were obtained, and are shown in figure \ref{0804_dist}. Clearly, time-lag estimates qualitatively agree between the different methods. Quantitatively, however, there are some differences: the centroid method results in somewhat larger values for the time-lag. This results from the non-symmetric nature of $\xi_{\rm CA},~\xi_{\rm AA}$ showing a more extended wing toward longer time-delays (Fig. \ref{0804}). Results based on the ZDCF approach result in errors, which are typically smaller than those obtained using the ICCF method. This results from the fact that the ZDCF peak is largely determined by a few individual bins (note the two bins around 200\,days that lie above most other ZDCF points, as well as above the ICCF curve; Fig. \ref{0804}). In fact, the resulting uncertainty on the ZDCF, being of order 20\% in this case, is smaller than predicted based on our Monte Carlo simulations presented in figure \ref{1sigma}. We conclude that estimating the uncertainty on the time-lag using the ZDCF results may be less reliable. Using the ICCF time-lag distributions, we determine the observed time-lag for PG\,0804 to be $250\pm100$\,days, which is statistically consistent with the spectroscopically measured value of $193^{+20}_{-17}$\,days \citep{kas00}. As far as systematics is concerned, we do not expect a bias in the measured lag to exceed $\sim 40$\% given that $t_{\rm tot}/t_{\rm lag,measured}^l\gtrsim 10$ (Fig. \ref{biases1}). Using the numerical bootstrapping scheme defined in \S4.2.2, we find that an input time lag of $100-200$\,days is qualitatively consistent with the data. In particular, much shorter or much longer time-lags, are inconsistent with our findings in terms of signal amplitude and shape. Interestingly, a weak yet marginally significant hump at around $200-300$\,days is implied by the data even when much shorter artificial time-lags are used as input. Therefore, one should be careful when interpreting marginally significant signals. For the specific case considered here we can conclude that the data cannot be used to detect short time-lags, which is of no surprise given that the cadence of the light curves is at $20-30$\,days intervals. To conclude, the time-lag estimated here by the photometric reverberation mapping approach is $\simeq 200\pm100$\,days, and is therefore consistent with the value deduced by \citet{kas00}, albeit with an expectedly larger uncertainty. \subsubsection{PG\,0844+349} This low-$z$, low-luminosity object has a potentially prominent ($\sim 8\%$) line contribution to the $R$-band from the Balmer emission lines. With $\lesssim 70$ points in its light curve, and what appears to be significant variations over 200\,days time scales, our analysis reveals a marginal ($\gtrsim 1\sigma$ using the FR-RSS method and $\gtrsim 2$ using ZDCF statistics) peak at $\sim 500$\,days (Fig. \ref{all_pgs}). The spectroscopically measured time-lag for this object is of order 50\,days and agrees with the value expected from the BLR size vs. luminosity relation (table 1). Nevertheless, the data do not show a corresponding peak at those time-scales. In fact, our simulations indicate that only a time-lag $\gtrsim 100$\,days, could have been detected with some certainty. Interestingly, simulations with an input lag of 500\,days (\S4.2.2) are able to qualitatively account for the observed signal (dashed magenta curve in figure \ref{all_pgs}). It is not clear whether a 500\,days lag, {\it if} real, is associated with a long-term reverberation signal of the BLR, since there is no corresponding signal in the \citet{kas00} analysis of the spectroscopic data. We note, however, that it is also possible that the measurement uncertainties are somewhat under-estimated for this object, hence the significance of our deduced signal over-estimated. That this might be the case is hinted by the normalized variability measure in both the $B$ and $R$ bands being similar (contrary to naive theoretical expectations) and the fact that contamination by the host galaxy in this low-$z$ quasar may be important (see below). \subsubsection{PG\,1229+204} This quasar, being at low redshift ($z=0.064$) has one of the most prominent ($\lesssim 8\%$) Balmer line contributions to the $R$-band among the PG objects in our sample. The analysis detects a significant {\it trough} at $\gtrsim 200$\,days, in both statistical estimators, and using both the ICCF and the ZDCF methods. This is unexpected since a peak rather than a trough is predicted by the simulations for an expected lag of $\sim 34$\,days (Fig. \ref{all_pgs}). Simulating cases with time-lags of order 200\,days also produces a peak, in contrast to the observed signal. The interpretation that the $B$- rather than the $R$-band is the line-rich band, is not supported by the optical spectrum showing rather typical quasar emission lines \citep{kas00}. We cannot positively identify the origin of the unexpected signal but note that, at its low redshift, and being a rather faint low-luminosity quasar, the host galaxy is likely to substantially contribute to the flux in the bands. This has a dual result: 1) the actual contribution of the emission lines to the bands is smaller than that predicted here, and 2) the data may be substantially affected by varying seeing conditions between the nights, thereby contributing to the variance in the light curves, and masking the true signal. Interestingly, the reduced variability measure in both bands is similar for this object, which may indicate the presence of an additional source of noise not accounted for by the reported measurement uncertainties. \subsubsection{PG\,1613+658} Analyzing the data for this object, a marginally significant signal (at the $1\sigma-2\sigma$ level) is detected in both statistical estimators, at $\lesssim 300$\,days (Fig. \ref{all_pgs}). Formally, $\xi_{\rm CA}$ statistics implies that the time lag is $300\pm130$\,days. At its redshift, the net contribution of emission lines is of order 5\%, and is therefore larger than the typical flux measurement uncertainty in this object ($\lesssim 2\%$). Simulations with an artificial lag of $\sim 300$\,days are able to qualitatively account for the observed signal. Much shorter time lags [e.g., $\sim 40$\,days, as spectroscopically deduced by \citet{kas00}] do not seem to be supported by our findings. Interestingly, for this object, spectroscopic time-lag measurements using the peak distribution function give very large errors for the H$\alpha$ line, and given its luminosity, PG\,1613+658 lies considerably below the BLR size vs. luminosity relation \citep[see also our table 1]{kas00}. Using our deduced lag, a better agreement is obtained with the \citet{kas00} size-luminosity relation. \subsubsection{PG\,1700+518} This object shows a statistically significant ($2\sigma-3\sigma$), wide peak extending over the range 150-450\,days. Assuming that H$\beta$ is the main line contributor to the $R$ band's flux, our simulations indicate that the signal is consistent with a time-lag of $\sim 430$\,days (formally, $\xi_{\rm CA}$ statistics yields a $300\pm160$\,days' lag). This value is statistically consistent with the time-delay measurement of \citet{kas00}, but our deduced lag is in somewhat better agreement with the expected BLR size given the source luminosity (table 1). Our simulations show that time-lags considerably shorter ($\ll 200$\,days) or longer ($\gg 800$\,days) are less favored by the data and cannot fully account for the observed signal (Fig. \ref{all_pgs}). \subsubsection{PG\,1704+608} This object, having $\sim 40$ visits in each band, shows a marginal ($\gtrsim 1\sigma$) hump at 200-300\,days. Formally, the deduced time-lag is $400\pm200$\,days, and is consistent with the values of \citet{kas00}. At $z=0.371$, the net contribution of emission lines to the line-rich $R$-band is only about twice as large as the mean measurement uncertainty. This, and the small number of data points, account for the low-significance of the observed signal. Further, simulations show that a marginal signal is indeed expected to occur at roughly the spectroscopically determined lag. \subsubsection{PG\,2130+099} There is a hint for a localized ($\sim 1\sigma$) peak around 200-300\,days in both statistical estimators ($\xi_{\rm CA}$ statistics yields a lag of $240\pm100$\,days) using both the ICCF and the ZDCF methods. At face value, this timescale is consistent with the results of \citet{kas00} who find a time lag of $\sim 200$\,days. Nevertheless, as discussed by \citet{gr08}, the cadence of the \citet{kas00} observations may not be adequate to uncover the true time lag in this object. Specifically, \citet{gr08} find a time lag of order 30\,days using better sampled light curves on short time scales. The fact that we do not find a significant peak on short timescales is not surprising since we are essentially using the \citet{kas00} data set. \subsubsection{PG\,0026+129 and other objects yielding null results} Like PG\,0804, PG\,0026+129 ($z=0.142$) is among the best-sampled quasars in our sample with $\sim 70$ points per light curve (see Fig. \ref{0026}). While the normalized variability measure is comparable to that of PG\,0804, most of the power is associated with long-term variability. In particular, when subtracting a linear trend from the light curve, the normalized variability measure drops by a factor of $\sim2$ in both bands. At its redshift, the $R$-band is the line-rich (Fig. \ref{wise}) with a net relative line contribution of $\sim 3\%$ due to H$\alpha$ (or $\sim 7$\% if iron is important). The statistical estimators calculated here do not yield a significant time-lag. Specifically, there is no discernible peak detected in either estimator using either cross-correlation method (i.e., ICCF or ZDCF). While de-trending seems to somewhat improve the signal (there is a $1\sigma$ hump at around 100\,days), it falls short of revealing a significant result. We attribute our failure to securely identify a signal to a combination of factors related to the smaller variability measure on the relevant BLR-size timescales, as well as to the lower net contribution of emission lines to the filters, compared to the case of PG\,0804. PG\,0052+251 has 72 points in each band, and a normalized variability measure of $26\%/20\%$ in the $B/R$ bands, partly due to a sharp feature extending over $\sim 200$\,days scales. The typical measurement uncertainty is $\lesssim 3\%$ in both bands. At $z=0.155$, the net line contribution is of order 4\%. i.e., comparable to the noise-level. Therefore, despite the relatively large number of visits, no clear signal is detected, in either statistical estimator, using either numerical scheme. The remaining quasars in the \citet{kas00} PG sample also do not yield a significant signal. This is however expected given the small number of points in the photometric light curve of these quasars being $\sim$40 (see Fig. \ref{1sigma}). The fact that such objects do have a spectroscopically-determined time lag is also not surprising: the photometric approach to reverberation mapping is less sensitive than the spectroscopic one. \begin{figure} \epsscale{1.2} \plotone{f16.eps} \caption{Light curve analysis for PG\,0026+129. {\it Left:} The $B$ and $R$ light curves show significant variability on timescales of the order the duration of the light curve. De-trended light curves were calculated by subtracting a $1^{\rm st}$-degree polynomial. The normalized variability measure for the de-trended light curves is a factor two smaller than the original data. {\it Right:} the calculated statistical estimators for the de-trended light curves (upper panel) and the original data (lower panel). See the caption of figure \ref{0804} for the meaning of different curves. While de-trending seems to be able to improve the signal, a statistically significant peak is not detected by either method.} \label{0026} \end{figure} \subsubsection{Ensemble averages} \begin{figure} \plotone{f17.eps} \caption{Ensemble averages for $\xi_{\rm CA}$ (upper panel) and $\xi_{\rm AA}$ (lower panel), for the PG sample of 17 quasars from \citet[see our table 1]{kas00}. Both the arithmetic mean (dark shades) and the median (light shades) are shown. Ensemble averages for both statistical estimators give consistent results, and peak around 180-280\,days. This is in agreement with the predicted time-lag based on the BLR size vs. luminosity relation of \citet{kas00}, for a typical $L_{45}=1$ quasar (see text). The median shows a less significant peak although the results are qualitatively consistent with the ensemble mean. This demonstrates the feasibility of the photometric reverberation mapping technique in uncovering the typical time-lag in an ensemble of objects for which individual time-lag determination is difficult to achieve.} \label{pg_mean} \end{figure} To overcome the limitations plaguing the analysis of individual objects, ensemble averages may be considered. As demonstrated in \S3, combining the results for many objects, allows the time-lag to be determined with good accuracy, and was a main motivation on our part to develop this approach to reverberation mapping. Clearly, large ensembles are required to reach a robust result when less than favorable light curves are available on an individual case basis. Nevertheless, with upcoming photometric surveys, statistics is unlikely to be a limiting factor, and defining ensemble averages over narrow luminosity and redshift bins will be possible. As our small sample of 17 PG quasars consists of a range of redshifts and quasar luminosities, we use the following transformation to put the results for different objects on an equal footing: $\delta t \to \delta t/[(1+z)L_{45}^\gamma]$, where $L_{45}\equiv \lambda L_\lambda (5100\,{\rm \AA})/10^{45}\,{\rm erg~s^{-1}}$. This transformation corrects for both cosmological time-dilation, and the fact that the BLR size scales roughly as $L^\gamma$. Following \citet{kas00}, we take $\gamma=0.7$ but note that similar results are obtained for $0.5 \lesssim \gamma \lesssim 0.7$ \citep[and references therein]{ben09}. Using the above transformation, all PG quasars are effectively transformed to a $z=0$ quasar with $L_{45}=1$. Figure \ref{pg_mean} shows the ensemble-averaged results for $\xi_{\rm CA},~\xi_{\rm AA}$ for the entire PG sample (table 1). We deliberately include all objects regardless of data quality issues (as in the case of e.g., PG\,1229). A peak in both (mean) statistical estimators is seen around 200\,days. This value is similar to a time-lag of 170\,days, which is the value predicted by the \citet{kas00} BLR size-luminosity relation for a $L_{45}=1$ quasar. Also plotted are the median results for both statistical estimators. Both the mean and the median provide qualitatively consistent results, although quantitative differences do exist. Given the small nature of the sample, we do not present estimates for the uncertainty on the mean/median values. To conclude, averaging the results for the PG quasar sample, we obtain a similar peak in both statistical estimators, occurring at times which are qualitatively consistent with the BLR sizes implied by the \citet{kas00} results. A larger sample of objects will allow to better quantify the mean time lag for sub-samples of quasars over narrow luminosity and redshift intervals. Alternatively, better quality data (higher S/N, and better sampled light curves) could be used to better constrain the time-lag for individual objects in the PG sample of quasars. \section{Summary} We demonstrated that line-to-continuum time-delays associated with the broad emission line region in quasars can be deduced from broadband photometric light curves. This is made possible by defining appropriate statistical estimators [termed $\xi_{\rm CA}(\delta t)$ and $\xi_{\rm AA}(\delta t)$] to process the data, and which effectively subtract the continuum contribution to the cross-correlation signal of the light curves. Using this formalism, the time-lag is identified with the time $\delta t$ at which these estimators peak. The proposed formalism makes use of the fact that line and continuum processes have very different variability properties, and that quasars exhibit large amplitude variations on timescales comparable to the BLR light crossing time. The formalism is generally adequate for cases in which few strong emission lines, having different intensities, contribute to the spectrum, as indeed characterizes the optical emission spectrum of quasars. To effectively apply the method, prior knowledge of the quasar redshift (using either photometric or spectroscopic means) is required to identify relatively line-dominated and line-free spectral bands. Together with knowledge of the filter response functions, these may be used to better implement the algorithm and interpret the results. Using numerical simulations, we demonstrated that photometric reverberation mapping is feasible under realistic observing conditions and for typical quasar properties. We showed, by means of numerical simulations and real data (pertaining to the PG quasar sample), that the measurement of the line-to-continuum time-delay is robust although somewhat less accurate than that obtained using the spectroscopic reverberation mapping technique. In addition, we quantify biases in the photometrically deduced time-lags, which are sensitive to the BLR geometry and the duration of the light curve. We identify their origin and suggest several methods to correct for them. In addition, we qualitatively study potential biases associated with the presence of multiple emission lines in quasar spectra, and with finite inter-band continuum time-delays. The main advantage of the photometric approach to reverberation mapping over other methods is that it is {\it considerably} more efficient and cheap to implement, and can therefore be applied to large samples of objects. In fact, any survey which photometrically monitors the sky with proper cadence and signal-to-noise in two or more filters, and for which quasars may be identified and their redshift estimated, can be used to measure the size of the BLR. We demonstrated that, with large enough samples of quasars, it is possible to beat down the noise, and deduce (statistical) BLR size measurements that are considerably more accurate than are currently achievable by spectroscopic means. By combining photometric light curves with single epoch spectroscopic measurements, which allow to estimate the velocity dispersion of the BLR, it may be possible to measure the black hole mass in orders of magnitude more objects than are directly measurable by other means. This would allow for the statistical measurement of the SMBH mass for sub-samples of quasars (selected according to e.g., their luminosity, redshift, or colors), leading to highly accurate results, and potentially revealing differences between different classes of objects, which, thus far, may have eluded detection. To conclude, photometric surveys that monitor the sky on a regular basis are likely to play a major role in the coming decades in the (statistical) measurement of the BLR size in quasars, as well as in determining their black hole masses. This will lead to a more complete census of black holes over cosmic time, and shed light on outstanding problems concerning the formation and co-evolution of black holes and their host galaxies. \acknowledgements We thank E. Behar, T. Holczer, and S. Kaspi for thought-provoking weekly meetings that triggered much of this work, and the referee for insightful comments. D .C. thanks C. Kochanek, D. Maoz, and H. Netzer for valuable comments on an earlier version of this paper, and S. Kaspi, A. Laor, H. Netzer, and C. Thompson for wise advice. Fruitful discussions with D. V. Bowen, D. Dultzin, K. Horne, Y. Krongold, S. Rafter, and O. Shemmer are greatly appreciated. We thank M. Vestergaard for providing us with iron UV emission templates. This research has been supported in part by a FP7/IRG PIRG-GA-2009-256434, and by grant 927/11 from the Israeli Science Foundation. E. D. thanks E. Behar for financial support. D. C. thanks Chuli and the people of Arlozorov caf\'e for continuous encouragement.	{ "redpajama_set_name": "RedPajamaArXiv" }	8,541
The Magnificent Seven (2016) Joshua Faraday/Vasquez Joshua Faraday Modern AU Bantering Boykisses Excessive amounts of fluff two idiots in love ← Previous Work Part 9 of the Doing It To Country Songs series Next Work → You Name The Babies And I'll Name The Dogs jdrush Wedding planning takes a detour when a unexpected addition joins the Varaday household. No, it's not what you think. RATING/WARNINGS: PG, at best. DISCLAIMER: Title comes from Blake Shelton—are you really surprised? I don't own these characters and made no money from this story. AUTHOR'S NOTES: Another story that wasn't supposed to exist, but as long as the boys keep talking to me, I'll keep writing 'em. Thank you to everyone who has left kudos and comments on the other stories in this series. I hope you enjoy this one, too. As usually, all Spanish comes from Google Translate. No betas were harmed in the making of this fic. Any mistakes are mine. Bill. Bill. Credit card solicitation. Bill. Junk. Uh-oh. . . Hammacher Schlemmer catalog. Better hide that from Rafe. Faraday finished shuffling through the mail before tucking it under his arm as he slid his key in lock and opened the door to the condo. He placed everything on the entryway table, along with a bag of bakery-fresh chocolate-chip muffins, and was just about to slip off his jacket when Maria and Ethel—alerted by the sound of clanking keys—came charging down the hall, barking happily, almost running him over. "Yes, yes, hello, I'm home," he chuckled, squatting down to pet them. "And no, I don't have any treats on me," he added, as Mimi started sniffing around his pockets. "You've confused me with the one who spoils you rotten." Said spoiler of dogs rushed up to him at that point. "What are you doing here?" Rafe demanded, a touch of frustration in his tone. "Um, last time I checked, I live here," Faraday remarked, as he stood up. "Oh, and you're welcome for the muffins." "You're supposed to be with Fernando!" Faraday's brow furrowed in confusion. "Why would I be with an ABBA song?" Rafe rolled his eyes in exasperation. "The caterer! We're sampling dishes today!" Way back when Faraday had proposed to Rafe, he had hoped the man was joking about having a huge wedding with 300 guests, but as the months went by, and the appointments—and the bills—mounted, it was obvious Rafe had been serious. The wedding was six months away, and they had only checked off about half the things on Rafe's 'to-do' list. Thankfully, Faraday had been able to talk him out of having the shindig at Rose Creek stables. Be grateful for small mercies, as his Grams would say. "I thought that was next week." "They had a cancellation, so they bumped us up but we our appointment is in 20 minutes," Rafe told him. "Didn't you get my text?" "You mean the plate and fork, clock, and 2-heart emojis? How was I supposed to piece that together?" "Seriously? How could you not?" "Sorry I don't speak emoji. What's the big deal anyway? Can't you just pull up a menu on-line and pick something?" Rafe clutched dramatically at his chest and gasped. "Mi amor (my love), it is a very big deal! Soup. Salad. Appetizers. Main courses. Next to the cake, this is the most important thing about the wedding!" "What about the groom?" Faraday asked, dryly. "Well, of COURSE you're important, too, mijo, but. . .appetizers!" "Right. How silly of me. Do we need to drop the girls off somewhere?" "They'll be fine," Rafe replied, grabbing his jacket off the coat hook. "We won't be gone that long." "You know, it's not too late to just elope to Vegas," Faraday suggested half-jokingly/half hopefully. In response, Rafe looked pointedly at his watch. "18 minutes." 'Lord, give me strength', Faraday thought for the hundredth time. "Got it. I'll drive." "So, caterer is settled," Rafe announced with a grin, drawing a checkmark on the 'to-do' list he had pinned to a clipboard. "I think we made a good choice, yes?" "At a buck fifty a plate, I hope so," Faraday answered, sliding into the driver's seat. "But worth it! Those bacon-wrapped scallops were sinful! I could have eaten a whole tray!" "I think you did." "Well, excuse me, Mr. Mini-Crab-Cake Hoarder," Rafe snarked, buckling himself in. "I didn't have lunch today, okay?" Faraday said, adjusting his own seatbelt. "I was hungry." Rafe pouted. "You could have saved me at least one." "I didn't notice they were gone until they were gone." Faraday put the key in the ignition and started the truck. Flipping on the windshield wipers, he added, "Besides, you were busy 'sampling' the entrees. Why do we even need three entrees?" "One meat, one fish, one vegetarian," Rafe explained, patiently. "We have to accommodate all our guests." Faraday lowered his visor, slid out his favourite Blake Shelton CD and fed it into the player. "Well, anyone who would pick rabbit food over sirloin tips is a fool," he commented as he pulled out of the parking lot. Rafe snickered. "I'll be sure to tell them that." "Please don't. Your friends will probably think I'm a big enough loser as it is." Faraday rarely felt insecure, but the thought of meeting Rafe's famous, glamourous friends made him more than a bit nervous. He had seen enough pictures, and heard enough stories, to fear he wouldn't measure up. The last thing he needed was to give them another reason to judge him, and find him lacking. "Why would you say that? They're going to love you." Faraday just snorted in disbelief. "It's true," Rafe insisted. "And if they don't, who cares? I love you. My family loves you. Our friends love you. The girls love you. No one else matters." Trying to cover up the burst of emotions that simple sentence did to him, Faraday muttered, "I still think it's too expensive." Rafe brushed his concerns away with a wave of his hand. "Stop worrying. We'll make it back with interest. My friends are very generous. And besides, we're saving money on a photographer." "You are NOT running around the reception taking pictures," Faraday informed him in no uncertain terms. "Of course not. Barry is." "Who the hell is Barry?" "An old friend." "Friend?" Faraday repeated knowingly, as he slowed down for a red light. "Boyfriend, if you want to get technical." "And you invited him to our wedding!?" Faraday squeaked. "He's an excellent photographer. Covered the last two royal weddings. Said he'd do me for free." "I'm sure he did." "Don't be jealous, guero. It was a long time ago." The light turned green, just like Faraday's eyes. "Not jealous," he mumbled, as he stepped on the gas. Trying to change the subject, Faraday asked, "So, what's next on your list?" Rafe checked his clipboard. "Invitations. There's a stationary store on Dexter Street that got excellent reviews on YELP." Faraday shot him an incredulous look before turning his attention back to the road. "You're kidding. Actual paper invitations?" "How else are we going to tell people about the wedding?" "I thought we sent out 'save the date' cards a few weeks ago." "Those are different from invitations." "Then why did we send them?" "To let people know they'd be receiving an invitation." "Why didn't we just send the invitations then?" Rafe laughed and shook his head. "Silly man! You don't send invitations eight months in advance!" This conversation was starting to make Faraday's head spin—pretty much like every step of the wedding planning so far. "Well, what's wrong with emails or texts?" "We're NOT texting invites to our wedding!" Rafe stated decisively. "Why not? Just send the church, calendar, and party face emojis. They can figure it out." "Oh, ha-ha." "Aren't paper invitations kinda old fashioned?" Faraday queried, as he made a left turn. "Nothing wrong with going traditional. I'll even let you pick the font." "Ooooh, lucky me." "What do you have against paper invitations?" Faraday shrugged his shoulders. "Nothing. I was just trying to make things easier for you." "And less expensive," Rafe deduced. "Hey, at this point, what's another couple hundred dollars, right?" Rafe fidgeted in his seat before admitting, somewhat sheepishly, "Couple. . .thousand." Faraday internally grimaced, but forced a smile. "Right. Put me down for the stamps." "You don't have to do that." "Yeah, I do." Rafe might have been picking up the bill for the big ticket items, but Faraday's pride was determined to pitch in as much as his bank account would allow. It grew quiet between them, as Rafe perused his check list and Faraday sang along to 'Neon Lights'. After a few moments, Rafe reached over and placed his hand over Faraday's on the steering wheel. "Gracias, mijo," he said, softly. "For what? Buying some stamps?" "For letting me do all this," Rafe clarified. "I know this isn't the wedding you wanted, and it means a lot to me that you're letting me have it." Faraday brought Rafe's hand up to his lips and brushed a kiss across his engagement ring. They had finally gotten around having it engraved, and it still gave him a thrill to see his initials, JCF, on Rafe's hand. "Babe, as long as I'm married to you at the end of it, I don't care how we get there. Just do what makes you happy." Rafe turned in his seat and flashed Faraday a smile. "You make me happy, querido (darling)." Faraday glanced over, and joked, "And you do me so well." They were both chuckling at that when suddenly Rafe expression changed and he cried out in alarm, "Joshua! Detener!" "Stop! Now!" Reacting automatically, Faraday slammed on the breaks and turned the wheel sharply towards the curb. Before he could say a word, Rafe had jumped out the truck and was running across the street. Faraday's breath caught in his throat as a car coming down the road swerved, missing Rafe by mere inches. He watched as his fiancé crouched down on the opposite sidewalk for a moment, then stood up. Checking the traffic this time, he waited until a car passed before starting across the street. He slid carefully into his seat, clutching something to his chest; faint, pitiful mewls echoed through the cab. "What is it, Rafe?" The curiosity and concern could be heard in Faraday's voice. "Are you okay?" Rafe uncupped one of his hands, allowing a little pink nose to peek out, and Faraday gaped at the small, dirty, gray fluff ball he was holding. When the head poked out further, Faraday suddenly found himself staring into a pair of big, frightened, bright blue eyes. "You shouldn't pick up stray animals. . ." At the sound of Faraday's deep voice, the tiny kitten pulled his head back and snuggled closer to Rafe's chest. "Mijo," Rafe scolded, gently. "Not so loud. You're scaring him." He resumed tenderly stroking the kitten, murmuring soft, calming nonsense. "Shhh, sweetie, it's okay. We're not going to hurt you. We're going to help you." Remembering to keep his voice low and nonthreatening, Faraday said, "What's wrong with it?" "I think its front leg is broken. It was limping pretty badly." "Babe, we can't take an injured cat home, especially not with Meems and Ethel. How will we care for it?" "Well, we can't leave it here," Rafe challenged. "It still gets cold this time of year once the sun goes down, and it won't be able to find food or defend itself. It's a death sentence." Faraday studied the sad, little kitten, realizing Rafe was right—the poor thing wouldn't last the night on its own. "Maybe it belongs to someone nearby. We could go door to door, see if anyone knows." "I doubt it. There's no collar. And look how skinny it is." Shaking his head, Rafe whispered, "I don't think it has anyone." Already knowing the answer, Faraday asked the question anyway. "I guess a shelter is out of the question, huh?" Rafe nodded. "They'll probably just put it down. They have a difficult time placing healthy animals, let alone an injured one." Faraday sighed. It was obvious Rafe's mind was made up. "I don't know anything about cats. Do you?" "We had a couple when I was a kid. I think I remember how to handle them." "The girls won't be happy." "Nonsense. They're friendly dogs. And we're just keeping it until it gets better, then we'll find it a nice home. Maybe Goody and Billy will take it in." Faraday sighed again. He knew damn well that once that kitten entered the condo, it was never leaving. "Well, if the invitations can wait until another day, there's a vet not too far from the office, next door to Mustang Sally's." Rafe laughed at that. "I didn't know that dive was still open." "It may be a dive, but it's got the best steak sandwiches in town." "Probably seasoned with a good dose of ptomaine." "Whatever their secret ingredient is, it works." "What are we waiting for then?" Rafe gestured towards the steering wheel. "Let's get this little one checked out, and we can grab some sandwiches." "We just ate," Faraday reminded him. "But we'll need SOMETHING for dinner, yes?" Faraday was still stuffed with crab cakes, but the thought of one of Mustang Sally's cheese-steaks was too tempting. "If you say so." Gesturing with his chin toward their passenger, he asked, "You're really sure about this?" Rafe looked down at the tiny, helpless creature in his hands and grinned. "Yeah. I'm sure." Faraday nodded, resigned to a new life filled with cat-hair covered shirts, although with all the fur Ethel tended to shed he supposed it wouldn't make much difference. "You know, this could take a while. Maybe you should call Red to check in on the girls in case we're late." "Good idea." Rafe pulled his phone out of his coat, then gently tucked the kitten in tighter to his chest. "And can you turn up the heat? Poor thing is shivering." Reaching over, Faraday gave the kitten a quick scratch under its chin before ratcheting up the heat. He glanced into his rearview mirror, then cautiously eased back into traffic. FOUR HOURS LATER: Faraday strolled out of the kitchen balancing a large food-covered TV tray, Maria and Ethel scampering after him. He sidestepped the many half-filled shopping bags from Pets-R-Us and new kitty toys littering the living room floor and placed the tray on the coffee table before flopping down on the couch next to Rafe. Grabbing the remote, he turned on the big screen, and flipped through the stations until he stumbled upon 'Road House'. He paused, waiting for Rafe to complain or wrestle him for the remote so he could change it to something—ANYTHING—other than 'Road House'. Rafe, however, didn't even notice, completely captivated by the small kitten in his lap. Leaning down to the dogs, Faraday joked, "I think we've been usurped." "Hmmmm?" Rafe replied, distractedly. Faraday began unwrapping the steak sandwiches, knowing if he waited for Rafe to do it, he wouldn't get to eat until midnight. "I said, dinner is served," he stated a little louder, sliding Rafe's meal closer to him. Rafe finally glanced up. "Oh. Gracias." He reached down, careful not to jostle his ward, and picked up half of his sandwich. "The little guy's settling in, I see?" Faraday noted, opening a large bag of chips. "Yeah, he's doing great." Rafe looked down at the bandaged—but now clean—kitten and smiled. "I owe you for this, guero." Faraday took a big bite of his sandwich and moaned blissfully. "You buy me a few more of these badboys, and we'll call it square." "These things are a heart attack on a bun," Rafe declared, even as he took a bite of his own sandwich. "Totally worth it," Faraday muttered around another bite. The kitten, alerted by the new, delicious aroma, lifted his head and noticed the sandwich Rafe was holding. He reached out with his unhurt paw and batted at Rafe's hand, demanding a sample. "Oh, no you don't," Rafe laughed, jerking his hand away. "You've already had your din-din." Faraday joined in with a chuckle of his own, remembering how the kitten had plowed through an entire can of tuna. "Who knew such a tiny animal could eat so much? Reminds me of his papi." Not to be denied, the kitten again batted at Rafe's hand, throwing in some sad, forlorn eyes for good measure. And Rafe—always a push-over when it came to animals—caved at the look. "Okay," he said, breaking off a small piece of steak and feeding it to the kitten. "But just a bite." Faraday rolled his eyes. "Great. Not even here an hour, and you're already spoiling him," even as he reached down and fed some steak to Ethel. "But of course! He's one of the family now. And after the day he's had, I think he deserves a little pampering." A tickle to the kitten's tummy produced a particularly loud purr. "Oh, you like that, don't you, bollo pequeño (little bun)?" he cooed in baby-talk. "Jesus, Rafe, have some dignity," Faraday groaned, snagging a few chips from the open bag. "It's just a kitten." "Oh, really? Then why did you insist on staying with him at the vet's office?" Rafe said with a knowing smirk. "Because I didn't want to see how much money you were going to spend at Pets-R-Us," Faraday deadpanned. "Come on, mijo," Rafe teased. "You can't fool me. You love him just as much as I do." "Is that a fact?" Faraday remarked blankly, fighting to keep a straight face. "Uh-huh. And you know what else?" Rafe looked over at Faraday, a soft smile gracing his stupidly handsome face. "You made me very happy today." Faraday answered Rafe's smile with one of his own. "That's all I want for you, babe" he murmured, leaning over and claiming Rafe's lips in a kiss. "A life filled with happy moments." "Dios mio (my god), guero," Rafe laugh/snorted. "If you were any sappier, I could pour you over my pancakes." "I'll try for something more porn-y next time." "Please do. Now, give me a REAL kiss." "Oh, yeah," Faraday sighed, pressing closer. His mission was thwarted, however, as the kitten swatted playfully at Rafe once more. "Oops, someone doesn't like being ignored." Breaking off another small bit of his steak and feeding it to the kitten, Rafe crooned, "Is that what you wanted, Shelton?" "You already named him?" "You got to name the dogs," Rafe countered. "Technically, I named only one of the dogs. Maria came pre-named. Why Shelton?" "You were playing a Blake Shelton CD when I found him. It seemed fitting." Taking a bite of his sandwich, Rafe asked, "Why? What did you want to call him?" "I was thinking maybe Sir Murder Britches." Rafe just gave Faraday a look. "We're sticking with Shelton." Faraday shrugged, nonchalantly. "Suit yourself," he said, taking a big bite of his sandwich. "I will. And besides, he likes the name, don't you, Shelton?" The kitten, in response, rubbed his head against Rafe's hand and purred. "Rafe, you're feeding him cheese-steak," Faraday pointed out. "He'd answer to just about anything right now." "You're just jealous because he loves me so much." Lapsing back into baby-talk, Rafe tickled Shelton under the chin and cooed, "Don't you, bollo pequeño? Yes, you do!" "You don't say?" With that, Faraday reached over and scratched the kitten behind the ears. "How's it hanging there, Sir Murder Britches?" he quipped, as Shelton's purring grew even louder Rafe watched as the kitten quickly changed allegiances and huffed, "Yeah, well, if you scratched me behind the ears like that, I'd probably purr, too." "I have. . .and you do," Faraday replied, smugly. "Cabrón," Rafe laughed, even as he curled up against Faraday's side. "Love you back," Faraday chuckled, swinging his arm around Rafe and pulling him close. Finally noticing what was on TV, Rafe frowned. "Guero? What are we watching?" " 'Road House'." "Must we?" With a laugh, Faraday handed him the remote. "No hoity-toity PBS crap." "Not even 'Bake-Off'? It's cake week." Because of course, the only thing that Rafe enjoyed as much as eating pastries, was watching people make them. Pressing a kiss to Rafe's cheek, Faraday sighed, "Sure, babe. 'Bake-Off's' fine." Shelton watched the two strangers bicker back and forth, but he wasn't afraid. Their voices were soft and friendly—they didn't seem to be angry at all. Indeed, the kiss at the end of their 'quarrel' confirmed they hadn't been fighting. Shelton was young, but he had been around. He had seen a lot of things and had met a lot of humans. Some were nice, some not so nice. These two were nice, of that he was certain. When he had woken up that morning, cold and hungry, he could never have guessed how adventurous the next few hours of his life would be. At that point, his only thought was to find someplace warm where maybe he could beg a few scraps. He was making his way to an apartment where he knew an old lady lived—she could always be counted on for a bite or two—when he had slipped off the wet porch railing. He hit the ground with a 'thud' and it took a moment to shake the stars out of his head. As he stood up, he instantly realized something was wrong. He had never known such pain and his front leg wouldn't work right anymore, but he started walking anyway, instinctively knowing he couldn't stay still. Still was bad. He was just hobbling down a familiar street when his path was suddenly blocked by a pair of worn cowboy boots, and a hand was reaching down towards him. Panicked, he tried to run, but his leg buckled, and he stumbled instead. The hand gently scooped him up and pressed him close to. . .oh, so much warmth!. . . as a soft, droning voice washed over him. He didn't understand the words, but the tone was soft and comforting, and while still afraid, he didn't feel threatened. He snuggled into the warmth, soothed by the scent of this pleasant-smelling stranger. After a few moments, he felt the stranger stand and begin to move. He soon found himself in a small, enclosed area, where he encountered another stranger. This one was bigger than the first one, and had a loud, rough voice. But the bright green eyes were kind, and his fear quickly disappeared. He snuggled against the pleasant-smelling stranger and fell asleep. Then, a bright room and another stranger, this one dressed all in white. He felt a sharp, quick pain in his hindquarters, then nothing more. When he awoke, he found his hurting leg was bound, and the stranger with the kind green eyes was talking to him softly. A bowl of warm soapy water was brought over and the stranger started washing him. He attempted to struggle but he was too tired, and settled instead for an air of bored aloofness that caused the stranger to make weird sounds. He had never heard chuckling before, but decided he liked the sound, and liked this large stranger, too. Soon after that, the other stranger, the tall thin one who smelled good came back, picked him up, and carried him from the bright room. . . and brought him here. His new home. He looked around the living room, surveying all the soft, cushiony furniture that he could lounge upon. There was a new 4-foot tall scratching-post tree set up in the corner, just waiting for him to climb. Numerous toys were scattered around that he could play with. Some of them were dog toys, but he didn't care. He could make it work. Speaking of dogs, one of them was sitting on the floor near the sofa, head cocked, regarding him curiously. He waved his bandaged leg peevishly at it, only to have a gentle hand carefully stop him. "Hey, be careful there, Shelton," the dark eyed stranger crooned. "Don't want to hurt yourself. Meems is just trying to be friendly." "I think she's trying to figure out who stole her favourite lap," the green-eyed man laughed. "C'mere, girl." He waved a piece of that tasty meat towards the dog, who happily trotted over to him, where another dog was already seated. Those two would make fun playmates—and perfect scapegoats for any mischief he managed to get into once his leg was healed. Yes, his injury was going to be an inconvenience for a while, but Shelton couldn't complain. He was warm and clean, and his tummy was full for the first time since he could remember. And best of all, he now had not one, but two devoted slaves, prepared to cater to his every whim. If cats could smile, Shelton would have been beaming. Instead, he snuggled deeper into the tall stranger's lap and yawned, contentedly. He was going to like it here. In case anyone was wondering, this is what I pictured Shelton looking like: https://thehumorbot.tumblr.com/post/627921124431413248/puss-in-boots-via-raww-httpsifttt2eduv8s Series this work belongs to: ruggerdavey, Anarchist_Queen, Savage2003, QuickSilverFox3, 1FiroLover, IAmYourWatson, and SubCapatin as well as 5 guests left kudos on this work! (collapse)	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	321
Plum Orchard, Georgia, is about to get even juicier... Notorious mean girl Dixie Davis is back in town and it's payback time. Literally. Dixie is flat broke and her best—make that only—friend, Landon, is throwing her a lifeline from the Great Beyond. Dixie stands to inherit his business...if she meets a few conditions: She's got to live in Landon's mansion. With her gorgeous ex-fiancé, Caine Donovan. Who could also inherit the business. Which is a phone sex empire. Wait, what? Landon's will lays it out: whoever gets the most new clients becomes the owner of Call Girls. Dixie has always been in it to win it, especially when it comes to Caine, who's made it clear he's not going down easy. (Oh, mercy.) Can Dixie really talk dirty and prove that she's cleaned up her act? Game on! Look for Dakota Cassidy's next novel Something to Talk About available soon from Harlequin MIRA DAKOTA CASSIDY Talk Dirty to Me For my agent, whom I lovingly call Agent Fab, Elaine Spencer. You're a gladiator, my friend. There are no words in the English language to adequately describe how dear I hold the notion that you have always believed. Also, to the many folks who've been involved in making this project a reality: My editor, Leonore Waldrip—for seeing this one little crazy idea/book in its earliest stages and passing it on. Add to the mix your amazing sense of humor and genius brainstorming, makes you a keeper. Emily Ohanjanians, your insight, attention to detail, and overall brilliance will forever influence the future words that flow from my fingertips. An enormous nod to the show Hart Of Dixie, my inspiration for writing Southern fiction. I love every "bless your heart, Lemon Breeland, Lavon Hayes, Annabeth Nass, Zade" moment spent with you each week. If you're a fan of the show, you'll know what I mean when I cry, ZADE forever! To all of my amazing readers—really who else can I count on to talk about anti-inflammatory cream and one's (ahem) nether regions (all in one whacky conversation that I swear didn't begin related at all) with me at three in the morning on Facebook but all of you? I treasure our conversations. I hold your thoughts and continued support in the highest regard. Thank you for always being so willing to laugh (and sometimes cry) with me! Contents Chapter One Chapter Two Chapter Three Chapter Four Chapter Five Chapter Six Chapter Seven Chapter Eight Chapter Nine Chapter Ten Chapter Eleven Chapter Twelve Chapter Thirteen Chapter Fourteen Chapter Fifteen Chapter Sixteen Chapter Seventeen Chapter Eighteen Chapter Nineteen Chapter Twenty Chapter Twenty-One Chapter Twenty-Two One "He looks really good, considering." Emmaline Amos sniffed, pushing her way past an enormous bouquet of white lilies standing by Landon Wells's casket at Tate and Son's Home Of Eternal Rest. She pulled Dixie Davis with her, away from Landon's casket and into the privacy of a connecting mourning room where she set Dixie on a couch surrounded by pictures of Landon. The scent of dark wood paneling, vanilla candles, and Old Spice invaded Dixie's nose, making her "ugly cry" hangover pulse in her temples with the force of a sledgehammer. Dixie lifted her sunglasses, thwarting another ambush of tears, so grateful for the opportunity to have had a few moments alone with Landon without the intrusion of the long line of people who'd shown up to pay their last respects. She muttered up at Em, "Why does everyone always say that, Em? Landon's dead. There's nothing good-looking about it. I always thought that was a crude thing to say." Em huffed, brushing the brim of her black sun hat, and sat down beside her. She gave her a nudge to make some room. "It's not crude. I was complimentin' him. New adjective, please," she drawled, her Southern lilt like macaroni and cheese to Dixie's homesick ears. Comfort food for the soul. "Crass?" "Crass is harsh, Dixie." Landon Wells, her best friend ever, was dead. That was harsh. Harsher still, Landon's other best friend, Caine Donovan, was just outside that door. Don't forget he's your ex-fiancé, too. Right. Dixie started to regret her terse words with Emmaline. She couldn't afford to alienate the one and only, albeit totally reluctant, ally she had left in her small hometown of Plum Orchard, Georgia. Maybe what was making her so snappish was exhaustion after the long drive from Chicago. Or the anxiety of returning to said small hometown where everyone knew her name and mostly wanted to throw darts at her picture. Maybe it was the precariousness of her life in financial semiruin that made her voice what she'd been thinking for almost two hours as mourner after mourner repeated Em's words while she'd waited for her private viewing of Landon's body. Or maybe it was the likelihood that a good portion of the female population of Plum Orchard High, class of 1996, were just outside this very funeral home with metaphoric stakes soaked in the town's specialty, homemade plum wine, just waiting for Reverend Watson to perform her public exorcism. Then they could seal the deal by driving their angry pieces of wood right through her despicable heart. It would be nothing less than she deserved. She'd been a horrible person in high school and beyond, and here in Plum Orchard where time seemed to stand still, no one forgot. You were horrible long after high school, too, Dixie. To Caine... Point. Most of her anxiety had to do with the fact that she had no choice but to see Caine Donovan again. Bingo, Dixie. The thought of seeing him left her feeling fragile and raw. To this damn day his memory still leaves you breathless. Acknowledged. Dark, star-filled nights under a scratchy army blanket in the bed of Caine's pickup truck, the scent of magnolias clinging to their sticky skin. It was just one of many of the images—both good and bad—she'd warred with since her return to Plum Orchard became a reality. She scrunched her eyes shut before reopening them. "Sorry," Em said, dragging her from her internal war. Her blue eyes held sympathy beneath her wide-brimmed hat. "I'm glad they gave you some time alone with Landon before the latecomers swarm in to pay their last respects. I can't even imagine how much this hurts." Em squeezed her shoulder with reassuring fingers. Dixie let out a shaky sigh, hooking her arm through Em's. "No, I'm the one who's sorry. I'm tired and on edge, and you've been so kind to me through this whole process when I totally don't deserve—" "No, you surely do not, Dixie Davis!" Em's voice rose, then just as quickly reduced to just above a whisper. She peered over her shoulder as though unseen eyes might bear witness to her bad manners. God forbid. "You were a mean girl back in the day. My high school years were torture because of you. And might I remind you, people don't forget, especially here in little ol' Plum Orchard. Why, you're lucky I even picked up the phone during Landon's last days, knowing it was you I had to talk to on the other end of the line," she finished on an offended harrumph. But Dixie knew better than to take Em's outburst personally. Em was as kind as she was generous, and nothing, not even a faded-around-the-edges grudge, would keep her good heart from beating selflessly. For all her leftover high school anger with Dixie, Em had called her religiously with updates on Landon's last days, because he'd asked her to. Em always did what was right. That was just who she was. Still, Dixie gave her a sheepish glance, and bumped her shoulder playfully to ease the lines of Em's frown. "This is about the cheerleading squad, isn't it?" Em's arm stiffened. She lifted her chin. "You told me my legs looked like sausages in that stupid cheerleading skirt, so I couldn't be on the squad. But my split jumps were better 'n Annabelle Pruitt's, and you knew it." True. Every last word of it. She'd been cruel, twenty or so years ago. Yet, comments like that, among the many she'd hurled at Em, obviously crept into a person's soul and hung around. From the moment she'd seen Em after being gone so long, Dixie had known she'd be met with extreme caution. Maybe some angry outbursts and plenty of tests to see if she really had changed. So Dixie's next admission was without hesitation. "I did." Dixie let her hand slide down along Emmaline's arm to thread her fingers through hers, giving them a light squeeze. "I'm not that person anymore, Emmaline. I'm really not. You were right then and now. Your split jumps were at least a hundred feet higher than Annabelle's. I lied back then out of jealousy. Your legs are long and gorgeous." They were. Em was undeniably beautiful. Em ran a self-conscious hand over her bare leg and said, "Don't you try and flatter me after all this time. Not after I spent four months' worth of babysitting money on the ThighMaster because of you." Dixie winced. "Then, if nothing else, you know, for every mean thing I did to you back then, I hope you'll remember, the Lord says to forgive is divine." "The Lord didn't go to high school with you." "Fair." Dixie let her chin drop to her chest, noting under the lights of the funeral home, the long curls of her red dye job were fading dismally. Em's nostrils flared at the pin Dixie'd effectively poked in her bubble of anger before her rigid posture deflated, and she let out a half chuckle. "Don't you be nice to me, Dixie Davis. I'm not one hundred percent buyin' this 'I've changed' act. You've done that bit once before, and we all fell for it ten years ago, remember? Not so fast this time. So just keep your compliments to yourself." It was obvious Em was trying to keep her resentments in check out of respect for Landon, for which today, Dixie was grateful. If not for Em, she wouldn't have been able to speak to Landon the one last time he was still coherent—nor would she have known about a single funeral arrangement. So Dixie nodded in understanding. "No rights allowed." The tension around Em's crimson-colored lips eased some, her expression growing playful. She fingered one of the lilies in a fluted vase on the table near the couch. "And as a by the by, Lesta-Sue and the Mags said they'll never allow you access to the Plum Orchard Founders Day parade committee, if you were hopin' to worm back into everyone's good graces, that is." It was a "take that" comment meant to hurt her—to remind Dixie, when she'd been head Magnolia, the town's decades-old society of women, and a rite of rich Southern girl passage, she'd once used her popularity and status to shun others via the town's elitist club. Especially Em. If Lesta-Sue was here already, that meant the rest of the Mags would be here, too. Terrific. Surely, Louella Palmer, Dixie's head Magnolia predecessor, wasn't far behind. Louella hated her, too. In fact, there was a special kind of hate reserved by Louella just for Dixie. Because she'd broken the girlfriend code ten years ago. Really broken it. But Dixie nodded again, and this time, if there was such an act, she did it with even less hesitation than the time before. "Lesta-Sue shouldn't allow me access to a public gas station bathroom after what I did to her. Stealing her high school beau of three long years by offering to let him get to second base with me was a horrible thing to do. So it's a good thing I'll be long gone by the time they break out the hot glue gun and crepe paper. I'm not here to stay, Em. I'm just here to say goodbye to Landon." The statement tugged at Dixie's heart. She'd missed home—even if it hadn't missed her. Em's dark brows knitted together while her gloved fingertips fluttered to the pointed collar of her black-belted dress with the flared skirt. "You're upsetting me, Dixie." "How so, Emmaline?" "You're still bein' sweet." Dixie flashed her a warm smile. "Aw, thank you." "Stop that this instant!" Em insisted. "It's unsettling. I should hate you just like every woman still left in this town who attended Plum Orchard High does." She stiffened again, as if her years of piled-up high school hurts caused by Dixie were warring with her naturally forgiving nature. Em had just wanted to fit in back then, and Dixie'd used it to her advantage at every outlet. She wouldn't forgive someone for treating her the way she'd treated Em, but if regret counted, she had plenty of that to give. Dixie shot her another smile full of more gratitude. "Yes, you should hate me. You still can, if you'd like. But I appreciate you, and everythin' you've done. So see? We balance each other out." "The only thing that keeps me from shunning you just like the others is I can't help but feel badly for you, Dixie. I, unlike you, have a conscience. You've had a horrible patch. I mean, first we all hear you lost your fancy restaurant—" "That was almost two years ago, Em." Two long years, scraping together a pathetic living with the degree she never quite got, and working odd jobs with her limited—really limited work skills. Em clucked her tongue. "Two years, two days. Does the amount of time since the descent into financial devastation truly matter?" Dixie had to nod her agreement. It only mattered to her. Her and the investors she'd let down. "Okay, but then, your best friend, Landon, asks me to keep you—of all people—up to date on his journey to the end, knowing darn well I'd never say no because one, I grew to love him, too, and two, for gracious sake, he was dying. Then that best friend in the whole world of yours, I'm guessin' your only friend left, dies." "You're a fine human being, Em. I mean that." Dixie refused to take the bait and let Em get a rise out of her. Em pushed some more. "Adding to all that misery, there's Caine Donovan. Your heart must be in an emotional tizzy about seeing him after, what is it now? Ten years..." Dixie remained stoically silent. About all things failed restaurant and especially all things Caine Donovan. "You remember him, right? One-time Plum Orchard High heartthrob and all-county track star, now one of Miami's biggest real-estate moguls... Oh, and the man you claimed to love but bet on like a Derby horse?" Em was dropping a line into Dixie's ocean with a juicy worm on the end of it to see if she'd rear up and bite. The bet. God, that damn bet. But the truth was the truth. Her restaurant had failed because she'd been too busy partying and running up her credit cards to bother with silly things like managing the restaurant she'd convinced herself, with absolutely no experience at all, was as good a place as any to escape her hometown and run away from the horrible thing she'd done to Caine. Her engagement had failed because at the time, Dixie Davis didn't know how not to turn everything into a three-ring circus. And yes, Caine was successful, and she wasn't. All ugly truths. Topping everything off, there'd been Mason—the beginning of her end. Dixie lifted her sunglasses once more and forced a smile, letting her eyes purposely meet Em's. "Sorry to disappoint, but there's no emotional tizzy here. Seeing Caine is part of the process of saying goodbye to our mutual best friend. That's all. He has as much right as I do. He was Landon's best friend, too." Liar. She'd practiced those words in her bathroom mirror hundreds of times before she'd left Chicago so they'd come off cordial and, above all, gracious. She'd almost convinced herself this imposed meeting was just that—two people who hadn't worked out, simply running into each other again and chatting niceties until it was time to go back to their lives. But seeing Caine meant remembering how madly in love they'd been for a time. It meant hearing his voice, a voice so warm it could probably still make her thighs clench. If they ended up in a close setting, it meant possibly brushing against his granite wall of a chest or watching him confidently smile while he arrogantly tilted an eyebrow at her. It meant that swell of clawing longing for him rising upward and settling in her chest. It meant reliving emotions that still ached almost as fresh as the day they'd happened. No one since Caine had ever touched her quite the same way. Caine Donovan was like a drug, and she was his junkie in need of a Caine Anonymous meeting. Dixie chose to avoid Em dangling the Caine carrot under her nose. Talking about Caine meant stirring up all the emotions that went with everything that had happened. Today all her turmoil was reserved for Landon and her gratitude toward Em. That Em had walked this far out on the ledge, offering to come with Dixie to Landon's funeral in front of all of Plum Orchard's very prying, judgmental eyes, was more than was her due. The ache of more tears tickled the back of Dixie's eyelids. "You know, even though I knew Landon's death was inevitable, it really is just like everyone says—you can never prepare for it." Em waved a hand around the room, chockfull of life-size pictures of Landon doing everything from zip-lining in Alaska over an icy glacier to cooking in Bobby Flay's kitchen. "Well, if no one else was prepared, this sendoff is a sign Landon was prepared. He knew how he wanted to go out, and he left strict instructions about it. You don't think his mother arranged those drag queens on stilts outside, do you? The Plum Orchard Bible study ladies nearly fell faint to the ground when they arrived." A glimmer of a smile outlined Dixie's lips, lips still chapped and peeling from her nervous habit of tugging them. "He wasn't shy, was he?" "Landon was whatever the antithesis of shy is." That Landon had been. Loud and proud. Just thinking about him always made Dixie smile. Yet, each time she thought she might smile, a new wave of loss washed over her, and it reminded her she'd never smile with Landon again. "I hate that he's gone." God, she really hated it. She hated even more the fact that she hadn't made it back in time to be with him when he'd passed. Everything had happened so fast, in a blur of urgent phone calls from Landon's hospice care nurse, Vella, and Em's updates, to the humiliating decline from American Airlines of her very last credit card. Em pointed to one of the pictures of Dixie and Landon on a nearby table, her eyes fondly roving it. "He hates it, too. Who wouldn't hate being dead?" She chuckled, eliciting a laugh from Dixie, too. Dixie's shoulders relaxed a little in her ill-fitting jacket. She leaned into Em and said, "Landon's probably pretty upset he's missing this." Em's hand strayed to her hair with a bob of her head. "Oh, you know better 'n all of us what Landon Wells was like. He had to have his nose in everything, or it drove him positively crazy. I'm sure wherever he is, he hates missing out on the circus outside these doors. Did you see the gentleman who looks like he just left the set of that movie Coming to America? And bless his heart, all those grief-stricken comments from parts near and far on his Facebook page made me tear up." Dixie let slip a fond grin of recollection. "The turnout would have tickled Landon's 'come one, come all' bone," she agreed, referring to the mass of mourners she'd witnessed on their way inside. Eclectic defined her best friend, or maybe, he'd defined it? Either way, it was what made Landon Landon. His joy in everything great and small—his wonder at the differences in people, cultures—his determination to experience anything he could get his hands on and celebrate it with gusto. His ability to collect people from all walks of life and turn them into lifelong friends. Shortly after college, he'd invested his trust fund wisely in several startup internet companies and was a self-made multimillionaire by the time he was twenty-five. Those companies continued to provide steady incomes to this day. And along the way, he'd added new ones—via winning bets on everything from a game of pool with a castoff royal to a polo match with some foreign politician. Because of his savvy business acumen, Landon was able to retire at twenty-six. Since that time, he'd been to exotic, sometimes isolated locales Dixie'd never heard of, had experienced the gamut of a world traveler, from a pilgrimage in an ashram in India to bobsledding with the Swiss Olympic team. Landon had lived and loved openly and freely, sharing his wealth wherever he went. Dixie gripped the edge of the couch, her heart overloaded with the empty beat of grief. She'd miss everything about him: his pushy late-night phone calls about her nonexistent love life, his questions about her financial security, his inquiries into her cholesterol levels, and anything and everything else Landon had pestered/mothered/nurtured her about in their lifelong friendship. The small room had grown oppressive with her sorrow in the last vestiges of the late August day. She reached into her bag and used one of her many overdue credit card bills to fan herself. "Mercy, it's hot in here." "Are your ears hot, too? Because I hear through the iPhone grapevine Louella Palmer's in the back row of this very establishment, sittin' next to Caine, and chewin' his ear off as we speak. You know, the man you're not in an emotional tailspin over?" Em showed her the text in yet another obvious bid to take her licks. It only made sense she'd think the subject of Louella Palmer would be the straw to break Dixie's back. Everyone in town probably thought the subject of Louella was a sore spot for Dixie. The real sore spot belonged to Louella, though, and she had every right to it. Louella had once been her right-hand, helping her lead the Mags as if they were the mob—Southern contingency. They'd been frenemies of sorts then, and in the end just before Dixie left town, bitter rivals. Not only was Louella currently the head of the Magnolias, she was almost as good at mob relations as Dixie had been. On the outside the Mags were refined and decorous, and they considered themselves the epitome of Southern grace and charm, but upon Dixie's harsh inner reflections these past few years, they were all nothing more than elitist snobs with Southern accents—and she'd been the biggest one of all. Of course Louella was sitting with Caine. It gave her plenty of time to remind him anew how Dixie was spawned from the loins of the devil. Caine was already here, too. Dixie's heart sped up as though someone had revved its engine, but her next words belied the storm brewing in her stomach. "You know what, Em? I hope Louella reminds him just how silly he was to ever get mixed up with the likes of Dixie Davis." Take that. She would not bite on the matter of Caine Donovan or Louella Palmer. The whole town had witnessed their messy breakup with Louella smack dab in the middle, and in a town as small as Plum Orchard, people were sure to speculate about their eventual meeting after all these years. It was only natural—expected even. So why was she so jittery about it? Because what you did was unforgiveable, Dixie. Then you ran away without so much as an apology. Em's expression was astonished, her eyes full of some good ol' Southern shock. "I can't believe you're not biting, Dixie. How can you even be in the same room with him after everythin' that happened between you?" "Technically, we're not in the same room. I'm in here, and he's out there in the foyer." Right out there. "And I no longer bite," she teased, snapping her teeth in jest. "For two people who were gonna get married and had the biggest breakup Plum Orchard's ever seen, in the middle of the town's square to boot, you sure are calm and collected." Her spine stiffened. Em just couldn't seem to choose to love or hate her, and while Dixie recognized it as her due, the reminder of her and Caine's breakup was still like a knife in the gut almost ten years later. There'd been rain, and thunder, and shouting, and accusations, and even a small fire and finally, the death of their preordained relationship, left splattered all over the whitewashed wood-stained floor of the gazebo in the town square. Dixie shivered. She would not revisit that horrific night today. "I bet your mother's still crying over all that money wasted on your fancy engagement party. Caine's mama, too." Poke, poke, poke. Dixie knew for a fact her mother, Pearl, was still crying. She'd told her so from her sickbed in Palm Springs when she'd made Dixie promise to pass on her condolences to Landon's mother. Though, her tears always had crocodile properties to them. Pearl Davis didn't cry genuine tears over human beings. She cried over investments lost, bank accounts in the red, and the merging of two prominent Plum Orchard families lost to her all because of Dixie. And Caine's mother, Jo-Lynne? She still didn't speak to Pearl. Regret, sharp and just as vivid as if their breakup had happened only yesterday, left Dixie fighting an outward cringe. Dixie, Landon and Caine's mothers were all best friends once—the belles of Plum Orchard's hierarchy aka the Senior Mags. So it was only natural their three children were virtually weaned from the same bottle. Just over two years older than Dixie, Landon and Caine had been her protectors since birth. While their mothers had played canasta every Thursday, planned church events at Plum Orchard Baptist, and been a part of every social organization a small town finds imperative to good breeding and proper social connectivity, they'd also planned Dixie would one day marry one of the two boys. Either one would do as far as Pearl, Jo-Lynne, or Landon's mother, Charlotte, were concerned. They were all as good as family, the women used to say. That hadn't quite worked out as planned after Landon confessed to their families he'd only marry Dixie if she had male parts. And Caine's male parts didn't interest him in the least. Caine and Dixie had always known their mothers' plans were fruitless where Landon was concerned, but as it turned out, the plan wasn't so far-fetched when Dixie and Caine's relationship took a turn toward romantic upon their simultaneous returns to Plum Orchard. "So has Miss Jo-Lynne spoken to Miss Pearl since the 'incident' or is there still bad blood after all this time?" Em prodded with a smile. Dixie shot her eyes upward. "Look, Landon, who knew you weren't the only busybody in Plum Orchard? Emmaline's going to carry the torch in your stead," she teased, warmth in her voice. Em swatted her with her plastic fan. "Oh, hush, and don't you worry. There's still plenty of busy to be had from Landon, Dixie Davis. Plenty." She shot Dixie a secretive look with her sparkling blue eyes. The same look she'd given her when Dixie had mentioned the phone call she'd gotten from Landon's lawyer, insisting she be at the reading of his will. That phone call still made no sense, and it would definitely hold her up. Her plan all along was to get herself in and out of Landon's funeral lickety-split because she desperately wanted to avoid running into Caine, and Louella and the Mags, junior or senior. Avoid running into them like she'd avoid a venereal disease—or hitting a brick wall at full speed, driving a Maserati. A foolish hope, no doubt. She should've known Caine wouldn't miss Landon's funeral, even if he was living in Miami now. Of course, Caine deserved to pay his last respects to Landon as much as Dixie did. He'd remained one of the best friends Landon had long after she and Caine had fallen out of one another's good graces. I will not pretend like neither one of you exist, Dixie-Cup. You're both my friends. Y'all will always be my friends, and that's just how it's gonna be, whether you like it or not. Landon had said those words with a sweet-and-sour delivery after dropping a fond kiss on her forehead. She'd loosely maintained her friendship with Landon around Caine, as well. After Landon's refusal to walk on eggshells, he relayed information on Caine's life and exploits. While Dixie would never admit it, she ate the scraps Landon fed her like a hungry stray dog. Dixie turned, folding her arms across her chest to find Em with expectant hope in her eyes. "Okay, this is me biting. Care to explain exactly what that 'plenty of busy to be had' means? You are Landon's attorney's secretary, so you must know something. You've been giving me the side eye since I got here yesterday." Em's eyes snapped back toward the doors, connecting the mourning room to Landon's viewing room. "I'm just a lowly secretary. I know nothing you don't know." Suspicion pricked Dixie's internal antennae, making her narrow her grainy eyes. "You do know something, Em. My spidey senses are dull from the long drive from Chicago and fraught with grief, so just spit out whatever it is that's made you so full you're gonna burst." "I assure you, there's nothing." Em crossed her heart with two properly gloved fingers, gazing stoically at Dixie. And she didn't even blink. "Now, I think we should get a move on before we're thrown outta here for loitering." Outside the door buzzed with activity from impatient mourners still waiting to say goodbye. On a deep breath, Dixie took one last glance at one of her favorite pictures of Landon. One with his sandy brown hair, wide gray eyes and a smile he'd handed out as if he was handing out Halloween candy, Landon epitomized handsome. Goodbye. How would she ever say goodbye to him? "If you want to keep avoiding the man who shall remain nameless and absolutely doesn't put you in an emotional tizzy, you know, Caine—you'd better step up your game. He's four mourners, one a stripper from Glasgow, away from us in the line just outside that door," Em whispered low in her ear, holding up her phone to show her the warning text message from Augusta White. Dixie's stomach dived toward the floor, twisting and swirling as it went. The temptation to take just one quick glance at Caine when they walked through those doors made her twitch. Don't you dare look, Dixie. Do not. Her curious eyes would not betray her by peeking to locate his face in the crowd. His delicious, handsome, chiseled face. No. She wouldn't allow it. She soothed herself with the idea that it had been close to ten years since she'd last seen him. He was almost thirty-eight now. Maybe he had a paunch and a bald spot. It could happen. Early senior onset or something. "Dixie, c'mon now. Let's go," Em urged with a squeeze of her hand. With one last glance of Landon's smiling face, she picked up the photo and whispered, "Please, please remember this—wherever you are." Dixie closed her eyes and recited the words they'd used before they hung up after every single phone call, before every goodbye they'd ever shared. "I love you like I love my own spleen." That's a whole lotta love, Dixie Davis, he'd say on a hearty chuckle. Landon's all-too-familiar response to her decades-old declaration of love echoed in her head, leaving her fighting back another raw sob. Landon Wells—protector of all things defenseless, smart, rich and the best friend any girl could ever have was dead after a short, but incredibly painful bout with pancreatic cancer. Everything was bad right now. The world was dull and pointless. The future was cloudy with a chance of lonely. Tears fell from her eyes, making her shoulders shudder uncontrollably. "Oh, Dixie," Em whispered into her hair, wrapping an arm around her waist in a show of undeserved sympathy. "He'd hate you crying like this almost as much as he hates bein' dead, and you know it." Dixie's throat closed and her shoulders shuddered, making Em grip her waist harder. "Stop this right now, Dixie Davis. We have an afterlife party to attend. Landon planned it all out. Rumor has it, Bobby Flay's gonna be there. You don't want to miss bacon-wrapped sliders made personally by Bobby Flay, do you?" Em's words made Dixie set the photo down and take a deep breath, preparing herself to face the crowd outside. She was right. Landon would hate her grief as much as he'd hated the pity showered upon him when he'd first been diagnosed. He'd told her to live, and while she did all that living, he wanted her to love again. Someone, he'd said into the phone during their last phone call, his husky voice deep and demanding in her ear even in the last throes of his illness. Love someone until it hurts, Dixie-Cup. And for everyone's sake, don't cry over my lifeless body. You're an ugly crier, girlie. A deep, shuddering breath later and she turned her swollen eyes to Em's compassionate gaze. "You're right. He'd hate to see me cry." When Em propped open the door to the viewing room, Dixie stumbled, forcing Em to tighten her grip around her shoulders. "You and your love of astronomically high heels. You'll break an ankle someday, Dixie." But it wasn't her heels that made Dixie stumble. It wasn't the endless rows of heads that shot up as they stepped into the chapel to join the mourners, skeptically eyeing their first glimpse of the Horrible Dixie Davis after so many years gone by. It was Caine Donovan and the momentary eye contact they made as Em pulled her away and down the seemingly endless candlelit aisle of the funeral home. The electric connection his deep blue eyes made with hers snapped and sizzled, sending blistering rushes of heat through her veins. It was everything and nothing in one short glance, hot and sweet, dismissive and breathtaking. Her heartfelt prayer he'd developed a paunch and had lost all that luscious chocolate-brown hair had gone unnoticed by whoever was in charge of aging. He stood beside a smug yet pretty, Louella Palmer, wearing a conservative black sundress and matching sun hat, her blond hair sweeping from beneath it. As Dixie and Em moved toward them, Louella's fingers slipped possessively into the crook of Caine's arm just as she turned her pert little nose up at them. A reminder to Dixie she'd once broken the mean girl's girlfriend code. Job well done. "Ladies," Caine said with an arrogant nod and an impeccably unmistakable impression of Sean Connery. Em whisked Dixie past him so fast she had to run to keep up. But she hadn't missed the subtext of his Sean Connery impersonation. Caine had once used that accent, and his uncanny ability to mimic almost anyone's voice, on more than one intimate occasion. His knowledge of just what a Scottish accent did to her naked flesh was extensive—and he was lobbing it in her face. Perfect. Em twittered in girlish delight, bright stains of red slashing her cheeks. "Oh, that man," she gushed, holding firm to Dixie so she wouldn't divert off their course to bacon-wrapped sliders. "He's so delicious. I can't believe he didn't take that gift and use it to make big money in Hollywood or somethin'." Dixie flapped a hand at her to interrupt. "I know. He's so dreamy when he does his Sean Connery impression." And Frank Sinatra, and Jack Nicholson, and Brando, and even Mae West. Caine's ability to impersonate not only movie stars but almost any stranger's voice was something they'd once laughed over. Dizziness swept over Dixie like a soggy blanket, clinging to her skin. But Em kept her moving to the end of the aisle and out the door. "Yes. That. All that dreamy handsome, well, it's dang hard to hate." Em's face was sheepish when they finally stepped outside into the hot August day. The darkening sky hung as heavy as her heart. Spanish moss dripped from the oak tree above them, drifting to the ground. Em crumpled some with her conservative black pumps. "Sorry. He's just such an honorable man. He makes despisin' him akin to killing cute puppies. Forgive me?" Dixie gave her a small smile of encouragement, moving toward the parking lot on still-shaky knees. "I'll forgive you, but only if you call him a mean name in feminine-solidarity. It's the only way to atone." Em pressed her key fob, popping open the locks on her Jeep. She looked over the top of the shiny red car at Dixie who stood on the passenger side and put her hands on either side of her mouth to whisper, "He's the shittiest-shit that ever lived. Shittier than Attila the Hun and Charlie Manson on a team cannibalistic virgin-killin' spree." She curtsied, spreading her black dress out behind her. "Forgiven?" Dixie smiled and let loose a snort, adjusting the belt of her jacket to let it fall open in order to cool off, if that was possible in the last days of a Georgia August. "Done deal." Em winked at her. "Good, right?" With a deep breath, Dixie let go of the restrictive tension in her chest. "You're a good human being, Em. Right down to the cannibals and virgins." Dixie paused, letting their light banter feed her soul. It was okay to laugh. Landon would have wanted her to laugh. She tapped the roof of the car with a determined flat palm. "All right, c'mon. Let's get to this shindig before I have to go to the reading of Landon's will. I really hope you weren't kidding earlier about the bacon." Dixie slipped into the car, taking one last glance of the funeral home in the side-view mirror where her last true friend in the world was housed. Her mentor, her shoulder to cry on, her life raft when everything had gone so sideways. And then Caine stepped off the curb and into view—his tall, hard frame in the forefront of gloomy clouds pushing their way across the blazing hot sun. Whether she'd admit it or not, Dixie watched Caine get smaller and smaller in the distance against the purple-blue sky until he was gone completely from her grainy-eyed vision. Déjà vu. Two "Phone sex. You mean like—" Dixie dropped her voice an octave "—'Hello, this is Mistress Leather' phone-sex?" "Correct, Ms. Davis. Phone sex. The act of engaging in verbal fornication." Dixie took a moment to process the entirety of the phrases "phone sex" and "verbal fornication" and what that entailed, but it was proving difficult. After so many sliders, she thought maybe not just her arteries were clogged, but her brain cells, too. Yet, she tried to let the words of Landon's attorney sink in as casually as if he'd told her she was now the proud owner of one of Landon's classic cars. So Landon Wells, the man Dixie was sure she knew everything about, right down to his preferred brand of underwear, owned, among various other assorted businesses, a phone-sex company he'd won on a bet in a high-stakes poker game in Uzbekistan back in 2002. Dixie tore her eyes from Landon's lawyer, Hank Cotton, Sr., and cocked her head in Em's direction, her eyes full of accusation while purposely avoiding the invasive gaze of Caine Donovan. He'd remained brooding and silent while Hank read the will, but Dixie knew Caine like she knew herself. He was just waiting for the right moment to pounce on her with his cutting words. Dixie chose to ignore Caine, turning to Em who'd known the whole time what Landon was up to. This was what her code-speak had been about back at the funeral home, and she'd held her tongue. Em, from her seat beside Landon's lawyer where she flipped papers for him to read, folded her hands primly in her lap and made a face at Dixie. "Oh, stop lookin' at me like I'm Freddy Krueger. Might I mention, I am a legal secretary for heaven's sake, Dixie. I couldn't tell you. So I'm callin' the cloak of—" "Client confidentiality," Dixie finished for her, lacing her words with bold strokes of sarcasm. "I know you're the last person I deserve common decency from, but at the very least, I expected more originality, Emmaline Amos. Something like, all memory of Landon's recently revised will was snatched from you by aliens, and no way in the world would you have kept this kind of shocking news from me as yet another form of payback had those despicable aliens not sucked your brains out through your nose with a pixie stick." Em shook her head, her silky dark hair semiflattened by the sun hat she'd discarded. Her ruby-red lips curved into a wince of an apologetic smile. "Mmm-hmm. You know, I almost went with that story, but then there were all the complications that come with the pixie sticks, and I just couldn't get it to...gel." She threaded her slim fingers together to articulate her effort to gel, then let them fall back to her lap. Caine sat in the corner, still silently sexy, his gaze burning a hole in the side of Dixie's head. As if this was all her fault. If the world came to a screeching halt, just before it did, the last words she'd hear before it all ended would be Caine declaring it was all Dixie Davis's fault. Gritting her teeth, Dixie clenched her hands together in her lap to cover the bloat from the Alaskan king crab and sliders they'd consumed and lifted her chin. "I call traitor. You were traitorous in your intent. It isn't like I don't deserve as much, but this?" Phone sex wasn't something you kept from someone—not even Satan. Em pouted, her heart-shaped face scrunching comically. "That's mean, Dixie, especially coming from you. And just when I thought you'd taken a turn, too. See why I was so hesitant to believe? I was just doin' my job. I do have children to feed. And a very large dog." "Did you just say Dixie's taken a turn, Em? A turn for what?" Caine finally inquired with that delicious drawl, his growly voice warranting an unbidden stab of heat in places along her body Dixie had to mentally beg to pipe down. His square jaw shifted, going hard as his lips turned upward into a smug smile. "Satanic worship?" If there was one person who could make her reconsider sidekicking it with Satan, it was Caine Donovan, making her heart race like a Kentucky Derby horse all while she hated him for still being capable of wreaking havoc on her emotions after ten years apart. Instead of reacting to him, Dixie turned the other cheek, narrowing her eyes at Em. While it was true Em should have no loyalty to her, she couldn't help being upset. "Is it your job to taunt me, too? Because that's exactly what you did back there at the funeral home. You hinted. You bandied, and you took pleasure in it to boot." Em slapped her hands on her lap, sending up a cloud of black material from her dress. "Bandied? That's a fancy Chicago word there, Miss Dixie, and I did not taunt. I was just tryin' to prepare you in a very roundabout, non-confidentiality-breaking way for—for this...And of course I was dying to tell not just you, but everyone in Plum Orchard. It's the most scandalous news ever. I can't wait to see what the senior Mags have to say about this. But in the end, I couldn't betray our client." Hank's nod from behind his glossy desk was of staunch approval. "That's true, Ms. Davis. We take our clients' confidentiality very seriously." Em's head bounced again. "We definitely do. That also means I couldn't tell you lots of things until the reading of the will. As a for-instance, a small village in some east African town I can't pronounce will now reap the benefits of books, teachers, and medical care because of Landon." "Africa isn't phone sex, Em," Dixie reminded. "Then guess what? Landon owned one of the most successful phone-sex companies in the world, and he left it all to you and Mr. Smexy. You know, with conditions. Surprise!" She smiled and winked at Caine aka Mr. Smexy, who was back to sitting stoically in his corner chair. He'd surprised Dixie when he'd shown up—surprised her and made her blood pressure pulse in her ears. Em had explained Landon's request Caine be present for the reading of the will, too. Something she'd also failed to mention while she was bandying and taunting. Dixie shifted in her chair, still absorbing what she'd just heard. Forcing her lips to form a question, her eyes sought Hank Cotton's again. "So just to be clear, when you say Landon had a phone-sex company, you don't really mean, 'Oh, Daddy, do it to me one more time' kind of phone sex, do you, Mr. Cotton?" Did he? No. That couldn't be what he meant. Yet what other kind of phone sex was there but the kind with ball-gags and chains and furry costumes? The palms of her hands grew clammy. "Say that again, Dixie—just like that." Caine antagonized, drawing out his words. "All that honey pouring from your throat, husky and full of rasp is hot. It's a voice made for sinning. The only thing missing is your accent. Where did that go, Miss Chicago?" The words he spoke were designed to hurt. Dixie knew he was taking pleasure in seeing the red stain of embarrassment flush her cheeks. Deeper and deeper Caine shoved the knife of their memories into her chest. Landon's lawyer, someone who hadn't been a resident in Plum Orchard when she'd left, sharply dressed in a dark suit and red tie, winced then straightened in his chair as though he realized control was needed. He cleared his throat, breaking the awkward silence in his overly warm office. "I'd like to get back to the business at hand. So yes, in fact, I do mean that, Miss Davis. And it's very successful, lucrative phone sex, I might add. After Landon won the company, he turned a sagging Call Girls into a multimillion-dollar corporation." A thought dawned on her just then, making Dixie relax into her hard seat. She nodded her head in sudden understanding. A nervous snort slipped from her throat. "This was Landon's idea of a joke, right? He told me before he died—" she puffed out her chest in Landon fashion "'—Dixie-Cup, don't you weep and wail long now, ya hear?' If you knew Landon, you'd know he'd go to any extreme to cheer me up." Even from the afterlife. Where she totally planned to, when time and hiring a psychic to locate him allowed, hunt him down and kill him all over again for mocking her this way. Hank shook his head with a firm sideways motion, his perfectly groomed, salt-and-pepper hair never moving. His vehement nod meant a resounding no. Not a joke. Hank leaned back in his plush leather chair and folded his slender fingers. "This is no joke, Ms. Davis. Landon Wells was very specific and quite detailed in his last wishes. He was the sole owner of Call Girls, and he hoped to pass that on to either you or Mr. Donovan in order to keep it in the family, so to speak. Clearly, his mother, Charlotte, wasn't an option. That left the two of you, his closest friends. And I warn you there's more to this. The will states that if you and Mr. Donovan wish to benefit from the entirety of the proceeds of his very unusual venture, both of you will have to earn it." Dixie looked away from Hank for a moment, focusing on an abstract painting on the far wall, full of slashes of color and streaked with gold edges. The tumble of emotions displayed in oil reflected her muddled thoughts. "Earn it? We have to earn a phone-sex company? Meaning?" "Meaning you'll both have to work the phones at Call Girls as operators. In essence, you'll be Call Girls employees for a two-month period with a general manager to train you, and watch your progress. As another stipulation, if you should decide to take on this challenge, you must both reside in Landon's house while you do—together or the offer becomes null. Landon had phone lines set up for you both at the guesthouse next to the other women he's employed. They're to help both you and Mr. Donovan learn the ropes of the industry, so to speak." Em's finger shot upward. Clearly, there was something in this madness Em hadn't been privy to. "Do you mean to tell me Landon's plan is to keep the business and those women in his guesthouse here in Plum Orchard for good?" She grabbed a stray file folder and began to furiously fan herself with it. "What will Reverend Watson say? Oh, the ladies of the Magnolias of the Orchard Society will not like this. Not one bit." Dixie actually had to fight a giggle at the thought of the Mags, especially Nanette Pruitt's face, the busiest busybody of them all, when she heard the news that Landon Wells planned to harbor harlots in what everyone in town, as far back as she could remember, lovingly called the "Big House." "I'm the only person who knew the complexities of Landon's will, and the people asked to help him execute it. No one in Plum Orchard knows the extent of it yet. Leastways, I haven't heard anything through the grapevine," Hank soothed. "But yes, that was his intent. After finding out he was terminally ill, Landon had his general manager, Catherine Butler, begin the move—they only left their old offices a couple of days ago. Landon wanted the ladies of Call Girls moved from a lush apartment in Atlanta into his guesthouse, where he had Catherine set up operations in order to keep what he called 'his girls' closer to home. As Emmaline may have told you, Catherine's now happily engaged to Emmaline's cousin, Flynn McGrady." Em's eyes widened, her hand immediately drifting to her cheek. "Cat knew Landon planned to keep Call Girls here?" She turned her gaze to Dixie. "Why, the two of them were just over for Sunday dinner at Mama's and not a peep about it!" "Catherine was bound by legalities to remain silent until the will was read," Hank reminded Em. "I hope you won't hold it against her." Dixie nodded her understanding and gave a tired sigh. "I don't know about Em, but I don't blame her. How do you say, 'I manage a phone-sex company, pass the fried chicken, please?' Especially with your mama in the mix, Em." Em's mother, Clora Mitchell, was a lot like her own mother. Controlling, and angry about something that had no label. Dixie handled her situation by running away from it, and Em handled it by taking exhaustive good-girl measures. In her later years, Clora had loosened her stranglehold on Em a bit, but she was still as proper as they came. Clora'd faint dead with the knowledge she was related, even loosely, to someone working for a phone-sex company. Hank cleared his throat. "We were talking about the guesthouse. That's where Dixie and Caine, if they choose to accept this challenge, will work during the course of their training. All of the appropriate permits are in place, and there's a formal letter to Reverend Watson and Mayor Hale available should there be any doubt this is all done within the confines of not just county regulations but state, too." Caine, who'd gone back to quietly brooding, cleared his throat and steepled his tanned hands under his chin. Dixie knew that look. It was the one where all the processing of pertinent information was done, and he was ready to play. In three, two, one... * * * Caine fought to keep his voice even while trying to ignore Dixie and her gravitational pull. He was still damn angry with her. As angry as he had been the day they'd broken up, and that made him angrier. After all this time, Dixie still had the power to make him feel something he didn't want to feel. "So let me get this straight. Landon left everything to Ms. Davis and I, but only if we actually work at Call Girls and live in his mansion together?" Em coughed to disguise her laugh before pressing her fist to her lips to suppress another outburst. Hank locked eyes with Caine, steady and sharp. "Yes. That's correct, Mr. Donovan. You each have two months to create your personas as phone-sex operators, and your, um...specialty, so to speak. Whoever garners the most calls at the end of the two-month time period wins the company. Full ownership. I have a list of what exactly specialty means in the phone-sex industry, and some other details to be hashed out, but that's the laymen's gist of it. You'll have full access to the house and staff, but I warn you, Landon left strict instructions that a court-appointed mediator will monitor your actions, so in his words, there'll be no funny business. Your reputations for one-upmanship precede you both." Son of a bitch. Landon had covered every base, hadn't he? Especially the base that kept him and Dixie from finding a way that led to the other's demise. If there was a way to manipulate herself as the frontrunner in anything, from a sack race to a hot-dog-eating contest, Dixie would do it, and like the ass he was, he'd take the bait. You knew us well, friend. But why had he done this? In this particular way? Putting them together in the big house? The house where there were a million memories of them as a couple. Why had he put them together for an extended period of time anywhere? Landon had known how dark those days after he and Dixie broke up had been for him. This contest was like rubbing salt in a bloody gash. Putting the two of them together after their shitty history was diabolical and possibly even homicidal. No way he'd survive being around Dixie for an extended period of time. He wasn't proud of admitting that, but it was the damn truth. But wait. Caine finally smiled. The bastard was messing with them even from the grave. Damn, he loved Landon and his balls-to-the-wall sense of humor—even in death, he was busting their chops. Dixie might have fallen for this act Em and Hank were putting on, but he wasn't. He barked an openmouthed laugh at the thought. "Hah! You son of a bitch, Landon," he said into the room. "Best prank ever, pal. This one even gives Dixie a run for her money. And great job, Hank. Really. You should consider Hollywood. So let's get to why we're really here. Did he just want someone to witness his last prank? Wait, did he have you videotape this?" Caine craned his neck to scan the room for a camera. "This will end up on YouTube, won't it?" He laughed again. And then he pulled up short. Hank gazed intently at Caine. Shit. He wasn't blinking. They were screwed. "Mr. Wells said you might say something like this. I'm not sure you really understand me, Mr. Donovan. I repeat, this is no prank. If you wish to review Landon's will with the attorney of your choosing, I'm happy to oblige." In his mind, he'd been busy sending Dixie back to Chicago where she belonged. Shipping Dixie and all the memories that came with her far away. Taking with her the dark circles under her eyes and the worry in her voice. Leaving. So he could do what he'd intended to do when he came back for the funeral. Stay a while. Catch his breath. Reevaluate where his life in Miami was going, or rather, wasn't going. There was something missing from it these days. Something big. Something important. He wanted to know what that something was. But now, he was back in the room with them all, hearing words like Landon figured he'd think this was all some joke. Which meant it was no joke. Damn, Landon. Dixie leaned forward, her beautiful face masked in more apprehension, and it made his chest tight, despite his wish that he could ignore it. She was thinner, almost fragile, maybe. Something she'd never been, but it wasn't just physically. It was in her posture, once straight and confidently arrogant, now a little slumped. Shit. Don't get sucked in, buddy. Don't you damn well do it. You know what it's like when she wants something. She could out-act Meryl Streep on an Academy Award–winning day if it meant she'd get what she wanted. Or have you forgotten all those tears she cried when you broke off your engagement? They looked damn real, pal. She's good. Too good. Caine shifted in his chair and forced himself to ignore any and all signs Dixie was suffering any more than he was over Landon's death—or suffering over anything at all. But there it was again, her voice a little small, a little hoarse when she asked, "What if I don't have an attorney because they cost money, ridiculous money, no disrespect to you—" She gave Hank an apologetic wave of her hand "—and there's no possible way I can afford to have someone review this? What if, as utterly shocking as I'm sure this will be for some, I don't want to work at Call Girls?" Dixie didn't have any money? Bullshit. He'd heard about her closing her restaurant, but she came from one of the richest families in the South. She'd just ask her mother for more. Wasn't that what all women like Dixie did? There was a game here. Caine just didn't know what it was. Hank's expression didn't budge when he gazed at Dixie. "If you don't want to participate, then you forfeit your ownership to Mr. Donovan, and he owns Call Girls and the profits from such in its entirety." Aha. Those words, so calm, so beautifully articulated tripped all the triggers Caine suspected Landon had counted on. He and Dixie in a hand-to-hand combat situation where, if it killed one of them, they'd do almost anything to win. As it once was, it always would be. Now he got it. Dixie slipped to the edge of her chair, drawing Caine's eyes to her legs. He snapped them shut and instead listened to her ask, "So he gets everything if I decide to bail because I'm not game to pretend I'm Mistress Leather?" "Mercy," Em muttered, letting her head drop to her chest, kicking up the momentum of her makeshift fan a notch. Hank rolled his tongue along the inside of his cheek. "That's correct. And Landon suggested you use the title Lady. I believe—" he shuffled through more papers on his desk, tapping one before putting his glasses on "—yes. There it is in my notes. Landon thought Lady Lana would suit you, Ms. Davis. My notes here say he thought it was the perfect name for someone with 'a voice meant for sinning'." Hank slid his thin index finger into the collar of his Brooks Brothers shirt, loosening it to clear his throat. Caine smirked, looking directly at Dixie. Lady Lana. Nice, Landon. Yet, his victory was short-lived. First, when he remembered, even after their ugly breakup, Landon had kept their friendships on equal footing for the near decade they'd refused to speak to one another. Second, when he saw Dixie's pretty eyes finally spark, he knew he was in for it, too. In the name of fair, Landon wouldn't play favorites. "Really," Dixie drawled, her Southern lilt reappearing. She leaned forward toward Hank, her gaze captivating his, her body language, a glowing halo of sexy. Just like the old Dixie. Caine relaxed a little. Nothing had changed. It was just another ploy. She let her eyelashes flutter to her cheeks in that coy way that made his pulse thrash. "And did Landon have a name all picked out for Mr. Donovan, too? It would only be fair." She smiled at Hank—the smile that was both flirtatious and subtle, one she'd used often to get almost anything she wanted back in high school. One she'd used on him. One you fell for, dummy. Caine eyed Hank's reaction, at first taken aback. Really, who wasn't when Dixie poured on the charm? But it was only a momentary lapse before he read her playful tone. "How well you knew him. In fact, he did, Ms. Davis." Caine gritted his teeth, bracing himself. Damn you, Landon. I hope you're getting your pound of flesh up there. Dixie cocked her eyebrow upward in smug anticipation. "You have Mistress Leather's full attention," she cooed, using her husky-honey voice to encourage Hank to spill. She swung her crossed leg and waited, smoothing her hand down along her calf to her ankle before pointing her toes. Jesus. Hank looked to Caine. "Landon's suggestion was Candy Cane, with a play on Caine, but he was also partial to Boom-Boom LaRue." How do ya like that for some boom, Boom-Boom? Three Caine gripped the arms of his uncomfortable chair. Damn her, after ten years, for not only still being so sexy it made his teeth grind together, but for possessing the ability to suck any man—even staid Hank Cotton, into her vortex of charm. Boom-Boom. The hell, Landon? Why wasn't he getting the hell up, forfeiting everything to Dixie, and going back home to Miami? He could reevaluate his life anywhere in the world. It didn't have to be here. He didn't need the money. He didn't want the money. He wanted Dixie to go home and Landon alive so he could take him back out. Worse, why was she still stirring things up in him better left unstirred? Just the brief glimpse of her with Em today at the funeral home dragged him right back to their short but tumultuous engagement. When they'd both come home ten years ago, and she no longer felt like his kid sister, their constant sibling antagonism turned to something much bigger than he'd ever thought possible. When he'd stupidly believed Dixie wasn't the reckless, cruel, entitled kid he'd left behind. He mentally dug in his heels while she sat in her chair, daring him with her flashing eyes to come play the game. Not a chance she was going to sucker him again. Which brought him back to the same thought as he watched Dixie watch him. Why wasn't he hauling ass outta here? "What's the matter, Caine Donovan? Are you afraid I'll beat you just like I did when you bet I couldn't spit watermelon seeds farther than you?" Dixie pointed to her pink-lipsticked lips, full and pouty-smug. "That's right—this mouth beat you by almost eight inches." Caine made a fist of his hand, flexing and unflexing the tense muscles to keep her from seeing she was getting under his skin. "Your mouth was as deceptive as the rest of you. And you stood on a chair, Dixie. Hardly fair." Dixie tilted her chin toward her shoulder, letting it nestle in her long red hair, gifting him that smoldering eye thing she used to do, knowing damn well it made him crazy. "Why, where in the rules for watermelon seed spittin' did it say I couldn't use a chair, Caine?" Caine's jaw tightened to a hard line, shifting and grinding. Resist. "I don't need Landon's phone-sex company, or the money it makes. No matter how much." No amount of money was worth being around Dixie again. No amount of money was worth the constant reminder that he was an asshole who couldn't tell the difference between the real thing, and the fake Dixie thing. Yet. Here you sit. * * * Dixie rose to her feet, hurling her large handbag over her shoulder. That settled that. "Good for you, Richie Rich. Unfortunately, I do." Wow, did she. After her drive here to Plum Orchard, her checking account was nothing but the kind of change you find in the cushions of your couch. She needed the money. But did she need it enough to become a phone-sex operator? Weren't you the one organizing an ad for your kidneys on Craigslist just three short hours ago? But what if she didn't want to play Mistress Leather to dirty old men and oversexed college boys as a way to get herself out of this mess? What if? What if you want to live the rest of your life never making the things you've done wrong right? What if you just sweep it under the carpet like you've always done? What if you just skip this part, the hard part, and fix something else you've broken instead? Something smaller, less difficult, maybe? No. She didn't have to do this. She could skulk back off to Chicago and continue to lick her wounds in her studio apartment with the peeling pink paint, a stove that had only one working burner, a shower that dripped exactly two drops of water per minute, and a punk neighbor who sold pharmaceuticals for someone named Dime. She absolutely could go right back to living just barely above the poverty level while she tried desperately to pay back money she'd charmed out of her mother's connections. Money she'd promised to handle with care—promised in the way the old Dixie promised everything. Loosely—offhandedly—with little regard for anything but what she wanted. No. This was a way to finally do something because it was right. Still, the more she played with the idea in her mind, the easier it was becoming to convince herself she could do this. If getting back on her feet meant spanking a chair with a fly swatter for effect while she whispered the words, "You must be punished for disobeying me," into a phone, she'd do it. It was either that or starve at this point. Food won. Food and a warm place for Mona and Lisa, her twin bulldogs to sleep. "So, it's settled? I win. You lose. Where do I sign, Hank?" Hank gave Dixie another "Hank look" translating to "not so fast." "Let's not be hasty. You have twenty-four hours to think about it, Ms. Davis. Mr. Donovan, too. Landon insisted upon a waiting period of sorts. In the meantime, Landon has offered his house and staff at your full disposal—to the both of you—while you mull this opportunity over. He wanted you both to be comfortable while you considered his offer." She'd already had two years of broke since her restaurant had gone bust. Why waste time? Dixie shot her hand upward to avoid more naysaying. "I don't even need twenty-four seconds. I'm in. Pass the pen." But Hank shook his head. "I'm sorry. Landon insisted that you both take the time to thoroughly think this through and get your affairs in order. He knew the two of you well, Miss Davis. His notes, and there were many, many notes—" Hank held up a stack of paper "—claim, on occasion, you're quick to jump before you think. Especially if it comes to any sort of competition with—" "With me," Caine interjected with confidence, quite obviously pleased with himself. Hank's lips pursed at Caine's interruption. He held up the ream of paper again and pointed to it with a short-clipped nail. "Yes. Landon did say that, but Ms. Davis wasn't the only one he left remarks about. He also mentioned you're quite easily baited by—" he looked down at the paper, shifting his glasses "—the lovely and irresistible Miss Davis. His words, right here." He tapped the mountain of white again. Dixie shot Caine a triumphant gaze. If there were notes to be had, she was grateful she wasn't the only one worth noting. Caine's fingers flexed and cracked, signaling his legendary simmer. "Thus," Hank continued, "he asked that you both take a hard look at his proposition. Landon was quite aware you both have lives and jobs elsewhere." Well, one of them did. "So please, each of you use the maximum time given, and we'll meet back here tomorrow at six with your decisions. Now, Landon had all the locks changed on the big house just prior to his death. I'll go get the set of keys he had made for each of you so you can settle in after such an emotionally trying day." Hank rose, whisking out of the office on expensively clad feet, quite obviously relieved to get away from Landon's tawdry business dealings. Em rushed to stand next to Dixie, peering down at her with an expression of guilt. "Before you rush to callin' me a traitor, yes, I was the one who had the keys made and called the locksmith to change the locks. But I maintain, I only knew Landon owned a phone-sex company and he was leaving it to you two to fight over. I thought Cat and the girls were going to show you the ropes temporarily. He left me a beautiful letter to thank me for facilitatin' his...his passin', but there was nothing about keeping Call Girls here permanently." Dixie's smile was as ironic as her tired nod. She patted Em's hand. "You don't owe me an explanation, and either way, I'm not staying at the big house." Not with Caine. Not knowing he'd sleep in one of the eight or so bedrooms—naked. He always slept naked. A fleeting visual of his wide chest with a sprinkling of dark hair and thickly muscled thighs spread wide to reveal his most intimate body part shuttled through her mind's eye unbidden. Dixie bit back an uncomfortable groan. "But the big house is so nice with every luxury available. Butlers and maids and a full-time chef," Em said, as though all those things in a gloriously opulent setting would make it easier to answer to the name Mistress Leather. "And bidets. He has bidets. Who can resist a bidet?" Dixie pulled her purse closer to her side, running her fingers over the surface. She knew everything Landon had. Scratch that. Almost everything. "Yes, I know Landon has a bidet, and a slide in the pool, and a screening room, and a camel named Toe he couldn't bear to part with when he left Turkey so he hired a zookeeper to care for him at the big house. He told everyone all the time what he had. I'm not interested in his possessions—just the predicament he's left me in." Dixie breathed deeply, pushing air in and out of her lungs to assuage her anxiety. "I don't want to stay at Landon's, and I don't care about the chef." "You just care about the money, right, Dixie?" Caine interrupted, rising from his chair to saunter with liquid grace toward them. As confident as ever, he'd added a dash more smug to his repertoire. Nice. Veiled innuendo. Fine. She deserved all of the mud he could sling. As she turned to look him directly in the eye for the first time in almost a decade, Dixie mentally reminded herself to stand strong and fight the bone-deep lust that never failed to consume her whenever Caine was in close proximity. The way he moved with the sensual grace of a panther, the light bronze of his skin beneath his white shirt and navy suit, the ripple of his thighs, pushing against his trousers, still affected her. But resist she would. Not an easy hurdle to jump when he moved in even closer and gazed down at her, waiting. No. He wasn't waiting. He was laying down a dare in much the way she had earlier, but his wasn't based on desperation. It was steeped in anger. Automatically, Dixie's chin lifted, her pride raising both metaphoric fists to the sky even as a wave of shivers covered her arms and the back of her neck. "Don't be coy about it, Boom-Boom. If you want to insult me then do it, but do it well. I'm not ashamed to say I need a job. So what?" "And you're willing to call men you don't know 'Daddy' for employment?" Her cheeks went hot, but her mouth flew open. "You're just shy of accusing me of hooking for cash, aren't you?" Caine's dark eyebrow rose while he jingled coins in his pocket. "Oh, I'm not shy, sweetheart," he reminded her. She swallowed hard, the room growing oppressive. No. Neither of them had been shy. Their chemistry was what legends were made of. Hot, sticky, soul-baring legends. Her legs wound around him while he drove into her with forceful thrusts until she screamed, was the hottest, rawest sex she'd ever had. Everything—everyone since was just lukewarm. She forced that to the back of her mind. "Well, I'm not shy either," she gritted, "as you well know. So here's the truth of the matter. The economy stinks. My restaurant went bust. I lost hundreds of thousands of some fine people's investment dollars. My 401K has tumbleweeds cohabitating in it, and I haven't been able to find a decent paying job in two years. So shoot me, Caine Donovan, for having the audacity to entertain the thought that this might answer a couple of long overdue prayers." There was nothing Caine would love more than to hear the opportunity she'd jumped on when she'd left Plum Orchard had failed. He deserved to roll around in her failure. Em stepped between them, casting Caine a pleading eye before turning to Dixie. "Suggestion? It's been a long, chaotic day. How about we go to Landon's and relax before someone says somethin' rash?" Dixie straightened, preparing to leave before she took the bait Caine dangled in front of her and things escalated between them. They were older—wiser—and their behavior should reflect that. She tugged her purse back over her shoulder with resolve. "I'm ready now. That Landon wants us to wait twenty-four hours is just enough time to grab a shower, eat some of Martha's infamous peach pie and Sanjeev's lamb curry, get a decent night's rest, and skip back over here to sign those papers." Her choice was made. "You do realize this is ridiculous, don't you, Dixie?" Caine's voice grumbled, still so sexy-rough. "Landon's really yanking our chains, pitting us against one another. You know, just like back in the old days when the two of us competed over everything, and Landon looked on fondly at his two foolish best friends making asses of themselves? He's having a good laugh, wherever he is. What I don't get is why he'd do something like this. It's not like Landon, especially knowing the way we feel about one another. I don't suppose he left the reasons he did this in all that paperwork, Em, did he?" Em's hands folded and dropped in front of her. "No. I don't know any more than the two of you know." It was clear Caine's anger with Dixie hadn't dulled after almost a decade, and he wanted her to know. Fair enough. "Then don't stick around for the five W's. Go back to Miami and sell some more million-dollar, oceanfront houses to leathery-skinned women who have pocketbook-size dogs. You don't need the money. I do. You probably couldn't handle the challenge anyway." Dixie was methodically inviting him to try and best her. It was silly and childish and unlike the person she strove so hard to be, but gravy. Ten years was a long time to still feel this much hate coming from Caine. The ripple of power Caine exuded reflected in his narrowed eyes. "Are you suggesting I let you have everything?" "I'm suggesting you go home and admit defeat. Because, as you've mentioned, you don't need the money." "And how is it that you've come to the conclusion I'll end up the loser?" "It's simple logic. Me—woman—with a hot voice, if all the compliments I've been getting all these years are any indication. You—man, probably not a key component when attempting to arouse a male who wants to be called Daddy by his little girl." Dixie had to fight the shudder those words evoked. That was most definitely not going to be her persona's specialty. "Ah, but you forget one little thing, Mistress Leather," Caine baited, gracing her with a smile full of white teeth. "What's that, Candy Caine?" Her eyebrow rose with total confidence. She hadn't forgotten anything. She had him by a landslide just by virtue of her gender. Caine leaned into her, the slightest hint of his cologne dousing her nostrils before she took an unsteady step back. "You're forgetting 'Bond. James Bond.'" The tip of Em's index finger went directly into her mouth. She nibbled the chipped end of her nail, her brow furrowing, her eyes flashing danger zone signals at Dixie. Oh, damn him and his Sean Connery bombs. Caine could create any persona he desired and melt the insides of millions of women into sticky goo. Dixie wanted to stamp her feet in frustration until she remembered one thing. Women didn't call phone-sex lines, or if they did, they sure weren't in the majority. Men were. Hah! Dixie was right back in high school when she said, "I think you're forgetting one little thing, Boom-Boom, name one woman you know who calls a phone-sex operator. One." Caine's lips flat-lined. Uh-huh. "I bet you don't have enough fingers and toes to count the men you know who've dialed a Mistress Leather, or variation thereof, do you, Caine Donovan?" More flat-lining and nostril flaring. She curtsied and winked. "Your serve." "Don't be so quick to call me dead in the water. The women of today are empowered, unafraid of their sexuality, bolder about their needs and about expressing those needs. Add in Sean Connery, Johnny Depp, maybe a little Sam Elliott or for that matter, almost anyone they'd like to, uh...verbally play with, and I'm your man." Then he grinned. Wide. Smug. Her nostrils flared. "So I'll tell you what, Dixie Davis, you go right ahead and rev up your sexy, because I dare you to top that." He'd used the word dare. Such a bad, bad word. Resist, Dixie. Fight it. Fight hard. Instead of reacting, Dixie gathered herself together, her body rigid enough to shoot an arrow and looked Caine Donovan square in the eye. The second gauntlet of the day she threw down was again silent, metaphoric, but it was no less meaningful. "Then I guess this is Donovan versus Davis. See you here tomorrow at six. Don't forget your thong and your flogging thingy." "Flogger," Em corrected. "It's just called a flogger." Dixie cocked her head at Em. "You know this how?" Her face flushed red as she backed away from them. "I'm gonna go check on Hank and see if he's found those keys," she said over her shoulder, her embarrassment painfully obvious. Caine rolled his tongue along the inside of his cheek, his expression once again arrogant. "You bet I'll be here, Dixie, and I'll see your flogger and raise you some latex and hot candle wax," he retorted, still so smug. Okay, conscience, fair is fair. I'm trying to be the best person I know how to be. I'm trying to leave my baggage at the airport carousel. But c'mon. He's baiting me. It's plain as the nose on my face. You can't expect me to take it and just lie down and die. Her blood pressure soared. "Funny you should mention the word see, Caine." Dixie paused, putting the tip of her nail between her lips, widening her eyes with mock exaggeration. "You know, I wonder if Landon's company provides live video chats? I bet he does in this age of technology. So, I'll see your ridiculous latex and raise you one hot Southern belle in a leather corset, fishnet stockings and some ruby-red stiletto heels. A real live Southern belle, not someone just pretendin' to be a celebrity," she sniped with a smirk. Caine leaned down, pinning her with his gaze, as though he were transmitting every last hot, lust-filled second they'd spent together to her mind's eye. He trailed a finger along her cheek, making Dixie fight a whimper for the weak-kneed hunger his touch left in its wake. It was all she could do to remain defiant rather than curl her jaw into the digit and sigh with years of pent-up yearning. His hand snaked around her waist, hauling her to him so their bodies were flush, his taut, hers softer but no less aware of the fire brewing beneath all that sinew. Her clit throbbed in reaction to the rigid line beneath his trousers, aching with familiar need. Her leg begged her to allow it to wrap around his trim waist. His hard fingers dug into her flesh, but Dixie didn't flinch. Instead, she issued what she was sure, if Caine actually decided to take Landon's offer, would be just one more of her many challenges. "What do you have to say for yourself now, Caine?" Leaning in farther still, his lips stopped a mere breath from hers, creating an all-over tremble of awareness. The scent of his cologne, sharp and musky, lingered in her nose. "I say you look hot in leather, Dixie. Your ass was the finest in all of Plum Orchard at one time. Maybe even in the entire state of Georgia." Caine emphasized that point by reaching around her and grabbing a handful of it, kneading it until she thought her lungs had stopped working altogether. Sliding his free hand along her bare leg, he traced his silken-padded fingers upward until they were under her skirt and had reached the edge of her panties, allowing his knuckles to skim the tender flesh where her leg met the apex of her thighs. Caine pulled away then, almost garnering a gasp of disappointment from her, only to run his index finger along her cleft, pressing the silk of her underwear against the heat of her achy clit. Shivers of need, desperate and wanton, made everything else fall away. Though her arms remained at her side, the all-consuming desire to twine them around Caine was a war she fought with steely resolve. He let his silken tongue dab at her lips, before he added, "Know what else I say?" Her breathing was choppy, there was no hiding it, but she was delighted to find, Caine's was, too. "What else do you say?" The delicious movement between her legs stopped as suddenly as it had started. He smirked down at her. "I say you don't have the guts. That's what I say." Just as Dixie was considering wiping the smirk off Caine's face with a good right hook, Hank and Em's footsteps sounded. She pushed at Caine, taking two unsteady steps away from the astounding effect he had on her body, away from the memory. Em held up the gleaming keys and shook them. Dixie snatched her set from Em and dangled them in Caine's direction with quivering fingers, and melting kneecaps. "I'll see you here tomorrow, Caine Donovan, and we'll see who has guts. Bring your impersonations. Bring whatever you think will help you win this. Just be sure to bring it, big boy." Dixie rounded on her heel with such fluid grace she owed herself a pat on the back for not collapsing. "Good night, Hank. I'll see you tomorrow at six sharp." She sashayed out of the office with the invisible words I dare you written all over the back of her suit jacket. When she reached the top of the stairwell, she had to grasp the banister to keep from pitching forward. The throb in her temple returned, matching the unmerciful throb between her thighs, beat for agonizing beat. She'd just consented to sell sex over the phone so she could win a new way to make a living, and in order to do it, she'd have to beat Caine Donovan, the one and only man who'd ever made her so insane with primal, wanton need, she would have done anything he asked. Crazy must have taken a global vacation, but not before making one pit stop in her small town in Georgia. Em skidded out into the hall, hot on her heels. As she reached the top of the steep steps she panted, "Don't do it, Dixie! I can barely afford to feed our dog, Dora the Explorer. I don't know if I can take Mona and Lisa in, too. And seeing as you have nothing left in your 401K, you won't be leavin' me anything to help." Dixie finally giggled, releasing her nervous tension. "I wasn't thinking about ending it all. I was just thinking about getting out of that room." "Where all that hot man sucks up every last ounce of air? I know. I get it. He's like a vacuum packer—or at least, when you're in the room he is." "That's not it either." The lie fell from her tongue like honey dripping from a bottle. "I was leaving before we ended up thumb wrestling till someone cried 'uncle.' You know what we were like, Em—always trying to one-up each other—fight to the death. That was years ago. I've grown up. So the last thing I want to do is engage in a pointless 'he said, she said' argument. I want to go back to my hotel and mull—plan—plot how in the world I'm going to pull this off." Em clucked her tongue. "First, we're going back to Landon's so you don't break the rules he's set forth and forfeit everything because you can't resist being difficult. My mama has the boys for the night, and I'm free. I can dine on cold, leftover crab and artichoke dip in Landon's hot tub, which runs at a warm ninety-eight degrees. And second, remember this—your voice is pretty sexy, Dixie. All raspy and Kathleen Turner-ish. No doubt, you've made a million foolish men fall at your feet without ever having seen you. All they needed to do was listen. Bet you could beat the pants off Caine Donovan in a phone-sex-off with a voice like that if you set your mind to it." If only his pants were the issue. Anxiety churned in Dixie's stomach. "But he can create thousands of different personas with his impressions, Em. He can be whoever a woman wants him to be. How can I ever top Sean Connery?" "I can't even believe I'm sayin' this. What do you think the ratio of male/female callers really is? Ignore the story Caine was sellin' you and focus. You could beat him with your mouth taped shut with those odds. Women might be empowered these days, but the truth is, they don't have to work as hard as men out in the real world." Good point. But... There was still Sam. "Have you heard his Sam Elliott impression?" Em waffled, probably because she had. And it was a thigh-clencher. Still, she shook a stern finger at her. "Then you'll just have to work harder." She paused then, her smile ironic. "Funny, isn't it? You actually workin' for what you want instead of everyone doing the work for you? And besides all of the obvious, we don't even know if Caine'll take Landon up on this crazy endeavor I'm hereby callin' 'Survivor, the Porn Edition.' So before you even consider feelin' sorry for yourself, just remember your new mantra—outwit, outlast, outplay." Em's words of encouragement warmed her. True enough. You didn't become a successful real-estate mogul by taking two months off. "You think?" Em nodded with a vehement dip of her head. "He has a successful real-estate business back in Miami, Dixie, employees and everything. He can't just up and leave for a long period of time. So I'd lay bets by tomorrow, he'll be on a plane back to the Sunshine State. Today was just him blusterin' like men do when a woman has the nerve to call them on their game." Dixie stood rooted to the top of the stairs while the phrase, "What can Mistress Lana do for you tonight, unworthy one?" ran like a stampede of elephants in her brain. Em roped an arm through Dixie's. "You're thinkin' too much. I can see it. Let's go to the big house and we'll talk it over." She stopped on the step for a moment, turning to Dixie, her eyes clouded with suspicion. "Wait a minute. Did Landon know what was going on with you financially? Did he know you were pushin' your last dime just to get here to be with him?" Tears began to flood her eyes again, but this time Dixie didn't stop them, she let them drip down her face and hit the steel steps. "No," she whispered. "I could never tell him...." "Because the first thing he would have done was meddle, and the second would be to set about making the boo-boo all better, and naturally, you have your pride." "So you know what happened?" That last bit of her pride floated upward toward the ceiling. "The grapevine is thicker than ever here, Dixie. Some took great pleasure in it when they read the papers and saw Dixie-Cup had gone belly up. Though I will tell you, I wasn't one of them. Honest." Facing Dixie, she held her right palm up. "I didn't want him to rescue me. I went in with my eyes wide open. I left Plum Orchard to open the restaurant with them wide open, too—definitely one of my more harebrained schemes. But I never told Landon a thing. I lied to him and told him everything was okay, because he was so sick and he had enough to worry about. I let him believe I walked away from all of my investors." "You're doin' this to pay back all those investors, aren't you? Because most of those investors were Davis family connections." Shame and humiliation tinged Dixie's gut, but she refused to let it dampen her determination. "If I have to sell an organ on Craigslist." Em let go of a heavy sigh. "That's what I figured. But it isn't like your mama's friends couldn't afford the investment, Dixie. They'll just write it off as a loss. And isn't that what bankruptcy is for anyway? So you don't have to pay anyone back?" Dixie shook her head sharply. No. That was the easy way out. No more easy. "It was the easiest way to keep the bank at bay, but I still owe a debt as far as I'm concerned. I'll repay it." Em's pretty blue eyes searched hers, a hint of admiration in them until they clouded back over with skepticism. "I just don't know what to think of you anymore, Miss Dixie," she said, her tone clear with conflict. "Then think about other things. Like how uproariously, ironically funny it'll be when everyone in town finds out Dixie Davis, reformed mean girl in deep financial debt, is selling sex." "You should've told Landon, Dixie. He'd have wanted to know. He loved you. He said that often to me durin' his last month. He said if he'd been hitting for the other team, it would always be you." He had said that on a million occasions. He'd said it when he admired the color of her hair or what he called the sexy half curve of her lip when she was thinking. He'd said it when she was singing along with the radio, and her sultry voice made every song sound dirty. Dixie smiled at the memory, and it grew wider. He'd said, The only person I'd change who I am for is you, Dixie Davis. You make this gay man pause from time to time. But then I remember I can't change, and you love Caine Donovan. Nothing can change that, girlie. Something had. Dixie shuddered a breath from her lungs and began to descend the steps one at a time, taking Em with her. Maybe it was Landon's spirit. Maybe it was just desperation, but an ember of hope sparked, and if she fanned it just right... "But he didn't know, and he didn't hit for my team, and now here we are. So let's go back to the big house and research phone sex, because I plan to be the best Lady Lana Call Girls has ever seen. Caine Donovan will rue the day he talks dirty to some lonely woman with Johnny Depp's voice." The pound of footsteps from behind them startled the women. Caine flew down the stairs past them, ruffling Em's hair on the way. "Race ya to the big house, ladies!" he yelled as none other than Christopher Walken, taking the steps two at a time as if he was twelve, and they were still walking the halls of Plum Orchard Middle School. "So we have some work cut out for us," Em squeaked. Dixie's eyebrow rose. "We? Won't that cause trouble for you with Louella and the gang?" Louella was going to have a kitten if she found out Em was helping Dixie Davis—once girlfriend-code breaker extraordinaire, now sworn enemy. Em flapped her hand, but her eyes wouldn't meet Dixie's. "Bah. They pay me little mind unless they need somethin' legal, so I pay little mind back. It's the same as it always was—just like high school. I wasn't born a Mag, so I'll never be a Mag. And since Clifton left me for that no-good woman in Atlanta, they only tolerate me because I can be of help from time to time in the legal area. I was always an outsider, Dixie. That's still just as true as it ever was." Dixie grinned. Em was bucking the system even though Dixie knew the lack of acceptance from the reigning queens of popularity and prominence stung. "Then we can be outsiders together." She tugged at her arm. But Em hesitated. "Wait. Before we go any further, there's one more thing." Dixie stiffened. "Now what? Oh, wait, I know. Landon owned a brothel, too, right? Is this the part where you tell me I have to get rid of my flannel pajamas for crotchless underwear, but you couldn't tell me before because it was confidential?" She accented the word with a roll of her eyes. Em's hand fluttered to her neck. "Why, Dixie, I almost think that would be easier." Hackles rose on the back of Dixie's neck. "Than?" "Telling you about the court-slash-Landon-appointed mediator. Remember Hank mentioned that?" Em's feet were suddenly moving down the steps at a rapid pace, the skirt of her dress flying behind her. Dixie followed suit, pushing the exit door to hold it open. "Vaguely. I was a little caught up in the 'oh, baby, I like it like that' at that point." Em stepped around her and held her hand out with a grimace. "Meet your court-appointed mediator." Four Dixie stood at the foot of the bed in her appointed room at Landon's house. The house he'd bought, expanded and renovated from top to bottom. He had instructed she stay in the aptly dubbed Princess room, the room he'd always given her whenever she'd come back home during and after college to visit the big house. Buttery lemon and pastel green leaves whispered across the wallpaper on the walls, surrounding the centerpiece of the room—a king-size canopy bed handcrafted in Italy of chestnut and ash and lacquered in a soft cream. This was the bed where she and Caine had spent the nights just before their engagement party, wrapped in each other's arms, contemplating their future. Caine would spread her out on the cool sheets while the sky outside grew heavy with stars. He'd rise up above her, running his possessive hands along her skin, paying special detail to the dip where her waist met hip, leaning forward and nipping at it while his hair grazed her shivering, frantic flesh. Her hands always rose to caress his thighs, loving the response he gave when he'd fall over her, taking her legs up around his neck and moaning the words with a rasp, You, Dixie. I need to lick you or I'll damned well lose my mind. Those decadent, raw sounds coming from his lips always made her press her hips upward, begging. When his head finally dipped between her legs, it was almost a surprise how the wondrous lust filled her up. Jesus, Dixie, you're all I can think about day and night, were always the last words he spoke before he parted her cleft with his thumbs and slipped his tongue inside her, drawing long passes around her clit, making her beg him to capture the bud between his lips and suck the hard nub until she was thrashing her way toward insanity. Rising up on his elbows, his glittering eyes held victory in them when they found hers. His raw power never failed to wrench the breath from her lungs when he demanded, Look at me, Dixie. Look at me when I— "Dixie?" The voice from over her shoulder jolted her with a yank from her memories and the indelible mark of Caine. Taking a shaky breath, she turned to find Sanjeev, Landon's trusted assistant, at the door with Dixie's lone suitcase. She quickly took the opportunity to hide her embarrassment by gazing around the room she'd helped to decorate. Her eyes scanned her surroundings and almost nothing had changed, from the thick carpet beneath her feet to the whimsical tea set on a corner table between the floor-to-ceiling windows, draped in shimmering silk, and overlooking the main house's pool. Despite the big house's lavish opulence, it was meant to enthrall those who stayed in it—not impress. Landon had never cared what people thought about his outrageous spending. He'd only cared that, should they grace his doorstep, they grace it with the utmost comfort at their disposal. Sanjeev, dressed in a traditional maroon kurta, put down her luggage then smiled at her. His olive-black eyes, set in flawless mahogany skin, gazed at her with warmth. "Landon said this should be your room for the remainder of your stay." He held out his long, well-defined arms and embraced her. He tightened his grip, as if he knew a hug was in order. She leaned back in his embrace so their eyes met, ruffling his thick thatch of midnight black hair with her fingers. "Yeah, about that, Sanjeev... Did Landon, that crazy prankster, say anything else about my stay?" His smile beamed wide. "He said I was to cater to your every whim, keep you well-fed, well-rested, and make sure you didn't spend wasted time mourning him." She gave him a look of admonishment, clucking her tongue. "Aw, come on, Sanjeev, you know what I mean, and it has nothing to do with your out-of-this-world lamb curry or your saffron rice or even your pillow fluffing skills. The phone-sex thing. You must've known." Sanjeev didn't miss a beat, though an erratic pulse throbbed at the base of his neck. "Of course I knew. I was his assistant. I knew everything." Dixie tapped him on the shoulder with a chastising finger. "So you knew Caine would be here, too." She didn't ask. His nod held no apology. "I did." "And a sneaky, late-night phone call, something along the lines of, 'Hey, Dixie-Cup, that guy who stomped on your dreams of marital bliss like he was stomping out a campfire is going to stay in the big house with you while you call men naughty boys' was totally out of the question?" Sanjeev's eyes twinkled. "First, I believe it was you who stomped first with that dreadful bet. And oh, no, it wasn't out of the question." The bet. She never, ever wanted to talk about the bet. "But it was disloyal to Landon?" She sighed in understanding. "I can't fault you for that, even if it wasn't in my favor." His smile gleamed playfully. "As per Landon's reminder, I was bound by the 'I saved your life' speech." Landon had found Sanjeev in the streets on one of his treks to India, dirty, infested with lice, homeless and alone at seventeen after he'd run away from an orphanage three years earlier. After living with Sanjeev for a year in India, Landon had acquired, via his multitude of connections, a visa for Sanjeev and brought him back to the States to live with him and manage the big house. That was eleven years ago, and never was there a better assistant to someone as whimsical and impulsive as Landon than Sanjeev. Dixie rolled her eyes, knowing Landon would have cut off his right arm before he'd have sent Sanjeev back to India. "Well then, I hope you gave him the 'If not for me, the big house would have collapsed by now, and Toe the Camel would have died of malnutrition' speech," she teased. And it was true. Sanjeev ran the big house like a well-oiled machine. Nothing, not even the tiniest of details went unnoticed under Sanjeev's watchful eyes. "I will always remain loyal to his memory, but above all else, his last wishes. Though," he said, cocking a raven eyebrow, "I did warn him, during the hatching of the conditions of this will, a war the likes of which no one in Plum Orchard had ever seen was bound to ensue." So Sanjeev knew the thought process behind Landon's last wishes. Interesting. But it wasn't the time to press. "And he said?" Dixie prompted, shrugging off her jacket and laying it across the bed. "He said, and I quote, 'I hope you videotape it and put it on YouTube because it'll probably get a lot of hits and become the YT's newest sensation,'" Sanjeev responded with his comical imitation of Landon's accent. Her head fell back on her shoulders as laughter, rich and free, spilled from her throat. It was so good to be where Landon's presence was strong—where his memory still breathed life into every nook and cranny—even if, in his memory, he'd left her between a rock and a hard place. "So he really has a phone-sex company?" Sanjeev's eyes were amused. "Indeed." "These women are in the guesthouse right now?" "They are. It's where Landon insisted they work." Dixie eyed him. "Did he give any thought to what will happen to these poor women when he decided to drop them in Plum Orchard? You know what they're like here, Sanjeev. How they all gossip. It can ruin your life if you let it." She knew. She'd stomped on a life or two in her time. "He gave it great thought. Surely, you know Landon did nothing without care, Dixie. He consulted all of them, and they made the decision together to come here, knowing how judgmental this town can be. However, when you meet the ladies of Call Girls, you'll understand why Landon left this earth at peace with his choice." "The Mags will find a way to make their lives miserable all while looking for a way to have this shut down, Sanjeev. Did Landon think about the fact they could lose their jobs?" "Have you thought about the fact that Landon has greased many a wheel in his time here on earth and in Plum Orchard—or that he was as careful about picking his lawyers as he was his locations for phone-sex operations?" Dixie gave a halfhearted laugh, rubbing her eyes. "Point in Landon's favor." "You look tired, Dixie. And I don't mean the kind of tired grief brings, or the kind a good night's sleep will fix. I mean soul weary. This worries me." Ah, leave it to Sanjeev to look beyond the concealer under her eyes. "It's been a long couple of years" was all she was willing to admit. He tugged on a strand of her hair, his eyes concerned. "And in those long years, you forgot to freshen your roots? Who is this Dixie?" This was the Dixie who was too focused on her goal to pay everyone back and didn't have time or money to go to the hairdresser. She shrugged, casting her eyes down at her feet. "This Dixie was just caught up in other things." "Then this assistant will fetch you some henna before you become too much more caught up. Pronto," he added with a wink. Dixie kicked off her heels, sinking her bare feet into the Persian carpet. She leaned her shoulder against the canopy post to fold her hands in front of her. "I just can't believe he's gone," she choked out. Those words would never sound right. "So what will you do next, Sanjeev? Will you go back to India? I imagine Landon left you plenty of money to return in style." Though Sanjeev leaving the big house and going back to his homeland left her heart as empty as a good bottle of wine after a long night of girl-talk, Dixie had always wondered if he yearned for the sights and smells of his native country. Much the way she'd longed for the comfort of her small town even with its irrefutable throwback to a simpler way of life, and its antiquated views on a woman's place in the world. Sanjeev's eyes flashed momentary confusion. "I will do as I've always done. Maintain the big house and handle the multitude of charities Landon was involved in." She cocked her head, her ears burning hot with new information. "So Landon isn't selling the big house?" He'd left the big house to Sanjeev and the numerous staff? His arms went around his back. "No, quite the opposite, in fact." Uh-huh. Suspicion pricked her spine just as it had with Emmaline back at the funeral home. "You know something I don't know, don't you?" Sanjeev's eyes shadowed. "I know only the things I know." "As clear as mud as always, Sanjeev," she said even though his evident secrecy made her grin. Sanjeev's chin lifted as it always did when he was disgruntled about the fact that he still didn't have a full understanding of the subtleties of the English language. "For as long as I've been in your country, I will never understand you. Mud isn't clear, Dixie." Dixie tilted her head, squinting one eye. "Know what else isn't clear?" He took a solemn stance, his expression serene as he waited. Dixie began to pace, a revived, caged energy freshly unleashed. Surely Landon had confided his reasons to Sanjeev for putting her and Caine together. "Why Landon would do something like this to me—to both of us? He knew where we stood with each other. Caine and I are in the worst possible place two people who broke up the way we did can be." Sanjeev's eyes shifted downward in subtle recognition before refocusing on Dixie. "A place entirely of your own making." Dixie nodded at his more than fair statement. "That's the absolute truth. You're right. But he's pitted us against one another like two children fighting over the last piece of Martha's peach pie. Why would he want to hurt me like this? He knows—knew—how painful the subject of Caine is for me." Sanjeev smiled as though he were recalling a fond memory. "He's also the man who stood by you even after enduring Louella Palmer's public accusation that you had a sexually transmitted disease, lest you forget." Dixie's fists clenched at her sides. "The clap to be precise." Sanjeev raised his hands and slapped them together, jarring her. "Still not funny." "Oh, Dixie. It was almost a lifetime ago. Surely you can see the humor in it by now?" "I'm not sure I'll ever see the humor in Louella Palmer, standing in line behind me at Lucky Judson's hardware store, randomly clapping while everyone was in on the joke but me." The memory of that still stung as freshly as if it had happened just moments ago. A mix-up in her pre-marital test results, tests both she and Caine had agreed to have administered before their marriage, had resulted in the "teetering-on-senility" Dr. Wade Johnson somehow allowing his onetime receptionist, Louella, get her hands on them. Of course, she'd told anyone who'd listen Dixie had the clap. "What is it your countrymen say about payback?" "While I see your point, that's not the point. This phone-sex business isn't about punishing me for being a mean girl, Sanjeev. Landon loved me when I was horrible, and he loved me after I wasn't so horrible. Anyway, we're off track here, friend." He pursed his lips, giving his cheekbones a hollowed look. "I'm not off track. There is no track. Landon didn't always have a rhyme to his reason. As you well know, he did many things on a whim—or because it simply pleased him, but never without the utmost caution. I don't know what would please him about seeing you suffer when he did nothing but indulge you almost all of your life, even at your worst, but I have no answers, only my orders to keep you safe, well-fed, and comfortable." "Nothing concerning Caine Donovan is safe," she muttered. Sanjeev acknowledged her words with a nod. "Be that as it may, we're here in this moment. Now, I have Mona and Lisa to bathe. They're as unruly as your hair, and I won't have them laying all over the bed I expressly freshened for you until I'm sure we're cleared for fleas. You, lovely Dixie," he said, pointing toward the equally opulent adjoining bathroom, "have an appointment at the guesthouse to meet your fellow employees. Freshening up wouldn't hurt you either. You're funeral worn." He chuckled at his joke, padding out of the room with a wave over his shoulder. The silence of the bedroom engulfed Dixie in its subtle hues of silk and throw pillows, leaving her a moment to hear the throb of her panicked heart. Meet your fellow employees, rang in her ears with a hauntingly Vincent Price–like quality. Sanjeev said it as though her new job was something as ho-hum as retail sales or file clerking. Which brought a thought to mind. What were the women of phone sex like? Did they have office parties or swingers' parties? Celebrate birthdays with a cake from the local grocery store and attend in pasties and a thong? Gossip at the water cooler about what a limp dick Dale in Idaho was for calling them from his mother's basement, and running up her phone bill just so he could get off to the sound of some imagined sex-starved woman who was just waiting for his dulcet tones to lull them into a pretend orgasm? Did they send each other the BDSM joke-of-the-day emails? Oh, Dixie, reckless and impulsive be thy name. The jingle of dog collars and heavy breathing startled her from her panic. "You're overthinking this, Dixie!" Sanjeev called out with a pant as he flew past her bedroom with Mona and Lisa dragging him down the long hallway. Sure. She, Dixie Davis, was overthinking. Not something often credited to her, but on this rare occasion, certainly applicable. Reaching for her purse, she made her sulky way to the bathroom, paying little attention to her lavish surroundings. She didn't notice anything but her purse vibrating the sound of a text message when she threw it on the countertop just under the gorgeous Venetian mirror she didn't want to look into. The only person who'd ever texted her was Landon.... Dixie took a hesitant step forward, the tile beneath her feet no longer soothing her with its cool surface. Instead, it magnified the apprehension sweeping along her nerves like an out-of-control firecracker left on the ground to spin haphazardly. With a trembling hand, she opened her purse on the vanity and snatched her phone out, stifling a shaky breath in order to read the text—from none other than Landon. My beautiful friend, your journey awaits. Today is the first day of the rest of your life, Dixie-Cup. Carpe phone sex! After freshening her makeup, brushing her hair into a ponytail, throwing on a cotton skirt and a tank top, impossible text message still on her mind, Dixie strolled along the winding path of arborvitaes and rosebushes to the guesthouse. Which wasn't really a guesthouse at all. It was a mini version of the big house with only five bedrooms instead of ten, a pool lined with white travertine along its sloping edges, and an island, complete with palm trees, chaise longues and a bartender in the middle of it all. As she made her way past the pool area, she noted not a single string bikini or Insanity Workout body to be had. The pool didn't have a ripple of activity swirling in the crystal-blue waters, dotted with solar lights beneath the surface where she'd expected to see a bevy of beauties playing volleyball on the shoulders of beefy men. Her images of sex goddesses scantily draped in bikinis, dangling their feet in the pool while they whispered, "I love it when you touch me there" fled and were replaced by the sound of a voice that couldn't belong to someone more than ten years old. She followed it toward the wide glass doors leading inside, scooting through the doors, and making her way across the terra-cotta tiled floor to the rounded entryway where the voice grew stronger. "Ohhhhhh, I'm so wet for you!" an enthusiastic voice cooed. "You're so big and hard, I just don't think I can stand it! Doooo me, Enzo," the little-girl voice—far too youthful for phone sex—purred. "Do me like that, you Italian stallion!" Dixie stopped all forward movement as if she was playing a game of life-or-death freeze tag, gripping the overstuffed chair in the twilight-filled foyer to keep her legs from collapsing. She couldn't do this. The woman's voice, coming from Landon's old office, belonged to, at best, a teenager. How could she possibly support anyone who wanted to talk to a child—even if she was a grown woman merely pretending to be a child? How could Landon have supported it? Disgust bloomed in the pit of her stomach, mushrooming until she couldn't breathe. This had gone much further than she'd gone in her head. It was one thing for two adults to consensually have make-believe sex with a phone as their barrier. That she could almost handle. But when a man wanted a child he could pretend to have sex with—that was well off her morality chart. Not to mention—Italians and stallions? That was her cue. Exit stage left. Five A hand clamped on her shoulder, a cool hand with a gentle yet firm grip. "I know what you're thinking, Dixie. You are the Dixie, right?" a soft voice asked. She stiffened, caught in the act of running away. "If I said no, would that mean I could escape from this madhouse, and you'd never be the wiser?" "Well, no. I'd be the wiser. I'd know you just as easily as if I'd run into you buying milk at the Piggly Wiggly. Landon talked about you all the time, and he must have showed us a hundred pictures of you." She paused for a moment, putting both hands on Dixie's shaking shoulders, forcing her to turn around. What met Dixie's eyes was a creamy-skinned, fresh-faced young woman of no more than maybe thirty, with long chestnut hair spilling over her shoulders and down her spine, and a pair of the widest, deepest green-blue eyes Dixie had ever encountered. Her coloring was naturally peach-inspired, and the clothes she wore, a T-shirt that read Georgia Tech and black capris, were as simple as Dixie's. "I'm Catherine, Cat for short, Butler. I'm general manager of Call Girls." "Gage's new fiancée, right?" Cat flushed a pretty pink—the kind of pink you flushed when you were wildly in love. "That's me. Em asked me to tell you she'd see you tomorrow. Something about the hot tub at the big house and cold king crab." Dixie suppressed a smile. As a single parent with a husband who'd just up and decided he deserved a midlife crisis a little early, Em deserved a good pampering. "She deserves it after today." "And you are definitely Dixie Davis. Landon always said you were even prettier in person than you are in your pictures. He was right. And that voice!" Cat said with obvious delight. "It's fantastic—so raspy and smoky. You're gonna give the girls a real run for their money." Dixie grimaced. "I think today I don't want to be Dixie Davis, and I don't want to give anyone a run for anything with my raspy or my smoky." Cat grinned, revealing adorable dimples. "If only trading lives with someone else was as easy as the words simply spoken, hmm? Now, before you set off to givin' someone hell—and yes, I can see that look on your face, Landon described your ire well—hear me out. The voice you hear in there on that phone is Marybell Lyman's, and she's not role-playing. It's just the voice our creator gave her. And it works for her, but we have strict rules about that sort of thing at Call Girls. I promise." Still shaken, though to a lesser degree, Dixie's tongue got the better of her. "Clearly, the rules for Italians and stallions escaped Landon." Cat chuckled. "What's the harm in making a small mob fish feel like a big ol' shark? That's why men call us, Dixie. To interact with women they've fooled themselves into believing are incapable of living without their magically lust-inducing words." Dixie exhaled a breath of regret, ashamed she'd jumped to the same conclusions people still jumped to about her. "I'm sorry. I heard...and I just assumed—" "Never you fear, Dixie. Landon wouldn't allow calls generated from men who wanted to talk to underage girls. He was a kind soul. In fact, it remains a strict rule. We entertain lots of fantasies here at Call Girls, but there are absolute no-no's, and if anyone's caught indulging a client in something that's off the table, it's cause for permanent termination." Another sigh of relief shuddered through her, leaving Dixie unsure how to respond to this woman who looked as if she'd just fallen off the pages of Seventeen magazine. She'd expected women who popped their gum, half-dressed in spandex catsuits, wearing six-inch stilettos and more eyeliner than Brugsby's Drugstore cosmetics counter could supply. Instead, a pretty, fresh-faced, articulate woman greeted her with a lovely smile and a lilting Southern accent. One of these things was not like the other, and two of these things weren't even kinda the same. Dixie squared her shoulders and pushed her hand toward Cat. "My apologies for my inexcusable manners. Yes. I'm Dixie Davis. It's a pleasure to meet you." Cat gripped Dixie's hand, curling her fingers around it to give it a firm shake before letting it go. "No, it's not. Not yet anyway. You look like you're ready to find the nearest pitcher of sweet tea laced with bourbon to drown yourself in." "Booze wouldn't go denied," Dixie confessed, dropping the tips of her fingers to the pockets in her skirt. Cat tilted her head, her eyes glittering and playful. "So you made it this far, right? That's a sure sign you're at least a little curious. Do you want to soldier on? Or do we end this conversation with a pleasant but cordial 'it was lovely to meet you?'" Dixie swallowed hard, her throat full of sandpaper, but she squared her shoulders. She was in. "We soldier. We definitely soldier. Battlefields and hand grenades ahoy." Cat's grin was infectious. "I confess, we all wondered what you'd do. I laid the biggest bet in the 'Dixie pool' by the way." "Bet?" Why, yes, Dixie. You're familiar with bets. Those crazy situations where you challenge some poor soul, not nearly as skilled as you, to race you for the win? Sometimes they involve money—other times? Hands in marriage. She shook off the voice of her past and repeated, "Bet?" "Well, yes. The bet that said you'd at least come see what you could see. You know, investigate what this was all about? Everyone else thought someone with the kinda means you come from would run away to your palace in wherever it is rich folk build their palaces. Not me, though. I just knew, from all the talkin' Landon did about you, you wouldn't turn tail. Knew it. So thank you kindly for the two hundred dollars I just won. Pizza night's on me." She let loose a breathy whisper of a giggle. Dixie managed to ignore the fact that this as yet unnamed group of women had bet against her and her palace and blurted something random. "You have a pizza night at Call Girls?" Phone-sex operators ate pizza? Next someone would tell her hookers had expense accounts. Cat grinned that contagious grin again. "Well, of course we do. We're not heathens, Dixie. Just because we call all parts southern on your anatomy words your mama would've washed your mouth out with soap for sayin', doesn't mean we blow up edible condoms and decorate them with whipped cream all the time. We're just like most everyone else. We have all sorts of things here at the office. Christmas parties—baby showers, 'Wear Your Pajamas to Work Wednesday.' You name it, and Landon insisted upon it. You know how much he loved parties, and impromptu parties were his specialty. It boosts morale if you can have a little party on the boss's dime, don't you agree?" she said with a conspiratorial wink. She'd like to have a little something on the boss, all right. She'd like to have a chokehold on him. "I..." Dixie held up a finger, putting it to her lips for a moment and shook her head. "I'm going to stop now so I don't come off sounding like an uneducated, high-handed ass. Something I'm sure happens to you a lot. With first impressions being everything, I'll just say this is unexpected." Her head swam from so much unexpected. "Your surprise is understandable, but I promise you, we're mostly all just average women who needed to find a way to make ends meet. Well, with the exclusion of LaDawn. She really was a—" Cat leaned in, leaving the lingering scent of jasmine and roses in Dixie's nose, and whispered, "a lady of the evening in Atlanta. Landon talked her out of the life and gave her a job here at Call Girls where she's been ever since." Everyone's knight in shining armor, weren't you, old buddy? "Some of us even have children, and Sheree has a husband who's out of work." Once again, judge not lest ye be judged, Dixie Davis. "I—I'm sorry... I just thought..." Cat crossed her arms over her chest as if she'd heard it all before. Yet, it didn't come across as a defensive gesture at all. "We know what you thought—or think. It's what everyone in this narrow-minded dink of a town still stuck in the 1950s thinks, and we've only been here just a few days. Some who call themselves open-minded think that. But I promise you we're not so different than the rest of the workforce. We're just more...er, colorful." "Ladies, I bid you good evening," a cheerful voice with a British accent called from the sliding glass doors. Dixie's limbs instantly froze even as her stomach heated. Oh, good. Candy Caine was on the loose. "Michael Caine, right?" Cat said on a tinkling laugh, her cheeks staining the color all women's cheeks stained when Caine did an impression. No one was left untouched by Candy Caine's charm. Dixie had to fight not to roll her eyes and whisper a warning to Cat to beware the Donovan spell. Instead, she stiffened her spine, lifted her chin, and activated her Caine-Away force field. He made his way across the tile with his pantherlike prowl, full of grace and a sensual glide of his cowboy boots. His legs, thick and muscular, worked under his tight-fitting jeans, flexing in time with his rhythmic walk. A familiar and unwanted clench, deep within Dixie's core, tightened as he drew closer. He stopped a couple of feet from the women and grinned, holding out his hand to Cat, showcasing his enticingly visible pecs beneath his fitted navy blue shirt. "I'm—" "Caine," Cat twittered, her free hand making a nervous pass over a long strand of her hair to smooth it. "Caine Donovan. I'd know you anywhere, too. We've heard a lot about you from Landon." "Sorry I'm a little late." Cat smiled at Caine. "I figured you might be. LaDawn said she heard at the diner you were over doin' Ezrah Jones's laundry for him. Is that true?" Caine shrugged his shoulders. "He's had a rough go of it since Louise died, hasn't been showing up for poker in the park with his buddies from the VA. Just thought I'd check on him, maybe offer some support. Louise used to make cookies for me whenever I won a meet. She was a great lady." Cat sighed a dreamy sigh. "You're as nice as Landon said you were. He told us all about your high school exploits, and how you three were thicker 'n thieves back in the day." "And now it looks like we'll be thicker than phone sex," Caine joked, eyeing Dixie with that penetrating gaze that asked as many questions as it had ever answered. "Damn. Guess I lost this bet, which might make pizza night a totally different ball game," Cat said to Dixie with a snicker. "Pizza night?" Caine queried, raising one eyebrow and wiggling it. Dixie's chin lifted defiantly, her eyes pinning Caine's. "Yeah, funny thing about pizza night... The women all bet I wouldn't show up today, but Cat. Cat had my back." Cat dipped her head. "But we definitely didn't think you'd show up, Caine. You know, as rich and successful as your real-estate business is back in Miami." Caine made a comically sad face, and in Daryl from The Walking Dead's voice, he said, "It cuts me deep you think I'd run away from the chance to talk dirty when I have the best Sean Connery impression ever. It speaks volumes about our future working relationship, ma'am. We're lackin' trust." Cat howled her pleasure, her slender shoulders shaking with laughter beneath her T-shirt. She pointed up at him. "Daryl—The Walking Dead, right? Lawd in all his mercy! Landon told us all about your celebrity impersonations. You really are as good as he said," she gushed. Hark! Who goes there? What was that she heard in the distance? Yet another woman fallen prey to Caine Donovan? Dixie fought another roll of her eyes. Turning her back on Caine, Dixie forced a smile to her lips and put her hand on Cat's arm to draw her away from the sexual napalm. "So maybe you could explain all of this? How Call Girls is run. What's expected of us? The thing about our chosen personas?" That troubled her the most, choosing a persona. "You mean our specialty kinks, right, Dixie?" Caine made a point of reminding her, stepping around both of the women so he could peer into the archway that led to the great room and the subsequent bedrooms. Dixie fought a scowl at his deliciously fresh, clean scent, but couldn't fight the pop of her lips. "Why yes, Candy Caine. That's exactly what I mean. I'm all about finding out what my kink is." "Um, we, in the business, that is, actually call them fetishes. Just an FYI," Cat interjected with another of her easy smiles. "Fetish." Dixie nodded, mentally making a note of it for future fetish exploration. "Got it." "Studious as ever," Caine remarked dryly, clearing his throat. The reference to her lack of interest in her studies back in her high school days didn't go unnoticed. "That's what got me that 4.3 GPA in college," she reminded him with a flash of her eyes. "If memory serves, you had a 4.2." So humph. "Studying was what got you a 4.3, Dixie? And didn't you leave college to cruise the seven seas on some rich guy's yacht?" It was only two seas, thank you. Her blood pressure soared. Just as Dixie was about to sling an arrow dipped in contempt back, Cat threw a hand up between, staring them both down with a matronly glare. "Okay, to your corners." She swished a warning finger at them, shooing them apart. "So let's just get this all out in the open, because even though I'm office manager, Landon was kind enough to allow me to take college courses online while I oversee Call Girls. So quite often, in between calls, I'm studying. Which means not only do I have other employees to protect, but my future career, as well. I can't do that if I'm breaking up petty disagreements between the two of you." Protect? As if they both had a penchant for serial killing? "Now, Landon told us all about the two of you and your ongoing love affair with a good war of words. He told us everything about your childhoods, Dixie's legendary mean-girl reputation here in Plum Orchard, your love of a good bet, your eventual engagement—the ugly ending to your engagement—the subsequent years you both spent hating each other over the ugly end to said engagement, all while he continued to remain friends with you both. Big yawn. Old news, right?" Both Caine and Dixie remained stubbornly silent. "Right?" Cat prompted, her expression stern and schoolmarmish. Their grating sighs were simultaneous. "Right," they responded in unison like two guilty children. "Good. So here's how this is gonna play out. I know there are hard feelin's between the two of you, and that's too bad, but they're absolutely not for the workplace. I run Call Girls, and I run a tight ship. If you decide to join us, I won't have the two of you taking potshots at each other, and making everyone around you uncomfortable while you do it. If you want to beat each other up over your history together, do it somewhere else. Do we understand each other?" Like two chastised children, they both let their eyes fall to the tiled floor. "And do not roll your eyes at me, Dixie Davis," Cat warned, planting her hands on her hips. Dixie stopped mid-eye roll and sighed, letting her shoulders sag and her chin hitch forward like the petulant child she turned into whenever Caine was around. Their bickering was bound to affect those around them, and that was unfair. "I'm sorry. We can really suck." Cat giggled. "Landon told us all about your brand of suck. We were locked and loaded." Caine's eyes were contrite when he shot Cat a sheepish grin after scrubbing his knuckles over his jaw. "I'm sorry if we made you feel uncomfortable, too." "Apologies accepted. Now let's let bygones be bygones and get to introductions and the business at hand, okay? The girls are dying to meet you both." Caine nodded his dark head. "Perfect. So let's set about finding our fetishes. Whaddya say, Mistress Taboo?" He didn't wait for Dixie to answer. Instead he held out his arm to Cat and smiled. "Shall we?" Cat giggled again, soft and as lovely as she was, but a quick glance at Dixie had her clamping her lips shut and frowning before she regained her composure. She roped her arm loosely through Caine's, keeping a visible distance between them. "C'mon. I'll introduce you to everyone and familiarize you with what goes on here." Dixie stuck her tongue out at Caine behind his back, and hurried to shuffle up to the other side of Cat, grabbing onto her free arm and winking. Her chuckle was throaty, but her words held the ultimate dare. "Let the games begin." * * * Back in her room, freshly showered and comfortable in an old T-shirt, Dixie snatched her phone with Landon's text from the nightstand and raised her fist to the ceiling with a shake. "You suck, Landon," she muttered, making Mona and Lisa stir. After an hour with Caine, Cat and the women of Call Girls, Dixie's head was still spinning. She'd thought she'd made her choice the moment she'd thrown down the challenge to Caine in Hank Cotton's office. Now? She was regretting her impulsivity. Once Cat had explained the inner workings of the phone-sex business, and only after Dixie was done mentally rolling her eyes at Caine, who'd smiled, joked and blatantly flirted with the ladies while making it appear this challenge was going to be akin to some leisurely stroll in the park, she'd waffled. As she processed bits of information such as, she was her own boss and her hours were flexible, but some of the best, most loyal U.S. clients called in at night between the hours of midnight and three. And it was up to her to create an interesting, yet alluring phone-sex operator pseudonym, a website for that pseudonym, and an area of sex she specialized in. Scripts on how to handle difficult client calls, calls that got out of hand, all kinds of calls, calls, calls were readily available to them. Shortly after meeting the women who ran the phones, and introductions, and all the details of the running of a phone-sex company, Dixie began to wilt, exhausted from the day's events. Cat, clearly intuitive, had handed her the Call Girls phone-sex operator package, and told her to go get some rest before she made her final decision. That was where she was now. Making her final decision. Her eyes flew to her bedside clock. And she only had eighteen hours and counting to do it. Tick, tick, tick. The only thing she had decided on, if she didn't chicken out, was the pseudonym Mistress Taboo. Caine had used it to taunt her, but it stuck like an earworm. Flopping on the bed, she absently flipped through the ream of papers Cat had given her while she stroked Mona's ear. Her eye caught the list of "specialties" Call Girls allowed, stilling her movement. "What, in all of heaven, do you suppose infantilism is, Mona?" "Oh, you know, the usual. Men in diapers, baby bottles," Caine said, strolling into her bedroom on bare feet, in a pair of cargo shorts and nothing else. The defined lines of his face almost always took Dixie's breath away. Tonight was no exception as the shadows cupped his strong jaw and enhanced his sharp cheekbones. Her heart thrummed with the inevitable longing it had since the day she'd set her sights on him in high school. Dixie forced herself to look directly into his eyes instead of at the chest she'd once brazenly sat atop as he... Dixie gulped. "How unexpected to find you're so in the fetish know," she drawled, digging for the old Dixie, the one who was cocky and capable of keeping her composure catty and aloof all in one sentence. Caine's eyebrow rose in that condescending way while his chest glistened in all its lickability in the dim lamplight. Coming to stand at her feet, he reached around her to give Lisa's broad head a scruff of his knuckles. As the skin of his arm brushed hers, she sucked in a breath of air at the tightening of her nipples. "Wanna see who knows the definition of more fetishes?" "Almost as much as I'd like to see my spleen advertised on eBay." Caine's eyes narrowed, glittering with amusement while his lips formed a sexy, cocky challenge of a smile. "That's because you know you'll lose. What's the matter, Dixie? All bet-out for the day?" "I'm all Caine'd out for forever. So what do you want, and why are you in my room? I don't recall hearing a knock." Rising to her feet, she brushed a strand of her wet ponytail from her face, stepping around his solid frame. "Door was open. And pillows," he said, jamming his hands into the pockets of his shorts as if he wasn't standing in front of her with no shirt on. "I know Sanjeev always has extra in here. I need another pillow. Please," he tacked on with syrupy emphasis. Dixie's throat grew dry and gritty. "There aren't a hundred people on staff who could find you pillows?" "Unlike you, I don't want to wake the staff for something as ridiculous as a pillow. I know you're used to having someone at your beck and call, Powder Puff. I, on the other hand, fend quite nicely for myself and wouldn't dream of waking them." "Look at you here in my room, fending," she mocked. His insinuation that she was selfish enough to wake an entire household over something as trivial as a hangnail infuriated her. In fairness, it wasn't exactly an untruth from her past, but it was no less infuriating now in the present. And that was exactly what Caine wanted. Rather than rise further to his bait, Dixie turned on her heel, hoping the sway of her backside made him salivate just like it used to. She threw the linen closet door open and peered inside, reaching for the chain to unsuccessfully turn the light on. The bulb was out. For all the fancy, highfalutin' gadgets Landon had in this house, he'd overlooked the simple things when he'd renovated. The heavy oak door snapped back at her, smashing into her hip with a hard thud, meaning the spring was broken. Dixie spread her legs to hold it open, using her foot to keep it in place while attempting to adjust her vision to see the interior. The space had a small entry, and was just large enough to house some shelving full of soft, fluffy towels and silken bedding. The door creaked when Caine came up behind her. Pushing her foot aside, he used his large hands at her waist to move her deeper into the closet. "I asked for a pillow. Not directions to the Fountain of Youth. What's taking so long?" he questioned, craning his neck upward to glimpse the top shelves. Distracted by the light press of his fingers and the sting of the fleeting memory when Caine's hand was never far from hers made her forget about the door. "Don't let the—" The door slammed shut behind them with a heavy thud, enveloping them in the quiet, Tide-scented darkness. Caine knocked into her, jolting her forward so her nose just missed the edge of a shelf before righting her with his arms. Which left his rocklike, warm body pressed tight against her back. Certainly a dilemma of her libido's highest order. Six "Uh, let the door shut?" Caine finished into her ear, leaving Dixie to fight the shiver his warm breath left in its wake. Dixie attempted to inch forward and out of his nerve-tingling grasp, but there was nowhere to go. "Impatience be thy name," she said between the clench of her teeth. "It's better than shithead, I guess," he murmured back. "Didn't I mention? Impatience is your middle name." "That's downright mean, Dixie." "It's downright true, Caine." "Viper." "Mistress Viper to you, thank you very much." Dixie twisted uncomfortably, bucking against Caine's hand in the process. "Now quit name-calling and open the door. You know how claustrophobic I am." Just the thought of how claustrophobic she was made the claustrophobia in her stabby and irritable. His sigh was a wash of raspy honey in the dark. "Stop wiggling around, woman, and let me—" one hand moved from her waist followed by the sound of the jiggling door handle "—open the damn thing..." Chalk it up to a long day, but locked in a closet with Caine was the final straw that broke her raw nerves' back. Though, the fight to keep from having any square inch of her body touching Caine's worked to distract her fear of the pitch-black closet swallowing her whole. "What is the problem, Caine?" she snapped. "I can't—" "If you use the words can't and open in the same sentence referring to that doorknob—" "You'll what?" he huffed, his chest pushing against her back. "I'll suffocate you with one of these fluffy towels." She heard him jiggle the door handle again. "Ready your weapon. I. Can't." Slapping his hand from her waist, Dixie managed to turn around in the tiny space, her nose brushing the springy hairs on his chest. "Let me try." She twisted the handle, her heart pounding out her body's awareness of Caine's. "It's locked, damn it." "Oh, Sherlock, still such a cracker jack," Caine cooed in another of his flowing British accents. "Oh, Holmes, still just a sidekick with a big mouth." "Move over, Dixie, and let me give it another try." Dixie snorted to the tune of the irritation in his tone. "You do that, Hulk. I'll wait over here in the two square inches of space, cowering weakly so the big, strong man can save me." They attempted to switch positions only to find themselves so closely fused their bodies were forced to make contact—delicious, heated, full-bodied contact. Her slip of a T-shirt left little between them, the material so worn over time it was like having on nothing at all. "So now what, Dixie-Cup?" he grumbled huskily, his chin brushing the top of her head. Dixie had to close her eyes to keep from swaying as the comfort of the familiar assaulted her. She would not allow her head to move just a hair forward and rest on his chest. She gritted her teeth. "Get us out of here before I claw my way past you to get to that door. And stop calling me Dixie-Cup!" Because pettily lashing out was going to make this situation better. Caine's fingertips twitched against hers. Knowing him the way she did, she also knew he was smiling into the dark. "But I've always called you Dixie-Cup, Dixie-Cup." "No. Landon called me Dixie-Cup. You called me a liar." Dixie's chest tightened with the familiar constriction of his taunts. Caine's fingers wound into the length of her hair, tugging her head back. "You were a liar," he replied smoothly, yet the edge to his voice was hard...raw. Rivulets of sweat began to form between her breasts, and she wasn't sure if it was panic because the closet was hot and suffocating—or because Caine was. Fear of both made her strike out again. "Move, Caine, or I swear I'll scream!" His response was to drag her to him, her spine arching, driving her against him, a moan rising to her lips when an aching rush of wet heat grew in her cleft. Her body's reply to him, to the gruff tug of her hair, and the once familiar command it wrought, infuriated her. "Go away, Caine. Better yet, go back to Miami." Caine's silky lips skimmed the darkness. "Like hell, I will. I was here first," he said, reaching a hand down to grip her hip, drawing her closer to the rigid outline of his cock, sharply defined against his cargo shorts. She gave him a shove only to have the sound of the thump of his back hitting the door cut into the darkness. "You don't want Call Girls. You want to best me so you can flip your middle finger up in the air in my direction while you tell everyone over a round on you at Cooters you whooped mean girl Dixie Davis." "Actually, I was going to buy everyone dinner while I did that. I'm disappointed to find you think me so damn cheap." Don't take the bait, Dixie. Be the adult. "The point is you want to win." His chuckle was thick to her ears, tipping her off to the fact that she wasn't alone in her arousal. "Oh, you bet I do. And in the process, adding a multimillion-dollar company to my portfolio won't make me sad." "A portfolio. Nice luxury if you can get it," she managed, stifling a breathy sigh when he let go of her hair and cupped the back of her head. Caine's body curved into hers even as his mouth continued its agonizing path upward. "Are things really that bad off, Dixie?" Were things really that bad off? Was the sinking of the Titanic just a little boating incident? But Dixie stiffened at his question—the question that sounded warm and sympathetic. Oh, no, sir. She wasn't falling for that old trick. The "draw someone into your web by being a kind shoulder to cry on, then wait for the moment you could use their misfortune to up your own game" trick. She was once the master. "Things are none of your business." "Pride is a sin, Dixie," Caine murmured into the darkness, his voice growing heavy, his body melting into hers. Fight the Caine charisma, Dixie. Fight it like you own a Justice League cape. "Falling for the notion that you're even a little concerned about me is a sin." Summoning what was left of her shredding will, she returned her focus to her claustrophobia. It was the lesser of the two evils. The mere thought they'd be stuck together like this until Sanjeev came to tell her breakfast was ready fed her fear. Her heart began a panicked staccato. The heat of their bodies coupled with the stifling lack of air served her focus on her claustrophobia mission well. "We have to get out of here, Caine!" She shoved at the solid wall of his chest again. Yet it only made him tighten his hold. "Dixie?" "What?" she yelped, her voice thin. "I'm going to do something that's probably going to piss you the hell off, but I want you to remember one thing after we get out of here." Her rising panic squeezed her throat, but she managed to sputter, "Like?" "Like this is for your own damn good." * * * The moment the words escaped Caine's lips was the moment he forgot he was trying like hell to remember she'd trashed him. He hauled her to him, planting his lips firmly over hers. His tongue sliced through the soft flesh, quieting her anxieties with the movement of his mouth. When he suckled her tongue, devouring it in slow sips, Caine forgot everything but his unquenchable thirst for Dixie. The way she angled her lips to fit his, stifling a needy groan. The whisper of a whimper that made him rock-hard, even now, ten years later. She made him feel things he didn't want to feel. She reminded him, everything after her didn't measure up. But that didn't stop him from tearing at her T-shirt, driving it upward until her breasts sprang free. The moan he emitted from his mouth, primal and raw, was predatory, possessive when he gathered her breasts in his hands, tweaking her nipples to sharp points. Christ, she felt like everything he'd been lacking in all the failed relationships he'd had since her. When she wound her arms upward around his neck, clinging to him in the way that had always sparked some primal instinct in him, he thrust a hand inside her wispy panties to touch her, sliding between the lips of her pussy. And all he could think was, here he was, lost again. Lost in the ultrafeminine vortex of Dixie. Powerlessly, helplessly lost. He wanted to punish her for opening this hellish box of feelings he'd kept shut tight by sheer will and the determination to never get caught up in her and her lies again. His breathing was ragged when he tore his lips from hers, as though he'd lose something of himself if he didn't. He couldn't see it, but he knew her penetrating gaze was eating him alive in the velvety darkness. "Damn you, Dixie," he hissed after a harsh pant. Damn her and this new fragility that left him wanting to fix the things that had left her looking so broken. Damn her for reopening the wound of his anger and spilling it all over him. Damn her for making him question himself, question whether this was all another game, whether she was just crying wolf, or she really needed help. He didn't want that. He didn't want to get tangled up in her again. But fuck. He did want her. * * * The zipper of his cargo pants was a vague sound compared to the rush of her pulse, sounding out the rhythm of her throb between her thighs. In a move so swift, even Dixie questioned the dexterity of it, Caine swapped positions with her, leaving her exposed back against the cool oak door. He yanked her arms upward over her head, fisting her wrists together in his hand. It made her chest crawl with white-hot heat, with a need so deep Dixie knew it would never be like this with anyone else. With his free arm, he lifted her at the waist until her legs circled his body. And Dixie went willingly, wantonly hooking her ankles behind his back, holding her breath when he let the head of his hard shaft slide between the lips of her wet sex. Her head fell back, exposing her breasts to his mouth, breasts Caine ran his molten tongue over until the head of his thick shaft sat at her entrance. In that suspended moment, Dixie forgot his harsh words the night they'd parted. She forgot how much she'd hated to still love him even though breaking off their engagement was what any real man worth his salt should have done. She forgot the sting of humiliation during their engagement party, when Louella Palmer, microphone in hand, had, instead of reading a lovely speech about their courtship, declared Dixie had a sexually transmitted disease, and she had the test results to prove it. All while the entire population of five hundred and fifty-six people in Plum Orchard, Georgia, were given a front row seat to her humiliation. She forgot the words of harsh reality he'd flung at her when Louella let Caine hear the voice mail Dixie had left her, crowing her victory. I'm not some damn race horse, Dixie! She forgot how desperately she'd wanted Caine to really believe she'd changed since her high school days, and how easily she'd slipped back into that mean-girl role because she could never resist the chance to win—at everything. She forgot how much she'd hurt Louella by pursuing Caine in the first place. She forgot how in the end, even after her apology, she realized it would always be like this. She would always be Dixie Davis, untrustworthy mean girl to him, and he'd always be fine, upstanding, kind-to-small-children-and-puppies Caine Donovan. She forgot that back on that dreadful day, he'd been right, and she'd been wrong. Dixie forgot all of that when the crown of his heated cock rested at her entrance, and her hips lifted, inviting him in. She forgot everything but the all-consuming desire tearing at her—a need that had to be sated or she'd die from the want of him. Nine years of memories resurfaced in a tidal wave of reality. Dixie let her hips slide downward in slow increments, sucking him deeper into her body until Caine drove upward with such force she wobbled in the strength of his arms, wincing when the pleasure-pain of his powerful entry stretched her. Each ripple of his abs pressed into her. Her nipples grew agonizingly tight, scraping against his smooth chest. He angled his hips and pushed upward again. The violent thrust of his cock, wide and thick, only heightened her need. The sweat accumulating between them allowed for a slippery glide of skin on skin. Dixie's clit raked against the flat plane of his abdomen as he took another possessive thrust, growling what sounded like surprise. "Has it been a while, Dixie? You're so damn tight and hot, so fuckably hot..." Caine's lips curved into her ear then, nipping her lobe and preventing her from striking back with an answer. Caine's response was a heavy chuckle. "Some things never change, do they, Dixie?" he asked, driving into her slick entrance again, filling the emptiness inside her until she melted fully into him. Harsh air escaped their lungs, full of rasp, heaving and choppy. Caine jammed his hands under her, cupping her ass, trailing his finger between her cheeks, forcing her to use his body as leverage to rise up and crash downward again. Their rhythmic thrusts were madness; stroke for stroke, Dixie grew wetter, hotter until there was no sound but that of their bodies connecting. Dixie's fingers went to his lips, pressing them to the fullness of his mouth to suppress his words, the fear of them slipping from his throat sliced through her haze of need. She couldn't hear the words he used to speak. Wouldn't. Caine reached for her nape, digging his fingers into the flesh of it. "So good, Dixie—so sweet," he said from a clenched jaw, tensing and flexing. Dixie didn't know if her response was to Caine's words or the incessant throb of her wish to be pushed over the edge—hard. And at that moment, she didn't care. She contracted around him, welcoming the intense fire burning her from the inside out, allowing it to be the only feeling in existence. She rocked hard against the tightly corded muscles of Caine, accepting his last plunge into her wet depths. Her orgasm exploded, ripping through her, eliciting a hoarse cry of completion. They slumped together, their chests crashing, the sheen of their sweat mingling. A scrape against the outside of the door and a low growl had them both scrambling. Caine pulled out of her with haste, but she didn't have time to mourn the loss before she heard Em's voice. "What is it, Mona? Or are you Lisa? Isn't it enough that you woke me? Stop scratching at the door, or you'll wake the entire house, young lady!" Still shaken, Dixie slid down Caine's body, dragging her T-shirt with her. Her underwear. Where was her underwear? Panic seized her. She heard Caine zip up just before he reached over her head and began banging on the door. "Em! We're in here!" There was a scuffle of dog paws scraping on the floor, and Em's gasp before the familiar sound of the jiggling doorknob drowned out everything else. Dixie didn't have time to regret. She didn't have time to be grateful she'd never missed a day of her birth control pills. She didn't have time to clean up. She didn't have time to despise her body's blatant betrayal just because Caine had whipped out one of his best assets and had driven her mad with it. She didn't even have time to see the look on Em's face when the light from the bathroom spilled into the dark closet, so damn invasive and harshly bright when she and Caine stumbled out together in a tangle of limbs and rumpled nightwear. Dixie shielded her watery eyes, gulping in the cool air as Mona pawed at her leg with a whimper, and Caine's hand went to her spine to right her. She swatted him away, angry with herself—angry with him—damn angry. The silence that greeted them when her eyes fully adjusted was a mixture of two things: surprise and shock. Em, wrapped in a blue terry-cloth robe, finally mumbled, "Mercy." Lord, please have. Em's gaze was pensive. She twisted one of the pink curlers in her hair. "I want to ask. Should I ask?" She shook her head, her brow furrowed. "No. I won't ask. It's ill-mannered." Caine, smooth and composed as always, patted Em on the back with a casual hand and a charming grin as though they hadn't just made torrid love inside a locked closet. "We just got locked in while we were looking for extra pillows." He turned his back to them and reached into the interior of the closet, pulling out a fluffy pillow and tucking it under his arm. "See? Thanks, Em, and g'night, ladies," he called in Paul Hogan's Australian lilt before jamming a hand into the pocket of his shorts and strolling out of the room. Dixie sagged against the wall in relief like the dirty whore she was. That they didn't have to discuss the whys and wherefores of what had just gone on was a blessing. Em plucked at her hair, her smile devilish. "Snookie bump?" Dixie caught a glimpse of her hair in the mirror and cringed, running her hands over the ends that stuck up in every direction. Her ponytail was smashed upward in a crazy likeness to the ultra popular bump. That's because you bumped—hard—against a linen closet door—with a man you tell yourself you despise more than you despise collard greens, Dixie Davis. It's the sex-bump. Wear it well, for it's your new tramp-stamp. Dixie cracked her knuckles while Em waited for an answer with a smug grin. "We got stuck in the stupid closet while we were looking for some pillows. The lock's clearly broken, and so is the spring on the door. So, long lost court-appointed mediator, maybe you could make a note to have Sanjeev fix that, huh? Now, I'm going to bed." She harrumphed, turning on her heel to make an indignant exit. "Hey, Dixie?" Em yelled. "I'm going to bed, Emmaliiiine," she bellowed back, yanking at the ridiculously lavish comforter and climbing in, hoping her cheeks weren't as red as they felt. "That's lovely," Em said, sweeping through the bedroom. "I remember how you always told us your beauty rest was important. You know, like the time all of us cheerleaders, and me, a lowly alternate, stayed up late sewing the costume for the football team's mascot while you got your eight hours of much needed rest?" Dixie dragged the covers over her head and huffed in exasperation. Yes. She'd done that. Yes. She'd been a dreadful excuse for a human being. Yes. Did no one ever let go? It had been twenty years. "Okay, Em. I get it. Mean girl in the house. How many more nails do you have left to hammer into my coffin anyway? Is there a daily quota you have to meet?" Em giggled. "Before you snuggle into that fine linen, I'd take a look at what Mona's chewing on right this second. It looks suspiciously like somethin' you wouldn't find in a linen closet. Niiight, Dixie!" She cocked her ears, noting the snarfing sounds her bulldogs made when they were eating something. Dixie threw the covers off and hung off the side of the bed. Mona stilled all motion. Her beautifully soulful brown eyes stared back up at Dixie's. Guilt. There was guilt in those big brown eyes. There was probably guilt in hers, too. For she and Caine had been caught red-handed by Em, fornicating. So much fornication. Dixie reached down and tapped Mona on the nose with narrowed eyes. "Give Mommy back her underwear right this second, young lady!" Seven Caine paced the length of Hank Cotton's short driveway, listening to the wooden sign announcing his law practice flap in the thick breeze. He glanced upward to the darkening sky. A purple cloud with fat black lines and puffy gray hues settled directly over the sharp peaks of the well-maintained mint-green-and-white Victorian, where Hank's office was housed. A storm was brewing, bringing with it the perfect setting for what he was about to do. Stay here in Plum Orchard instead of going back to Miami where he knew damned well he should go. He'd come here thinking this was a good time to reassess his career, grab a break from his hectic life in Miami. He was burnt out, bored, lacking—lacking something he just couldn't figure out. Plum Orchard, home, was the perfect place to do it. Now, with the messed-up mix of emotions he was feeling after last night, he knew he should go. Yet, here he was. Jamming a hand into the pocket of his jeans, he fingered his phone with the text he'd found from Landon bright and early this morning, and narrowed his eyes. Bawk-bawk-bawk, Candy Caine! it read in Landon's typically comedic voice. Translation—his best friend was razzing him from the great beyond, daring him to give in to the worst label one man could give another. Chicken-shit. He didn't even bother to wonder how Landon had set up timed-release texts before his death. He was too caught up in how the hell he'd survive being around Dixie for two months. "Damn you, you son of a bitch," he muttered, still madder than a coon cornered in a barn. Landon knew the two of them well, and he'd sure as hell known that throwing down the chance to whoop Dixie's pretty backside was like last night in the linen closet. A temptation Caine couldn't resist. To leave Plum Orchard now would be as good as admitting he couldn't beat Dixie. Admitting he couldn't beat Dixie was akin to he-man suicide in this off-the-wall imaginary competition they'd created through the years. Ridiculous? Absolutely. Shouldn't he be long past their childhood rivalries? Wasn't he enough of a man to take the high road? No. Because when he veered off to the high road, it wouldn't be long before Dixie'd come along on her pretty pink Schwinn with the matching woven basket, honk the horn with the stupid frilly purple streamers at him and kick dust up in his face to remind him who'd won this round of Donovan versus Davis. The memory made Caine smile—a smile he didn't even bother to fight. There were plenty of good memories involved with their legendary rivalry. Those were the memories that had shaped his childhood. They mingled with the familiar scent of magnolias in his mother's carefully tended yard, bag after bag of pork rinds and six-packs of Dr. Pepper. And Dixie. No matter where he went, whom he dated, Dixie had always been the one woman he couldn't exorcise. Last night, locked in that closet, her luscious body open and willing, had turned what Caine hoped was nothing more than an idealized memory of her into his worst nightmare come true. Dixie Davis still had the ability to get to him, burrow under his skin until she was so deeply rooted, he didn't know where she left off and he began. Nothing about that had changed. Not the long talks he'd had with himself about how he was romanticizing the ghost of their relationship, forgetting the bad and remembering only the good. Not even the bottle of Jack he'd indulged in when he'd learned Landon had passed and he knew he'd have to see Dixie again after all these years had been able to talk him out of the realization that her presence in his life wasn't ever going to let him have any peace. From the moment he'd seen her at old man Tate's funeral home, her wide, blue eyes puffy and red from crying, her long hair sexily tousled and now a faded red, the soft curve of her full lips chapped from her familiar habit of tugging at them, that voice in his head told him to get the hell out of Georgia. Because she'd never looked sexier. The years they'd been apart had been kind to her—even in grief. Knowing he was able to see past her weary exterior and still find the woman he'd have once given a major organ up for was a sure sign it was time to hightail it out of here. "But you're not gonna let me, are you, you interfering pain in the ass?" Caine, my friend, if I've said it once, I've said it a thousand times. This thing between you and Dixie, this imaginary fight to the death at all costs, is like the chance to travel back in time and watch a half-naked, Russell Crowe–ish prisoner do battle with skilled, equally as half-naked gladiators. Not gonna happen on this watch, pal. It's too delicious. Caine barked a laugh right there in the middle of Hank's cobbled driveway when he thought back on that conversation they'd had after Dixie had pulled one of her stunts. He ignored the curious glances of Nanette Pruitt and her pack of nosy Plum Orchard seniors staring at him from across the street, out on their customary after-dinner stroll. Damn. His eyes scanned the lay of the land with a desperate glance, spotting a sugar maple he could probably hide behind if he sprinted. The last thing he needed was a lecture from Nanette Pruitt, and no doubt, if word had leaked about Landon's will, lecture she would. "Caine Donovan!" she called out with a flap of a stark white handkerchief she held, crossing the street in her signature militant strut. He halted all action. Caught, Donovan. If he were to turn his back on one of his mother's oldest friends, she'd hear about it, and Jo-Lynne Donovan would make him pick his own switch for his whippin'. Caine threw a smile on his face, the one he reserved for the old money of Miami real estate, and turned around. Nanette stopped dead in front of him, the rolls of her neck lavish with her customary pearls. Her modest sundress decorated with swirly flowers in yellow and blue floated in the humid breeze just below knees that matched her neck. Behind her trailed her faithful comrades in piety, Essie Guthrie, Kitty Palmer and Blanche Carter. He noted the absence of Bunny Taylor who, last he'd heard from his mother, was visiting her new granddaughter in Atlanta. Nanette reached up with a hand that was beginning to show the signs of age and pinched his cheeks. Just like she used to when she taught him history in the second grade. "Why, it's our very own Caine Donovan. In the flesh, ladies, out in the middle of our brand-new lawyer's driveway, talkin' to himself. Have you come home to handle the riffraff that nuttier-than-a-pecan-pie Landon left out back in his guesthouse? May the good Lord rest and keep him." She paused, raising her eyes heavenward in good Christian honor before continuing. "If anyone can take care of that lot of giggling Jezebels, it's Plum Orchard High's answer to Paul Newman, don't you think, girls?" she called to the three women behind her as though their spoken approval actually made a lick of difference. Everyone in town knew Nanette didn't need the endorsement of her cohorts. She'd long ago taken up residence on the throne of moral high ground and decency. "You were always a good boy. So what will you do to rid our charming town of those...those..." "Ladies of the non-biblical persuasion was the title you decided on at last night's meetin' of the Senior Magnolias, Nanny," Essie Guthrie offered with a nod of her pin-curled, fading brunette head and a wrinkle of her bulbous nose, smack full of distaste. Caine mentally flung arrows at Landon. He was probably sitting on some cloud upstairs, sipping what he lovingly referred to as "champs" with his personal hero, Liberace, having a hearty laugh over the uproar he'd left behind. But Landon poured some serious money into the town, and while the town didn't necessarily love his lifestyle, they loved the money he dumped into improving it, hand over fist, and it kept them from openly making their displeasure known. But it had never stopped them from talking behind their hands about him or Dixie and Caine for accepting who Landon was. Landon knew what they said, and he didn't care. When Caine found himself bent out of shape over a snide comment from one of the Senior Magnolias, Landon had laughed at his raging. I don't do what I do for Plum Orchard because I need the acceptance of a bunch of hypocrites. I don't do it as a way to shut them up. I do it because this is my home, and I love my home. I love it more than I hate their ignorance. So the ladies must at least have an inkling about Call Girls by now. But his salvation, for the moment anyway, was the hope they didn't know everything. He shaded his eyes when a stray band of sunlight shot through the thickening clouds. Gazing down at Nanette, he summoned something he could latch onto from his Sunday school days to keep their fake saintly wrath at bay. "Now, Miss Nanette, that's not very neighborly of you, is it? What would Reverend Watson say? Wouldn't he want you to welcome those who've strayed from the flock? Maybe make them some of your famous lemon meringue pie? Weren't we always taught not to judge?" The group of women rippled with muffled laughter before almost simultaneously clamping their lips tight after Nanette cast them the eyeball of fire and brimstone. No one preached the Word to Nanette Pruitt. "Well, I can't get to forgivin' if they're still sinnin', now can I? They're a disgrace, Caine Donovan. Delilahs, every last one of 'em! They make no apologies for speakin' the devil's words and collectin' money for it to boot. Why, just last night while I was enjoying a lovely cup of tea at Madge's Kitchen, that awful La-Someone—" "LaDawn," Blanche Carter provided in a meek whisper, tilting the white umbrella she held back down over her fretful eyes. Nanette's sigh expressed her impatience, but her cool smile never faltered. "Thank you, Blanche. LaDawn was swishin' her way through that diner like she's a local. Lookin' like that, they're bound to corrupt the fine young men of Plum Orchard. One of them has tattoos and piercings, and her makeup looks like somethin' straight out of a Halloween parade. What are they doin' here, Caine?" "Now, Miss Nanette," Dixie's husky voice chided from behind the group of women. She sauntered over the sidewalk in her cute cutoff shorts and red T-shirt, the heels of her wedge sandals striking against the pebbled driveway. "Do you really think anyone could corrupt the fine young men of Plum Orchard quite the way I did?" Her eyes strayed to Caine for only a moment, a dark, sultry reminder, before she rounded on the group of women and smiled. That smile that was meant to take you off guard, make you forget whatever your beef was with her just before you were sucked into the charismatic vortex of her charm. Dixie hooked an arm through Nanette's and didn't bother to wait for an answer. Instead, she smiled brighter, if that was possible, winking at the Senior Magnolias with a sweep of her long lashes. "Ladies, it's so good to be back home," she cooed, taking a deep, appreciative breath before tilting her coppery head in Essie's direction and patting Nanette's arm with affection. "Essie? It's always good to see you. How's Jackson? Still conquering the New York stock exchange?" Caine took a step back and grudgingly reveled in the magic that was Dixie. Not twenty-four hours ago, these women had spent the better part of Landon's memorial gossiping about her return. Yet, when face-to-face with her, they'd rather bite their own tongues off than be considered impolite. His mother chalked it up to some Southern good breeding, cliquish thing he neither understood nor cared to understand. It was plain old hypocrisy. Period. But Dixie knew this game, and she knew exactly how to play to Essie. There was nothing Essie loved more than her boy Jackson and her daughter Shelby, except maybe her bloodhound Bowie. Essie's round face flushed with instant pride, her return smile fond if a bit hesitant. She let a tentative hand stray to Dixie's, giving it a light squeeze before yanking it back and tucking it into the pocket of her skirt with a look of guilt. "Aren't you just the sweetest thing for askin'? Jackson's well, as handsome as ever. Shelby's good, too." Dixie smiled and nodded her head in approval. Her blue eyes flitted to Blanche who'd remained quiet since revealing LaDawn's name. "And, Miss Blanche? How's Henry feelin'? Last I heard from Mama, he was nursing his shoulder and some bursitis? She suggested a liniment she's used when I talked to her this week. Why not give me a call, and I'll pass on the name of it? Mama swears it has miraculous properties." Magic. Caine had to give it up to Dixie. She'd managed to cast a spell over the hierarchy of Plum Orchard, diverting their attention and turning it into a gabfest. No doubt Dixie was good at making you feel as if you were the only person in the world, he mused. Maybe it was how she paid such close attention to the small details of another's life, or maybe her concern, completely fake, of course, was what made people believe she was genuine. But Caine knew better, and as he watched her play her flute, leading all the women in a dance they didn't even realize they were dancing, his eyes narrowed. Leave it to Dixie to launch herself right back into the fold with nary a reminder of how she'd once talked Essie's son Jackson into toilet papering Nanette's entire house by promising to make out with him. Or how she'd locked Kitty Palmer's daughter Louella in a porta-potty to keep her from riding atop the Miss Cherokee Rose float after being disqualified from the town's biggest pageant for getting caught drinking plum wine in the bed of Gordy Hansen's truck. Dixie took a quick glance at the watch on her slender wrist, her lips thinning before they curved into another pleasant smile. "Ladies, I hate to cut our catching up short, but I have an appointment with Mr. Cotton I absolutely can't miss." Her eyes strayed to Caine's once more, flashing him a fiery glance. Clearly, she understood why he was here, and she didn't like it. He rolled his tongue along the inside of his cheek and grinned at her over the heads of the women. Yep. That's right. He was here, and he was here to win. Boom. "An appointment?" Nanette's question was sharp, maybe even sharper than her keen knack for even a hint of gossip. Each of the Senior Magnolias' ears virtually stood at attention. "Why ever would you have to see Hank, Dixie Davis? Didn't you sort out that mess of a bankruptcy back in Chicago?" The Senior Magnolias stiffened one by one, twittering to the tune of their shuffling feet. Blanche cleared her throat while Essie leaned so far forward, fiddling to adjust her hearing aid, Caine was sure she'd tip over. Dixie widened her eyes, mocking surprise at Nanette and the women. "You mean you haven't heard, Miss Nanette? I thought the entire population of Plum Orchard would know by now. Must be a slow day 'round here. Bless your hearts, but it has been an entire day since the reading of Landon's will." She paused for effect, the kind of effect that had each of the women waiting on the edge of their seats and left Caine fighting an exasperated sigh. Aw, hell. She was going to do it. Right here. She really did have bigger balls than any man he knew. Which just went to show, Dixie was still the same old Dixie he'd known back in high school. Their short run as a couple, when he'd thought she'd changed but later found out it was all just a part of her endgame. Everything was about the coup. Kitty Palmer's hazel eyes jutted upward at the rumbling sky just as a plump raindrop splattered on the brim of her floppy, orange hat. She stomped an impatient foot. "Well, tell us, Dixie, before I can't stand it any longer. Besides," she remarked with a squirm, casting her eyes downward, "rain always makes me have to use the facilities." Dixie leaned into the group real slow, gathering them together as though she were going to share a classified government secret. She turned her back to Caine, purposely, of course. Though when she did, he wished her supple back end wasn't so well encased in those shorts or her long legs weren't so shapely. Caine fought the instantly sharp reminder Dixie was all woman, and waited for the land mine to explode, one he'd stepped right into. Dixie shook her head in astonishment. "I just can't believe you Magnolias, the backbone of Plum Orchard, don't know." Nanette's irritation exploded in the way only the most God-fearing of them all would—tight-lipped and narrow-eyed—yet still the picture of decorum. "Just spit it out, Dixie Davis, and stop beatin' around the plum tree, young lady!" "It isn't just me who has to see Hank Cotton. It's Caine, too. We're in this together, right, Caine?" Dixie shot him a smirk, her pink lips curling upward as a light rain began to fall. "In what together?" Blanche asked just above a whisper filled with drooling anticipation. Dixie's round eyes went demure as if she wasn't enjoying every second of this game of cat and mouse. "You know all those ladies at the guesthouse? The ones you haven't even given a chance just by virtue of their clothing and makeup choices?" Blanche and the rest of the ladies bristled. "They're working," Dixie whispered from behind her hand. "Workin'?" Nanette squawked. Dixie nodded. "Uh-huh. The ladies in the guesthouse are just doing their jobs. As phone-sex operators." Nanette gasped, her hand at her throat. "That's not true, Dixie! You're playin' one of your horrible pranks again. Same old Dixie!" Dixie put her fingers to her heart, making an X. "No! Swear it on Mama's Hermes bag collection. Landon owned a very successful phone-sex company. It's worth millions and millions," she emphasized. Nanette backed away from Dixie as though she'd seen the devil himself possess her. Her eyes went wide with outrage while the Senior Magnolias rushed to rally behind her. Dixie's face went sympathetic. "But that's not all. Guess what else? I have the chance to inherit Landon's phone-sex company—all of those luscious millions, as long as I can talk dirty, that is." "You have to speak fornicatin' words on the phone to strange men?" Out of nowhere, a fan appeared in Nanette's hand. She liberally batted at the air near her pudgy face before she spoke again, but when she did, it was as though it physically pained her. "It's unseemly, Dixie Davis! How could you?" Dixie's heart-shaped face softened, yet her tone held light reproach. "Now, Miss Nanette, how could I not? The good Lord and Landon, in what I'm sure was full heavenly cooperation, have opened a door for me. One that will help me dig my way out of my financial ruin. Would a good woman turn her back on what's clearly the gift of opportunity from above?" Dixie frowned and shook her finger. "It'd be like turnin' my back on all those morals y'all tried so hard to teach me." Caine shook his head. Oh, she was good. So very good. Nanette harrumphed her displeasure while Kitty held her hand to the spot where her heart beat. Caine saw the wheels of her razor-sharp, fueled-by-the-morality-police mind try to combat their very own words. "I can almost understand you doin' unsavory work. You were always a wild one, young lady. But what does that have to do with one of Plum Orchard's finest?" She waved a hand in his direction. And the final bomb... "I can't believe, while y'all were over here shootin' the breeze, one of Plum Orchard's finest didn't tell you himself." She gave Caine a "shame on you" eye roll. "So here's the story. Now, you girls be sure and pass this on to everyone proper. We wouldn't want any misunderstandings. Deal?" Four heads nodded their consent while four mouths struggled to pick their jaws up off the ground. "Caine and I have to compete against each other if we want to win Landon's phone-sex company. That's why he's here at Mr. Cotton's, I assume. To accept Landon's crazy challenge. Whoever has the most calls at the end of two months wins all of it. That means our town hero will be speakin' those unsavory words—to—" she leaned in conspiratorially—"are you ready for this?" Every woman nodded her head, eyes round with horror. Great. "To women!" Dixie whispered on a giggle. Caine closed his eyes and sighed. His mother would know about this in fewer than twenty minutes. "It'll be just like old times, don't you think?" Dixie asked. "Me and Caine tryin' to best each other to the bitter end? So many good memories." She sighed and clapped her hands together with over-the-top glee. "Anyway, I have to run now, ladies, but I sure hope you'll remember to save me a seat at the next Magnolias tea, seein' as I'll be in town longer than I planned. I've missed those pretty sandwiches with the crusts cut off." With a wave of her hand, Dixie gifted them with one last innocent wink of her eye and pushed her way past Caine to strut up Hank Cotton's driveway. Just as she escaped into the stark white door of Hank's office, the sky opened up and let loose its fury, pummeling them with fat splotches of water. While he held umbrellas and helped the Magnolias back across the street to the dry confines of Madge's Kitchen, he did his best to avoid Nanette's glare and dodge the shocked stares of her crew. Son of a bitch. * * * Dixie ran her fingers over the face of her phone before sticking her tongue out at it. At six o' clock on the dot, she'd received another text message from Landon that read, Are you ready to rumble, Dixie-Do? She shoved the phone in her back pocket and strode past the pool toward the guesthouse with purpose, determined to hang on to the victory of her earlier coup and her conviction she would take no prisoners with this phone-sex thing and beat Caine. She was convinced he'd shown up at Hank's extra early just to rub her nose in his acceptance of this bizarre contest. A smile flitted across her lips when she remembered Caine's handsome face changing from smug to furious as she'd told the Senior Magnolias the costarring role everyone's golden boy played in Landon's game. Which was as good as grabbing her old cheerleading bullhorn and shouting it from the gazebo in the middle of the square. By this time tomorrow, Caine would be as tarnished as she was.... Then she straightened and kicked herself for slipping back into the ways of her ugly past. Not only had she wanted the Mags to know she wasn't going away any time soon, she'd wanted to hurt Caine for making love to her. For hitting refresh on her closeted emotions. But after signing a stack of papers at Hank's office, agreeing to the strict rules of the phone-sex game while Caine simmered in a puddle of rainwater, she'd promised herself she would not gloat. It was ugly and childish, and while no one brought out the ten-year-old in Dixie like Caine did, she was going to do everything in her power to ignore him and her inner elementary-school demon. That last bit of one-upmanship in front of the Magnolias was the absolute end, and while it was a fine way to go, it was also a promise she was sticking to. For real this time. She had to stop clinging to a pain she herself had created. Her focus lay solely with clobbering her opponent, and in doing so, it wouldn't do to spend hours masterminding ways to best him. Caine was banking on two things—that she'd want payback for the end of their engagement, and on the fact that she'd lose her "eye of the tiger" while she plotted his doom. That Dixie no longer existed. It was going to take grit and determination and a lot of dirty, dirty words to win this without resorting to some sort of scheme. The dirty words. They made her flush hot from chest to forehead. Mercy. How was she ever going to say those words out loud? She jammed her hand in the pocket of her shorts, remembering the second text she'd had from Landon today. The kinkiest girl gets the worm, Dixie-Cup! Dixie let her chin fall to her chest, closing her eyes. She paused just outside the doors leading her into Landon's den of iniquity and repeated the mantra she'd been saying over and over in her head since she woke this morning, determined not only to forget every luscious moment in that closet with Caine, but to win. Because as hurt as she still was about losing Caine—maybe always would be—he still made her burn white-hot from the inside out. Caine's scent, still fresh on the shirt she'd burrowed her nose in just before she'd drifted off into a fitful sleep, had left her aching and empty. Full of so many memories of their short-lived romance, she'd had to war with herself not to run straight back into poverty's arms simply to avoid reliving that moment when she'd known, no matter how many corners she turned, she'd always be the old Dixie to him—the girl who'd chased after him until she'd finally caught him, then ruined everything by doing something awful. That one moment in time at their engagement party had shattered her soul, pulling the carefully woven fabric of the rug that had begun to make up the new Dixie, the Dixie who'd come to terms with what a deplorable human being she'd once been, right out from under her. It was, and always would be her own doing, but she didn't need constant, agonizing reminders. If Caine, and Louella, and everyone else who hated her knew karma had taken a chunk of her back in Chicago on their behalf, maybe that would satisfy them. If they knew it wasn't just her restaurant that had blown up in her face, if they knew her life had blown up in her face, if they knew about Mason... Tears bit the back of her eyelids. Not even Landon knew about Mason. No time for tears. Rolling her head on her neck, she straightened her shoulders with resolve and ran directly into Caine's wall of a chest. His hands reached out to steady her, but she brushed their strength off, attempting to step around him. "Evenin', Dixie. You ready to get your sexy on?" Caine tumbled into the night, blocking her entry. At eye level, his thick chest covered in a soft, faded black T-shirt, made her mouth water. His abs rippled when he lifted his arms and spread them across the doorway, bracing himself by pressing those magical hands against the frame. Her gaze moved past the hard-angled planes of his body with conviction, her weak legs stiffened in order to keep her stance menacing. You are a cucumber, Dixie. Cool as such. "Move. Please." Caine dipped his dark head low, bending at the waist so he was almost eye-to-eye with her. "Aw, Dixie. Are you disappointed to find I'm cleaning my chain mail and greasing my arrows in preparation for our phone-sex battle?" Her chin lifted with righteous indignation—their eyes meeting squarely. She didn't budge. Instead, Dixie launched her words at him full throttle. Anger was the costume best suited to hide all other emotions. "Don't be silly. When have I ever backed down from a battle? But were I you, I'd add a little something you seem to have overlooked on your medieval shopping list." Caine grinned, infuriatingly, cooing his question in the gravelly velvet whisper of Sam Elliott. "What's that, Dixie?" Heavens. That low, slow drawl left her almost as girlishly giddy as Sean Connery's did. He knew how much she loved a rough cowboy who sold big, manly trucks. Remain strong. Dixie ran her hands along her arms to hide her goose bumps, keeping her tone even, she replied, "A chest plate. You'll need it to protect you when I shoot that final arrow right through your heart, the one that leaves Landon's Call Girls millions to me." She made a face and used her forefinger to jab him in the ribs. Caine surprised her when he threw his head back and laughed. "Still just as headstrong as always, huh, Dixie-Cup? I know you, Dixie. You're a wild woman when I'm inside you, but as bold and as brash as you are, as manipulative and cruel as that pretty mouth with the flaming-hot voice slitherin' out of it can be, it won't ever be able to play the dirty games. We both know that was my thing." He growled down at her with a chuckle. Caine's reference to being inside her made her jiggle with butterflies. "I'm more determined than I ever was. Maybe even more so than that time we played those six rounds of strip poker at Dwayne Hicks's house, and I was losing by three hands. Just you remember who ended up naked, Donovan, and minus almost five hundred dollars in his wallet." Caine's lips brushed the shell of her ear, and Dixie froze in order to ward off the inclination to lean into them. "You cheated then. Will you cheat now, too? Because if you play the game fair and square, I'd be happy to see you have to take your clothes off," he said so low and husky it made the tips of her ears burn and her nipples pucker against her flimsy bra. Don't bite, don't bite, don't bite. Stop biting, Dixie! "Why didn't you just go home to Miami, Caine? Don't you have a successful real estate-empire to run? Don't empires crumble when their leaders go missing?" Aw, Dixie-Cup. You bit... His expression was the standard cocky. "I have a whole office full of staff for that, Dixie. Besides, I was due some time off. What better way to spend it than being back home again where everyone knows my name, catching up on old times with some of the guys from high school, visiting my mother and driving you right out of your mind?" "Well, at least you're honest about your motivation." His already hard face went harder, the deep shadows of the evening playing across his clenched jaw. "Someone has to be. Honest, I mean." Caine's caustic remark, meant to dig up her lying, cheating past, would hurt more if she weren't already at peace with her past. Not even Caine could manage to jam that knife any deeper into her gut than it already was. Dixie'd made a promise to herself when she'd finally turned the mean-girl corner. She would own her past misdeeds. All of them. Every ugly, manipulative, hurtful one of them. But there would be no dwelling or groveling. She'd simply set out to right her wrongs with as much kindness as those scorned would allow. And those scorned who refused her kindness would end up in the pile of her "regret wreckage" while she let them continue to hate her. She'd been well on her way to redeeming herself when she and Caine had become engaged. He'd gotten to know the Dixie she aspired to—the Dixie that struggled daily to be good enough for someone so honorable, so revered by everyone who crossed his path. Yet, when the true test came, when she'd slipped up once, he'd chosen to believe she was always going to be the worst human being in heels, hell-bent on making Plum Orchard's hottest man hers at all costs. He hadn't given her the opportunity to apologize. He'd believed that she would never change. How could she marry a man who left no room for mistakes? It was then that she'd known she'd never be good enough for Caine Donovan. So she'd shown him, hadn't she? By leaving him and never looking back. And she was going to continue to show him. Right here. Right now. As the humid air picked up, sending her hair flying in every direction, she gave Caine an intentionally haughty gaze. "So here's the deal, Donovan. You can poke, taunt, bring up, remind me over and over about how utterly dreadful I was back in the day. In fact, I welcome it because it's true. But let me make one thing clear. Your innuendo about what a scheming, dishonest, manipulative man-eater I was rolls off me like water off a duck's back. It's old and unoriginal at this point. I'm here for one thing and one thing only—to pay off my debts and hopefully, find a new career...somewhere in this mess. So you stay on your side of the playing field, and I'll stay on mine. Don't cross that line or I swear, I'll stomp all over your Italian leather shoes with my ten-dollar Payless pumps." His nostrils flared at Dixie's order, but he remained silent. Seeing she had him and his sense of honor under her thumb, Dixie backed up and planted her hands on her hips. "Now move. I have a job to learn and a website to launch. Mistress Taboo is in the house!" When he didn't budge, she planted a hand on his chest, trying to ignore the warm skin covering his well-structured muscles. "Move, Caine." Caine faked her as though he were stepping out of the way, but instead, his arms snaked around her waist like bands of steel, hauling her to him with a grunt. "So we're going to just pretend nothing happened in that closet last night?" He ran a firm hand over her backside to cup a cheek with a hot groan. "Like this didn't happen?" he asked, his strong, magical fingers reaching under her T-shirt to stroke the hot skin of her belly. Dixie forced herself to go limp in his embrace. She hung there, staring up at him with a dead gaze. "That's exactly what we're going to do. For not just the good of you and me, Candy Caine, but everyone else who has to suffer our ridiculous Hunger Game–ish behavior. What happened last night happened. Now it's over, and in moving forward, it's going to stay over. Because really, Caine, why would you want to sleep with a woman with so few morals as me? What does that say about you?" Their eyes met, glaring, scanning, searching. Using every ounce of will, Dixie kept her stare blank but determined. She had to or her heart would never survive the next two months. "So let go of me. I have men to entertain. So many, it'll be like it's rainin' 'em." Caine set her down, his eyes in the coming darkness perplexed when she brushed him off. "You're really going to do this. Huh." Dixie's lips thinned. "I'm really going to do this, and I'm going to do it so well, my mean-girl legend will be virtually forgotten, and in its place, Mistress Taboo will reign supreme!" Shaking from his embrace, she managed to scoot around him, stomping into the lavish entryway. She gave a rebel yell to all the Call Girls ensconced in their bedroom offices. "I offer myself as tribute to the phone-sex games!" There was a bark of Caine's amused laughter before his return shout echoed in her ears. "May the odds be ever in your favor, Mistress Taboo!" As she left Caine outside, more handsome than he'd ever been—more sex on a stick than even her tortured dreams had accurately remembered, she left him knowing that no matter what Caine thought of her, no matter the awful things he thought her still capable of, it didn't matter. Because Dixie Davis was still mad about Caine Donovan. Eight Marybell Lyman of the infamous little-girl voice sat across from Dixie at her appointed desk, her saucerlike eyes wide after scanning the website Dixie had managed to design with a template she'd found online. Dixie winced, twisting a strand of her freshly dyed hair around her index finger in nervous anticipation. This had to be right, and soon. Doubt seized her. Maybe all those sparkly, torch-size candles in the shape of a heart surrounding her make-believe online bed were overkill? Or maybe the words of the slogan itself, mimicking dripping ice cream cones filled with vanilla confections and sprays of confetti-colored sprinkles, were too over the top? Mistress Taboo— for those who like "All Things Missionary." "Too much?" Marybell shook her head, the spikes of her red-and-green Mohawk stabbing the air. "No. Not at all, it's—it's pretty daggone sexy. Maybe we should consider another name? Missionary and vanilla isn't exactly taboo." Dixie rolled her eyes. "Some people think talking about any kind of sex is taboo. Just ask the people in this town." Marybell laughed, sweet and tinkling. "I really thought I could pull off a fetish, but when push came to shove, I had to rethink. But I love the pseudonym. So maybe my specialty is not having a fetish at all? Do you think it'll scare potential customers off?" That was the last thing she could afford to do. "I think you should be whoever you want to be, Dixie. Damn what other people say." She swiveled the screen toward Dixie and pointed to the cartoon caricature of a redhead, dressed in a white corset with satiny red ribbons cinching it together so tight, her enormous, animated breasts spilled over the top like a freshly popped can of crescent rolls. "Is that you?" Dixie winced again, marveling at the fact that the cartoon hadn't tipped right over with only that tiny waist to hold up her bodacious hips. "If only I could get my hair to hold a curl like that," she said on a self-conscious chuckle. Marybell pinged a stiff strand of her Mohawk with a black, glossy fingernail and smiled. "Yeah, me, too." Dixie's nervous laughter filled up the bedroom turned office, annoying even her. "You're nervous." "Like the last weaponless soul alive after a zombie apocalypse." "It's the dirty words, right?" She blew out an anxious breath, twisting her hands together in a fist. "It's the everything." Despite her contradictory appearance, despite her short-cropped leather jacket, leopard leggings, multiple piercings, three tattoos, and one partially shaved eyebrow, Marybell was as sweet as her voice. When Dixie was paired with Marybell as her official Call Girls liaison, she'd made every attempt to hide her reservations and her relief LaDawn wouldn't be her mentor. LaDawn Jenkins was one tough cookie, and frankly, she scared the breath out of Dixie. She was surly, short with her words, and the dismissive glance of disdain she'd sent Dixie by way of the end of her nose was enough to make her want to pack up Mona and Lisa and long-distance run back to Chicago. To say LaDawn was less than thrilled two new phone operators would be cutting into her profits was putting it lightly. She was the gold medalist of dirty, and the tiniest of threats to her burgeoning call list were grounds for some scathing remarks aimed at Dixie, who'd barely left their introduction with her skin intact. So in phase one of this contest, she was glad Marybell was her leader. Not in a hundred years would she have put Marybell Lyman's youthful voice with the young woman who looked no more than twenty-five or so sitting before her. Though, according to Marybell, or MB, per her request, she was actually almost thirty, proving one should never judge a book by its cover. Judge not lest ye be judged. Another lesson learned. Marybell reached out a hand and patted Dixie's arm with a thump. "You're letting LaDawn get under your skin. In no time at all, you'll be sayin' the P word like it was hyphened on your name. So let's give this a practice run. You up for that, Dixie?" Again, she silently wondered how Marybell had ended up here. She was incredibly bright and articulate and full of encouraging words. The world was her oyster. Unlike Dixie, whose world was more like stinky, week-old dead fish. Dixie stared down at the cheat sheet—the one filled with all matter of festively colorful words. It wasn't like she didn't recognize most of them—on the contrary. Maybe one or two were dicey, but she knew what they meant. Saying them out loud was another story altogether. "Dixie?" "Sorry. Ready." Marybell wrinkled her nose, the light of the desk lamp catching her gold nose ring. "Breathe." "Can't." "You'd better. It's a deciding factor when pretendin' you're having the big 'O.'" Oh. Dixie snorted out a breath, easing some of her tension. "We're just practicin' right now. Loosen up, Dixie. Relax." "Says the pro..." "I'm no pro. I've only been doing this for a couple of years. Now LaDawn? That's a pro. She has more callers than Facebook has users. She's my phone-sex idol." "Wasn't she—?" "A hooker." "Companionator!" LaDawn's thick Southern accent called out from one of the adjoining bedrooms where the constant beep of phones ringing drifted about. "I gave my companionship to lonely men. You'd do best to get that right, Marybell. I won't have you ruinin' my unsoiled reputation!" Marybell's eyes rolled. "Companionator. Got it, LaDawn," she shouted back. Her eyes caught Dixie's again, warm and easy. "Either way, she knows how to lure tons of men in and keep them on the hook for hours. And LaDawn can do it all. So don't go comparing yourself to her just yet. She's the Jedi master of phone sex. You're just a youngling." Dixie's fingers drifted to the edge of her note pad, smoothing the corners. "Well, I guess I won't ever get my light saber if we don't get the show on the road." She shoved the squares of paper away and with one last deep breath, urged, "Go. Just do it." "Ring-ring—horndog calling!" Marybell chimed, pointing to the earpiece Dixie was supposed to click to the On position when it buzzed. On a gulp, she pretended to flip her earpiece on and read from her cheat sheet, "Hello, this is Mistress Taboo. Are...you...um, worthy?" Weak. Dixie's voice was weak and watery to her ears. Ugh. Marybell flipped her an encouraging thumbs-up and fluttered an engaging grin. "Okay, that was good, but don't ask if your caller is worthy like you're unsure. He has to prove he's worthy enough to talk to you. Demand that. Be firm, Dixie. You're the one in control, no matter how out of control your client would like to be. Always remember you're the captain of the Good Ship Vanilla. Vanilla sex can be just as hot and nasty as a virtual BDSM session. Don't let anyone steer you off course." She waved a finger at Dixie to commence their practice call. Vanilla. She'd decided on sticking with what she knew. If Dixie'd learned nothing from her tainted past, she'd learned if there was a will, she'd find the way, and her way was the good old-fashioned way. "Let's go again before I have to start my shift, Dixie. Ready?" When Dixie nodded, she called out, "Ring-ring!" Pay back every penny, Dixie. Her chin lifted. "This is Mistress Taboo. Are you worthy?" she husked out in a growl, finding a focal point on her desk to concentrate on. To help put her in the mood, she let her eyes smolder seductively in the way she once had when she was on the boyfriend hunt. Marybell's hazel eyes widened in surprise beneath her heavy black eye shadow. "Well, hell-to-the-lo, Mistress Taboo...." Dixie closed her eyes, swallowing hard. If she could just transport herself back to high school Dixie—if she could just summon up all those wiles she'd flung about at the boys as if she was lobbing pennies to peasants. If... "Well, hello to you, too. Now that you know my name, I think it's only fair you share yours, don't you? So I know what to call ya when we're doin' the do." Her words were suggestive, breathy, swishing from her tongue like the Dixie from days of yore. "I don't think I should tell you my real name." Marybell's impression of a male voice bordered comical, coming from the throat of a woman whose dulcet tones conjured up colorfully winged fairies and Disney princesses. Marybell rested her chin on her hand and smiled again with a nod of encouragement to continue. Dixie nodded back as though she actually knew where Marybell was going next. "Then who would you like to be tonight? Tell Mistress Taboo." Her cheeks flushed. Had that really been her? Marybell picked up a manila envelope from the desk and fanned herself with it to indicate she approved of Dixie's hot response. "Why don't you just call me Bob?" "Bob? Ohhh, that's sort of hot—and anonymous. It's perfect. Mistress Taboo approves," she purred, twirling her hair around her finger. "So, Bob—what's on your mind tonight? What are you in the mood for?" "Heaven be," a voice stuttered from the lavish white oak doorway. Dixie's head popped up, breaking the sensual cocoon she'd somehow managed to immerse herself in. Em stood frozen just beyond the wide door leading to Dixie's assigned workspace. Dixie cocked her head in Em's direction and motioned her in. "Look who finally came up for air from all that leftover crab and hot-tubbing. Marybell, do you know Em?" Marybell smiled and waved in Em's direction. "Nice to meet you." Em's crimson lips attempted movement, yet no sound came out. Dixie jumped up from her chair and crossed the small space to hold her hand out to Em who tucked her fists firmly behind her as though Dixie were offering to escort her into the bowels of Hell. "Give me your hand, Em. Come and sit with us. It's okay, I promise." Em swatted at her, her voice suddenly found if not a bit rusty. "Oh, you hush your mouth, Dixie! You just caught me off guard is all." "Off guard?" "Stop behavin' like you don't know what I'm talking about. I mean, good gracious, I've never heard anything quite like it. It was like you were meant to do—to do—this." She made a sweeping gesture with her hand at the desk and every last sin it encompassed. Dixie rocked back on her heels with a knowing nod. "Ah, yes, the devil's work. I should think that wouldn't surprise you, Em. Wasn't your nickname for me back in high school Satan's Sidekick or S.S. for short? Don't think I didn't hear through the grapevine." Marybell choked out a laugh while Em flushed a pretty pink. "You were mean." "I was. Seeing me reduced to this should go a long way toward filling up my 'payback' account. Marybell was just givin' me a practice run. Do you want to see my cheat sheet? It has all sorts of interesting words—" "I most certainly do not!" Yet, her eyes strayed to the stapled papers on the desk, curious and wide before snapping back to Dixie's. "I heard all I needed to hear. I was just checkin' in on you to reassure myself you're stickin' to all those rules Landon made before I left to go home for the night. It's my job." So there. Dixie internally cheered Em's new backbone, fighting a grin. "Then job well done, Em. Landon couldn't have picked a better, more efficient mediator had he been the one mediating himself." Em relaxed a little, finally inching across the room and perching on the end of a chaise longue positioned far enough away from command central for her comfort. "Don't you try and win brownie points with me, Ms. Sidekick. I know you and your low-down dirty tactics, all complimentin' me while you steal my number two pencils right out from under my nose because I was stupidly greedy for acceptance. You won't pull the wool over these eyes. No, ma'am." For emphasis, she pointed to her eyes. Dixie grinned. "I'm going to buy you ten packs of pencils to make up for that incident, in any color your heart desires. No wool. All on the up-and-up. Cross my heart." "You can't cross a heart you don't have," Em said, relaxing a little more. Dixie placed her hand over her heart and slumped back in her chair. "Well, if I had one, you would have just pierced it, and rightly so. So where are you off to tonight? Home to the boys?" "First I have to check on Mr. Smexy and be sure Catherine's got him under control, and then I have a personal mediation with my own snake to attend." Dixie's eyes went sympathetic. The rumors she'd heard about Em's divorce were few and far between, but they weren't any less painful to hear knowing the breakup of her marriage left two boys without their father. "Clifton?" Em's intake of breath was broken up and choppy. "You'd think after thirteen years of marriage to a woman who cooked and cleaned like she was born to do it, two fine young boys, and a pretty little ranch house, the old dog would at least be willin' to let me have my two favorite things in the world besides my boys." "What are your two favorite things? If you don't mind me askin'," Marybell said, her coal-covered eyes genuine when she cast an interested glance in Em's direction. "Our iguana Beauregard Jackson, the boys just love him to pieces, and—and..." She put a fist to her mouth, lines of worry crowding the sides of her sincere eyes. Dixie slid her office chair toward her and patted her thigh, sensing Em's discomfort. "You don't have to tell us, Em, if it's too personal." "My dresses!" she shouted with a burst of words rocketing into the room. "Dresses? Why would Clifton want your dresses?" Dixie paused and caught herself, letting out a dramatic, Magnolia-worthy gasp. The cad. "He doesn't want to give them to his new harlot, does he?" Em waffled, her eyes driving nails into the lush carpet at her feet. "Not exactly..." Marybell's head shot up while Dixie was just plain confused. "Did your husband Clifton like to wear your clothes, Em? Was he a cross dresser?" Emmaline's whimper confirmed Marybell's statement. "Yes. Clifton liked to dress up as a woman, and now that my shame is lying all over this house of debauchery's thickly carpeted floor, I'm going to sneak off to the woods out back at Coyne Wilkinson's and look for a cave to spend the rest of my life in. I hope my boys learned how to build a campfire after all that boy-scoutin' they do. We'll need warmth come winter." "Clifton likes to wear women's clothes?" Dixie repeated, fighting a squeal of disbelief. Clifton Amos, descendant of one of the founding fathers of Plum Orchard, a flannel-wearing, coon-hunting, tobacco-chewing, six-pack guzzling, all 'round good ol' boy liked to wear women's clothes? Em's face went instantly defensive, her eyes hot with more emotions than Dixie was able to count. "Yes, he likes to wear women's clothes, Dixie Davis. Shoes and nylons, nail polish and pretty underwear, too! I didn't know about it, and when I found out, I tried to understand it while I prayed no one in town would ever find out. It was a tryin' time, to say the least. But for all my prayin', Clifton just knew I couldn't find my way." "Oh, Em," Dixie murmured. "Now don't y'all misunderstand me. I don't hate the idea. In fact, it doesn't bother me almost at all. What I hated was that Clifton did it behind my back. I hated that he didn't think I was worthy enough to share his secret with. A secret he says is so deeply personal, it's only meant for someone who truly supports him. I hate that he thought I'd love him less." "And where does the other woman in Atlanta figure into this?" Dixie asked. Em's sigh was pained. "He'd go off to Atlanta on weekend trips to a secret world I didn't know about, and he didn't want me to be a part of. That's where he met the woman he left me for. He left me for her because she understands him," she said with gooey disgust. "Oh, that word! How could I have possibly understood him if he didn't ever tell me what I was supposed to be understandin'?" Dixie's heart clenched for her frenemy. Left here in judgmental Plum Orchard to fend off unwanted inquiries about her husband's whereabouts had to be worse than anything she'd endure short of death. Add in babysitting Caine and Dixie, and it made for a recipe with "stressful" as the number one ingredient. "Oh, Em. I'm so sorry, honey. How can I help?" Em stood ramrod-straight, defiance marring her full lips. "You can help by not helping. Don't use it against me when it suits you. Because you have plenty o' times before. And I'm sure you will again... I think..." "Wow." Marybell whistled, her eyes wide. "Wasn't this whole thing with you two twenty years ago? No disrespect, but that's a long time to hang on to a high school grudge, don't you think?" "You have no idea what acceptance means to a soul as insecure as I was. It stays with you a long time, but mostly, I just like remindin' Dixie about it to get my licks. It's a self-preservation reflex. Sort of like the sign of the cross to ward off demons," Em teased. "But it's also true," Dixie admitted. "I manipulated people to my advantage, I teased, I cheated and I lied. I might not have been stealin' people's lunch money and beatin' 'em up in the schoolyard, but there wasn't much I wouldn't do to come out on top." Em's eyes honed in on Dixie. "Someday, we're gonna talk about what motivated you to be so mean, Dixie Davis. What made you tick." "So what did make you turn over a new leaf, Dixie?" Marybell asked. "It's all people talk about. Dixie Davis the man-eater. Dixie Davis, Plum Orchard's answer to all things rebellious and wild. Bad Dixie. I've heard them talk about it all, when they're not talkin' about the rest of us anyway." Dixie's stomach clenched. Mason. Mason brought this on. Instead, she said, "It just happened. I'm living a clean life free of man-eatin' manipulative deceit. You'll see." "Did you have an intervention an' everything? Like, did they take your makeup and sexy push-up bras away so you couldn't woo unsuspecting men back to your lair?" Marybell choked out before resorting to stifling her squeals of laughter. No. She'd taken them away from herself. But that enormous change in her life was deeply personal, and she wasn't willing to open up about Mason. Dixie forced a straight face. "They didn't take my makeup, but I did have to give up my lace panties. That pained me like nothing else. And flirting? I went on a flirting fast. Probably the hardest part about a mean-girl intervention. You get lots of speeding tickets without the superpower of the flirt in your arsenal's purse. It was a dark, dark time...." She finished, finally cracking and joining Marybell in gales of laughter. Marybell wiped a tear from her eye, smudging her eyeliner. "Someday, I want to hear all the stories. I'm countin' on it." Em shook her head, pinching the bridge of her nose. "I don't know what's happened to me since you landed on Plum Orchard soil, Dixie, but I don't like it. Here I am, spillin' my guts all over Landon's house of ill repute. Sanjeev will never get it all out of the carpet at this rate." "That's the least of what's been spilled here," Marybell snickered, winking at Em. Dixie raised her right hand, palm forward, her eyes latching onto Em's. "No one, not a Plum Orchard soul, will ever hear a word of it from my lips, Em. Thank you for trusting me enough to tell me about it." The turmoil on Emmaline's face instantly melted. Dixie noted Em's shoulders didn't quite sag the way they had when she'd entered the room. "C'mon, Em. I'll walk you out to your car before I have to start my shift." This time when she held out her hand, Em took it with much less reluctance. But trust, total trust, was still far off. As they strolled out of the guesthouse, entering the soft glow of the pool area, Em yanked her close and whispered in her ear, "I swear to you, Dixie, if one soul finds out, I'll know it was you or Marybell. No one, not even my mama knows. This would crush the boys at school, not to mention Clifton's parents. They have no idea. I don't care how progressive Plum Orchard claims it is. We both know this town's still stuck somewhere back in the 1950s. Why, if Landon hadn't been so important and had all that money and connections and done so much for this town, they'd have run him and his crazy ideas right out of here." "I get it, Em. I—" "I mean it! I won't have my children or my in-laws mocked like I was when I was a child. They know nothing about what their daddy's up to, and I still don't know how to explain it. Don't you hurt my boys or I'll borrow old Coon Rider's gun and shoot your kneecaps right out from under ya, you hear?" The mother bear in Em stole Dixie's breath for a moment. A well of admiration sprang from deep within Dixie's gut. She gripped Em's clammy hand hard. "As I stand before you, not even the jaws of life could tear this secret from my mouth. I'd rather be toe-up in Purgeeta's Cemetery." Em's hard edge softened a bit, but her words were still cutting. She gave Dixie's shoulder a poke. "Just you keep that in mind. And remember, I'll be the sole attendee at your funeral 'cause Louella Palmer's gonna be too busy plannin' a party the likes of which Plum Orchard's never seen. A big gala that'll make Landon's off-to-the-afterlife bon voyage look like a kiddy party at Chuck E. Cheese's." "Is that my name I hear being used in vain?" Dixie's eyes swung past the string of colored lanterns lining the awning of the pool house to catch her first glimpse of none other than Louella Palmer, sashaying her way toward them through the huge silver palms along the cobbled path. Her white sandals clacked against the stones. Each step she took marked Dixie's long-overdue dressing-down. Dixie tucked Em behind her, but not before she tugged on a strand of Dixie's hair and muttered a fierce warning in her ear, "You remember what I said. Not a single word." Dixie squared her shoulders as Louella approached. Incoming. Em's stiff body fairly hummed with tension from behind her with maybe even a little fear. Dixie reached around and gave Em's hip a reassuring pat as if to say, "Not on my watch." Still as ethereally pretty as ever, Louella was immaculate in a lime-green frock, one that hugged her figure just enough to show off the easy swell of her flowing curves, yet still met the confines of good Southern decorum. She strolled toward them, confident and lean, and her sure strides almost didn't betray her lingering anger. But Dixie knew better. A humid breeze lifted the soft tendrils of her blond hair, artfully arranged to fall loosely around her face. The barrette, partially holding her locks up, matched her dress, and her nails, a pretty pink, mirrored her toes. Dixie flashed her a warm smile, opening her arms wide for a hug. "Louella Palmer! It's been too long." Louella bent from the waist, ignoring her offer of a hug and instead offered up an air kiss somewhere in the vicinity of Dixie's cheek. She straightened, folding her arms across her chest. "So look who's back," she said, showcasing the kind of restraint that was the hallmark of etiquette and good breeding. It was in her tight smile and glittering, uptilted eyes. In the way she lifted her chin and stood so straight, her spine just might crack. Dixie, in return, ignored Louella's cool reception and dragged her unwilling frame into a hug. She gave her a tight squeeze, then grabbed her hands, entwining her fingers with Louella's to spread her arms wide. Her sweeping glance was full of approval. "Look who's still as pretty as a picture. You positively glow, Louella." Dixie was surprised to find it had taken almost two whole days before Louella came to pick her way through the remaining debris of the train wreck that was her life. Aside from seeing Caine for the first time, this was the meeting she'd dreaded the most. Dixie decided to keep things light, whether Louella liked it or not. The best way to bury the hatchet was to steal the weapon from your opponent's hands before they could even lift an arm to swing it. "So what brings you to Landon's?" she asked on a smile. "What kind of leader of the Magnolias would I be if I didn't come and pay you a visit, Dixie? We welcome everyone new and those returning. I didn't have the chance to give you my condolences personally. The Mags send their condolences, too, of course. I'm sure they'll be dropping by soon." There was a brief moment of genuine sympathy in her eyes, but they turned cool quick. "It's just that I was so caught up with Caine as my escort to the funeral, it darn well slipped my mind." Dixie flapped a conciliatory hand at her, giving her another welcoming smile. "I know firsthand just how charismatic that man can be—even at something as somber as a funeral. Don't you concern yourself with something as silly as courtesies. Manners just fly out the window when it comes to the charms of Caine." She used the same hand to fan herself. Em poked her in the ribs with a dig of her red fingernail, clearly to serve as a reminder that Dixie was egging Louella on by openly reminding her who the first of their once tight-knit group had been to experience Caine's charms. Damn. Old habits died hard. Louella folded her hands together, propping them under her chin, and tilted her head as though she was genuinely interested in Dixie's welfare. "So, how's the restaurant business treating you?" Her first instinct was to shoot back. Louella knew as well as anyone else how life was treating her. She was here under the guise of her leadership of the Magnolias, but in reality, it was the first warning of an all-out attack. The Mags stuck together, and when you were out, you were out with a vengeance. She should know. If she were truly reformed, not even the woman who'd accused her of having a sexually transmitted disease would bear her wrath. So Dixie held up her palms in defeat, her gaze purposely humble. "Not nearly as well as I hope the phone-sex business will." Em poked her head around Dixie's. "You should see how good she is at it, too, Louella. She took to it like a duck to water." Her voice had a certain amount of pride in it, and it might have been wonderful, even supportive. Em was just trying to ease the palpable tension—except Louella wasn't going to let an opening like that slide. Dixie winced in preparation before Louella went straight for her jugular. "Now that doesn't surprise me at all, Emmaline. Not in the least. We all know Dixie's good at ropin' men in like cattle. She's left her panties...I mean mark from here to Atlanta." That was only semitrue. "Well, to be fair, it wasn't quite as far reaching as Atlanta. Maybe just Johnsonville, and I never left a good pair of panties behind." Em jumped to Dixie's defense, startling her. She waved an accusatory finger at Louella. "That's unkind, Louella, and you know it! Dixie's in a bad spot right now. Takin' such pleasure in someone else's pain makes you ugly and cruel." Louella feigned a look of wonder, her eyes hardening as they zoomed in on Em. "Would you look at the two of you? Bondin' over your bad spots. How cozy. Who would have ever thought you and Dixie would end up the best of friends in your despair. How quickly we forget," she added dryly, glaring at Em who was no doubt now doomed to have the title traitor added to her list of Magnolia betrayals. By tomorrow, no one would speak to her for associating with Dixie. Dixie heard Em's sharp gulp and knew her fear of everyone finding out about Clifton was where it stemmed from. She absolutely wouldn't risk exposing Em. Not even if it meant begging and pleading. "Look, Louella. Why don't we just call it even? Let bygones be bygones, and leave Em out of this. She has nothing to do with what's passed between us. She's only doing her job working for Hank Cotton. Surely we can behave like ladies after all this time? It's been almost ten years." Her tone was pleading as she faced Louella, begging her with her eyes to let Em alone. Louella's ironic laughter met Dixie's ears. "Oh, Dixie. It's funny, but I remember saying almost those exact words to you several times in high school. I think I might have cried once or twice while I did it, too. But probably not near as much as you did at your engagement party, I'd bet." Em stepped out from behind her and loomed her extra two inches of height over Louella, her lips tight, and her face full of outrage. The swift rustle of Em's feet behind her caught Dixie so off guard, she was unable to stop her from saying what she said next. "You stop this right now! When will enough payback be enough for you, Louella? As if telling everyone Dixie had a horrible disease and ruinin' her future marriage to Caine didn't fill your revenge cup to the very top, you went and crawled right between his grievin' sheets and took advantage of a bereaved man not hours after he dumped the love of his life, for mercy's sake!" Nine Dixie's head tilted in time with Louella's surprised gasp. What? Emmaline's face distorted with regret as she realized what she'd just done. "Oh, gravy." Dixie imagined that revelation was just what this was to Louella. Gravy on her chicken-fried steak. To have slept with the man Dixie was supposed to marry was better than a cherry on top of one of Martha's ice cream sundaes. Dixie gritted her teeth, trying to remember the rule about karma. Even as she imagined Caine's muscular, tanned limbs wound around Louella's leaner ones. Her stomach lurched in response, threatening to let loose Sanjeev's carefully prepared dinner of curried chicken and jasmine rice. She'd broken the girlfriend code ten years ago then she'd run away from the consequences. Whatever Louella dished out now had been fermenting for a long time. Louella didn't remain silent for long. Her lips popped in a smack of pleasure. "Well, now that that cat's out of the bag, I'm going to go collect my date for the evenin'." Dixie's blurred vision caught sight of Landon's camel, grazing on a thorny twig at the far corner of the big house's vast acreage. "Toe's dating now?" she quipped before she was capable of stopping herself. Louella's laughter tinkled, mocking and satisfied. "No, silly, but Caine is," she drawled, waving a hand at something over her shoulder. Dixie and Em's heads swiveled to catch sight of Caine, stepping out of the doors of the guesthouse, his smile warmly aimed in Louella's direction. She pushed her way past Em, giving her a searing look of contempt before gushing the words, "There you are, Caine. Ready when you are!" followed by the sound of Louella's girlish laughter and Caine's heavy footsteps leading them away from the guesthouse. Touché. Em's head fell back on her shoulders before she lifted it up and let her chin drop to her chest. Her blue eyes peeked up at Dixie, shame and sorrow rimming them. "I'm dreadfully unpredictable these days, aren't I?" "Woefully so." "Do you hate me?" "How could anyone ever hate you?" "Even I'd hate me after tellin' someone their intended, and the love of their life, slept with someone else but an hour after those two someones broke up." It was only an hour? Talk about jumping in her grave. Dixie pressed her trembling fingers to Em's lips to prevent her from repeating the unspeakable. It cut too deep. "I don't hate you. You're unhateable." A sigh of despair slipped from her lips. "I'm sorry, Dixie. Truly. It was an awful way to make Louella hush her mouth, but I just couldn't stand her bein' so mean to you, and now look. I'm just as mean. I got on a roll, and it just slipped out in the way all my foolish attempts to defend do." Dixie slapped her hands against her hips in her second gesture of defeat tonight. "I would have found out eventually, Em. If not from you, then one of the Magnolias would have let it slip when Louella sounded the warning bell to attack. There are no secrets in Plum Orchard. So what better way than to find out in front of the perpetrator herself?" Em gnawed her bottom lip. "My moods are so unpredictable with my life in such a quandary that I forget myself. I'm edgy and angry one minute, fragile and weepy the next. I'm just a babbling mess." "You're going through a devastating time with this divorce. You're allowed to have mood swings." Even if those mood swings would now result in visuals filling Dixie's head she'd never be able to purge. "Besides, if you'll recall, I broke the code first. Why shouldn't she?" Em's eyes rolled. "Was it really girlfriend code, Dixie? Caine never asked her out again, and that thing she keeps calling a date doesn't really count as a date. If you pay for your own coffee—that's 'Dutch,' not 'date.' She has her 'D's mixed up is all." "It's the rule, though. She saw him first and staked her claim. Out loud. He might've asked her out again if not for me." Em flapped her hands dismissively. "He would not have. It lasted all of fifteen minutes if what Essie Guthrie said is true. Which says to me, he couldn't wait to get away from her." "And yet, he slept with her..." God, that hurt. Em's face flooded with sympathy. "You do know I don't actually believe a word of it, don't you? Louella never lost her penchant for embellishing the truth to suit her. So don't you worry, S.S. I'll pay dearly for my mouth workin' overtime. I'm bettin' she wanted to tell you that nasty piece of business herself so she could drive that knife deeper into your already painin' heart." The idea they would launch a good ol' Mag attack on Em sharpened Dixie's protective instinct, a loyalty she hadn't felt toward anyone but Landon. She planted her hands firmly on Em's shoulders and looked her right in the eye. "Don't you even think about Louella and the other Magnolias. I won't let them cause you or the boys any trouble. Not even if the old Dixie has to pay them a visit." Her evil could just as easily be used for good. Em's gaze narrowed. "I'm still not sure the old Dixie's really gone." "Louella still has her hair, and she left the house with Caine, even after you told me she'd...they'd...well, you know." The hard lump in her throat prevented her from actually saying the words. "Surely, my restraint says it all." She shook her head as if to say Dixie's declaration of restraint wasn't convincing. "But if we're bein' honest here, you have to admit, the old Dixie was supposed to be gone ten years ago, too. And you were real good for a little while. I was almost convinced you really had turned over that new leaf you kept talkin' about. Then boom!" Em's fresh, new spine had come out to play, and it wasn't ready to go home for supper yet. Dixie's head dropped. She couldn't meet Em's eyes. Em's pause of silence was a sure sign her mind just wasn't able to process a kinder, gentler Dixie. Letting the handle of her purse slide to the crook of her arm, she crossed her arms over her chest. "Mercy be, I just don't know what to believe about you anymore. All I do know is if I wanna be right with myself, I have to do right. I wouldn't want my feelin's to be crushed so callously while someone stood by and let it happen." Tears stung Dixie's eyes again. "Thank you," she whispered. "I'll tell you this. The Mags have been gunnin' for you since they heard you were comin' back, and Louella's been the captain of that Hate Boat. It's like she picked right up where you left off." "In more ways than one." Every fiber of her being was still in the process of rejecting Em's statement while every curious bone in her body wanted to know everything that had happened between Louella and Caine. How had it happened? Why had it happened? Em glanced at the watch on her wrist and frowned. "I really have to go, Dixie. You gonna be okay? Or will you be plottin' Plum Orchard domination to keep you warm tonight?" Dixie laughed, glancing back at the lights of the guesthouse where phone calls filled with untoward sexual hijinks awaited. "No plots. Promise. That's not who I am anymore. Bygones, right?" Em rubbed her arm. "All right then, but you call me if you need me, hear?" Dixie gave her a quick hug and wished her luck before shooing her off, escaping back inside the guesthouse on sluggish legs. Making her way back to the bedroom where Marybell waited for her, she clenched her fists and bit the inside of her cheek. But it wasn't in anger. It was in anguish. Anguish she'd rather be skinned alive than reveal. She might be reformed, but she still wasn't above pride, sin or not. Louella had learned from the master, and her crush on Caine, while maybe not as lengthy as Dixie's, had been just as valid. Making Caine hers had been just as much her dream as it had become Dixie's. If Louella wanted sloppy Caine seconds, she was welcome to them. So much for making amends, Dixie. That attitude is exactly the kind of reaction one would expect from the former you. That thought deflated her, making her steps swift and guilty when she had to pass Caine's office, conveniently lodged right next door to hers. She scooted inside, shutting her door on the memory of Em's revelation with a shaky hand. Flopping down in her office chair, she closed her eyes, letting the release of tension flood her limbs. Yet, she couldn't block out Em's words. Couldn't stop the endless loop of torturous visuals. She focused on the picture of her and Landon located on the right-hand corner of her desk, clinking their champagne glasses together at the opening of her restaurant. His wide smile, one that always hinted a mischievous thought, soothed and rankled her at the same time. She traced the sterling silver frame with a fingertip. "How dare you not be here right now when I need you so much? Did you know Caine slept with Louella, too?" But Landon's voice, full of the candidness he was known for, taunted her. How dare you have an ounce of anger left for Louella when you snatched Caine right out from under your former friend's cute, upturned nose to begin with, Dixie-Cup? * * * Alone. She was alone with a phone. She'd laugh at how comical the rhyme was if it weren't for the fact that her stomach had decided to take up residence under the desk, and her tongue felt thicker than molasses. Letting her head drop to her folded arms, Dixie rested her cheek on the cool desk and forced deep breaths in and out of her lungs to help subside her panic. Waiting for the phone to ring was like knowing the grim reaper would knock, but you had no time or date for his arrival. She peeked up at her computer screen, letting her eyes roam the numbers indicating the visitors to her website since it had gone live on the Call Girls' main site. Ten. No doubt a nice, round even number. Ten visits in four agonizing hours? Not as nice. With four more hours to go until her shift ended. Ugh. The constant jingle of Caine's phone through the wall behind her set her tired, frazzled nerves on edge. His Michael Douglas impression, wherein he'd asked a caller if they'd like him to find her Jewel of the Nile, was the last straw. She'd taken a lot of heat since she'd come home, but listening to Caine speak to other women—she just couldn't do it. So she'd thrown on her headset and turned up the volume on 98 Degrees. If a call actually came in, it would interrupt the soundtrack. What did he have that she didn't? Where were all the stats in her favor tonight? All she had to do was look at his website to see. Her fingers toyed with the smooth surface of the mouse then pulled away. She hesitated for only a moment before going to the Call Girls home page to click on Candy Caine's website. An image of a man with ripped abs, more hard definitions than a dictionary covering his body, and bronzed skin, popped up on her screen. His face was in the shadows but for his lean jaw covered in dark stubble. Caine knew how much she loved stubble. He wore a low-slung pair of jeans, his right thumb hooked into the waist, and a lone candy cane dangling suggestively from the left right by that indentation in his hip that led to all things sweet. The tagline read, Come and Get Your Candy from Caine. Damn Candy Caine and his money. A live model was out of the question for her. Caine's date with Louella hadn't lasted more than an hour, not that she was counting. When he'd reentered Call Girls, he'd strolled past the bedroom door she'd reopened with a wave and a wink before settling in to address his many admirers. The bastard. The light of her desk lamp, though dim, shone on her computer screen, giving more life to Caine's website than he deserved. With a grunt, Dixie flipped the button on the lamp's base, blanketing the room in darkness, leaving only the muted twinkle of the lights from the big house to shine through the lone bedroom window. As she settled in for the next round of "How Many Ugly Adjectives for Caine Can Dixie Come Up With while She Prepares to Board the Train Called Penniless," the gentle chimes of her earpiece rang, cutting off the music. Her head popped up. She had a call. A real live call! She froze. Mercy, now what? Her hand shook, so much so, she almost couldn't press the button on her Bluetooth. "This is Mistress Taboo..." Dixie cleared her throat, fighting the thin wobble of her unsteady words. Confidence. Marybell had said confidence was critical to keeping control. She squared her shoulders and lifted her chin. "Are you worthy?" "Of?" the refined, definitely Southern voice asked. If her caller didn't know, she surely couldn't supply the answer. "Of my attention, of course," she whispered with a semiconfident delivery. There was a rustle of something that crinkled, and then he asked, "What does a man have to do to prove he's worthy enough for you?" Not sleep with Louella Palmer moments after crushing her heart? Dixie... "Why don't you tell me how you go about provin' yourself when you want a woman." She drew the sentence out, pausing slightly between words, letting them roll off her tongue with suggestive inflection. "First," he drawled, cultured and so honey-thick sexy, Dixie's nerve-endings fluttered. "I have a question." "I have one, too." "Ladies first," he offered, grumbly and smoky. She let the dark interior of her office envelop her, allowing it to hide her embarrassment and forged ahead. "What's your name?" "What's yours?" "You already know mine." "I know what your website says your name is." Did this man want to talk dirty or not, for gravy's sake? He must be what Marybell had titled the "reluctant caller." The kind of man who was angry that he'd resorted to phone sex to get his kicks, yet was still excited by the prospect. Kid gloves were in order here. He needed to feel comfortable. Though, how she'd make him more at ease when she was so uneasy was a puzzle. "Then you have your answer, silly man," she shot back playfully, deciding it was time to divert. Most men, no matter how tough on the outside, were easily diverted by sweet, submissive words and praise. "Or would you rather I don't call you anything at all? Because I can do that, too, you know... You can be whoever you'd like to be, darlin'." Her eyes widened at how easily the endearment slipped from her throat. "Walker." "Like a zombie?" she teased, twirling a strand of her hair as though he actually could bear witness to her flirtatious gesture. His laughter filled her ear, thick and sensual, oddly raising an unbidden goose bump or two at the back of her neck. "That's what you can call me—Walker." She squirmed in her seat before catching herself. Whoever he was, he had a masterful command she couldn't, or rather, refused to define. He was calling her for sex, for goodness' sake. What other definition was needed? Control, Dixie. And sex. They had to get to the heart of this phone call's matter. She had to, or she'd turn tail and hang up. "Very Texas Ranger. So, Walker...do you fancy yourself a big, strong lawman?" She rolled her eyes at how ridiculously porn movie she sounded. But Walker didn't seem to mind. "Do lawmen call phone-sex operators? I'd find that questionable." Dixie giggled openly before she remembered she was supposed to be sexy, not eight. Clearing her throat, she prodded, "Only naughty lawmen, I suppose. Are you a naughty lawman?" "Not today." The line between them crackled with a hiss and a spit while he paused, and she searched for a sexy response to his very unsexy answer. Now she wasn't sure if he fell into the reluctant caller category or the just plain ornery. She was, however, in the "need your bread buttered" category. "Then who are you today? You can be whomever you want to be with Mistress Taboo." No fantasy was going left untapped on the Good Ship Vanilla. "Taboo, huh? Vanilla sex isn't very taboo, you know. It's a definite contradiction to the meaning." He had a point. "Ohhh, Walker," she cooed. "I disagree. Vanilla sex can be just as hot as any sex with floggers, and chains, and all of that control. If you do it right, that is." "Define right," he ordered, yet it wasn't a harsh demand. The underlying tone to his voice was gentle. That meant they were getting somewhere, and it frightened her. Yet, she fought the impulse to put the brakes on like she would have back in the day when a suitor thought she was genuinely going to allow him to get somewhere she'd never intended to go. She would do this. Once more, Dixie let the darkness of her office envelop her, imagining it was an intimate atmosphere rather than threatening. She hunkered down in her chair, cupping her chin in her hand. "Well, everyone has their own definition, I suppose. For me it has to do with giving yourself up completely to your partner. Sexually, I mean. It's getting lost in the sounds of your lovemaking. It's reveling in the taste of your partner's skin, knowing their hot buttons and exactly when to push them." "And have you done that, Mistress Taboo? Have you given yourself up to someone that completely?" Dixie didn't know what made her answer truthfully. Maybe it was her heart, raw and still bleeding after tonight. Maybe it was the anonymity of pretending to be someone she wasn't. Maybe it was desperation, but she sensed if she didn't at least reveal a little of herself to her callers, if she allowed her fear to seep into her words, she'd come across as a fake. And "Walker" was the only call she had right now. Dixie swallowed hard before answering. "I have, Walker." She stopped short for a moment when her voice hitched and more tears threatened to fall. "Utterly and completely." The admission came out breathless and unexpected before she remembered this phone call was about him. "Have you?" "Oh, I definitely have," he rumbled into her ear, soft, sincere. "Only once, but it's something I'll never forget. Not ever." She sighed into the mouthpiece, forgetting he was a client, forgetting that she was supposed to be the one in control of their phone call. There was the wet muffled sound of something or someone snorting before Walker cut her thoughts off. "I have to go for now, Mistress Taboo. But I'll call again. Count on it." "Wait—" her protest was cut off by the sound of him hanging up. The strangest mixture of giddy and disappointed settled in her chest. Dixie laid her head on her arm to bury her strong reaction to this stranger with no face. Walker's words, so intense and soulful, gripped her heart with their seemingly genuine honesty. Ludicrous, of course. Who was honest in phone sex? Walker was probably married with three children and had a wife who didn't understand him much in the way Clifton claimed Em didn't understand. And his real name was decidedly not something straight from a romance novel. It was with that thought her silly, romantic bubble burst. Dixie yawned, fiddling with the mouse to bring her computer screen back to life in order to check the time. She hadn't been up past midnight in ages, and it was already pushing one o'clock and the end of her shift. As silly as it was to consider Walker was anything other than a middle-aged husband, bored with his lot in life, the amount of time she'd logged in with him on the phone wasn't so silly. Forty minutes times four ninety-nine a minute wasn't anything to sneeze at. It wasn't like a LaDawn payday, but it was better than the nothing she'd collected the first half of the night. Removing her earpiece, Dixie put her hand to her forehead and saluted her minutes logged. "Thank you kindly, Walker of the sexy voice and romance novel name. You've brought me a small step closer to digging myself out of debt." Ten Dixie woke with flushed cheeks and a groan of frustration. She'd been dreaming about Caine. His hands on her, his words in her ear, his hard length pressing her into the bed. Scrunching her eyes shut, she gave them a good rub to rid herself of Caine Donovan. Whether Em believed it or not, the idea that he'd slept with Louella Palmer had been planted, and her self-esteem was taking an enormous hit dreaming about him even after such an ugly revelation. Her cell blared, startling her from the bed. She reached blindly, skimming the surface of the nightstand for her phone. She sighed when she saw the caller. Her mother. Yay. She swallowed the dread from her dry throat. "Hi, Mama. How are you feeling? How's Palm Springs?" "Dixie Davis, am I hearin' things right through the grapevine? Haven't you disgraced our good name enough?" There was always room for improvement. Dixie pushed the covers off and slid to the edge of the bed, pulling Mona to her side. It didn't shock her that her mother didn't ask how she was, or that she wasn't concerned about how Dixie was feeling. She'd learned long ago that her mother was concerned with two things—the Davis name and her money. "So you heard what Landon's done?" "Of course I heard what that crazier 'n a bedbug fool's done. What I don't understand is why you're playing his crazy game. How could you, Dixie?" "Do you mean how could I even consider talking dirty on the phone so I can pay back all those friends of yours who invested in me so you can still attend their social events without the embarrassment of your daughter's mistakes hangin' over your head?" The moment she spoke the words, she regretted them. This wasn't the Dixie she wanted to be. Accept what you can't change. Love without strings attached. No more grudges. Over and over she'd repeated those words in her head. Words spoken by someone whose memory she cherished. Her mother was never going to love her without conditions attached. It wasn't in her DNA. Pearl didn't nurture, she demanded. She connived. She orchestrated everyone and everything. Acting out wouldn't change that—ever. "I'm sorry, Mama. That was wrong of me to say. Yes. I'm talking dirty on the phone in the hope I'll win Landon's company. I'm sorry if it embarrasses you with the other Mags." Pearl's outraged sigh crackled over the connection. "You'll make a fool of yourself and the good Davis name!" "Better a fool than a pauper, right?" she joked, knowing it was futile. "You don't have to be a pauper, Dixie. The Donovan money is still good." No. Never again would she let her mother encourage her to find the easiest way out. In Pearl's superficial world, all Dixie had to do was find another way to lure Caine back in and everything would be right as rain. "No, Mama. I'll do this on my own or not at all. I think you'd better get used to the idea that the Donovan-Davis merger is off." So off it hurt. "It didn't have to be, Dixie. You were careless." Disapproval—Pearl's specialty. Even though she was always prepared for it, it never stung any less. Dixie kneaded Mona's soft fur. If only her mother meant she'd been careless with someone's feelings rather than just carelessly getting caught. "It was a long time ago, and it's over now. It's always going to be over. Now, I have to go because I have a date with Emmaline and her boys." "Emmaline Amos? Why, in all of heaven, would you have a date with her?" Because she's the only person who'll associate with me? But something in her mother's tone struck her as odd. Why would her mother care if her date was with Emmaline? "She's not in the same class of people as you, Dixie. You'll do well to remember where you come from." Long live the ultimate snob. "If you mean the class of people who are honest, genuine, and kind, you're right. I'm not in the same class. Now, I have to go. I hope you feel better every day and that Palm Springs is treatin' you good. Bye, Mama." Dixie clicked her phone off before her mother could protest. She wouldn't hear a single bad word about Em or another word about how she'd messed up the marriage merger of the millennium. * * * Dixie's blurry eyes reread the newest cryptic text message from Landon and clenched her hands into fists of sheer frustration. Don't believe everything you hear, Dixie-Doodle... Whatever that meant. She wondered for the hundredth time since this had all begun, what Landon had set out to accomplish by sending her text messages. It was as if he'd known how their battle for his company would play out, and he was enjoying the role of incorporeal commentator. Despite wanting to kill him all over again for poking her at her lowest, mostly, it soothed her to see his name pop up on her phone. The ringing of the bell over the door at Madge's brought with it Em and her two boys, instantly quelling her lust for Landon's blood. Em and her sons, Clifton Junior, and Gareth strolled in with Em's mother. Clifton Junior, his lips a thin line, the slump of his shoulders screaming disinterested, stood beside Gareth, who beamed a smile at her, his front tooth missing. He poked his head out from behind Em's hip. "Hi, Miss Dixie." Dixie grinned at him and crooked her finger. Em's children warmed her all over—even sullen Clifton Junior, struggling so with his father's absence. "Well, if it isn't Gareth Amos. Your mama's shown me all your pictures. It's nice to finally meet you in the flesh. What brings you to Madge's on this fine day? Shouldn't you be off at your big, important job so you can pay for your big, important apartment? How will you keep the lights on if you're here eatin' donuts?" Gareth giggled, scratching his dark head. "I don't haf a job. I'm only five," he lisped a protest that turned to a squeal of joy when she handed him a jelly-filled donut. Dixie looked at Clifton Junior. "Boy, five years old, and your brother still doesn't have a job? How will I ever collect social security if you don't have a job? I suppose next you'll tell me, at eight years old, you don't have one either, mister?" When Clifton stoically refused to join in on her joke, Em nudged him, passing him a stern look. "Miss Dixie spoke to you. Surely, you have an answer?" But Dixie held up a hand with another chocolate donut—Clifton's favorite. "No answer necessary. I like my unemployed men strong and silent. Both of which Clifton Junior is." Like the tiniest bit of hope still existed, Clifton smirked, taking the donut. Dixie smiled at Clora. "Clora, good to see you again." Clora's chin, square and tight, lifted along with her eyebrow. "So you've come back." Dixie bit the inside of her cheek. Clora had never liked her. She probably liked her less now that she was so closely tied to Em. Her disapproval was evident. "Like the dead risin'," she joked. "Or trouble stirrin.'" "Mama! I warned you." Em hissed over Gareth's shoulder. "Please be gracious." Dixie flapped a hand. "It's all right, Em. You go right on and disapprove, Clora. I'd disapprove if my daughter was mixed up with Dixie Davis, too." Em clapped her hands together, effectively quieting a response from Clora. "Say thank you to Miss Dixie, then off you go with Grandma. Mama's got work to tend to." She gathered each boy up in a warm hug, one Clifton remained stiff in, and thanked her mother for taking them for the afternoon before ushering them off. "I'm sorry, Dixie. You know what my mother's like." Dixie smiled. She did know what Em's mother was like. She'd grown up under the same kind of control and endless unattainable perfection Em had. "Just like mine. You know, I was thinkin' about that last night after what you said to Marybell." "You mean about how pathetically insecure I was." "And I was out of control. Isn't it funny that our mothers raised us with the same iron hand of disapproval, but we each reacted so differently? I just could never win with my mother." And clearly, after today's phone call, still couldn't. Em scoffed. "Right, except you had a daddy and piles o' money. I had no father and I wore clothes my mother hand sewed. But I think I can almost understand why you were so mean. I tried harder to please everyone, you spit pleasin' everyone in the eye." Dixie nodded. "It's funny what shapes us—motivates us." Made you act out at every possible turn. "How about we forget our controlling mothers and talk about you and Caine?" Em teased, sliding into the booth and latching onto the coffee Dixie had waiting for her. She peeked over the top of her oversize coffee cup and grinned. "There is no me and Caine, Emmaline." She buried her nose in the vinyl red-checkered menu—one whose items hadn't changed much for as far back as she could remember. "Oh, Dixie, who do you think you're kiddin' here, petunia? You still love Caine Donovan." She flicked Em's menu, the delicious fried scents of home mingling with strong coffee called to her. "Focus. I invited you to breakfast—not a therapy session." All she'd thought of since waking from that dream was how much she wanted to run right back to Caine, burrow her face in the hollow of his neck, beg him to let her back into his life. To recapture those perfect moments before she'd ruined everything or even before she'd found out he'd slept with Louella. Being in Chicago had dulled those emotions—muted the sharp edges of them. Being back in Plum Orchard had continually plucked them. Em tapped the table. "Have I apologized enough for what I blurted out last night, Dixie?" "You have." Dixie grimaced, the pain still as sharp as it had been the night before. Em reached across the table and squeezed her hand, remorse still lining her creamy skin. "I was just sick over it last night. I'm sorry again and again, S.S. And I still don't think I even believe a word that comes out of that woman's mouth." Dixie smiled at her nickname and blew a strand of messy hair out of her eyes. "Forget it. Let's forget everything but a big plate of greasy bacon while you tell me how things went with you and Clifton last night. How did the meeting go?" Em plunked her coffee mug on the white Formica table and sat back in the red cushioned booth, fading and cracked. "You want to talk about me?" Dixie's frown portrayed her confusion. "Of course I do," she said gently. "I was worried about you in the state you were in last night. Seeing Clifton couldn't have been easy." Em's surprise filled her gaze. "I can't remember the last time anyone wanted to talk about me or even to me unless it was my mama and the boys. But to have you want to talk about me? It's like baby Jesus just dropped by for a playdate." "Then tell me all about it, or I'll be forced to make you break out the lighted manger you put in your yard every Christmas." Her joke fell flat. Rather than laughing, Em's blue eyes skirted hers. Shame cast them downward to her lap. Dixie's heart clenched into a tight ball seeing her struggle with her emotions. As people scuttled around them, Madge's finest and oldest waitress, Tammy, moving furtively from table to table across the yellowing floor on orthopedic shoes, Dixie sat silent. When Em lifted her head, her eyes were swollen with unshed tears. "Can you believe he brought that woman with him to mediation, Dixie? Worse still? She's absolutely gorgeous. Forgive my horrible thoughts, but I wanted to pull her hair and knock her long legs right out from underneath her." "But why? You're gorgeous, too." Reaching into her mint-green purse to pull out a tissue, she shook her head. "I'm not. I'm a used-up ol' housewife who's been tossed out like week-old biscuits." It struck Dixie like thunder right then and there—she knew that look. She'd seen it a hundred times on Em's face in high school. She'd had a hand in Em's low self-esteem—her and the Mags. A big one. Twenty years later, and the damage was part of who Em had become. And she had to make it right. Change it, fix it, own it. Dixie shook her head in denial. "Please don't say things like that about yourself, Em. None of them are true. I don't care what anyone's said—what I said, you're beautiful, and funny, and smart. And you're a housewife who still has plenty of life left for livin'! The next night I have off—or any loose change I find in Landon's couch—whichever comes first—I'm going to treat you to a girls' night out just to prove as much to you. You'll see. The men'll line up." Em was about to protest again when the tarnished copper bell on Madge's door rang out a new entry. Or several of them. Louella and the rest of the Mags wandered in on a cloud of sweetly scented perfume and a colorful array of modestly heeled pumps. As they swished past her and Em's booth, Dixie was dismayed to see Em sink deeper into the seat, obviously hoping to fly low under their radar. Dixie knew better. There was no flying low or otherwise. If the Mags had their sights set on you, secret-government-agency-issued invisibility packs couldn't keep you hidden. Unlike Em, she sat straighter, setting her menu aside and waiting until the group of her former henchwomen made eye contact. "Sit up, Emmaline Amos," she scolded. "Always look your enemy in the eye. It's one of the first mean-girl rules ever written." "If you'll recall, I wasn't privy to the rule book." "I wrote it." "Then to thine own self be true. As for me, I'm just gonna make myself as small as possible and hope they find a moth to pull the wings off of in my stead." "Dixie Davis—look who's come home to roost!" Em groaned out her misery from across the table, reopening her menu and setting it in front of her to hide behind. Dixie plucked it up and away from Em's face, whispering, "Watch and learn, grasshopper." Sliding out of the booth, she pivoted on her chunky wedges, likely considered much too high for this time of day by the Mags, and smiled at Annabelle Pruitt and Lesta-Sue Arnold as they weaved through the assorted white tables. "Annabelle, Lesta-Sue! So good to see you both. I see life's been treating the two of you well. You both look fresh and vibrant." Annabelle, petite in her knee-length white skirt and blue-and-white polka-dotted silk shirt, let a polite smile tease her frosted lips. Her rounded chin grazed the large bow of her blouse. "I see your Southern charm's all but escaped you, and your appearance, too, Dixie. Where'd your good fashion sense and accent get to? Too much Chicago in you to ever come back to your roots?" Dixie's denim miniskirt, clingy red top, and denim shrug vest was as close to her native land of Plum Orchard as she'd come in a long time. She gave them both air kisses before inviting them to slide into the booth beside her and Em. They naturally declined with polite protests. They weren't here to sit. They were here to stir the pot. So Dixie decided to give them the spoon. "So what have you girls been up to since I left? Lesta-Sue, you married now?" Dixie knew she was married. Married to the man she'd once stolen right out from under Lesta-Sue's nose, a shameful act she'd be willing to redeem herself for, if only they were willing to let her. But that wasn't what this visit was about. Lesta-Sue's pinched face and thin upper lip wrinkled. "You know I married Grover. I sent you an invitation to our wedding. I know it was after the 'incident,'" she whispered, hazel eyes wide and full of innocence, "but we hoped you'd come back sooner than this." "Ten years is a long time to stay away." Annabelle, no longer blond but chestnut-haired, twisted her thick, side-swept braid between her fingers. And it was game on. This was called the bait and bait. Wherein, they baited Dixie then baited her some more until she became so frustrated she exposed herself and did the Mags' dirty work for them without them ever having to lift a dainty finger. Her cool reply said the ball was in their court. "There were other pressing matters keeping me in Chi-Town." Annabelle's smile grew glib. "And we heard there are newer, more pressing matters keepin' you here." A quick glance around Madge's told Dixie no small children were present. She tapped Annabelle on the arm playfully. "Oh, stop implyin', Annabelle. Let's all just be adults here and say it. In fact, why don't we all say it together? Just to get it out of the way. Ready?" She smiled at them as though they were two toddlers she was teaching a new word. "Phone sex." She let her voice rise an octave, enough to catch everyone's attention. "I'm engagin', in Miss Nanette's words, 'in the devil's acts,' over the phone, no less. I bet you're not surprised at this unexpected turn in my life, are you, girls?" Heads turned in the group's direction as Annabelle's cheeks went pink and Lesta-Sue took an uncomfortable step backward, righting the wobble of her sensible turquoise pumps. "So, let's discuss, shall we? Then we can get all the awkward moments right out of the way. You know the moment I mean, too. It's the moment where you set out to humiliate me publicly for being such a sad sack of a human being all those years ago. How would y'all like to do that? Maybe over some pie? My treat," she cajoled. Lesta-Sue cracked first, her manicured hands balling into fists at her side. Her sidelong gaze in Louella's direction bellowed help. Louella must have flashed her the go sign. Her words were meant to push Dixie's buttons. "I can't believe your lack of shame, Dixie! Your daddy'd do a back flip in his grave if he knew what you were up to." Dixie fought the urge to lunge at her by brightening her smile. The Mags knew any mention of her deceased father was a sore subject. She'd been a daddy's girl through and through, and she'd missed him terribly since he'd died unexpectedly of a heart attack during her first year of college. "I'd like to think Landon met Daddy at the Pearly Gates and told him in person. About the phone sex, I mean." And as if on cue, Annabelle cracked next, embarrassed by the rise in Dixie's voice. Her stiff words came out in a stream under her breath. "You're just as disgraceful as you always were, Dixie. You should have never come back to Plum Orchard. No one wants you here." "As well they shouldn't," she agreed, totally compliant, completely cool—on the outside. "I'm a bad person, Annabelle, and even as I apologize to you both for the million and one things I did to you in high school, like stealin' Lesta-Sue's boyfriend, Grover, and treating you like my lackey, I'm still not going to let that stop me from doing what needs to be done. When has someone's disapproval ever stopped me?" With yet another attempt to cast public stones at her out of the way, Dixie waited. Come hell or high water, it was surely coming. As Dixie suspected, the women turned their venomous attention to Em, still hunched in the booth, her face a mask of despair. Lesta-Sue launched the first grenade. "And you, Emmaline? How could you consort with the likes of Dixie and her phone-sexin' when you have two young boys to think of? It would serve you right if those poor children were taken from you and given to Clifton to raise. It's repulsive!" There was an unsettling stir in the crowd, heads turned with a rustle of discomfort, and all eyes fell on Em. Silence prevailed but for the jukebox playing an old Hank Williams tune in the background. Dixie felt Em die a little. Felt it like she'd feel her own heart stop dead. And that was all Dixie could stand. She'd fought her sharp tongue and scheming long and hard over these past couple of years. She'd made as many amends along the way as she could, but dragging the only victim the Mags could find into this, one who'd never dream of hurting a soul just to make a public spectacle out of her, wouldn't stand. Like the lighting of a sparkler, Dixie's infamous temper flared, dipping and rising to a new height before zeroing in on her targets in homing pigeon fashion. The red haze of her anger sizzled, almost blurring her vision. "Speaking of the word consort, Lesta-Sue, a word you're very familiar with, before you cast disparaging, not to mention judgmental, stones at Emmaline—" Dixie addressed the crowd trying their best not to gawk with open mouths "—would you like to share with everyone in this fine establishment what you did the summer just before our senior year?" Lesta-Sue sucked in a hitched gasp of air, her eyes sending warning signals at Dixie. "You wouldn't dare...." Dixie's smile was cunning when she crossed her arms over her chest. She leaned into Lesta-Sue, her eyes full of mischief and menace. "Aw, you know better than that, Lesta-Sue. I would dare. Em's just doing her job, and y'all know it. So if, in fact, you so much as whisper a hurtful word about Em or her boys in the same breath as something so wicked ever again, I'll make it my mission to share the consorting. The video of it." Em's hand snapped out, reaching for Dixie's arm, her fingers bit into her flesh with a trembling grip. Her face was paler than normal, her voice shaky. "Please." With Em's plea, the angry haze filming her vision cleared almost as swiftly as it had arrived. "Pass that along at the next Mag meeting, would you, Annabelle? Just in case Louella missed it from all the way over there." Dixie pointed directly at Louella, then gave Em a quick tug upward, grabbing her purse and hooking it over her shoulder. She ushered her out, head held high until they stood just beyond the front window of Madge's. The sweltering heat of early September clung to Dixie's already hot face in a wave of cloying slashes. On fast legs, Dixie moved down the sidewalk, ignoring the questioning gazes of Plum Orchard. Familiar faces all wondering, What has she done now? Em ran behind her and thrust a hand to her shoulder, curling her fingers into it. "Dixie, hold your horses!" Flinging her around, Em's eyes searched hers. Dixie threw her hands up in disbelief, furious with herself. "I just blew my stack as sure as Johnson Ridley blows up at least one box of fireworks by mistake every Fourth of July. It's wrong, Em." That manner of retaliation should have been below her. For many years, her impulse to strike had been quelled by what she'd learned from one of the most painful experiences of her life. But it had just all gone to hell, and Dixie didn't like the way it made her feel. Low-down dirty and out of control. "I'm sorry I embarrassed you. I know you hate when the attention turns to you, but I just couldn't stand it." "Girl, I realized you were just practicin' what you preach. It was pure genius the way you turned the tables on them. Hide out in the open, right? Expose yourself before they can expose you?" Yes. That had been the exact strategy—to expose herself, not Lesta-Sue. Tears of frustration began to well in her eyes. "I learn something new and useful from you every day, S.S." She stiffened, swiping a thumb under her eye. "It's not something I want to teach." Em gave her a shake. "You said you'd only use your evil for good, right? And you did." "How is threatening to expose Lesta-Sue using my evil for good, Emmaline? It's what the old Dixie would've done, and it was wrong and selfish. What I wanted to do was win and win big." She spat the word as though it were covered in filth. "I wanted to grind her into the ground in just the same way they wanted to humiliate and knock you down a peg." That particular competitive gene was the bane of her existence. Her hot temper was a close second; it made her reckless and foolish. Em smoothed her hands over Dixie's bare arms in a soothing fashion. "No, Dixie, don't you see? You said those things to protect the boys and me so the Mags wouldn't dig around about my troubles in front of a whole restaurant full of busybodies. Only good people do that, and I won't hear nothin' to the contrary. Good people speak up for people who're too afraid, or in my case, too pathetic to speak out for themselves. You saved me from those perfectly dressed, pink-and-blonde bullies. I, fair maiden Dixie, dub thee a hero." Dixie gave her head a shake, her lips a hard line. "I'm no hero, Em! Don't you confuse the two. I feel like I stepped right back into my pink stiletto shit-kickers like I never took them off. The real problem with that is they felt sooo, so good." "Well, you did wear them for a long time," Em conceded. Dixie shook Emmaline off with brisk impatience, moving away from her to sit on the bench in front of Brugsby's Drugstore under the shade of the dark green and gold overhang. The cheerful topiary beside her, one that had been there since she was a teen, was a painful reminder of all the times she'd sat on this very bench and plotted. She let her head fall back on the peeling bench, huffing a tired sigh. "Plum Orchard was no place to come back to test my mean-girl overhaul, Em. It was easier in Chicago. I had less history to battle. Too many things have passed in ten years, and that's not including what's passed in just the last two days. I don't want to be that person anymore. I won't be that person anymore." Em shoved her over, sitting alongside her. Crossing her feet at her ankles, she said, "Sure nuff, Chicago was easier. It's a big place full of more strangers than folk you know personally. You can't get by with that here. A test isn't really a test if it isn't hard, Dixie." Yet another reason she never should have agreed to this phone-sex-off. "And here's somethin' else to chew on, Dixie. I'm beginning to think you're not that person anymore already. Color me as shocked as anyone, but I'm this far from callin' you a new leaf. Really, Dixie, when was the last time you took up for someone weaker 'n you? There was no advantage to you in lookin' out for me and the boys." Dixie gave her a weak, heat-deflated smile and coyly batted her eyes. "Does this mean you trust me now, Em?" Her snort ripped through the humid air, full of laughing sarcasm. "Oh, no it does not, miss. One kind act does not eradicate all your sins. But it's a foot in the right direction, Dixie. Yes, ma'am, it is." Em gave her a playful nudge to her shoulder before patting hers and inviting Dixie to rest her head on it. Dixie did so with a mournful sigh and a million emotions warring with her heart. Surprisingly, the least of which was malicious joy at besting the Mags. What troubled her was how little time it had taken before she'd exhibited signs of her former path of destruction. Em plucked at Dixie's denim shrug. "So, want to tell me about this mysterious video of Lesta-Sue?" "No, I do not." "You don't play fair. How could you bring up something so delicious and not share?" "Because the former me would do exactly that. I forgot myself there for a minute, and every resolution I've made, every truly good thing I've tried to do since I made all those promises to myself went right out the window. I will not allow the Mags to try and convict the people who've chosen to give me another chance just to get to me." "Is that what this is with me, Dixie? Another chance?" She hadn't thought of it like that when she'd come home. She'd only given thought to getting in and out with the stealth of an F-16. But she liked Em. She enjoyed her company, and the more Dixie liked her, the more she wanted Em to forgive her and accept her friendship. "I suppose it is. You have something to say about it?" Em leaned her head against Dixie's. "Not a word, Dixie. Not a word." They sat like that for a time, Dixie pondering how she was going to keep Em from the Mags' angry wrath and anticipate their next move. She thought about how she was going to keep herself afloat and pay her debts off if she lost this phone-sex challenge. She thought about Caine and all the painful longing he dredged up. Mostly, she thought about how nice it was to simply sit with Emmaline on a bench across from the whitewashed gazebo nestled in the neatly manicured square, smelling the scents of summer nearing its end, and watching Plum Orchard go about its day. Eleven "This is Mistress Taboo. Are you worthy?" Dixie's voice bled through the walls to Caine's office, so husky and sultry it was ruffling his already ruffled feathers. He threw his pen down, leaning back in his chair. His desk was cluttered with a collage of sticky notes he'd made an array of designs with in order to keep his mind off Dixie. Erasing Dixie's memory was more difficult than he'd anticipated. Especially when he thought back to her trapped in that closet, clinging to him as though he were the last life raft on a sinking ship. Hearing her heartbeat against his ear, feeling her slinky thighs wrapped around his hips—all things distinctly Dixie. When it came to Dixie, he was helpless—hopeless. All she had to do was give him that wide-eyed look of anguish when he taunted her unmercifully for all her scheming and lying, and he was lost. It seemed the only way to curb his insatiable need to make her pay with his cruel words was to haul her body to his and make insane love to her. Losing her once had hurt like hell. Losing her twice wasn't going to happen. He'd dug in his heels this time. There'd be no convincing him she was changed. There'd be no convincing him she'd dug her conscience up somewhere in the landfill of her devastation. Huh, pal. I don't remember Dixie trying to convince you of anything. She just showed up, somehow very different than the Dixie you once knew. Landon in his head. Again. What made this doubly hard was the encounter he'd witnessed between her and the Mags today. He'd never, in the entire time he'd known her, seen her take up for anyone unless it meant the means to her endgame. Yet today at Madge's, she'd gone at Annabelle and Lesta-Sue as if Em was hers to protect. That side of Dixie, fierce and so primal, had chipped away at the ice forming around the part of his heart that had once belonged to her. He'd watched that go down from behind a potted plant and a plate of eggs, shocked as hell. Stealing glances at her defiant eyes and defensive gestures almost made him like her. Like her? Where the hell had that come from? He didn't want to like her. Loving her was hard enough. To like who Dixie wanted everyone to believe she'd become would be to forget it hadn't just been her life, her future that was smashed to a million pieces by the end of their engagement. He'd fallen hard for her back then, heedless to the warnings of everyone around him, heedless to his own internal warnings. He'd just fallen. Leaving Plum Orchard without Dixie as his wife had been like a knife to the gut. To forgive her? He'd never given it much thought. Leaving her unforgiven was what fueled his macho fire. It was all he had, and he'd be tarred and feathered before he wasn't on the ground blowin' on the flames that kept that fire burning. Emmaline poked her head around the corner of his office space and waved at him, halting his dark thoughts. "It's my job as mediator to pay you drive-by surprise inspections of your work. So, you stickin' to the rules, Mr. Smexy?" Caine grinned at her, stretching his cramped arms. "You bet. Did you expect anything less?" She grinned, smoothing her dark hair with a pale hand. "From Plum Orchard's golden boy?" There was that damn moniker again. How had he missed that catch phrase in reference to him all these years? "The one and only." Her glance was one of reproach. "Well, if I was mediatin' anything else, I'd expect nothing but good form from you. But, we are talking about you and Dixie. That'd give Jesus himself pause." The mention of Dixie's name made Caine tentatively inquire, "You okay?" For a fleeting moment, concern crossed her face, marring her creamy ivory beauty. The wary look in her eyes shifted, and she let her face relax into one with no expression. "Of course I am. Why would you ask such a thing?" He shrugged indifferently, as though the gossip around town after Dixie and the Mags' little confrontation wasn't the talk of every dinner table. Em was hiding something, something she was desperately afraid of. Which made Dixie's defense of her even more significant. Whatever Em was afraid of was hers alone. He wouldn't add to her anxiety by saying anything more, but the Mags' anger just because Em had chosen to dine with Dixie, couldn't be the crux of what troubled her. Still, Caine kept his reply light, his fingers busy making a colorful Christmas tree from the sticky-note pad. "Just checking." Emmaline's posture deflated. "You heard." Caught. "Who hasn't?" he replied honestly. "Then did you hear about Dixie, too?" The intensity in her question rang like a church bell, making him look back up. Em's loyalty to Dixie stunned him. Of all the people she'd dragged through the mud, Em had been her most utilized target. "Well, did you?" she pressed, her grip on the door handle white-knuckled. He was a crappy at playing innocent, but he gave it a go anyway. "Hear about Dixie?" "Yes, about our Dixie! She took on the Mags on my behalf like some kind of avenger." Emmaline's inquisitive eyes waited for his reaction to such a strong statement. "And what did it get her? Bragging rights? See Dixie take out a Mag at a hundred paces? As if she hasn't done that a hundred times before, Em? It's just the first step toward regaining her rightful place on her tarnished throne." Em slipped inside the room, closing the door as she did, coming to stand at the edge of his desk. She lowered her voice, tapping a painted fingernail on the wood to demand his attention. "Don't you be so callous, Caine Donovan. I won't hear you talk about her that way. She was like some kind of warrior today—for me—and the boys. Just you remember that when you go judgin' her by using her past as your weapon." Caine threw up his hands and slid the chair away from his desk. "Not another word, because I'm afraid of you," he teased in his best Geico Gecko impersonation. Left unmoved by his attempt at humor, Em warned, "Just a reminder that some things can change. Dixie being one of those things." "If you say so, Em." Not happening. He wasn't ready to concede out loud he, too, had been impressed by Dixie today, because there had to be a catch. Somewhere. There'd be no more making him look like an ass if he had anything to say about it. Finally she smiled, her limbs relaxing. "I do say so. Now get to work, Mr. Smexy. You have sex to make." "Hey, Em? Before you go... First, did I ever thank you for keeping such close tabs on me during Landon's last..." Damn, it was still hard to say. "His last shining moments here on earth?" she finished for him, her smile a mixture of fond and sad. He'd only just left Johnsonville, where Landon's hospice care was located, when Em had called to let him know the doctors said Landon had taken a turn for the worse. Caine nodded, his chest tight. "Yeah. That was hell. Everything just kept falling apart. My plane was delayed, damn rental broke down." "You know something? If I didn't know better, I'd almost think Landon planned that, too. He often said he didn't want anyone seein' him in his last moments. But if it makes any difference, he knew right up until the end how much y'all loved him." Caine blew out a breath of more regret. "From the other end of a damn cell phone." The phone Em herself had held to Landon's ear. "But you were still with him until the last second he was with us, Caine. He smiled when he heard your voice. He might not have been able to answer you because of all the pain medication, but he knew. I know he knew." She said it with enough conviction that he had to believe or let the guilt eat him alive. "In all this, I never asked. Tell me he got to talk to Dixie, too." No matter what was between him and Dixie, she loved Landon. Em smiled again, but this time it was beaming. "She absolutely did. She was havin' her own troubles gettin' to him, too. Which is why I'm convinced Landon somehow orchestrated his own death just like everythin' else." That made Caine laugh. Yeah. That was Landon. He nodded his head. "Good, and listen, I know things have been rough for you and the boys lately, but if you need help with anything, around the house—maybe take the boys for a burger on my day off to get them out of your hair, whatever it is, you let me know." His mother had raised him alone after his father died when he was thirteen. No one knew better than he did how hard that could be. Em's big blue eyes grew watery. She crossed the room and hugged him hard around his neck. "Thank you," she whispered, patting his cheek before heading out the door. "Now back to work, Donovan!" "Aye-aye, Captain!" he repeated Scotty's infamous Star Trek line to the sight of her whisking out the door in a flurry of blue material and the scent of gladiolas. Caine cracked his knuckles and rolled his shoulders, uncomfortable with this new Dixie emotion that was neither lust nor love. Like. Hah. For just a minute, you liked Dixie, buddy. Don't deny—testify! More Landon. "The hell," he muttered back under his breath, flipping on his earpiece. The hell. * * * "This is Mistress Taboo—are you worthy?" Dixie injected as much sexy into her greeting as was possible at midnight. Curling her legs under her, she settled in to answer her third call of the night with a mixture of nerves and the almost debilitating fear of the unknown. Her first call had been a complete bust, and the second caller hadn't made it past the first couple of minutes. Day two's numbers so far weren't going down in the phone-sex history books. "Hi," the distinctly male voice forced the word out. There was the sound of a loud buzzer in the background, resembling one you'd hear at a sporting event, the tinny drone of some sports announcer's voice, then nothing but the inhalation of air. A breather. Dixie instantly recognized the profile described in the Call Girls manual. Shy, unsure, embarrassed. They'd make a perfect couple. "I'm new to this," he finally said. Honesty seemed the best policy. They could be cohorts in newness together. She flexed her icy fingers and rolled her shoulders, willing herself to relax. "You know what? Me, too. So we'll be new at this together. Deal?" She kept her voice light, easing off the sex-kitten tone to it. She couldn't afford to scare off a possible long-term relationship with someone simply because she came on too strong. Slow and subtle won the race. "You're new, too?" "I am." A sharp scrape against the phone was followed by an exasperated huff of more words. "Jesus. I can't believe I'm doing this. This was a stupid idea." Stupid idea meet stupid idea, Dixie mused, clearing her throat. "Calling me, you mean?" "Calling anyone for sex. I don't know what the hell I was thinking. I just don't do this sort of thing. I've never done this sort of thing. I'm sure you'll hear that a lot, newbie." Dixie smiled at the teasing quality to his tone while she drew imaginary circles on her desktop with her finger. "Well, I don't know what I was thinking when I thought I could sell sex over the phone either. But here we are. Wanna try and make the best of it? The first two minutes are only two bucks." He laughed with a notable release of tension. It was a pleasant laugh, too, deep and full. "I'm..." he faltered then righted himself, "Dan. In case you were wondering what to call me." Dixie smiled to herself. Ice broken. "A pleasure. So what do you do for a living, Dan?" "I'm an accountant." "Ah, so you're good with numbers. I'm not very good with numbers." Unless they had to do with racking up a mountain of haters. Then she was a superstar. Dan cleared his throat and coughed. "Good with numbers, not so much with the ladies, if you know what I mean." She shook her head in disagreement as if he were in the room with her. "I don't believe that, Dan. You have a good, strong voice. It's clear and downright pleasing to the ear. At least, it is to my ear. You're articulate and pleasant. So tell me, how can any truly smart woman resist that?" He made a noise faintly resembling a snort. "Because I'm a complete idiot face-to-face. I stumble over my words. I never know what to say. I'm much better at formulas and fractions and summing up expense accounts than I'll ever be at pretending I'm some kind of Prince Charming." How ironic to find Dan was better on the phone than he was in person, and she was better in person, where her body language and coquettish smiles did all the work for her. "Who says all women want a Prince Charming anyway? He's overrated, if you ask this girl. Wasn't he the one who didn't even recognize Cinderella when she wasn't wearing her ball gown? I want real. I want genuine. I want someone who'll hold my hair out of my face when I'm throwing up." Throwing up? Dixie cringed, cupping her hand under her chin to keep from bringing her head to the desk. Vomit. Ugh. "Well, that was very real. I think I like you, Mistress Taboo." Dixie's cheeks flushed hot. "Really unsexy was what that was." Dan chuckled. "Maybe so, but it kind of brought things into perspective." "The kind of perspective that has you hanging up and never calling Mistress Taboo again?" She put as much pout into her voice as she could, but it rang false even to her ears. "No," he blustered then cleared his throat. "Not at all. But I figure if someone like you can get a job selling sex over the phone, you know, with the vomit talk, maybe there's hope for me." Dixie didn't even bother to hide her laughter. "Exactly my point. How hard is it to inquire about the well-being of a woman who's caught your eye instead of wowing her with your high-falutin' prose?" Dan's next words told her he had his doubts. "You make it sound so easy, but we both know it's not that easy." "It can be, if you let it." "Every time I see her, I just clam up." Her. "Freudian slip?" she teased gently, hoping to find insight into the real reason he'd called. Dan was lonely and needed a friend who wouldn't lend him the sort of ear a man was likely to get over a six-pack in a Neanderthal's man cave. His gruff sigh confirmed her suspicions. "Okay. There's this woman...." "You don't have to give me her name, of course," Dixie rushed to assure him. "Her name's JoAnne. She works in HR. I've been watching her for a long time, and I don't mean in the creepy stalker sense, so don't freak out. I don't want to hack her up and put her in a wood chipper." "My relief, can you hear it?" Dixie joked. "I just mean I've appreciated the view for what seems like forever." Dixie's smile was full of dreamy envy for JoAnne, having someone so smitten with her. "And she's got you all kinds of tongue-tied, huh?" Dan's laugh was sharp. "In ways you can't begin to imagine. She's pretty, and smart, and outgoing. She's everything I'm not." "Oh, don't be silly, Dan. I bet you're very pretty." His laughter filled her ear, and now it was warm and full. "I meant outgoing, though I'm sure no one would call me pretty. I'm not good with a lot of people in one room, but JoAnne always has people around her. People are drawn to her like a moth to a flame. She's amazing, and I'm dull and boring. We're...we're just very different." His genuine admiration for JoAnne rang in Dan's voice, almost making her sigh wistfully. To have someone feel that much pride in someone they hadn't even had one date with was the epitome of romantic. It was the Cyrano de Bergerac of romances. Dan didn't want to talk dirty with Mistress Taboo. He wanted someone to listen to him. Someone to offer him a sympathetic ear. Leading her to wonder out loud, "But sometimes, don't you think it's those differences that make a whole package in a couple? As long as you can learn to appreciate those qualities you lack rather than resent them, of course." "Don't get your meaning." "I mean, maybe JoAnne can teach you to navigate a roomful of people, and you can teach her to listen in the silence? You can sit back and revel in her people skills, and she can learn that communication doesn't always have to involve the spoken word. Sometimes communicating a feeling is just in the way you hold a woman, or offer to rub that sore spot between her shoulders to ease the ache of her workday. You don't have to be Cyrano and look like Brad Pitt, who's highly overrated, if you ask me, to convey your feelings." "Hah!" he barked. "There's no fear of that. Promise." His reply, rife with insecurity, made her wince. The wheels of Dixie's sentimental heart began to turn. "Oh, hold that thought! I have an idea, Dan. Have you paid attention to where she eats lunch? Maybe what she likes to drink?" "Coffee. She's always in the coffee shop at lunchtime in our office building. I see her there a couple of times a week grabbing a salad. I don't get out of my office much, but when I do, I always go to the coffee shop because it's close." "So why not bring her a cup of her favorite coffee and open with telling her just how amazing you think she is in the very words you used with me? Tell her you need a little sparkle to shine up your dull, Dan. The worst she can say is no as long as you're respectful and polite. And if she does, then you'll know whether she's waiting for Prince Charming with his fancy lines or just some nice, reliable guy who drives a sensible Honda." "Hey! How'd you know I drive a Honda?" Dixie visualized his eyes, wide with suspicion. "It must've been the fact that you're not pretty. Only men who aren't pretty drive Hondas," she finished with a chuckle. "Oh, and also, you said you were a numbers guy. Most of the numbers guys I know drive Hondas and Toyotas because the numbers for longevity of the life of the car are pretty good." The pause on the other end of the line was significant. Dixie relaxed into it, hoping Dan was processing her words, not preparing to hang up because she'd gone too far. When he spoke again, it was with quiet reserve. "This is probably going to go in the books as a prime example of epic failures in phone sex. I'm going to be honest with you when I say, you stink at it, Mistress Taboo." A snort escaped her lips at his very accurate assessment. "That I do, Dan. You're only my third or fourth phone call ever, and I haven't spoken a single dirty word yet. You're not helping." "But you're very good at something else." "What's that?" "Listening. You're one helluva listener, Mistress Taboo. I guess sometimes that's all a guy needs, someone to listen who isn't another guy. Er, wait. You aren't a guy, are you? Because I don't know if I could live with myself if the voice that's been making me a little hot under the collar belonged to a man, but maybe that's a totally different conversation?" Her laughter echoed in her office. "I'm not a man. I'm all woman. Swear it on my mother's homemade banana bread. So I guess my work here is done?" Dixie left it up to Dan. "Can I call you again, Mistress Taboo? Maybe just an update on how it works out with JoAnne?" "I'd love that, Dan." Oddly, strangely, she found, she'd be delighted. "Until then," his warm, hearty voice assured. Dixie nodded her head, almost sad to lose the connection. "Until then," she whispered, clicking her earpiece off and marveling at how oddly pleased she was. Was it wrong to take pride in the most unsuccessful phone-sex call ever? One that had lasted, according to her time log, almost an hour. Maybe. Or maybe she was onto something. Twelve "I knew it, Dixie. I damn well knew it!" Caine roared from the bottom of the marble staircase of the big house. "Get your pretty backside down here right now!" Sanjeev approached him, hands behind his back in typical respectful fashion, his black kurta immaculately wrinkle-free. "Ah, Caine. I know this face, this tone of voice. This is the face of a man who's been, yet again, cuckolded by Dixie Davis. Surely you know this path. You've traveled it often. Your return trip shouldn't come as a surprise." His lips lifted at the corners to obstruct a full grin. Almost two weeks had passed since their closet incident. A torturous two weeks where he'd suffered the giggles and small, breathy moans Dixie made from beyond the office walls as she sucked more unsuspecting men into her web of flirty. There was some guy named Heath, and another one named Dan. And two who went by the name Mike, or was it three?—all calling—all the time—all fooled by her load of crap. Fourteen days since he'd begun avoiding her, refusing to buy into the idea that she was a changed woman. Three hundred and sixty-eight hours since she'd blown his mind with her soft mouth and sexy groans of pleasure. Their silent standoff was about to end—big. Caine tucked his laptop under his arm with a grunt and eyed Landon's most trusted assistant with his "don't push me" look. "Where is she, Sanjeev? And don't hide her from me, or I'll tear this place apart, slab of marble by insufferably pretentious sculpture until I find her." Sanjeev's lips warred with another smirk, making Caine respond by clenching his jaw. He cleared his throat and stood straighter. "Now, Caine. I'd be so disappointed in you if you brought discord to my housekeeping. You don't want me to have to stay up for two days straight, cleaning marble chunks and sculptures with broken limbs, do you? Is the life of a man who chooses to serve so cruelly dismissive to you?" Caine shook his head, his lips thinning. "Oh, no. Don't you pull that self-sacrifice bit with me, buddy. There's going to be no keeping me from my mission." "And what is your mission today, Agent Caine?" His mission, should he choose to accept it, was to throttle a hot babe. "Where is she, Sanjeev?" He followed Sanjeev's serene eyes toward the back of the house, lined with ceiling-high windows and heavy hunter-green and ivory drapes made of silk. Just beyond lay the second pool Landon owned. A sprawling Olympic-size square of sapphire-blue water surrounded by various shades of blue, clay and green mosaic tiles. In the middle of it, down in the shallow end, right near the stone fountain of a naked man spouting water from his genitals, sat Dixie on the luxury liner of floats with a fruity drink in the cupholder. Lolling the day away, while she sunned and dreamed up new endearments to label her many phone-sex callers. Her red hair was wrapped into a loose ball on the top of her head, exposing the nape of her long neck. The sunlight danced on the soft swell of her breasts, clad in a surprisingly modest, yet incredibly hot, chocolate-brown bathing suit. Sanjeev placed a cool hand on Caine's arm. "A warning. I will not have wet feet in this house. I've just spit-shined the floor with my own saliva. Keep your battle of the sexes out of doors, please." Caine barked a sarcastic laugh. "No worries there, friend. I'm going to drown her all nice and neat right there in the pool. No fuss, no muss." "There will be no unsightly yellow police tape marring my view of the freshly vacuumed pool, Caine Donovan. Understood?" "You suck the fun out of everything, Sanjeev. Buzzkill!" he called over his shoulder, jogging down the steps leading to the sunken living room and past Landon's collection of Buddha statues. He pushed open the wide French doors and exploded out into the pool area. "Dixie! Get out of the damn pool. Now!" His roar of a demand echoed off the white pillars surrounding the cabana area, but still she didn't budge. The water sparkled with the heat of the late afternoon sun while Dixie floated on it as though she didn't have a care in the world, twisting a stray strand of her hair around her finger like she'd always done when she was deep in Plum Orchard domination thought. And she was blatantly ignoring him. Caine narrowed his eyes until her temptingly curvy body became a blur of smudged lines. The hell she was going to get away with this—not as long as he was in the game. He set his laptop at the end of the pool farthest from Dixie, scanning the area through the thick imported palm trees and assorted fringe of grassy plants. The incessant heat of the sun beat down on his head, pissing him off. Shit, it was hot. Gritting his teeth, Caine stomped over to the tented blue-and-green cabana where, under Sanjeev's instruction, Landon kept an array of toys for the assorted children who attended free swimming lessons in the pool each week. Mona and Lisa lay in the shade of a lounge chair, lifting their heads at his entry. He pressed a finger to his mouth and dug in his pocket for the treats he always kept with him in case he came across the chance to indulge them when Dixie wasn't looking. Mona settled back instantly while Lisa's soft eyes shone with gratitude when he held out the Snausages. He gave them each a quick pat on the head as they licked his palm clean and whispered, "Be very quiet, girls. I'm hunting a cheater...er, your mommy." He scanned the area under the tent, locating the enormous red tub that would hold his weapon. Popping the top open, his eyes landed on the first colorful gun he could find, still full of water. Nice. He scooped it up, kicking his shoes off and shrugging out of his shirt and pants. Lining up the mark of his prey with the red tip and a vindictive smile of satisfaction, he broke into a run, and howled, "Cannonball!" just as his finger pulled the trigger. Caine doused the top of Dixie's head with the super soaker. The stream of water made an arc of perfection as the waves his body created hitting the water shook the float. It wobbled until she toppled over with a scream of surprise. He smiled with deeper satisfaction when the remnants of her frosty drink spread out in globs of orange bubbles along the surface of the pool. So did her casualty-of-war iPod with the earbuds still attached. Well, damn. She hadn't been ignoring him at all. But Caine consoled himself with the notion that it wasn't as if she didn't deserve it anyway. She was right back to doing what she did best. Manipulating the rules of the game and cheating. Cheaters deserved to go down. In big, brilliantly blue pools, with cannonball-like splashes and squirt guns. Dixie's arms flapped wildly as she struggled to rise to the surface, giving Caine enough time to ditch the evidence and swim back to grab his laptop. Hair plastered to her face, she pushed it out of her eyes—eyes that shone bright like angry chips in ice blue. He loomed at the edge of the pool by the steps, arms crossed at his chest, waiting. And three, two, one... "Caine Donovan!" she bellowed at him when her vision was free of hair and dripping water. Adorably wet, her bathing suit clung to every enticing spot on her body. He grinned. As a response, she lobbed her soaked iPod at him, hitting his shoulder with the pink square. "You've ruined it! What the hell was that about?" She dragged her body through the water to the shallow end where he sat on a step. As her rounded hips sliced the rippling waves, Caine had to force himself to avert his eyes and focus on the pool's intricately woven deck. He flipped his laptop open and pointed to her new tagline for Mistress Taboo. "What the hell is that? Care to explain, Mistress Taboo?" Dixie's eyes focused on the screen for a moment, then she snickered, squeezing the bun of hair on top of her head free of excess water. "Oh, that." "Yeah, that. That got you three hundred hits today alone." He emphasized again by tapping his finger against the screen where it now read: "The Mistress of Taboo Subjects—No Woman Trouble Too Small—No Problem Too Big. Let me help you solve all of your female mysteries." Mistress of Taboo Subjects, his damn eyeballs. This wasn't therapy. It was phone sex. This woman and her way of bending a set of rules should be bottled and sold. "Catchy, right?" she curtsied and shot him her infamously flirtatious, over the shoulder smile. "Cheating, right?" he ground out even as his tongue ached to lick the beads of water between her rounded breasts. Look away, Caine. Dixie waggled a pink-tipped finger at him in admonishment and teased, "Oh, no, no, no, Candy Caine. I didn't cheat at all. Nowhere in Landon's rules did it say I had to actually do the talkin' dirty to the clients. I made it my duty to check with Hank first. And he says, Landon's rules only claim I had to acquire the most new clients to win. And look," she purred, pointing to the screen as the counter dinged another hit. "More acquirin'. Hmmm." Caine fastened his eyes on her face in order to keep his focus off her luscious legs. Not that it made much of a difference. It was either ogle her legs or witness the smug look of satisfaction in her eyes, both of which left him hotter than hell. "So, you plan to win by being some sort of sex/dating guru to lonely men who need a guide on how to properly get a woman into bed?" She grinned, the cute dimples on either side of her face making an appearance. "Guilty." Dixie reached up to tug the ribbon holding her matted hair, yanking it free and shaking it out with a scrub of her fingers so it fell around her shoulders in wet waves. She shrugged her shoulders in the way she used to when she was using indifference as her weapon of choice. The more indifferent she was, the harder the poor soul Dixie was indifferent to tried to get her attention. It made his hands itch to haul her from the pool and show her all about indifference. Caine slammed the laptop shut and shook his head. "I should have known you'd find a way around it. You couldn't dirty talk your way out of a paper bag." "Well," she said, "I guess I don't have to worry about that, now do I, master impressionist?" "At least we were on a level playing field then." "Level, how? You can be a hundred different men. I'm but one woman." Yeah. One. Hah! "So what you're saying is, you're not enough?" Her eyes, full of fight one minute, shuttered, and he didn't like what he saw in them. "No truer words," she muttered before squaring her shoulders. "What I'm saying is, I don't offer the variety you do. Who wouldn't want to talk to the famous Denzel Washington instead of some nobody like Dixie Davis? I had to find something that would catch a man's attention." And that was fair. And he didn't like that it was fair. Screw fair. Moving off the step, Caine moved in closer until they were chest to chest, planting his hands on his hips. "I'm going to beat you, Dixie, no matter what kind of con you pull. Only I plan to do it fair and square." Dixie clamped her fingers together, making the shape of a duck's bill. With a snapping gesture, she opened and closed them under his nose. "Blah, blah, blah, Golden Boy. I don't know why you're surprised. This is what you expected of me, isn't it? To find that one, teeny-tiny loophole that would make you want to tear your hair out because I thought of it first? Again." He snatched her wrist, bringing it to his chest, forcing her body to lurch forward until they were hip to hip. The taut pull of his boxer-briefs told him he was playing with fire, but his anger with her and her silly games he just couldn't resist playing, would damn well trump his hard-on. "What happened to the new and improved Dixie? I thought she was all sunshine and lollipops these days. Miss Honesty and Light." Dixie's eyes clouded over, but only for mere seconds before they were full of fire again. "She's fighting for her life, and to make a living. You don't need to make a living. You have one. In Miami." "But it's so good to be home." Her lightly tanned throat worked a visible gulp. "I'm glad it's good for somebody. For others, it's an effort just to watch out for pointy objects." Show no mercy, Donovan. "Nobody to blame but yourself for that." She rolled her eyes. "Well, thank heaven if I happen to forget, you're never far away for the remindin'. If I need someone to pass judgment on me or call me a liar, worry not, I have you on speed dial. You know what this is, Donovan? You're just jealous that my Mistress Taboo numbers are beating Candy Caine's, even with your stupid Patrick Stewart impressions." Captain Jean-Luc was popular, damn it. And judgmental? Hold the phone. How was the truth a judgment? Whatever. He wasn't falling for her reformed act again or the recent nagging feeling he was riding his grudge too hard. Caine hardened his expression, not only to remind Dixie he wasn't falling for it, but as a reminder to himself. "And you're just reaching here. How can you possibly give advice to anyone about anything when it comes to relationships, sexual or otherwise?" "Maybe I've learned a thing or two about them since we broke up." Her voice was smaller than he found spiteful comfort in, forcing him to pause. Dixie played coy and sultry as if she'd been tutored by a skilled madam, but weak and vulnerable? Never. Stand your ground, Caine. Do not believe. The hot babe with the mind-numbing body and breasts that fit with ease and precision in your hands is good at this game. You must be better. "And what have you learned, Dixie Davis?" Instead of fighting the clasp Caine had on her wrist, Dixie moved in closer to him, letting their bodies almost touch. The smell of her perfume, something soft and peachy, made his nostrils flare. The fruity scent of the drink she'd been sipping was still on her lips. She traced a circle around his nipple, making it pucker. "I've learned that you, for all your goody-two-shoes, holier-than-thou, impossibly high expectations, are not the only fish in the sea. There are some men in the world who'll cut a girl a break when she makes one stupid mistake on her road to salvation." "Really? How many of those have you been through before you got it right on the way to salvation?" Dixie's chin lifted defensively. "Oh, I've got it right now. And just in time, too, so don't you worry." "Does this 'just in time' have a name?" A flash of her smug smile almost made him forget her breasts pressed against his chest. "Jealous?" "Actually, I was hoping to get the poor guy's number so I can do him a solid and warn him off you. Maybe save him from the fall." "Walker loves me just the way that I am. Faults and all." Caine's ears perked. "Walker?" "Yes, Walker. His name's Walker." "Where does saint Walker hail from, and why isn't the entire town of Plum Orchard talking about Dixie's new beau?" Her eyes strayed to a point just beyond his shoulder before returning to his, full of conviction. "He lives in Chicago, and he's wonderful and kind, and no one knows about him yet. I'd like to keep it that way for now. At least until I have the opportunity to throw it in Louella's face when she's throwing you in mine." Now that was the Dixie he knew—but wait. "Louella's throwing me in your face?" That statement puzzled him. Her lips clamped shut before she said, "Forget I mentioned it. All you need to know, Candy Caine, is that Walker and I are an item now. As in, I'm all his and he's all mine." How interesting. "And how does this Walker feel about you hooking up with me?" Dixie's tongue darted out over her lips. "We hadn't committed as of that point. But all this...tension...with you was what made me decide to make the final leap." Caine's eyebrow rose with amusement. "I made you decide to commit to Walker? Does he know that?" "Yep... Well, not yet, but he will know everything as soon as I tell him. And I mean I'm committing fully—unequivocally—forever. Maybe even marriage committed." Caine wrapped an arm around the curve of her waist, lifting her from the water until Dixie had no choice but to look him directly in the eye. He smiled at her before capturing her lips, kissing her until she relaxed against him, pliant and soft. Until her fingers dug into the skin of his flesh and her soft moan, one he recognized as a sign of her will breaking, slipped from her throat. He pulled back with a sudden jolt, taking pleasure in her pink cheeks, noting she still had a small trail of freckles across the bridge of her nose. The surprise on her face was only matched by the defiance in her eyes. He smiled again. "Dixie?" "What?" Her voice, hoarse and thick, was one of many things that turned him on about Dixie— especially when it was as a result of his lips on hers. "Send me an invitation." "For?" "Your nuptials to Walker," he said with a chuckle, before dropping her into the pool with a resounding splash and the sounds of her screeching protests at his back. * * * Dixie stomped out of the pool, grabbing her towel, and chucking her ruined iPod on a lounge chair. She'd done such a great job of avoiding Caine for an entire two-week stretch. Hiding in the shadows of the various potted plants Landon had scattered about the house. Ducking into doors when she heard his footsteps or his voice. Eating her dinner microwaved so they wouldn't be tempted to use the sharp dinner knives as tools for killing. Walking Mona and Lisa on the south side of the house instead of the north where Toe spent his days and Caine often spent time with Landon's very friendly camel, had all been in the effort to stay as far away from temptation as possible. All of her covert operations had proven successful until today. In the meantime, she'd hatched this new plan to become her callers' advisor rather than their playmate. Her ridiculous attempts to talk them into having phone sex with her had become a joke— especially to LaDawn, who lorded her numbers at their first weekly meeting over everyone as if she'd invented phone sex. But when Dan had called back two days after their first encounter to tell her he'd finally gotten up the nerve to ask JoAnne out on a date, the seed his call had planted sprouted wings. Even if it wasn't conventional phone sex, it worked. Dan had left her a glowing review on the Call Girls main site, and since then, her numbers had jumped from dismal to decidedly less pathetic. Dixie'd double-checked the rules with Hank to be one hundred percent sure she wasn't breaking any part of her contract. Despite what Caine thought, she wanted to win this fair and square. After adding a disclaimer on her website about her lack of training or certification in the hallowed halls of therapy, she'd begun to average four calls a night. Since her meeting with Hank and Dan's recommendation, word had begun to spread about Mistress Taboo, and she'd gone a little bit viral. Something she was extremely proud of. But now she'd dragged Walker into it. She hadn't heard from him since that first phone call. Though whether he liked it or not—or knew it or not—he was officially her fictional boyfriend. His voice had made an impact on her, one she couldn't quite define—or was maybe a little afraid to define. Yet Walker had clearly lingered in her subconscious if the way she'd thrown his name at Caine was any indication. The upside to Walker? Being so full of all that integrity, maybe now that Caine thought Dixie was spoken for, he'd keep his inconceivably delicious body to himself. Being back here where she'd created so much damage was easier to navigate when Caine wasn't in the mix. It was easier to focus on keeping her promises to herself when he wasn't around to pass judgment on her every move with his angry eyes. Though, each time they came in contact physically was another time she died a little more inside. Her body's reaction to him meant plainly she was weak. Worse, in a desperate moment or two, she almost didn't care that Caine didn't like her. She just wanted him to make love to her with his signature relentlessly forceful passion. Thankfully, her self-esteem had kept her desires in check. But today, the image of him in the pool, his skin lightly bronzed and glistening wet, his arms a tangle of corded muscle, the dark line of hair that wound under his belly button and into his boxer-briefs always stole her breath, made her wet with wanting him. Even in the ungodly heat, Dixie shivered and peeked around the pool area, embarrassed by her wicked thoughts. Mona and Lisa slept contentedly under the lounge chair while the floor fan Sanjeev had set up for them misted their chunky bodies, keeping them cool and comfortable. Annoyed, she yanked her towel from the blue-and-green chaise, wrapping it around her lower body. Now she had a new lie to contend with. The Walker lie. Idiot. "Walker? Really, Dixie? What kind of fool declares she's on the verge of marrying a man who calls to engage in phone sex?" "The kind that wishes to make Caine the Neanderthal jealous," Sanjeev provided, handing her a frosty glass of sweet tea with a pink umbrella in it. Her sigh was exasperated, but she accepted the glass with gratitude. "You heard?" "I believe Wylan Landry three miles down the road by the county line heard." Dixie offered him a sheepish look. "I'm sorry, Sanjeev. I was angry. No one gets my goat like Caine." Sanjeev cocked his head. "I don't understand how Caine can get your goat. You have two dogs, no goats. You don't have a goat somewhere, do you, Dixie? Isn't it enough that I clean up after Toe, Mona and Lisa? I don't have enough misting fans for a goat, too." Dixie chuckled and shook her head, scrubbing her hair with the towel. "No goats. It just means he pushes my buttons." "You don't have buttons either." "They're internal. It's a metaphor. Never mind. The point is Caine makes me say things out of spite that should never be said in public." "Someone can make you perform a task you don't want to perform? I assume there's money to be made in this," he teased with an easy grin. Sipping her tea, Dixie wondered out loud, "What is it about him that makes me behave like a vengeful teenager? I've managed to keep my temper and my scheming in check for a long time now. I come back home, and in a matter of seconds, it's like I never stopped being a blight on humanity." Sanjeev nodded his sympathy. "Love is indeed a puzzle, isn't it?" "I don't—" He held up a hand to prevent her protests. "Oh, but you do, Dixie. To deny you love Caine would be to deny the entirety of your youth. It would be to deny some of the most monumental pieces of your past, and the incidents that have shaped this new, improved you." These days, it felt as if her heart was in a vise grip for all the pangs it suffered. Yes—there was truth in that statement. Back then, when she'd vowed Caine would rue the day he'd broken up with her, most of her supposed changing had all been empty, internal promises based on revenge. Once she'd gotten past the initial hurt over the end of their engagement, everything she'd done had been based on showing Caine what he'd missed out on rather than the real work she'd needed to do on herself. And then, one day, one ugly, agonizing day, the tables were turned on her, and everything changed.... "It doesn't matter anyway, Sanjeev. Caine's never going to see me as anything other than a manipulative liar. It's not without reason. But for heaven's mercy, I get it already. Every single thing I do is an excuse for him to remind me that I'll never be good enough for someone so filthy rich in integrity." "Why not prove him wrong then?" Hah! "Because even if Jesus himself dubbed me new and improved right in front of Caine's very eyes, he still wouldn't believe it. I'm not here to prove anything to anyone. That's not how it works." "How what works?" "Redemption. It's not about how many apologies you make. It isn't about begging for forgiveness. It's about being actively involved in turning your life around. Actively owning the things you've done to hurt people and stopping the hurt. For good." She bit back the next piece of her journey. It was still too personal to share with anyone yet. To speak of it was to diminish the impact it had on her life so long ago. For now, that would stay where it belonged—in her heart and mind. "Besides, you know what I discovered while I cried my eyes out and ate enough fried food to feed a small country, Sanjeev?" "I'm all ears, as you say." And here it was, the entire crux of the matter. The eternal weight of loving someone as moral and forthright as Caine, and what had kept her from begging him for a second chance, boiled down to one thing. "There's no room for error with Caine. I was always afraid I'd mess up back then, and when I did, I realized mistakes just weren't allowed where he's concerned. It's damn hard to impress perfection." Sanjeev wrinkled his nose. "Ah, but, Dixie, Caine is as far from perfect as you are imperfect." "He's a lot closer to perfection than I am, buddy." "He fights his own demons where you're concerned." Dixie grinned, pulling off her towel and laying it over the lounge chair to dry. "Demons and me in the same sentence. How unexpected." "Caine has his faults, too, Dixie," Sanjeev insisted. "They're not as big as mine. He wins." "This isn't about winning," Sanjeev said stubbornly. "It's about what should be." "What should be? Who's talking in riddles now, Sanjeev?" Sanjeev threw up a hand at her and waved her off. "We have no time for this discussion. You have to nap before your shift, and I have to restock the pantry and somehow manage to squeeze in laundering your delicates before the Waltons Mountain marathon." Dixie grabbed his arm. Sanjeev wasn't known for his comfort with displays of affection. Still, she latched onto his hand anyway and squeezed. "Don't be silly. You don't have to do my laundry, Sanjeev. I can do it. You don't have to cook for me either. Even though I owned a restaurant, lately, I've become an expert at microwaved meals. Please don't go to any trouble for me. Though, I appreciate it—and you." Sanjeev's eyes assessed her for a moment. His head tilted at an angle as though he'd never seen her before, and then he grinned—a rare and wondrous thing. "That you won't make me suffer the indignities of my continued confusion when identifying a thong and French cut panties makes me appreciate you back." Her throat tightened at Sanjeev's silent understanding before she teased, "Then John-Boy awaits." Sanjeev squeezed her hand back before shrugging out of her grasp and calling to Mona and Lisa with a sharp whistle. "Come here, you bottomless pits! Dinner and a movie are at hand!" Mona and Lisa snorted their way out from under the lounge chairs, stopping to nuzzle Dixie's hand before they followed behind Sanjeev, their wide hips wiggling in anticipation. She stood, watching Sanjeev and the dogs retreat into the shadows of the house until they disappeared. The sting of tears, born of Sanjeev's forgiveness, a small sign her efforts were genuine and from the heart, threatened to overwhelm her. Smiling to herself, she gathered her things, warmed from the inside out. * * * "This is Mistress Taboo—" "Yes. I'm worthy." Her breath caught in her throat as her heart raced. Walker. "Well, hello, Walker." "Howdy to you, too, Mistress Taboo. How are you this fine evenin'?" Dixie fought to keep her words calm when her insides were all kinds of tangled. "I'm very well, thank you." "So I see you're counselin' men now." Walker had seen the change on her site, meaning, he'd thought of her since the last time they'd spoken. Whoa. Dixie pinched her arm hard enough to make her bite the inside of her lip. A dose of reality was in order. Wasn't it bad enough that she'd used him as her make-believe boyfriend to hurt Caine, but now she was hoping he'd been daydreaming about her, too? She repositioned herself in her chair in an effort to get hold of her spiraling emotions and began arranging the sticky notes she'd accumulated since her phone line had begun to pick up. She kept track of the callers she advised by writing down their identifying traits. "I don't know if counseling would be the correct word. It's more like just offering helpful advice from a woman's perspective." "Who calls a phone-sex operator for a woman's perspective on love and dating?" Dixie tilted her head. He sounded just like Caine. Damn him. Her heart rate slowed down appropriately. "You sound just like everyone else." "Who's everyone else?" "No one specific. Just everyone else." LaDawn, the Naysayer. She'd mocked Dixie's change of strategy, even after seeing the jump in her numbers. "You have someone specific in mind. I can tell by the sound of your voice, Mistress Taboo. You don't have to tell me his name. Give him a fake name just like yours, if you want. We do need a point of reference," he coaxed sweetly. She stuck Mike number three's note in the "Mike pile" where four more "Mikes" just like him who'd called in the last week sat, and asked, "How do you know it's a him?" "Isn't it always about a man?" At least one man, yes. "How's Golden Boy?" she blurted out. His laughter, so inviting and gravelly, made Dixie laugh, too. "And what's Golden Boy got against you counselin' men on their sex lives?" Walker had begun to tread into personal territory, making her drop her notes on Heath, who used a fictionally tragic character's name as his phone-sex caller pseudonym. "It's kind of a long story," she said carefully. "Suffice it to say, I'm not very good at phone sex, so I found a way around it in order to put food on my plate." His voice held approval. "So you're scrappy. I like that in a woman." That Walker liked the part of her personality everyone else detested made her stomach tingle. Right or wrong, his words of approval left her with a forbidden glow. Maybe it was the hunger for support, or maybe...maybe it just was. "Or something like that, yes. I'm doing what I have to do to become fiscally stable again." "That's pretty scrappy, in my opinion." "You say scrappy, others say greedy," she let slip. "Well, now who'd call Mistress Taboo greedy?" he asked, incredulous. "Do you have an extra ten sets of hands to count with? I'm sure you'll need at least that many." "Why's it greedy to do what you have to do to get work? In this day and age, it should be considered an asset." Walker sounded genuinely interested, and at this point, there was nothing she'd like more than to have someone on her side. Though she reminded herself, this was about getting herself out of debt, not collecting minions to cheer her name in Dixie vs. Caine. "Another long story, but we're not supposed to talk about me. We're supposed to talk about you. So let's do that. Do you have a problem with a woman?" Was it crazy to hope Walker's problems didn't involve another woman? Like his wife? Or, heaven forbid, his mother? She squelched that thought. "I'd rather talk about you." His answer was definitive in all its silkiness. Dixie let out the breath she didn't realize she was holding. "You don't want to pay four ninety-nine a minute to talk about me, do you? I'm supposed to be helping you solve your female problems." "I don't have any female problems. We broke up a long time ago. Female problems solved." Her ears perked to the tune of Walker's apparently womanless life. Bad ears. "Anyone you've been interested in since then?" "No one quite like her." Something in Walker's voice made Dixie push the envelope a little further. "Was that a no one like her as in 'she was the love of my life and all things incredible,' or was that a 'no one like her, she was the worst thing to happen to me since crabs?'" Walker's pause was painfully pensive, his answer, carefully constructed. "It was maybe a little of both. Wait, that's not fair. She was more good than bad, now that I reflect. Frustrating as hell, kept me on my toes, but I loved the hell out of her anyway." "Was she the love of your life type or just the fondest memory you have of love and it's now become bigger than it really was?" She asked the question only because she'd asked herself that question over and over about Caine. Walker coughed, but his eventual reply was throaty. "She was, and remains to this day, the love of my life. She was the best and the worst thing to ever happen to me." Dixie held in a wistful sigh. "I understand that completely." Caine had been the best and worst thing that had ever happened to her. And she still loved him anyway. "So you had someone like that in your past, too, Mistress Taboo?" "I did." I still do. "What happened?" "We broke up, of course." "Why?" "It was my fault," she was quick to reply. "How could anyone as cute as you are...uh, as cute as you sound be at fault?" Ugh. The constant explanations of her dirty deeds was like living Groundhog Day. "I just did something that made him change his idea of who I was—or who he thought I'd become." "You didn't cheat, did you? I'd have to take back my scrappy compliment, if you did," he chastised, though his tone was still light and teasing. She resumed organizing her sticky notes for her callers. "No. It was nothing like that. Though, I'm not proud to say, I once cheated, too... He was a high school boyfriend, and I was young, not that that should excuse it. Anyway, I didn't cheat on him. I would never..." Dixie sighed into the phone. Trying to inject her personal experiences into her brand of "therapy" without exposing herself was proving more and more difficult. For all she knew, Walker was a serial killer who got his kicks off of getting to know his victims in unusual ways before hunting them down, throwing them in a pit of dirt and flaying them alive while he screamed the word infidel. "And then what happened?" "I did something really stupid and petty, and then I behaved abominably, which led him to believe I'd always behave abominably." Walker scoffed. "He sounds like a judgmental ass." Dixie's feathers instantly became ruffled on Caine's behalf. "It wasn't without reason," she defended. Wait. Why was she defending Caine? For all the flak he gave her, she should be willing to throw him to the wolves. Yet, ultimately, no matter how he'd handled their breakup, he'd been right. "Which begs the question, had you behaved badly before?" Her eyes went wide. "Why would you ask that?" "Well, you defended him like you were defendin' your mother, and you were quick to do it, too. I figure, with my superior detecting skills, if he wasn't a bad guy, or a judgmental jackass, you must've given him reason to believe you might slip up again." Score one for Walker. "Fair enough. Yes. I'd behaved badly before." "How bad is bad?" "Hah! I was probably a nominee for the worst human being on the planet. In fact, I'd wager I could have given some of those villains on soap operas a run for their money." "Like Victor Kiriakis, Days of our Lives bad?" Dixie grinned, leaning back in her chair and throwing her feet up on the desk to stretch her cramped calves. "I'd like to think more along the lines of Joan Collins from Dynasty bad. She was definitely more glam than Victor." Walker whistled. "Wow. That's bad." Dixie nodded her head, forgetting once more that Walker was the client. "Yeah. Superbad, in fact. But I learned the most valuable lesson of my life. Since then, I've been trying to mend the error of my ways." "What was the lesson? What turned Mistress Taboo into a fence mender?" Her fingers, flicking the tip of a ballpoint, froze. She gulped back her discomfort, massaging her chest, suddenly so tight she couldn't breathe. This was too deep. She'd never told a soul about Mason. Not even her Landon. So, she sidestepped his very personal question by brushing it off. "It was very A Christmas Carol. You know, bad girl reflects on her life and realizes she's all alone in the big bad world with no friends and a family who's all but given up on her ever being anything but a lying, manipulative, self-entitled brat? But without ghosts," she added with a chuckle. "I'm familiar with the story." "Then you know how it goes." "So you don't want to tell me what really happened?" What she really wanted was this introspection to end. "It's a little personal." "And here we hardly know each other, right?" She smiled her relief at his lighter tone, sinking back in her chair. "It takes at least four phone calls deep before I reveal all my secrets. But I warn you, you'll need a shower and clean underwear when I'm done." "There she is, the Mistress Taboo I know." His voice showered her with warm approval. "It's been real nice chattin' with you tonight." "Is our conversation over already?" Again, she found herself wanting to ask when or if he'd call again. Walker sighed, the phone close to his lips. "I think it is for tonight, Mistress Taboo. A man's gotta work to pay those phone-sex operator bills." Her giggle echoed in the dark bedroom. "Well, all right then, Walker. I hope we can talk again soon." "Me, too, Mistress Taboo. Me, too." Walker hung up then, leaving Dixie to sink lower in her chair, resting her head against the back of it. In the deep velvet of 3:00 a.m., she gazed out the window into the inky early morning in the direction of Caine's window at the big house and wondered what he'd say if he knew what had happened to her in Chicago—what she couldn't bring herself to tell Walker. Or anyone. With a yawn, she prepared to leave, stretching before gathering her purse, and that's when she saw the shiny gleam of an object at the end of her desk, buried under her sticky notes. Dixie scooped them up, dumping them in a completely incorrect pile in an effort to clear the mounting debris. Her heart skipped a beat when she identified the shiny object. A brand-new iPod touch. As she was marveling over Caine's decency for replacing the iPod he'd drowned when he'd been so angry with her, her phone vibrated. She dug in her purse, pulling it out and sliding her finger over the screen. Darling Dixie—I'm sorry I haven't told you I love you lately. This being dead thing is a real bitch when it comes to communication. But who loves you more than his spleen even from all the way up here? Me. J Dixie couldn't help but laugh, even as the yearning to talk with Landon one last time, to hear him say those words rather than just read them, stole her breath. She held the phone close to her chest and smiled at his picture on her desk. "I miss you, you intrusive pain in my derriere," she whispered. So, so much. Thirteen "You do know I'd rather be dipped in batter and fried than be here right now, don't you, Em?" She spread her arms wide to indicate the square in the middle of town where so many familiar faces milled about, exchanging handshakes and idle chitchat. "You do know that there are people just linin' up at the mere thought, don't you, Dixie?" Dixie rolled her eyes at her. "Of course, I do. That's why I don't want to be here. Fried isn't a color I wear well." Em burst out laughing, her mood lighter than Dixie had seen it in almost two weeks, and she was working hard to keep from spoiling it with her own sour disposition. "Wasn't it you who said to face your enemies?" "Whoever said anyone should take advice from me?" "Apparently the men who call Mistress Taboo seem to think they should. Did you see the end of the week report for your numbers? They were incredible, Ms. Davis! Your calls are up by thirty-eight percent. Of course, I say that with absolutely no bias on your behalf—being the impartial mediator that I am. But the way this is going, I think that ornery LaDawn's gonna have to take lessons from you soon." "Or kill me in my sleep." Yes, sir. Her numbers for calls logged were way up. So far up, she was working sometimes an extra two hours on her six-hour shifts, and her mountain of sticky notes was going to need a desk of its own soon. It was exhausting and exhilarating. More importantly, it was working, and in only two weeks' time since she'd instituted her new plan. "Stop that now. Don't be so melodramatic. She was just teasing you." Dixie muffled a snort with her hand. "Have you ever known LaDawn to just joke? Wasn't it her who said she'd run off all the other 'companionators' on her old block in Atlanta by threatening to pick them off one by one with a sawed-off shotgun her granddaddy left her in his will?" Em's eyes twinkled sapphire blue under the strands of lights hung to perfection from tree to tree in the square. "You know, she's a little like you minus the violent tendencies, don't you think? All manipulatin' a situation to her advantage." "She's a little like a serial killer." Em licked her finger and swiped at the corner of Dixie's mouth. "You have lipstick on your face. Oh! Speaking of, look who's over in the corner, chatting up Ray Johnson." Dixie kept her fear on the inside, and stared straight ahead. "LaDawn?" Em clapped her hands excitedly as if it was Christmas morning. "Yep! And Marybell, too. I'm so glad they agreed to come." "Please say you reminded them this wasn't a place to drum up new business? I'm all for advertising, but I think if some of the women in town got wind of LaDawn's magic flogger, they'd hog-tie her and leave her on the train tracks." "Of course I did, Dixie. You don't think I'd let them go into something like this without tactfully suggesting they leave their sexy apparatus at home, do you?" Dixie shook off the bad feeling she'd had all night since, at Em's insistence, agreeing to attend the annual Plum Orchard Autumn gathering and potluck. Em didn't know the Mags the way she did. She didn't know what they were truly capable of when they wanted to hurt someone. Her giving nature, the generosity she granted everyone, would be her downfall. Inviting the Call Girls was a kind, Em-like gesture. Her hope to integrate the operators she'd come to refer to as close acquaintances into Plum Orchard was naive at best, disastrous at worst. Em's blind eyes and deaf ears clearly only heard the gossip everyone spread about Dixie. The Call Girls operators had been the subject of more than one overheard conversation, and the conversations were never about welcoming them into the fold with open arms and a basket of freshly picked peaches. They were more along the lines of scripture, damnation, and "how-to" videos on performing exorcisms. "Are you having those misgivings again?" Em asked. "I'm having many things." "Don't be so cynical, Dixie. Louella said it was a wonderful idea to invite the girls. And this is a perfect opportunity for them to get to know everyone. It has to be hard to get out and meet folk when you work the graveyard shift." "Harder still to get out and meet people when you sell sex over the phone in the graveyard," Dixie quipped. "I know the town isn't in love with the idea of Call Girls being here, Dixie, especially the Senior Mags. But Landon did everything right. Trust that. He got all the permits he needed from the county and the state. He knew his stuff, and he made sure Hank Cotton did, too. So they all might as well get used to it and make nice, because there isn't much anyone can do about it." Except make the women of Call Girls feel so uncomfortable and out of place, they'd give a Mag their personal gas money to run them out of town. Em's hopeful face, brighter than it had been since Dixie returned, was fading, pulling her up short. She rubbed Em's arm. "You're right. I'm sorry. Ignore my sour mood. I'm just tired and cranky from all those calls I've been takin.' I'm sure everything will be fine." Em nodded her agreement with a happy smile. "I know it will. You'll see." Dixie didn't want to blast Em's Mr. Rogers moment on the off chance Louella and the Mags would mind their manners. Yet the possibility existed they were merely laying the groundwork to humiliate the Call Girls for having the nerve to think a phone-sex operation was acceptable in Plum Orchard. Dixie tugged on the bow at Em's waist. "But hold that thought, and just walk to the other side of this equation with me, okay? Wasn't Louella also the one who thought it was a wonderful idea to jump into bed with Caine two point two seconds after we broke up?" Em wrinkled her nose. "Still don't believe it, and two very different things, Dixie. Now, before you ruin a perfectly good reason to wear that beautiful black dress, let's go have some punch and enjoy a nice night out." "Punch. That sounds good. Let's have lots of punch with lots of alcohol." Dixie followed behind Em, pushing their way through the crowded grass of the square. She admired the Kelly-green dress Emmaline chose for tonight; soft and flirty, it covered only one of her shoulders, and brought out every good feature she possessed. Judging by the appreciative glances of some of Plum Orchard's most decidedly un-bachelors, they admired Em's dress, and what was filling it, too. If only Em knew how beautiful she really was, inside and out, she wouldn't be single long, Dixie mused. Pulling her up the stairs of the whitewashed gazebo, Dixie momentarily flashed back to the night of her engagement party. The air had smelled the same then—the end-of-summer heat easing to welcome in the crisp scent of falling leaves and fireplaces burning fat pine logs. Then she sucked in a breath of air to fend off the horrible images that always followed that happy memory. The fire she'd started by knocking over the candles that matched her wedding colors while she'd raged. The screaming. Essie Guthrie's updo singed to a crispy frizz. The rain that had made a tsunami look like a mere shower. The buffet table, lined with assorted dishes brought by everyone in town was the same, too. The only difference was all the food was on the table, not splattered all over the floor after being trampled by three hundred people all vying to get out of the rain and under the safety of the gazebo. Glasses of plum wine, Herbert Fox's specialty, sat on a far table in the corner, glistening with the deep purple liquid. Dixie grabbed a much-needed glass for both her and Em while scanning the familiar faces of the crowd and the people she'd grown up with. Some were older; some were in the next stages of their lives. All of them, whether they felt the same way about her, were welcome sights to Dixie's homesick eyes. She stood in the back of the gazebo against the rail, absorbing her surroundings, inhaling the smell of Essie's homemade corn bread and Jeter Orwell's mother's fruit salad surprise. Dixie sipped her wine, wistfully watching couples gather in the center of the square where a dance floor had been laid. The first notes of one of her favorite songs, "We Danced," by Brad Paisley, filled the air, a slow intimate song about finding everlasting love on a barroom floor. She swayed with it, closing her eyes with a smile. A hundred years ago, she and Caine had danced to this tune at an old bar they'd frequented during their engagement. The memory transported her right back to the dusty hardwood floor littered with peanut shells and hay and the odor of stale beer and sweat. Caine's hard chest beneath his old college T-shirt, pressed to her cheek, his chin resting on the top of her head. The smell of his aftershave tingled her nose, his cowboy boots occasionally scraping her ballet slippers. The heat of their bodies flush with one another's, their hands clasped together at their sides, oblivious to everyone else as they swayed. "I see you decided to venture out into the pack of hungry she-wolves tonight? What gives, Dixie-Cup. You lonely for a public lickin'?" Caine asked from behind her. And while his tone was light, she wasn't up for his kind of lickin', one he'd surely lavish happily. Her eyes popped open when he positioned himself behind her, placing his hands on her shoulders. She stiffened, then shrugged him off. It took every ounce of her will not to fall back against his hard chest in search of shelter. No more touching. No more kisses that set her soul on fire. "I'm just enjoyin' a nice evening in a hometown I've missed more than I thought I did. You do the same, Caine. Oh, and thank you for replacing my iPod," she replied, attempting to skedaddle away from him without looking him in the eye. Caine caught her arm, pulling her back to his side. "Wait, Dixie. Please." The harsh grit in his tone left her helpless to deny him. That and his muscled goodness dressed in an ice-blue fitted shirt, blue jeans, and the very same cowboy boots he'd worn when they'd danced at Junior's—all so acutely male, it physically hurt. Her legs trembled while she silently damned him. She'd never been so aware of any other man as she was Caine, and it infuriated her. Lashing out at him was the only thing keeping her from begging him to take her back. "Is there something you'd like to point out to me that you haven't yet? Have we moved on from my character flaws list already? That was fast." He took the glasses of wine from her hand and set them on the table. Cupping her chin, he ran his thumb over her lower lip, one that trembled in response. "Stop, and just listen to me, okay?" Caine's eyes reflected something she'd never seen, rooting her to the spot. Sure, she was used to anger, amusement at her expense, lust even, but this—this softer, almost gentle gaze? That she didn't understand—or maybe it was just that she hadn't seen it in a long time. He hitched his jaw to the big maple where a park bench sat beneath its sprawling limbs and thick leaves. "C'mon. Let's go sit." Sit? To ensure she was an easy hit? Uh-uh. This was just another trick to get her back for finagling her way out of phone sex. Since the Call Girls' weekly meeting, he'd had plenty of time to plot her payback—or at the very least, work up a good simmer. "What's this about, Caine? Are you angry that my numbers beat your numbers this week?" She looked at her wristwatch, tapping the face of it with a fingernail. "Or is it time to call me a cheat again? Can't you just tell me what horrible offence I've perpetrated right here? You know, in front of everyone? You're so good at that. You want a microphone, too? Let's ask Louella if we can borrow hers." When Caine smiled, it wasn't with malice or arrogance, it was easy and kind. And it was freaking her out by the second. "No tricks. Swear it on my old stack of Playboys." The corner of her lips almost lifted. "You had Playboys?" "Stacks of 'em." "And everyone calls you the golden boy? Why is it that if Dixie Davis is making the nasty on the phone she's the town pariah, but if Caine Donovan does the same, he's excused from the fire-and-brimstone speech because he wears a cape and saves drowning puppies?" Caine lifted his wide shoulders and grinned. "Stop exaggerating. I only saved a puppy. No plurals in there." Yes, yes, yes. He really had saved Aida-Lynne Gorman's puppy that'd wandered off from her backyard by jumping into the overflowing creek and swimming after it. Dixie rolled her eyes at him. "And on the day that you were born the angels got together, and decided to create a dream come true. I've heard it all—in a song, I think. Maybe I ought to let Kitty Palmer hear you invite a woman to discover how you Pirate her Caribbean in Johnny Depp's voice? I hear you lay it on thick almost every night from my thin office walls. You get zero points for originality, mister, after you've used it three hundred times. And what is it with the Johnny Depp fixation? Does every woman want to talk to Depp?" "Depp's pretty popular," he said in Depp's voice, following that with a crooked, endearing grin... Damn him. "That's neither here nor there. The point is..." She frowned. "I forget what the point was." Caine's laughter, so genuine she was tempted to laugh with him, made her heart skip. "So come with me?" Caution being the better part of valor, Dixie cast suspicious eyes on him. "Okay, but one cross word and I'll find a can of gasoline in Landon's barn and set your pile of dirty women with big boobs in Technicolor on fire. It'll be a blaze like Plum Orchard's never seen. Trust that." Pulling her behind him, Caine took her down the steps, crisscrossing their way through the crowd and across the square. For a moment, her hand in his, Caine leading them off somewhere, brought back a time when she would have followed him into the bowels of Hell if he'd just promise to be hers forever. "Sit." He motioned to the bench directly across the street from Madge's. Dixie looked around with suspicious eyes then swiped the bench with her hand and held it up under the lamppost's light. "If you ruin this black dress, Caine, I'll steal your credit card and order a hundred more. It's the only one I have left from my old wardrobe in Chicago, and even if it's a little tight after all the food Sanjeev lavishes on me, it's still designer." "I dunno, it doesn't look too tight to me. You look really pretty tonight, Dixie." Heat rose in her cheeks. "Take that back." "I won't because you do look pretty. Very pretty." Her hands went to her hips. "Okay. I know you've got something up your sleeve, what is it?" He shoved the sleeves of his shirt upward over his arms. "No tricks up my sleeve." Stubbornly, Dixie refused to sit. "I don't trust you. I'll stand." "And that's the root of our problem." "We don't have problems, Caine. We have catastrophes—tsunami-like issues of devastation." "And we can't have a relationship like that." A relationship? Her throat went dry, making her wish she'd brought her wine with her. Did he want to...No. He wouldn't be so cruel as to make her believe he wanted her back so he could... No. "What relationship are we talking about? We don't have one unless antagonism is the basis for a relationship," she croaked. "But we do, Dixie-Cup. We have a working one." Hopes dashed appropriately, she hid her disappointment and gave him a questioning glance. "We never see each other during our shifts. You're in one room, and I'm in another. Strategically planned, I'm sure, by Landon, in an effort to keep us from lopping each other's heads off with industrial staple guns." "We don't have a relationship because we do just that—avoid the hell out of each other. Every time you hear me leave to grab a water or some coffee, if you're in the hall, you run the other way." "I do not run." To imply she was a coward was ungentlemanly. "Okay, you walk swiftly." "I prefer to call it speed walking. It's good for your heart." "Call it whatever you like, it's still avoidance." "And that's been working, hasn't it?" "No. Yes." His lips thinned. "No. I don't want to avoid you, Dixie. I don't want LaDawn and Cat and the others to cordon off portions of the break room by groups so they don't have to worry they'll witness a murder on pizza night. Everyone avoids us if we're even within two feet of each other. Sure, we're civil, just as Cat threatened we'd better be, but the tension is still there. I don't want everyone to act like one wrong thing said could set us off, do you?" Caine was right. Last week on pizza night, Sheree had seen the two of them each grabbing a slice of pizza from the same box just as she'd reached for one herself, but she'd jumped back when Dixie had tried to pull it from his grasp and teased Caine he was taking the slice with the most cheese on it. She'd only been joking, but it had stilled all movement. She saw his point. The last thing she wanted to do was alienate the only people who willingly spoke to her, even if it was through clenched teeth à la LaDawn. "We're dividing them. Making them choose between us." "You bet we are. We're killing their comfort with just our bad vibes. We've got some time left to this competition, Dixie, but eventually, one of us will go home. For the girls, this is their home now. I don't want to shit all over that just because we acted like asses. Or more importantly, I acted like an ass." Perplexed, she murmured, "Okay. Will the real Caine Donovan please stand up?" "Why doesn't the real Dixie Davis come sit down?" He patted the place beside him with a convincing, heartbreakingly adorable smile. It made her want to bracket his jaw with her hands, press her lips to his until he swept her up in his arms like he used to. Instead, Dixie fought the impulse and took a cautious seat, finding the very edge of the park bench and sitting on it as if it was nothing more than a sliver of wood. "So what are you proposing here? Do you want to have more group hugs? Maybe prayer circles? What will make the girls feel less like they have to choose between us?" "All we have to do is be friends again, Dixie. And that means I have to lay off razzing you every chance I get." Dixie's lips pursed and her eyes narrowed until Caine was nothing but a delicious hunk of a blurry image. "But it's how you're able to breathe. It's your life force. You can't do that any more than I can stop buying lip gloss and conditioner." Brushing her hair from her face with a tender finger, Caine said, "But I can, and I will if you will. I've been pretty hard on you, Dixie. I'm not saying it isn't without cause, but that reason's grown older and apparently outdated. Almost ten years older. It's time to let go of it and behave like adults for the sake of those around us. It's time. I keep hearing about how you've changed from Emmaline, and it occurred to me, I haven't even bothered to attempt to respect those changes by giving them a chance, let alone paid attention to them. We're always too caught up in knocking each other's knees out from under one another." Dixie's jaw dropped open. "Something's very wrong here. We've always done this, Caine. It's just what we do." Or it's what I do so I don't have to really face what I've done. Caine was the person she'd hurt the most, yet he was the one person she was most afraid to make amends with. "No, Dixie," he said, his voice that odd gruff again. "We didn't always do this, but maybe it's time we change what we do. I will if you will." She went for one last poke at him, to test his commitment to this cease-fire. "And if I don't?" "I'm guessin' my stack of classic Playboys are set for a bonfire, and your makeup's going to find the trash compactor." The inability to breathe kept her silent for a few moments while the music drifted to her ears and the conversations of the surrounding people floated over her head. Now, Dixie. Now would be the perfect time to tell Caine you're sorry.... "O...kay. It's a deal." "Besides, with you marrying Walker—" Dixie blanched. Damn her and her need to win everything. "Committing to him." He flashed her another genuine smile. "Right. Committing. I'd think you'd want to let what's happened go and head into your marriage with an open heart, free of old hurts." Right. That's what she'd want if Walker were real, and she wasn't still madly in love with Caine who clearly was managing just fine without her. Oh, Dixie, what have you done? She plastered a fake smile on her face. "Right. Free of old hurts is cleansing. So let's try it. But I promise you—one crappy word, one prank, trick, whatever you want to call it—you go down, Donovan." He nudged her with his elbow and winked. "Same goes for you, Davis. So, friends? Maybe we'll grab dinner together, or whatever you call what we eat in the break room at three in the morning." Sure. Dinner. How did you go about being just friends with the one man you'd loved all your life? Yet the idea of a kinder Caine, one who wasn't calling her a liar and a cheat, spread a healing balm over her bruised heart. "Okay. Deal." He stuck his hand out, and Dixie took it without hesitation, letting the warmth of it envelop hers, wishing she could cling to it, pull it to her cheek so he'd draw it back and kiss each of her knuckles the way he'd once done. Caine surprised her when he pulled her to his side, pressing a gentle kiss to her cheek with warm lips. Her eyes closed of their own accord as she savored this moment—a moment that felt like goodbye. Caine let her go and slapped his hands against his delectable thighs, rising with a tilt of his head in the crowd's direction. "Good. I'm glad. I'm gonna go call up Jo-Lynne, and tease her about missing Kitty's potato salad." Dixie's head bobbed its agreement with a slow nod even as her throat clogged. "Checkin' on your mama is important. Say hello to her for me, would you, and give her my best?" Oh, and also tell her I'm sorry. So I can avoid chickening out with her just like I did with you. "You bet," he murmured, looking fifty pounds lighter now after agreeing to their truce. As Dixie watched him push his way through the throng of people, his tall frame navigating it with ease, hot tears stung her eyes. She had to lean forward and brace her elbows on her knees to keep a constant flow of air to her lungs. Suddenly her dress was even tighter, her push-up bra constricting. Who were they as friends? What would it be like when Caine no longer sneered at her, but instead, greeted her like just another fellow employee? Who was she if she wasn't cultivating some sarcastically flirty response to his taunts? It was as though a piece of her had just died. She didn't know how to breathe if Caine wasn't pushing one of her buttons. At least when they were at each other's throats it was an excuse to interact. Now she was relegated to the friend pile, and somehow, that didn't seem nearly as intimate as being his nemesis. So this was closure. Myriad emotions, now all tied up into one neat package with a pretty bow. Closure well and truly sucked. And hurt. Really, really hurt. The tap of someone's hand on a microphone startled Dixie from her pity party. She sat up just as Louella's voice touched her ears. "Fine people of Plum Orchard, I'd like to thank y'all for joinin' us tonight. Lots of announcements to be made here, folks, so if you'll just bear with me, we'll get right back to dancing in just a few minutes...." Dixie tuned her out, kicking off her heels and scooping them up to let them dangle from her fingers. She scrunched her toes up in the cool grass, unsure where to go next. She had two hours until her shift began. Yet the last thing she wanted to do was rejoin the festivities. Watching Caine give a test run to their new friendship was not on her list of most desirable things to do. With a grunt, she rose, and with a lump in her chest, began to pick her way through the crowd to head back to the big house. "And let's all put our hands together for some new additions to Plum Orchard!" Louella breathed into the mic. "First up, Miss LaDawn Jenkins. A former prostitute turned phone-sex operator, currently running amok with our young children, y'all, helping to shape their minds one foul word at a time." Dixie's head shot up to the sound of all of Plum Orchard sputtering and gasping in horror. Her shoes fell to the ground. She knew it. Knew it as sure as she'd known Em was a fool to believe the Mags were anything but heartless bitches. Her stomach turned, sour bile flooded her throat. Breaking into a run, she sprinted across the square, pushing people out of the way like some designer-clad linebacker. The freshly trimmed clippings of the boxwood bushes surrounding the gazebo cut into her bare feet with stinging slashes. She crashed into Howard Fordham's back, yelping an apology when she knocked his baseball cap off, making a break for the stairs leading up to the gazebo where, from a distance, she saw Annabelle and Lesta-Sue waiting, modern-day Southern sentries, guarding their posts. "Heads up, ladies!" she yelled in warning, familiar faces whizzing by her bobbing head in a blur, her pulse throbbing in her ears. She had one task—get that microphone away from Louella Palmer and wrap the cord of it around her neck until she strangled her to death with it. She no longer heard Louella's voice. Instead, it had become like the voices of the adults in a Charlie Brown cartoon. She didn't hear the outrage of the disgruntled crowd or the angry outbursts of protests as she knocked people out of her way, tripping and stumbling. When Dixie became aware Annabelle and Lesta-Sue were in it for the long haul, staunchly standing their ground, she decided it was all or nothing. If the Mags were looking for a good rumble, they'd chosen the worthiest opponent. With a warrior cry, Dixie barreled toward them full speed, knocking them onto the steps with a grunt at the force of their bodies slamming against the wood. Dixie fell on top of Annabelle and wiggled her way up, using Annabelle's long torso for leverage. She didn't think about the fact that her damnably tight dress had risen almost to her hips by the time she reached the final step. She didn't hear the scream Annabelle let go of when she discovered her nose was bleeding. She heard nothing but the roar of her anger, screeching through her veins and throbbing in her ears, the huff of her lungs trying to produce enough air to keep them inflated long enough to get her hands on Louella. Scrambling to the floor of the gazebo, Dixie stumbled to her feet and directed her wild gaze at Louella who still had the microphone in her hand—whose mouth was still moving—whose venom was still pouring from her lips. That was when everything inside of her, every tightly wound ounce of restraint, uncoiled like the head of an uncontained garden hose. Dixie reared up, her eye of the tiger the microphone Louella held so smugly in her hand. Dixie lunged at her with a rebel cry, knocking Louella and her pretty gold and chocolate-brown fall ensemble to the ground, making the crowd below them gasp again. Louella, dazed on the floor of the gazebo, gave Dixie the opening to straddle her and snatch at the microphone. The screech of feedback ripped through the air, the muffled sounds of Dixie's hands clawing at Louella's to wrest it from her mingled with their grunts and eventually, Louella's piercing scream when Dixie latched onto her hair and yanked hard. She tore the microphone from Louella, pushing off her opponent's body and struggling to her feet. With a howl of rage, she threw it out into the dark, using every last ounce of energy she had left. A magnified thunk signaled it had landed on something hard and unyielding. Dixie mentally brushed her hands together while gasping for the air wheezing from her lungs in harsh puffs. She bent at the waist, resting the heels of her hands on her knees, heedless to the fact that her underwear was the subject of much chatter. Heedless to the fact that this also left her weak and vulnerable. In hindsight, as her feet flew out from under her, Dixie realized, she'd forgotten the golden rule of a girl-fight. Never leave your opponent conscious. Fourteen "Ow!" Dixie moaned with a flinch. "This from a woman who took out not one, but three Mags in the space of two minutes?" Caine quipped. "Hike up your warrior panties and hold still. I can't see if you need stitches if you keep pulling away, honey." He tried to stay calm while looking at Dixie's battered face, running his fingers over it to reassure himself nothing was broken. He tried to remember his mother's words about always being a gentleman, even as he considered driving over to Louella's and beating down her door so they could end this all now. If Louella was doing this because of the rumor his mother divulged tonight on the phone when he'd told her about the argument Dixie had with the Mags at Madge's—that Louella had once been in love with him and he'd chosen Dixie without knowing Louella was even interested, he had a record to set straight. He'd never even been a little interested in Louella, and he never had a single clue she'd been interested in him. It had always been Dixie. Shit. Dixie flapped her hands at him, pushing him away as if he was making something out of nothing. "I don't need stitches. I need aspirin and maybe a bag of peas. A steak, too, if you could manage it, and I don't mean some cheap steak for my eye. The bloodier the better." Caine pushed her hands out of the way, running his fingers over the planes of her face to check for breaks, wincing when he got a close look at her swollen eye, bruised to a deep purple. "You need boxing gloves and a cage." He winced. Jesus, she had some shiner. But what was that tingle in his chest? Pride. It damn well was. It was probably wrong, but seeing her tonight, flying up those steps to shut Louella up in defense of the Call Girls, had been a thing of beauty. This side of Dixie, the side he was seeing more and more of lately wasn't so different than the old Dixie. Her goal tonight was the difference. "Well, if this phone-sex thing doesn't work out, maybe I'll see if Craigslist has cages." LaDawn and Marybell rushed into her room then, skidding to a halt in front of the bed he had insisted she lie down on. "Are you okay, Dixie?" Marybell reached for her, shoving Caine out of the picture and tilting her chin up to pop one of her eyes open with two fingers. She gazed down into Dixie's blackened eye until she began to squirm. Dixie jerked her head away, swiping at her eye with her thumb. "I'm fine. Stop fussing, Marybell. It's just a black eye." "Just a black eye? Look at you, talkin' like the boxing ring is your home base," LaDawn crooned, her lips forming a smile. "And on behalf of lil ol' me, too. I do declare, I'm all kinds of flattered," she drawled, playfully punching Dixie's arm. "Hah!" Dixie snorted. "Don't kid yourself. I was doing it to keep you in the game, LaDawn. If your numbers don't make me aspire to do better, then whose will?" "Good thing you got there before I did," LaDawn assured her, arms crossed at her chest. "I was gonna show that stuck-up, frilly, tight-ass what a whoopin' feels like. I put on the nicest dress I have for this party, too. One that keeps my best companionator assets from fallin' all over the men of this backwoods town." She hiked up the front of her overflowing maxi dress to rearrange her assets. Dixie rubbed her temple, trying to sit up, but Caine settled her back in. "Next time, you do the whoopin'. I'll just eat and watch the chaos from the safety zone, okay?" "No. No, sir. There's not gonna be a next time for me." LaDawn shook her mane of platinum-blond hair, still tangled up in her big hoop earrings, as she dipped a washcloth in more water, wiping at the scratches on Dixie's legs with gentle swipes. "No way I'm goin' somewhere where nobody'll just let my past be. To think, they were all laughin' and pointin' at me behind my back, but smilin' right in my face and offerin' me home-cooked food on top of it." Dixie grabbed LaDawn's hand, wincing when she looked up at her. "You will so go back. I'll take you back, tied and gagged if I have to—and you'll like it. You live in Plum Orchard now, LaDawn. Marybell, too. This is our town, not just Louella Palmer's. As long as I'm here, no one's going to treat you like anything else but the taxpayers you are. I'll make sure of it." She groaned, leaning back on the bed when Caine swung her feet upward, repositioning her. LaDawn shook her head. "It's not your fault, Dixie. Landon told me it might be like this, but I took the chance anyway because I wanted off the streets for good—and he sure can talk good game with all the fancy stuff he offered. I think Landon coulda talked Jesus into the second coming if he could have gotten face time with him." "Tell me about it," Dixie muttered. "But I'll tell you this—I didn't think there was anything I couldn't take on. I was a companionator, for Jesus sake. Those of us who companionate, we have our share of haters, mind you, but they ain't like the people in this town. This small-town bullshit is fierce. At least the haters back home hate ya right up front. These people fed me baked beans before they attacked, Dixie. That's just not right." "I'm so sorry, LaDawn," Dixie said, while Caine pretended not to listen to the real sincerity in her voice. It wasn't like the gooey crap she'd laid on with a trowel years ago. It was raw. And real. LaDawn's round face fell. She was one tough cookie, if all the stories Caine had heard from the other girls were true, but she appeared genuinely hurt that she'd been tricked into believing the Mags would allow the women to blend in with everyone else. LaDawn shrugged her shoulders. "Just seems like they don't want anybody to have a chance to make good without bringing up your misdeeds every chance they get. I might not be sellin' dictionaries on the phone, but I ain't spreadin' diseases and stealin' women's husbands anymore either. That should count for something, right?" Marybell shook her head, her Mohawk flopping forward and back. "You were amazing, Dixie. A-maz-ing. All ragin' and yankin' that microphone from Louella's hand like you were some kind of thug. I know I was next on her hit list, so thanks, Dixie. I'm not as tough as LaDawn. I don't think I could have listened to her say those things about me." "Looks like you won't have to worry about it, for a little while, at least," Em chimed in, crossing the floor in bare feet, her stained dress brushing her bloodied knees. "Seems Annabelle's on bed rest—she has a concussion. And Louella needs to pay a visit to Johnsonville to see a specialist to get her broken nose set." But Dixie didn't look as though she was taking pleasure in any of it. She looked worried. "Is Lesta-Sue okay? I landed on her pretty hard, Em." Caine cocked his head. Now she was inquiring about the opponent's health? Em nodded. "I saw, and why should you care? Those women were out to turn somethin' perfectly lovely into a circus. They don't deserve your pity, Dixie!" "Everyone deserves a second chance, Em," Dixie said groggily, making Caine worry she might have a concussion, too. Dixie preaching about second chances was like LaDawn preaching abstinence. Em's lips thinned to a fine line of red. "I should have listened to you. After all, you were once head of the Mags. Butter wouldn't melt in their mouths while they assured me it was a wonderful idea to invite the ladies. Oh! I could just march right over to Louella's and give her a piece of what-for." Dixie shook her head. "We need to let everyone simmer down now, Em. Don't go pokin' the beehive." "I want to poke her eyes out!" That Dixie wasn't plotting Louella's ugly revenge right now was blowing his damn mind. Maybe she was plotting on the inside. But something about that notion just didn't sit right with him. Dixie's face scrunched up. "Hush. Take that back. It won't solve anything," she scolded, accepting a cold pack Sanjeev handed her before he escaped on silent feet. But Em wouldn't be pacified. She paced, rubbing her bruised arm. "I say we strike back, Dixie—strike now while the iron is hot!" "Listen to me, Darth Vader, no striking. That's what feeds this childish behavior. We're behaving no better than I did back in the day. I'll handle the Mags. You'll stay out of it—for the sake of your boys," Dixie reminded Em pointedly. What did Em's boys have to do with the Mags? Appropriately chided, Em crumbled, her eyes full of tears when she turned to LaDawn and Marybell. "I'm so sorry, girls. I truly believed that piranha Louella when she said it was a wonderful idea to invite you. If I'd known you were virtually walking into the lion's den, I'd never have insisted y'all come." Marybell gave Em a hug, squeezing her tight. "You were just trying to be a good person and make us feel welcome in a place where it feels like I don't even speak the language. Besides, the fried chicken was almost worth it. And those biscuits Ben Johnson's wife made—wow." She held two thumbs up. LaDawn clapped Em on the back. "I'm not much of a hugger, but it was nice you thought of us. Next time, though? Don't think of us. I'd rather just live out my days in Landon's backyard with Toe and the pool." Marybell pinched one of Dixie's toes. "We gotta go, Dixie. Our shift's comin' up. Cat said she'd drop by to see you tomorrow and check on you. She was off doin' her online classes and making goo-goo eyes at Flynn." Dixie reached for the nightstand, trying to lift her legs up and off the bed. "Wait for me. I'm coming with you. I have a shift tonight, too." Caine straightened and pressed a hand to the flat of her chest, trying hard not to think about how good it felt just to touch her skin again. He'd drawn the friend line between them tonight and here he was already thinking about how good it'd be to cross it. "No. No work tonight, Dixie. You need to rest, and I'm seriously thinking of calling in Dr. Johnson just to have him give you a once-over." "Why, so he can mess all the tests he's so fond of up and tell everyone I have the clap? Not on your life, Candy Caine," she teased him with a half grin. "Ray took over his father's practice several years ago. I think you're safe," Caine assured her, keeping his hand on her chest, his concern heavy. "You mean the Ray I was talkin' to while enjoyin' some baked beans was a doctor?" LaDawn mused in disbelief. "Shoot. Just figures. I think Ray liked me." "And on that note, we're off to work," Marybell said with a snicker, blowing Dixie a kiss before ushering LaDawn out of Dixie's room, their chatter drifting along the hall until he couldn't hear them anymore. Em motioned for Caine to move over. With gentle fingers, she traced the outline of the bruise surrounding Dixie's eye, her eyes welling with more tears. "I'm sorry, Dixie. I'm so sorry this happened. I should have listened to your misgivings." "Or ten," Dixie reminded. "I remember I said approximately ten." So Dixie had been suspicious the Mags would mess with the ladies. Damn pious lot. It was time he did some intervention. Landon never would've stood for this kind of harassment. He wasn't going to sit back and let those women shit all over Dixie and the girls. Em shook her head, her eyes on fire. "You're all banged up, Dixie. We can't just let this slide. Next thing Louella'll think it's okay to launch anthrax attacks. She's going to do whatever it takes to get you outta Plum Orchard. We can't let this stand. Besides, wasn't it you who always said the one with the most toys wins?" Dixie's voice sounded thin and so sad, it hurt his ears to hear it. "That was before I lost my toys. All of them. No, Em. No more return attacks. I didn't take up for you at Madge's because I enjoy the fight. Hard as that is for everyone to believe." Dixie gave Caine a pointed look. "I did it because I was afraid they'd try to attack your personal life. This isn't a war, Em. This is a petty, spiteful woman who hates the very ground I walk on." Em flapped her hands as she rose from the bed. "No matter now. For now you need rest. I'll do enough fumin' for the both of us." Dixie grabbed at Em's hand, her eyes soft. "Hey, before you go—thanks for having my back, Em. The bits of it I can recall as I lay on the gazebo floor, looked like you were scattering bodies and taking names." Em's smile was full of the devil. "I couldn't let Louella take advantage of your semiconscious state, could I? Especially after I created this fiasco." "You really slugged Louella. That's some right hook you have," Caine interjected with a wink of admiration, wrong or not, manners be damned. Em turned and gave Caine a prim smile over her shoulder. "I did not slug. I merely thwarted. Louella was gonna dump potato salad on a defenseless woman. I'd thwart again if the need arose. Now, I have boys who need a bedtime story." She leaned in to hug Dixie and whispered, "Rest, my friend." Em's hasty retreat left them alone on the bed with Mona and Lisa, and his questions. Staring at her bruised face, swollen and red, Caine couldn't stop looking at her. He couldn't stop looking for her ulterior motive. He couldn't stop thinking about how crazy it was that Em was the one who wanted to strike back and Dixie didn't. Dixie slid off the bed, her hands reaching out as she swayed. Caine caught her, tucking her to his side, his worry she was seriously hurt far outweighing the things swirling around in his head. "Honey—you need to let me take you to the hospital, for Christ's sake." "I have no insurance. No hospitals. I'm fine. Now you go do Candy Caine things. There must be a hundred calls backed up for Sam Elliott." "I'll pay for it, Dixie." His offer of money made her stand up straighter. "Friends don't let friends pay their medical bills. I'm going to take a shower. No more talking. It hurts my head." Caine stopped her, putting his hands at her waist. "No way I'm letting you take a shower in this condition all alone." Dixie shook her head with a grunt. "Friends don't let friends see them naked in the shower." "What do friends let friends do?" he asked in exasperation. Dixie gazed up at him, her eye puffier by the second, her cheek purple and yellow. She'd never looked more beautiful. "I'm unclear on our particular kind of friendship and its classifications. For now, I know for sure we're not the kind of friends who see each other naked. We're just friendly friends." "Well, this friend isn't letting you take a shower alone. Not after that crack to the head. Do you want to slip and fall? And friends do let friends see them naked. Girls do it all the time." He scooped her up in his arms, carrying her toward the bathroom. * * * She wanted to summon the will to make him put her down the entire trip into the bathroom. She really, really did. Mostly. The half of her that was determined to let him go wasn't nearly as strong as the half of her whose limbs felt as if they'd been softened like butter in a microwave. He set her on the vanity, sliding her backward until her spine was against the wall. "But you're not a girl, and we can always ask Sanjeev to help me." Dixie leaned against the mirror, afraid to look at her battered reflection while Caine filled a glass with water and dug around in the medicine cabinet. He handed her two aspirin with a grin. "Nope. I'm not a girl. I'm Golden Boy, remember? And Golden Boy isn't leaving this bathroom until you do. Sanjeev, at this hour, is probably knee deep in his nightly prayers. So you got me, babe," he joked in Sonny Bono's singing voice. "Now, lift your arms up," he ordered. Dixie did as he commanded, too tired to refuse. He slipped her dress over her head, and popped the clasp on her bra between her breasts. The cool rush of air against her hot skin made her hiss. Or was that Caine's touch making her hiss? "Dixie?" Her eyes popped open at the sound of his voice to find their noses a mere half inch apart. She smiled at his handsome face, his clear eyes fringed with dark lashes, the stubble on his jawline—him. Her hand went to his jaw in a familiar gesture, forgetting their friendship truce—forgetting her pact with herself to keep her hands off him. "What?" Caine ran a finger down her nose and smiled back before dropping his boots and socks on the vanity next to her. "Keep your eyes open. If you have a concussion, you can't sleep." Dixie frowned. She knew there was a reason for that rule—she just couldn't remember what it was. "Or?" "Or you might not wake up in time to try out for defensive tackle. You don't want to miss an opportunity to hone your skills with the big boys of Plum Orchard High, do you?" he teased, fingers sinking into her waist to lift her to her feet. Dixie's arms went around his neck to steady herself again, her nipples scraping the fabric of his shirt. The promise she'd made to herself to have no physical contact with Caine was well on its way to crumbling. She gave a halfhearted shove at his chest, longing to rest her head on it instead of push it away. "Let me take a shower. Go handle your shift. All the Connery groupies need you. Plus, you don't want to lose the opportunity to find some new clients, do you? If I take a personal day, your chances of beating me just improved. I'll be fine. Really." His arms tightened about her waist, the slide of his hands against the skin of her back making her shiver. "That's not going to happen. So if you're desperate for that shower—it's while I watch, or it isn't going to happen." She chuckled at the irony of his words. "Kinky." Caine hooked his thumbs inside her panties, sliding them down along her legs to her feet. She used her hands to brace herself on his shoulders, loving the feel of his fingers brushing her thighs then her calves, completely comfortable with shedding her clothing. He nudged her with the top of his head to lift her foot then pulled her panties off. Maybe it was only her imagination, but his breathing sounded hitched when he rose to his feet and slipped another arm around her waist to lift her off the vanity and walk her toward Landon's custom-made shower. She'd once jokingly referred to the extravagance of it as a shower fit for an orgy—party of ten, please—it was so spacious and decked out. Caine flipped the faucets on, adjusting the temperature before picking her back up and setting her in the water under the first set of showerheads, stepping in behind her, clothes and all. Dixie's shock was bigger than her protest. "You'll ruin your clothes," was all she could manage. "I'll send you the bill," he grumbled, leading her deeper into the enormous tiled space. He placed his hands under her forearms, guiding her to the long bench where he sat her down and set about adjusting the remaining showerheads. The water sprayed both sides of her body, soothing and warm. Dixie closed her eyes, not caring that she was naked, and Caine was still fully clothed. It felt too good to have Caine take care of her, to see the way his jaw rippled while he concentrated on the task. She continued to ignore those promises she'd made to herself regarding Caine and touching when he settled between her knees, shower gel in hand. The sting of water on her scrapes made her wince, but Caine's hands, smoothing the gel over her skin, massaging her aching muscles as he went, distracted her enough to forget everything but the soothing motion. "Tip your head back, honey, so I can wash your hair." Dixie did as directed, letting the pulse of the water and the scent of her pear shampoo wash away the grime and sweat of her kill. Nothing mattered but Caine's fingers in her scalp, easing the tight tension, rinsing the thick ropes of her hair by squeezing out the excess water. She bit back a soft moan at the gentle pleasure his hands wrought, one that was sure to echo in the cavernous space. Caine finished rinsing her hair, and then his fingers came to rest on the area surrounding her black eye. "You were really something tonight," he murmured, so soft and low, she almost didn't catch what he'd said with the rush of the water. Her eyes were still closed but her grimace was ironic. "Yeah. Something." Tipping her head back up, he thumbed away drops of water from her face with tenderness before cupping her chin. "What I meant to say was, you were awesome tonight." Dixie didn't understand his concerned gaze. She didn't recognize this Caine—one who had sympathy in his voice and touch. And still her promises to herself continued to slip away. "Thank you," she replied, her voice gruff, her throat tight. "I mean that, Dixie." She began to shy away from what Caine meant as a compliment. "I was angry." "You were incredible." She shook her head, casting her eyes to the buttons on his soaking wet shirt as remorse began to weigh heavy and ugly in her gut. "I was a spectacle. Seems I suddenly forgot to use my words. I was wrong to approach the situation like that." His fingers went to her chin so her eyes were forced to meet his. "No. No, Dixie. Louella created the spectacle. You did the honorable thing and tried to stop it before it went too far." Seeing Louella now, so much like she'd once been, left her overwhelmed with such sadness. That she'd been responsible for hurting Louella so much, she'd turned cruelty into a defense mechanism, made her bones ache with regret. "I used to be Louella, Caine. Or Louella's turned into me. I'm not sure anymore," she whispered. Caine's eyes pierced hers. "No one is you, Dixie. No one." Tears stung her blurry eyes, warm water cocooned her and exhaustion left her limbs shaky and her heart vulnerable. This Caine—the one who was looking at her with understanding, the one who was calling her something other than the dregs of society—was breaking her resolve. She didn't know what that meant. She didn't want to know if it meant the exact opposite of what it usually meant. Dixie shrugged her chin from his hand, her shoulders caving inward as a small sob escaped her lips. She could take almost anything. She could take his stinging words. She could take his indifference. She couldn't take friendly understanding. And then Caine was reaching up, pulling her close, and capturing her lips tenderly, delving between them with his silky tongue. Her fingers automatically sank into his wet shirt, relishing the hard strength beneath the soaked material, pressing her palms against the bulges in his shoulders until she wrapped her arms around his neck, bringing him as close as humanly possible. Her mouth opened beneath the weight of his, accepting his kiss, twisting her neck to get closer, and ignoring the sharp stab of protest in her sore tendons. She moaned when he pulled away, moving along her cheek with care, gliding his mouth to the hollow of her neck, nibbling, tasting. Her nipples grew tight, hardening with need. Caine sensed that, and cupping the underside of her breasts, he kneaded them, dragging the tips of his thumbs over the points until she was slippery with desire. Her head fell back, exposing the column of her neck. Her fingers drove into his scalp, clutching handfuls of his wet hair, her hips lifted, and her legs went around his waist, drawing him flush to her. Caine licked at her neck, letting his teeth lightly graze the skin, dipping lower and lower until his hot breath teased her nipples. When his lips wrapped around one tight bud, Dixie thought she'd explode, the sensation so intense, heightened by this new connection between them. Water rained down on them, pelting the hard tile, creating a steamy shelter that soon blocked everything out but Caine's mouth on her, devouring every exposed inch of her slick flesh. Her back arched as his free arm went around her waist, and his hand splayed across her ass, forcing her hips upward. Caine's hand slipped between her thighs, spreading her swollen lips open, dragging his fingers up and down until he made a wet path of heat. "So good," she whispered. Dixie placed her hand over Caine's, guiding his finger to her entrance, lifting her hips upward until his finger slid inside her. She hissed her pleasure, rocking against his thumb now rasping against her clit in delicious circles. The wet slide of his tongue licking at her nipple, along with his finger driving into her greedy body brought her orgasm, fierce and sharply sweet. Her chest tightened and her thighs clenched together, trapping his hand between them. Caine found her mouth again, drowning out her whimpers as she came. Dixie gasped for breath when another dizzy spell left her clutching at his arms. "I've got you. Just hold on to me, honey," he murmured, placing her hands at his waist. She gripped his clothes, soaked through and through, hearing the crunch of his wet jeans as he brought her with him to turn off the faucets. Dixie sighed when he lifted her back up and hiked her legs around his waist to exit the shower, leaving puddles of water on the floor. "Sanjeev'll have your head." Caine chuckled, the deep vibration of it from his chest making her sigh with contentment. "You don't think this is the first time there's been water all over the princess bathroom, do you? Remind me to tell you about the three-way we had in here two summers ago." Dixie giggled. "Two summers ago you were in the Baltic with Landon." "You keeping tabs on me?" he teased, his eyes crinkling at the corners. "Like some creepy stalker," she joked back. Through weary eyes, she watched as Caine somehow found a fluffy towel and dried her from head to toe, slipping her nightgown over her head then stripping off his own clothes and managing to struggle into her old flannel bathrobe. The brief glimpse of Caine naked, still hard from their encounter, made her mouth water and her heart speed up, but her body wasn't cooperating with her libido. Caine took her by the hand and led her back into the bedroom. "Bed. Now." Dixie climbed under the covers, grateful for the cool luxury. Mona and Lisa burrowed next to her, knocking the assorted pillows to the floor as her eyes began to droop. Caine brought her fingers to his lips. "Hey, tiger, no sleeping until I talk to Ray and make sure you're cleared." He settled behind her, spooning her, sliding her nightgown upward until the heat of his cock pressed into her back. Caine's hands roamed over the planes of her body, teasing. Lifting her leg, he spread her wide, flattening his palm against her swollen heat, smoothing his hands over the lips of her cleft until she bucked against him, writhing and needy. Her arms went up around his neck from behind her, thrusting her breasts forward so Caine could slide his hand under her and cup one. "I want to bury my face between your thighs, Dixie. Taste you on my tongue when you come. Lick you until you scream." Dixie shivered at his words, arching up and inward against his hot length in response. She reached behind her, finding the solid length of his cock, and pumped it, moving her ass up higher until Caine had no choice but to answer her body's question. Right now, in her dazed state, the only thing that was clear was she needed his cock in her, driving away her fears, putting out the fire of revenge and replacing it with his hands, his mouth and his glorious tongue. "Please, Caine," she begged. "Please make love to me. I need this so much more than you'll ever know." He stiffened behind her, his thick thighs pressing into hers, the crisp hair of them brushing against her smoother skin. "I want to touch every part of you, lick every inch of you, drown myself in you. But you belong to someone else. Walker," he grated the name against her ear. "I don't make love to someone else's woman, but Christ if it isn't killing me to keep from doing it." She twisted her neck, pulling his head in for a wet, deep kiss. "I made Walker up. Make love to me. Now," she begged with a whisper. He growled a chuckle, gripping her tighter, stroking her tender clit. "Say it, Dixie. Tell me what you want me to do." She was so turned on, so fragile, so vulnerable the words flew from her mouth with ease. "I want your cock in me. Hard. I don't want you to ever stop. I want you to make me come like you're the only person who can do it. Do it, Caine, please, please do it." "I want to watch, baby. I want to see you touch yourself," Caine ordered, cupping her ass, rolling his hands over her skin, dipping between her wet thighs, dragging that wetness to his lips and licking his finger. "No one tastes like you, Dixie. No one," he said before positioning himself between her thighs and driving upward. Dixie buckled then, stifling a scream of raw, uncensored pleasure, coming almost instantly—her need was so deep—with Caine not far behind. And then he was moving away, running a warm cloth over her, toweling her dry with hands that nurtured as she burrowed beneath the warmth of the comforter. Some of the last things she vaguely remembered were Caine signing off with Ray, a brief kiss to her temple, and his secure arms around her with her cheek resting on his chest. But the very last thing she remembered was she and Caine lying next to each other just the way she'd always meant for things to be. Fifteen Caine's feet pounded against the pavement, sweat pouring in familiar rivulets down his chest and along his belly. Fall was coming, but summer wasn't leaving without a fight. Thick humid air rushed at his soaking wet skin, his thighs and calves straining to meet his goal even though they burned. His iPod blaring Nine Inch Nails in his ears, he ran the track at Plum Orchard High. It was too early for the kids to be there for practice, leaving him with the track and his thoughts to himself. Running helped him think. And he had a shitload of thinking to do. Dixie was going to kill him—no question about that, and rightfully so. Caine didn't know what made him do it the first time, or the second time, or even now as he was contemplating the third. He just knew he was eating Dixie up under the cloak of anonymity. He'd gone along with this damn contest with the idea he'd nail her with his anger at every underhanded, low-down dirty opportunity afforded him. Not very golden boy of him, he mused, but the lingering effects of the end of their relationship left seeing her again a bitter pill to swallow. One he did his best to remind her of like some kind of damn tyrant. It was his fail-safe against falling for more of her bullshit. It was his wall. Because of the bet. She'd called him her judge and jury, and the arrogant ass he'd been kept him from seeing that until just the other night when he'd finally apologized. Dixie had once shredded his idea of what his life with her as his wife could have been. She'd taken the woman he'd created in his mind and blown her to shit. That was when he realized some things never changed. Dixie was always in it for the win, and in the process, fulfilling a long overdue wish from their two families. She didn't love him—she loved to brag about winning him. She always got what she wanted, but back then, he'd decided, he'd be damned if she'd get him. Not a chance he was marrying a woman who wanted to hang his head on her wall like some trophy. No matter how much he'd loved her. He'd kicked himself over and over after he left Georgia and moved to Miami. He'd watched Landon indulge Dixie all through her teenage years—through all of her heartless shenanigans. They'd argued about it more than once. But Landon had seen something in Dixie no one but him saw, and he'd done what he'd always done, remained friends with both of them. When they'd both come back ten years ago, and the older, allegedly changed Dixie caught his eye and every other aspect of him, he'd finally become a believer. He, like everyone else, began to worship at the altar of Dixie. He'd called himself all sorts of asshole for not seeing through her bullshit in the same way he had when they were kids. She'd been the same old Dixie—cruel and manipulative, hiding behind the word changed. His pride and lack of good judgment had taken an ass-whooping, and there was nothing he liked less than feeling as if he'd been had. So he'd licked his wounds far enough away to stay sane and suffered through any mention Landon made of her when he had to. But since they'd met up again, the cat had grown tired of toying with the mouse. Dixie wasn't the same damn mouse anymore anyway. She was a gentler mouse, a mouse with a conscience. One who protected her friends and stuck up for the underdog. He didn't know what had brought about the change, but change she had. Though he'd sure like to know what made someone as vindictive and calculating as Dixie turn into Gandhi. He'd watched her take on the Mags, defend Emmaline, make dinner for the women of Call Girls without asking—or rather, expecting—a single thing in return. She'd put up with the jokes and constant reminders of some of her more widely known hijinks with a smile and a good-natured nod of her head. Sometimes he wondered if she did it because she felt as if she deserved some sort of continual punishment. As though Dixie thought it was everyone's rite of passage to take a bite out of her as payment for her cruelty. Yet she suffered the wrath of her past everywhere she went like a trooper. She strolled into Madge's every day, head held high, back straight and sat on a stool where she ordered Danish for the girls coming off the night shift with a smile on her face and a kind word for anyone who bothered to ask how she was. If Dixie heard what everyone was saying about her, and most didn't even bother to say it behind her back, she ignored it. It took guts to come back and face that kind of ridicule. Desperation and her sad financial state aside, Caine had to give it up to her for lasting as long as she had. And then there was last night. Hell if he understood that Dixie. Not a sign of retribution. No defense in place for attacking Louella and the other Mags. Just her firm resolve that she'd been wrong to solve the problem with her fists instead of her words. And tears, tears that ripped his guts out. Bruises that made him want to make everything better with his hands—his mouth. Scratches he wanted to make disappear. The vulnerability in her eyes and in her body language said she needed someone to lean on—even if it was only for the moment. That Landon's death, this crazy phone-sex challenge, the worry her debt had brought had all begun to crush her, and she was fighting to keep her pretty head above water, said something about who she'd become. That was the Dixie he was crazy about. The Dixie he couldn't keep his hands off. He slowed to a more manageable pace, dragging a hand over his jaw and kicking up dirt. Shit. He'd just admitted he was still crazy about her. That brought him to a slow walk toward the bleachers. He dropped down on the hard metal, wiping his forehead with a towel before draining his bottle of water. Leaning back on his elbows, he gazed out at the track, one he'd spent hundreds of hours at with Landon and Dixie. In the stands, just hanging out, drinking their first beers, watching Dixie cheer. They'd been an odd dynamic—the three of them, both of them sharing separate friendships with Landon. But Dixie had always been a part of his memories. Now a piece of that dynamic was gone, and today, when he needed someone to throw his thoughts at, he missed that dynamic more than ever. His phone chirped, signaling a text. Digging it out of his duffel bag, he hoped like hell it wasn't his secretary back at the office in Miami. It was getting harder and harder to keep everything on track being so far away. Caine shaded the phone with his hand, scrolling to the latest message. Buck up, buttercup! Decisions, decisions to be made, my friend. Choose wisely. Choose for keeps. Caine smiled at the phone, his chest tight with missing his best friend. "You son of a bitch, Landon." * * * Catherine rushed to her the moment Dixie stepped foot into Call Girls. "Dixie! Mercy, girl, you should be at home resting." She made a face and groaned when Cat cupped her cheek to assess Dixie's black eye and swollen cheek. Dixie attempted a smile, though the muscles in her face didn't much like it. "Hey, stranger. How've you been? Studying hard so we can all call you Miss Bachelor's Degree?" "Forget about me. What are you doing here?" "Working." Catherine made a face at her, distorting her pretty features. "Oh, Dixie, nothing's more important than your health. Two days off won't break you." "Nay, I say. Two days off could be the deciding factor between survival and poverty. You don't want to see me out there livin' under the old bridge by the creek, do you? Besides, I don't need my eye to talk." Though, it would certainly help if her tailbone wasn't on fire. Catherine grabbed her hand and led her into her office, motioning her to sit on one of the chairs next to her. Her eyes were full of sympathy and concern. "So the girls told me everything about last night. I was worried something just like this would happen. I told Landon as much, but I'll never forget what he said when I expressed my concern." Dixie knew what Landon would say. No one knew the kind of discrimination and disapproval that PO could throw at you like Landon did. Cat puffed out her chest, raising one eyebrow and smiling a cocky half smile in Landon fashion. "He said, 'Kit-Cat, never you worry. If those buncha nosy old biddies with sticks up their hemorrhoid-filled asses give you or the girls any trouble, Dixie'll look out for you. She'll have your back. Nobody knows how to handle those women like my Dixie-Cup.'" Dixie's chest grew tight again for the umpteenth time. Landon had had faith, more faith than she deserved. She gulped, looking away from Catherine's intense gaze and focusing on the pictures on her desk of a beautiful, chubby little baby boy with sandy brown hair. "He said that?" "You bet he did. Landon would be so proud of you, Dixie." Dixie deflected another rush of tears by snorting a laugh, dropping the bag of Tupperware filled with homemade split pea soup for the girls on her lap. "You think he'd be proud I wrestled Louella to the ground like some kind of caged animal in front of Plum Orchard all over a silly microphone? Oh, and let's not forget my panties. Most of the town saw those, too. I think Landon would call me unladylike, not heroic." Catherine shook her dark head, her eyes glittering. "Don't diminish what you did, Dixie. Why do you do that? You did something good, something right, and it makes me mighty glad you're on our team." Good Dixie. Brave Dixie. Caine's words last night, coupled with LaDawn's and Marybell's, and now Catherine's, were too much. All this praise made her uncomfortable. She'd done two whole things right in her life to this date. That didn't deserve praise—it deserved an about time. She needed to escape before she couldn't breathe. "Is that all you needed? My shift starts in five minutes and I'd like to catch up on what I missed yesterday." Cat leaned forward and grabbed both of Dixie's hands, her eyes deep and serious. "Look at me. You're a good person, Dixie Davis, and I don't give a coon's ass if you don't want to hear me say it out loud." Letting go of her hands, she smiled and leaned back. "Now, before you go, first, holy smokes. Did your line ever ring off the hook last night. Yay for you—lots of lonely men lookin' for advice. Second, you have a visitor. If you're not up to visitors, say the word, and I'll send her on her way." Dixie's stomach reacted with a jolt. "A visitor? Friend or foe?" she asked weakly. She'd need to take kickboxing classes before she could go another round with a foe. "I'm callin' friend, because she was lovely and sweet, but then after last night and that mean old Louella, you never can tell with the people in this town, can you?" "Who is it?" Cat hesitated for only a moment. "Jo-Lynne Donovan. She's waitin' on Caine, but she asked to see you while she did." Perfect. Ding-dong, Caine's mother calling. Yet, her own mother would never forgive her bad manners if she didn't speak to Jo-Lynne, even if they weren't actually speaking to one another because of her. She blew out a rush of air and gathered her things. "Where is she?" "Waiting in your office." Even more perfect. Her ex-almost-mother-in-law was sitting in the office where, as a joke, LaDawn had turned Dixie's screensaver into a floating penis shaped like a rocket ship. She closed her eyes for a second to get a second wind, then popped them open with determination. "Then to the lion's den." Cat's gaze was filled with concern again. "You gonna be okay?" "Could be I'll have matching black eyes," she joked on her way out the door. She hadn't seen Jo-Lynne since the night of her engagement party. Another one of the many regrets she had. Not only had she ruined the friendship between her and Caine's mother, but she'd screwed up having one of the best mothers-in-law a girl could hope for. Jo-Lynne represented some of the very best things about her childhood. She was kisses and Estée Lauder-scented hugs, gooey slices of pecan pie, tater-tot hot dish, and a million Band-Aids on knees torn and scraped from trying to keep up with two boys for your closest friends. She'd loved Jo-Lynne like a second mother, and she'd spit in her face as surely as she had Caine's by not even apologizing for her part in their engagement fiasco. One deep breath, two silent prayers later, and Dixie was face-to-face with the woman who'd given birth to the man she couldn't manage to stop loving. Jo-Lynne rose at the sound of the office door opening. Her features, a softer version of Caine's, were serene as always. She wore her hair in a silky, sandy-brown bob that hugged the shape of her face to perfection. A smile flitted across her red lips before it tipped upside down, and she was putting her navy purse on Dixie's desk and rushing to her side. "Dixie! Oh, sugarplum, what happened?" More tears? Really, Dixie? Who are you? The concern in Jo-Lynne's voice, the way she dropped everything to examine her scrapes and bruises, unlike her own mother, had Dixie in the grips of another display of her out-of-control emotions. She'd called Dixie that all her life. Sugarplum. In fact, she could only remember two or three times when Jo-Lynne had actually used her given name—and one of those times was at her engagement party. Just after everything had gone sideways. Dixie held up a hand to keep her at bay. A hug filled with warmth and sympathy was far more than she deserved. "I'm okay. Really. Just some scrapes." Jo-Lynne pointed to the garish chaise longue. "You sit down this instant, young lady, and let me look at you. You're an absolute mess!" Dixie did as she was instructed because you just didn't cross Jo-Lynne. Her perfume, classic and as elegant as she herself was, drifted to Dixie's nose while her gentle hands, aged only slightly, smoothed Dixie's hair back. "So you gonna tell me what happened?" "You don't know?" "I certainly do not. I just got in tonight from Atlanta and came straight here to meet Caine." Dixie shrugged in embarrassment, putting her hands in her lap. "I got into a fight." Jo-Lynne nudged her over and sat beside her, holding Dixie's hand under the ceiling lamp to inspect the deep scratch Louella's heel had left. "With?" "A gang of wandering thugs?" "Try that again, miss." "Some people?" "One more time." "Flash mob brawl?" Jo-Lynne's lips thinned. "Dixie Cordelia Davis..." That was the warning. All three given names spoken at once. "Louella and some of the Mags. Two, to be precise." Jo-Lynne's eyes flashed, angry and outraged. "They did this to you?" "I sort of did it first. Then they did this back." Jo-Lynne gave her back her hand, folding her own hands in her lap. Here came the disapproval. "I'm guessing that spoiled Louella deserved it?" Had the entire world gone mad? Had Plum Orchard moved to some sort of alternate dimension where Dixie was always right and everyone else was at fault? She shook her head. "She didn't deserve to be mowed over like I was driving a John Deere and she was an acre of unkempt land. I shouldn't have resorted to physical violence." Jo-Lynne pursed her lips. "I want to hear every detail." Dixie explained only that she was defending a friend then waited for her lecture on violence with quiet apprehension. Instead of giving her hell, Jo-Lynne nodded her head in agreement. "Well, it doesn't surprise me that Louella was a part of pickin' on someone. I'm here to tell you, I never liked Louella. I know I should because she's the daughter of a Mag, and we're all supposed to be like family, but sometimes you're forced to tolerate family you don't like." Dixie's astonishment bled into her words. "Wait. You believe me without anyone backing up my story?" "Of course I do, sugarplum." She patted Dixie's hand. She'd been far worse than Louella once. It was only fair to point that out. "But I picked on people, too—all my life, Miss Jo-Lynne. I coerced, connived and in general mowed down anyone or anything in my path, especially Louella." In fact, had Landon brought the Call Girls into town just ten years ago, she probably would have behaved just as reprehensibly. Jo-Lynne's eyebrows rose. "So Louella gets a free pass because you did it, too? I'd like to own a yacht, so I should steal one just like all the other criminals? You were a child. A rebellious, out-of-control one, but still a child. Louella's no child, Dixie Davis. She's a grown woman who should be long past this kind of judgmental behavior." Dixie stubbornly refused to agree. "I wasn't a child when I—" Jo-Lynne's finger flew upward, cutting her off. She pressed it to Dixie's lips, giving them a pat. "No. You were still a child back when you were engaged to Caine, sugarplum, whether you knew it or not. Life hadn't taught you your lessons just yet, so you'd appreciate what you'd been given instead of taking it just because you wanted it. You had to learn those things in Chicago where no one cared that you were Dixie Davis, didn't you?" After all the humiliation and heartache she'd caused both she and Caine, Jo-Lynne was defending her as though Dixie had never left her watchful eyes. As though she still thought of Dixie as the daughter she never had. The revelation cut her to the quick. "Yes...ma'am..." was the best she could manage. "So tell me, this fight with Louella, was it over this phone-sex business you and Caine are wrapped up in because of my dearly departed Landon?" Dixie couldn't hear the words phone and sex come out of Jo-Lynne's mouth. It was like hearing her tell you about her private bedroom matters. Rather than clap her hands over her ears, she kept her reply simple. "Yes. Louella called one of the other operators out in front of everyone at the annual autumn dance, and it was cruel and humiliating. I was determined to make her stop. I just didn't stop her in the way reasonable adults do." Jo-Lynne was pensive for a moment as though she was tempering her words. "You know, the Mags don't like what Landon's done here, Dixie, and I'm not saying that I'd suggest the youth of Plum Orchard consider it a step toward a solid future in telecommunications. I realize some of the women involved haven't been afforded the kind of lifestyle the girls here in town have. They're makin' their way, is all. But I say who's it hurtin'? If it wasn't for some good old-fashioned makin' love, not a one of us would be here to begin with. It isn't for us to judge if it's right or wrong." If one could call flogging and foot fetishes old-fashioned. Jo-Lynne's justifications for the Call Girls were just shy of blowing Dixie's mind. "I'm not sure what to say. I didn't expect you to be so understanding. Mama certainly isn't." "Your mama and I didn't always see eye to eye on everythin', especially about raisin' you. I think you know what I mean by that." "Mama's a hard woman to please. We don't talk much anymore. Not since the restaurant fell apart..." "There's somethin' I've always wanted to say, and because I don't give a hoot about what anyone, especially a Mag thinks, I'm just gonna say it. Your mama's a mean woman, Dixie. Cold and controlling and as superficial as the day is long. But you're a strong woman. You were a strong child with an even stronger will. Your way was to rebel against that iron fist of hers. She demanded you wear your skirts below the knee, you ripped the hem out and cut them to midthigh." "I wanted her to love me." She cringed as her voice cracked. God. She'd really wanted that. Just once, she'd wanted her mother to love her instead of always reminding her she owned her. Jo-Lynne smiled a watery smile. "I know that, Dixie. You're not the reason I'm angry with Pearl Davis. It has nothing to do with connections or money or any of the things your mama's so angry about. Those are all superficial things. It has to do with how she pushed and pushed you until you exploded." Trapped. That's how she'd felt when she'd stupidly left college out of boredom and came back to Plum Orchard with no degree and no purpose. Her mother reminded her every day how time was wasting. If she were deeply truthful with herself, at first, Caine had been an answer to all her problems. Marrying him would get her mother off her back and her approval—approval Dixie was hungry for, all in one fell swoop. But he'd become so much more. Suddenly, he wasn't just to-die-for gorgeous, or kind and honorable— he was everything she'd never had. In the process of finding the answer to her mother's constant badgering, she'd found love without conditions and judgment, and she'd fallen head over heels for him. But it had been too late to put the brakes on the bet. A sharp rap brought their eyes to the door. Caine popped it open and smiled when his gaze connected with Jo-Lynne's—that warm, loving smile he'd always reserved for his mother. "There you are." He scooped her up in a hug, kissing her cheek, while Dixie hoped to blend with the chaise. "We still on for dinner?" Jo-Lynne squeezed his arm affectionately. "Give me two more minutes with Dixie, and I'll be right there." Caine's hesitance was reflected in his reluctance to leave. He put a hand under her elbow, his expression playful when he used a teasing Darth Vader impression to persuade her. "Mother, if we do not hurry, Madge will be all out of fried pork chops and gravy. How can we have enough energy for total universal domination?" Jo-Lynne almost cracked a smile then waved him out with a determined motion. "Shoo, Caine. I'll be right there." Caine sighed, running a hand through his hair. "Mom, Dixie's had a rough couple of days. Give her a break and let her be." Caine's attempt to prevent what he clearly feared would be a confrontation was sweet, but falling on deaf ears. "I can see that, Caine. Now out, young man!" Poor Caine. He didn't stand a chance. Dixie put a hand on his arm, almost incapable of looking him in the eyes after their lovemaking last night. "It's okay, Caine. You go." His fingers threaded through hers for a moment, dangling there for seconds, the connection making her heart pump harder, before he gave his mother a pointed look. "I'll be right outside this door." He hitched his unshaven jaw in the direction of the door. "Play nice." The moment Caine left, Jo-Lynne pulled Dixie up by her hands and looked her square in the eyes. "There's something else we need to discuss." Dixie swallowed hard. Atone, Dixie. If Jo-Lynne wanted to purge, scream, yell, call her names after what she'd done, she'd roll out the red carpet for her. She stared straight ahead, bracing herself for her due. "Yes, ma'am." But Jo-Lynne's eyes were soft. "You were a horror, Dixie. From the day you were born you were cranky and colicky and difficult to please. I know, because I rocked you at many a Mag meeting when your mama needed a break. And I loved you like you were my own. Not much changed with you for many years. You were ugly to people. I knew it. Everyone knew it, but I still loved you. And then you did somethin' so awful to my son—" "I'm sorry," Dixie rushed out the words she'd waited forever to speak, steeped in more remorse than she'd ever be able to properly express. "I should have said that the night it happened, but I'm so sorry I ruined everything. I'm sorry I embarrassed your good family name. Most of all, I'm sorry I hurt Caine." Jo-Lynne cupped her cheek. "Never you mind about family names and all the other silly ideals they feed you here in the Orchard. My name doesn't matter nearly as much as my son's heart does—his happiness. So I'm here to tell you true, Dixie Davis—you hurt him back then, bad. I don't want that to happen again. But mostly what I want you to know is I love you, I always knew you'd work past it one day—and I can see you have. But if you do it again, if you hurt my boy like you did last time, I'm comin' for you. Understand, sugarplum?" She managed a smile, keeping her sigh of relief to herself. "Understood." "Now, give Miss Jo-Lynne a big ol' hug. Don't know if I'll still be standin' after I eat one of those greasy pork chops Madge makes with my cholesterol bein' so high. So let's hug it out just in case." Dixie let Jo-Lynne pull her into an embrace, savoring the familiar hug scented with remnants of her childhood. She rested her head on her shoulder, fighting another slew of tears. When Jo-Lynne pulled away, she held her at arms' length and granted her a smile straight from her childhood. "Look at you, even all beat up, you're still just as pretty as you ever were." She didn't feel pretty, but she did feel less dark on the inside than she had in a long time. "Now, you make sure you come over in the next few weeks for some of my chili and corn bread. I won't take no for an answer. I'll make sweet tea just like always, and—" Jo-Lynne pulled her in close "—I've forgiven you, Dixie Davis. All was forgiven, a long time ago, sugarplum. Now forgive yourself." She dropped a warm kiss on Dixie's cheek, gathered her purse, and she was gone, leaving Dixie to war with more acceptance, more generosity of spirit. Forgiven. Used in reference to Dixie Davis? Yes. The world had become an episode of Fringe. This had to be the "other" Plum Orchard. She paused for a long moment, looking at the clutter of messages on her desk and the unopened emails in her inbox from clients without really seeing them. Her muscles ached, as did her mind, full of Jo-Lynne's words of forgiveness, and more—words of acceptance. As she sat in her chair, ready to attack her inbox, it was with a lighter load on her shoulders tonight. Lighter than she'd been in almost ten years. Sixteen An hour later, her phone chimed in her ear, reminding Dixie she was supposed to be working. After Jo-Lynne left, she was finding it hard to concentrate. Her face was on fire, her eye throbbed, and her mind was a vacuum of mixed emotions that left her all tangled up and simultaneously warmed to the core. Stuffy of nose, she grabbed a tissue and answered, "This is Mistress Taboo. Are you worthy?" "It's Walker." Her heart skipped that ridiculous beat it had the last time she'd heard his voice, so smooth and silky against her ear. Over the past few days, she'd toyed with what part of the country he came from. His accent said Louisiana; it was very like Harry Connick—cultured and sigh-worthy on the ears. "You sound like you've been crying. Are you okay?" When he asked personal questions about her, she found she had to remind herself, it really wasn't personal. Those questions were more about the client becoming familiar with her and building a rapport. They were questions that were typical, a way to strike up a conversation much the way one would in a real life circumstance. Except Walker's questions didn't quite feel as if they had anything to do with striking up conversation. They left her feeling a million things. Sometimes probed, sometimes cornered and exposed, but most times they left her hungry for more contact with him. Could be, her emotional state had her reading a whole lot more into this than was realistic. Maybe she was just overly sensitive these days, and talking with Walker hit her hot buttons. "Mistress Taboo? Were you crying?" His question sounded almost possessive. As though he'd take out whoever had made her cry. Another ridiculous notion. It had to be or she'd have to question her mental state. Callers easily became attached to the operators, but the operator probably shouldn't take girlish pride in the idea or worse, cultivate it. Still her cheeks warmed, betraying her better judgment. "Nope. Just my allergies." "You never mentioned allergies." Why would she mention them to some man hiding in his mother's basement, making phone calls to a cartoon figure on a computer screen? And why couldn't she keep her perspective with this man? She'd talked to plenty of men lately. Why did this one man make her question his every motive? "I don't think we ever touched on the subject of medical afflictions," Dixie offered, attempting to lighten the mood. "So how have you been?" "How have you been?" Again, Walker probably didn't intend for it to sound as if it held meaning, but there it was, that intensity, that gravelly lilt in his deep voice that made her feel as if he had a stake in her well-being. "Oh, I'm real good." "You know, I've noticed something strange about you." What about this bizarre attachment to him she'd acquired wasn't strange? Had he noticed it, too? "What's that?" She tried cooing the words to come off flirty, yet it sounded contrived and forced. "Your accent." She sat up straight, forgetting her spine felt as if someone had run a train over it. "What about it?" "It sort of comes and goes. Sometimes you're all Southern, and then all of a sudden, it goes away. What's that about, Mistress Taboo?" Dixie laughed, almost relieved to hear Walker wasn't going to try and pinpoint the dialect of her accent. "That comes from the time I spent in Chicago. I lived there for almost ten years. It sounds silly, but I was trying to fit in so they wouldn't think I was some country bumpkin who didn't know her backside from a cornfield." "Who are they?" "Some business partners." Aka, the men she'd tried so desperately to convince she was serious about her failing business long after it was already too late. "What did you do in Chicago?" "I owned a restaurant." "Shut the front door," he said on a whistle. "A real, live restaurant?" "It was very real." "So were you some kind of fancy chef?" Dixie fought a derisive snort. "Hardly. It was nothing like that. In fact, I hate to cook. I just invested in it, talked a bunch of other people into investing it, and put my name on it with the promise I'd take good care of it." Then I skipped off to shop like the spoiled brat I was while everyone else did the work. And while I was off partying and running up my credit cards, I lost everything because it was too late to save it. Yep. That summed it up rather nicely. "So there was a Mistress Taboo's Fine Dining somewhere? That musta been somethin' to see." Dixie barked a harsh laugh. "No. No Mistress Taboo's anything." "So, the million-dollar question. What made you leave the glamorous life of restaurant owning to become a phone-sex operator?" "Yet another of my long, sad stories." Bad-ump-bump. "You seem to have a lot of those," he remarked, though she didn't detect that it was offhand or cruel. Dixie slumped in her chair, reaching a hand into the drawer of her desk to look for some aspirin, wishing she had an ice pack for her eye. "If you only knew." Ugh. No more conversation about her. "Hey, I know. Let's try something new. Why don't we talk about you for a change? What does Walker of the sexy accent do for a living?" "Hey, I have another question." "Your quota for questions about me is up. This is all about you," she reminded him playfully, popping two aspirin into her mouth and sipping at her bottled water. His low chuckle relieved her. "Walker of the sexy accent does a lot of things for a living. But more recently, he's not so much making a living as he is behaving badly." Dixie cocked her head. That seemed to be viral in the male species as of late. "We've established that I'm the queen of behaving badly. Nothing you could do short of murder would allow you to take my title, mister. But so funny you should mention that, I just had someone say almost those exact words to me." "What a coincidence. Who would behave badly with someone like Mistress Taboo?" She smiled into the phone at his teasing tone. "You show me yours, and maybe, just maybe, I'll show you mine." "Okay, me first. I behaved badly with a woman." She rolled her eyes at how little information he gave her. "Aw, c'mon now. I need details, Walker! I can't offer advice if I don't have the details. Unless they're gory details. I don't want to have to testify against you in court. You know how that goes. I end up all over Inside Edition as the woman who caught the Phone-Sex Strangler serial killer. I can't leave my house without the paparazzi hounding me. People tweet about me. We can't have that, can we?" she teased. He laughed into the phone, which in turn made her insides a little like Jell-O. It caught her off guard. She'd never had a reaction like this to anyone except Caine. "Uh-uh. No details until you show me yours." Would it hurt to share just a little so they had a common bond that might allow him to open up? Would it hurt to recognize that you enjoy talking to this voice on the other end of the line? You're probably not the first phone-sex operator guilty of it.... "Someone apologized to me for behaving badly recently, too, but he wasn't doing anything I didn't deserve." "How did you deserve to have someone treat you badly?" His seemingly genuine interest in her softened her resolve to keep their phone call about him. Maybe just a little evasive confession wouldn't hurt. "I did an awful thing to this person a long time ago. Something I regret to this day. He was sort of taking out his anger about that incident on me. You know, poking me, reminding me of just how ugly I'd been over and over at every possible turn. So he apologized the other night." Then he made mad, incredible love to me, making me love him even more. "So whadja do to him that was so bad he had to keep rubbing it in?" She rose from her desk and headed for the chaise, hoping it would be kinder to her sore body than her office chair. Clicking off the light as she went, Dixie skirted the desk and tried to piece together an answer. How did you explain the bet she'd made for a man's heart? Dixie blew out a breath of air before she began. "I guess the easiest way to explain it is this—I bet someone I could win his heart, and when he found out, he thought I didn't really love him, and that all I really wanted to do was win the bet." The pause Walker made was deafening. Dixie held her breath until he asked, "Was there a reason he thought that? Seems kinda silly to marry a man just because of a bet." Relief washed over her when she didn't hear judgment in his voice. "At least a million. We had a combative...Wait, maybe competitive is the better word. We had a competitive relationship throughout our childhoods. Ca..." She bit her tongue. "He always played fair, though. No matter what we were competing for, because he's honest and decent. I, on the other hand, cheated more often than not. Like I said, I wasn't a very nice person for a very long time, and my reputation for getting what I wanted at all costs came back to haunt me because of it." "So who'd you make the bet with and why'd you do it?" "I made a bet with a frenemy, I guess. I used her. She used me. We were always trying to best one another." "Sounds like you did that a lot. Reputations, huh?" "I'm the champ. Anyway, at first it was just a joke—a way to poke at this frenemy. I didn't mean it mean it. Though, I did know she was nuts about this guy. I just wanted to get her goat. She said there was no way someone as honorable and decent as him would ever fall in love with me. Because I was the person I was, I bet her I could not only make him fall in love with me, but get him to ask me to marry him before she batted her falsies." "Falsies?" "An eyelash." "And then?" Dixie sighed. "Well, then because I can't resist bragging about a coup, I called her up and left her a victory voice mail, announcing that I was engaged. To the man she claimed to love. She played that voice mail at our engagement party—in front of an entire town." Walker whistled into the phone. "Ouch." She closed her eyes. "Pass the morphine and sutures ouch." "So, did you really love him, Mistress Taboo, or was he just a bet won?" Walker's question made her chest burn and her skull throb. "You can tell me. Swear on my Willie Nelson albums, I'll never tell a soul." But she nodded her head in response, grateful for the dark cocoon of night hiding her undying shame. "When we first started dating, he was a way out from under my mother's thumb, but it turned much deeper the more I got to know him. The more time we spent together, the crazier I was about him. So yes..." She stumbled on the hitch in her voice. "I didn't meant to upset you—" "Yes!" she blurted out, cutting him off before realizing she'd lost her composure. "Yes," she offered more calmly. "I loved him. I just didn't have enough self-respect to tell him about the bet before he was humiliated in front of his family and friends. Every single day, I wish I could take it back. I'd do anything to take it all back." When she'd finally realized just how much she'd wanted to be Caine's wife, she'd thought about telling him. She'd even tried once or twice, but back then, she'd been selfish enough to never want him to look at her with the brand of disgust everyone else had when she pulled a stunt. The longer she waited, the worse the unspoken threat Louella would snitch had become until it turned into her nightmare come true. Louella had waited until their engagement party for a reason. Impact. It was the final coup—the big win against Dixie, and it was exactly how Dixie would have played it had the roles been reversed. Total public annihilation. She'd have let Louella think she'd won only to rip the rug right out from under her at the last possible second, too. That was when the grasshopper had become the sensei. Dixie yanked herself back to the present when she realized no sound came from her earpiece. Damn it all. She'd let herself get too carried away, and way too intense. "Walker? Are you still there?" she asked, hoping he hadn't hung up because she didn't know when to shut up. The thought that she'd driven him away sent her into panic mode. "I am, Mistress Taboo. I am." His tone concerned her, making her heart stick to her ribs. "TMI?" she squeaked, with a wince. Walker cleared his throat, his next words gruff, as if she'd hit a sore spot for him. "On the contrary. It helped a great deal." "With your situation?" She heard what sounded like Walker swallowing and then, "Could be. I have to go now, Mistress Taboo. It was a real pleasure." Before she had the chance to protest or apologize, Walker was gone. Just like that. The dial tone in her ear signaled his obvious disgust. Clicking her earpiece off, she rolled her head on her neck and stretched her sore arms. She'd blown it with a client. A client whose voice she'd grown attached to hearing nestled against her ear in her dark, cozy office. One who asked inquisitive, if not personal, questions which were smart and well-articulated. And made you say things you were better off not saying, Dixie. It only reopens the wounds and exposes what should remain private to a virtual stranger. Despite all that, she still hoped Walker would call back. * * * "Emmaline?" On her 1:00 a.m. break, Dixie plunked down beside Em whose legs were dangling in the pool outside of Call Girls. The warm water felt particularly good on the soles of her feet, shredded from her barefoot run across the square last night. "Mmm-hmm?" Em asked on a wobbly sway, leaning into Dixie. "What are you doing out here so late? Everything okay? The boys all right? Where are they anyway?" she asked, spying droplets of spattered pink wine on the deck of the pool. "Mama and Idalee have been lookin' out for them while I mediate you and Mr. Smexy. Gareth said to say hello." Dixie smiled. She'd spent a little time with Em's boys recently, and she was falling madly in love with them. Even sullen Clifton Junior. "So, are you okay?" "Everything's good, good, good," she chirped, too high and too forced. "You sure? How would the good Reverend feel about you out here by the pool, drunk on wine, young lady?" "Jesus drank wine." She leaned back and hiccupped, her chest rising and falling with a heave. Dixie swallowed a snort. "Why, yes, he did, lightweight. But did He drink two and a half bottles of strawberry Boone's Farm? Nay. I think not. I think he drank the yucky kind made from his blood. The kind they have in church that makes you gag and gives you a heinous headache?" Em giggled then took a hearty swig of the culprit, dribbling some on her pretty blue top. Her hair was mussed, her cheeks were flushed, and she looked more adorable and relaxed than she had since Dixie had been back. "It's not really Jesus's blood, Dixie. It's just cheap wine at church." "Glad we cleared that up. So why are you out here drinking like you're still in college?" Em snorted, spitting at a strand of hair caught in her mouth. "I took my college classes online, silly Dixie. There was no partying. Just me and a computer." "In that case," she held up one empty bottle, "this means you're well on your way to righting an egregious wrong." Em ran the tip of her finger over the bridge of Dixie's nose. "You're silly, Dixie. Silly and so pretty and too prideful to tell the man you love more than anyone ever how sorry you are. That's just plain dumb. Maybe you should be the one drinking?" Maybe Em was right, but... "I'm not ready." "To drink? Since when? As I recall, you were never afraid of a six-pack." Dixie rolled her eyes at her. "No. To have—" she threw her fingers up to air quote "—the talk." Em drove a finger into Dixie's arm and wrinkled her nose. "You know what you're not ready to do, Dixie-Doodle? You're not ready to get this sleeping with Louella thing all out in the open. That's what you're not ready to do. Know why? Because it hurts. Louella claims to have cuckolded you. Take charge of that. Face your demons is what you told me, right? So why not just ask Caine why or even if he did somethin' so vile and get it over with? The Dixie I knew would never let this fester." "The Dixie you knew would have done something awful to Louella by now as a way to ease her pain. Trouble is, I don't fully know the Dixie sittin' here before you yet, my friend. A Dixie who'd rather hide than break out the heavy artillery? Who is that? I'm either knocking people down like some sort of deranged linebacker or looking for my turtle shell to burrow into. I just can't seem to find my middle ground." Or any ground. Everything felt soft beneath her feet. As if this tenuous grasp she had on her bad behavior would cave in if she didn't tread carefully. Em's finger waggled under her nose. "Communication is the key. Just ask me about how wrong everything can go when you don't communicate." "Speaking of communication, have I ever thanked you properly for helping me talk to Landon that one last time?" Em had called her the second the hospice nurse informed her it wouldn't be long until he was gone. She'd been driving all night, praying her crappy car would make it the last bit of the trip, praying the little money she had would be enough for gas. Em shrugged. "There's nothin' to thank me for. It was one of his last wishes. We only spent a month together, but to be invited into his beam of sunshine is like bein' invited into a warm hug wrapped in a Snuggie. He had the best heart I've ever known." Dixie forced back tears. Landon hadn't been able to say much, but she'd filled that void with a promise to make him proud. "That he did. Anyway, thank you, Em. I wanted to be there more than I wanted to take another breath." "He knew that, Dixie. I need you to know, he knew. Now, no more thank-yous. Let's focus on you before I get to cryin'." Her sadness made Dixie wrap an arm around her and give her a squeeze. "Let's not talk about me anymore, please. I'll just say this, I'm not ready. Okay?" Em nodded with a sigh. "So what brought the drink on? 'Cause this sure isn't like my girl Em." Em sat up straight with a jerk. "Am I your girl, Dixie?" she asked, pouring wine into her discarded glass and handing it to Dixie. Dixie held up her glass to clink the bottle. "How about we're each other's girls?" "You mean each other's person—like on Grey's Anatomy? Like Mer and Cristina?" Dixie cocked a half smile, swishing her feet in tiny circles to ease the cramps in her calves. "Minus the surgeon part. Though, if we stuck in a McDreamy, I wouldn't be broken up." Her glassy eyes focused on Dixie with fire in them. "You're the devil, Dixie. Is anyone the devil's person?" "I dunno. Wanna find out?" Em touched her bottle to Dixie's glass, her aim slightly off. "I'll drink to that." Dixie winked. "Then to each other's person. Hear-hear!" Em's head tipped before the wine bottle was actually on her lips, sending the pink liquid gushing straight down her chin and onto her chest. "Hey, person?" Dixie grinned as best she could with her eye half hanging out of its socket. "Yes, other person?" "Cleanup in aisle ten," Em slurred, with a lopsided smirk. Dixie giggled, digging around in her purse for a tissue. She dabbed at the splotch of wine in a hopeless attempt to wipe it away. "So, person, I have a question for you." "I don't know if the 'person handbook' requires that I answer said question, but I'm all ears. And wine." "What's the real reason you're out here drinking, Em? You wanna talk about it or do you want me to hush and leave it alone?" "I met a man today." She dropped the statement between them as if she was dropping a hand grenade. It was a very un-Em-like admission. One Dixie was almost positive she didn't mean to make and certainly held unnecessary guilt. "So soon?" Em's lips thinned. "Are you judging me again, S.S?" "Me judge? Nope. I was just making an observation. Wasn't it you just the other day who told Marybell men were akin to locust and the black plague?" Em scrunched her nose and waved an unsteady hand at her. "I was just spoutin' off because I was angry with Clifton, is all." "So, person, you feel inclined to tell S.S. all about this man?" Em sighed and smiled. "I didn't really meet him-meet him, I guess. He met Louella, actually. I was sort of eavesdropping on their conversation. Despicable, I know, but there was just somethin' about him...." "Has Louella gone and had his name tattooed on her arm yet? Just so he's properly branded?" Dixie teased. Em rolled her eyes, reaching out to clutch Dixie's arm to steady herself. "Well, you know Louella. Any new man in this podunk town is fresh meat. So can't say as I blame her for makin' a move as fast as she can." "Forget Louella. Who's this man?" "I only heard his first name. He was here lookin' at real estate, of all things. Lawd knows there isn't much of that to be had in the PO. He's some kind of software developer or something smart. Like I said, I was just fringing the conversation like some kind of stalker right there in front of Brugsby's. But we did make eye contact...." As Em's words trailed off, her smile became a little like Dixie's once had over a particular Backstreet Boy. "So I'm guessing you thought he was handsome?" She shook her head in absolute, alcoholically infused disagreement. "Oh, no." Dixie kept her response relaxed as she looked up at the twinkling clusters of stars and watched the palm fronds from the surrounding trees sway with the warm air. "Explain?" "He wasn't handsome. To say as much would be to deny his very essence. He was hot, Dixie. So very, very hot," she whispered, licking her index finger and making a sizzling sound. "Handsome is meant for a more distinguished man, in my humble opinion. This man—ohhhh, this man made me think all sorts of dirty things. The kinds of things the girls talk about on the phone with their clients." Dixie gave her shoulder a playful nudge, thrilled Em was beginning to feel comfortable enough to admire someone of the opposite sex. It meant she was considering moving forward. "Wow, huh?" "Wow-wow, for sure." "So he's why you're out here drinking?" Dixie clucked her tongue. "Because if it was me, and I saw a man who was wow-wow hot, I'd be out there givin' Louella Palmer a run for her money." "That's because you're a shameless flirt. Not a well-mannered lady like me," Em said on another inebriated giggle. "Reformed shameless flirt, thank you." She tipped an imaginary hat at Em and smiled. Em let her head drop. "I have no place thinkin' about other men when I couldn't manage to keep the one I had." The mention of Clifton reminded Dixie of something she needed to clear up. "Speaking of the one you had, why did I hear Nanette telling Essie Guthrie that you were the one responsible for the breakup of your marriage the other day?" Em made a face. "Because I've done nothing to right that wrong, and why're you listenin' in on someone else's conversation, oh thee of the reformation?" Dixie gazed at her with astonishment. "Oh, no, person, I was not listening in. I couldn't help but hear her. I was in Madge's, picking up some of those yummy lattes for you and the girls before my shift. She doesn't hide the fact that she loves to gossip, and I reminded her of that on my way out the door." Em sighed. "Better that go 'round town instead of the truth then." "Hold on there. Why are you taking the blame for Clifton's misdeeds? While I admire you taking one for the team, I absolutely do not agree with him shifting the blame. Where's that spine of yours you're always shoving in my face?" "What else was I supposed to do, Dixie?" she hissed, pressing a finger to Dixie's lips. "Tell my children and his parents that Clifton moved off to Atlanta because he wants to wear makeup and high heels and live with the most understandin' female this side of the galaxy? He's a descendant of one of Plum Orchard's founding fathers, for heaven's sake. You know how much pride Clifton's parents, Harlow and Idalee, take in that. How could I let them be humiliated that way? And Harlow's health isn't good. I won't have his heart attack on my hands." No, damn it. This wasn't okay. How dare Clifton leave Em to clean his mess? "Will it be any better if Harlow ever learns the truth, Emmaline?" "For now, this is the way it's gonna be, Dixie. It's better everyone think I was a terrible homemaker and wife than know the truth. Now, I won't hear any more of it." "Fine," Dixie said between clenched teeth. "But for the record, Clifton's no kind of man if he'd let you take responsibility for his dirty pool. It's wrong to make you keep his secret and take the blame, too. I don't like it, Emmaline Amos." Em's face relaxed again, the fiery anger in her eyes subsiding. "Well, I appreciate your concern. It's over and almost done now, and all that matters is everyone stay out of the line of fire. Which means I shouldn't be thinkin' about other men or Nanette and the senior Mags will surely have something to gossip about then." Dixie leaned back on the palms of her hands, swirling the water with her feet. "Wanna know what I think?" "About Nanette's gossip?" "No, about how you're feeling when you think about this man and starting new relationships." "Because relationships are your specialty—yours with Caine bein' so successful and everything." She stuck her tongue out at Dixie playfully. "Here's what I think. Maybe you weren't meant to keep the man you had, Em. The one you had just wasn't good enough for you, in my opinion. So maybe there's someone else you're supposed to be keepin'?" Em's gaze was faraway, full of guilt mixed with excitement. "His name is Jax. The hot-hot man, that is. I heard him say it to Louella while she was makin' those big round moon-eyes at him. How ridiculously soap opera, melt-your-knees sexy, is that?" A lot like ridiculously soap opera, melt-your-knees sexy as Walker was. "Look at my knees—all melty," Dixie teased. Em took another long swig of the now almost empty wine bottle and laughed. "Hah! Your knees don't count. They're just scarred from sittin' on 'em so much while you begged the man upstairs for forgiveness." "Touché. So is this Jax why you're drinking?" "No," she said on a sniff. "Though maybe he's just a small part of it. He just made me really think. I'm at a crossroads, one foot in my old world and the other somewhere undefined. I'm still stuck in the confines of marriage to Clifton—on paper anyway. Still sad I couldn't keep the promises I made the day I married him, but I'm itchin' to move on, too, you know? I feel so dirty bein' cheated on. I know it's absolutely ridiculous, but I feel like once we have a piece of paper that says it's over, I'll be clean again. Like I'll have a fresh start, and his sins won't be the boys' or mine anymore." Dixie grabbed her shoulders and gave her a light shake. "Listen up. As your person, it's my responsibility to tell you, you didn't break those promises, Em. You did not. Clifton did. This was about Clifton's insecurities, not yours. It was easier to find another woman who already knew all his secrets than be man enough to tell you about them." "He never even gave me a chance to decide how I'd feel about him wearing women's clothes, Dixie. But the truth is," she whispered, low and ragged, "I just don't know how I feel. The only thing I'm sure of is I don't know what I know anymore." Dixie squeezed her arm then patted her shoulder, lifting it up and pointing to it. Em laid her head there. "I know, honey. But you know what else I know? I know that you're loyal, and generous, and probably the kindest human being I've ever met. I also know you won't always feel like this—so lost, so sad. It sounds meaningless right now, but this, too, shall pass." "Like it did for you?" she asked in a whisper. Dixie nodded in agreement because it soothed Em, not because she really believed anything would ever pass for her. "You bet," she whispered back against her hair. "Just like me." Seventeen Dixie sat on a bench under a big maple whose leaves were just beginning to turn while Caine watched her from behind a real-estate magazine like some kind of town-square voyeur. She chatted with some of the senior men of Plum Orchard as they played chess and she ate her dinner out of a take-out box from Madge's. The sunlight danced on her hair, casting golden highlights on her long, loose curls and leaving her cheeks dusted a pretty pink. Her words were filling the air, and her laughter filled his head. The days were finally cooling off a bit, allowing her to wear jeans and a light pink button-up sweater that made his mouth water for the way it hugged her curves and his fingers burn to pop the small pearl buttons right off it. His mother had ordered him to locate and escort Dixie back to her house for a date they'd set to indulge in some pecan pie, coffee and girl talk. Because he couldn't resist, knowing she'd be here in the park with the seniors like she was most evenings lately, he'd arrived twenty minutes early to get his fix of her. Since the night she'd decked Louella, she'd been purposely avoiding him again. This time, he knew why. Their relationship, the complexities and reasons they were at each other's throats all the time, had changed. He'd apologized to Dixie for his crappy treatment of her, and now neither of them knew what to say if they weren't searching for an available artery or a bed while they said it. It was crystal clear neither of them knew what the hell to do with this fragile declaration of peace. So neither of them said anything. Add to that, Dixie's new vulnerability. She'd been vulnerable in that shower with him—to him—and it scared the shit out of her. It scared the shit out of him more. She was no longer just the flirty, sexy woman he'd fallen in love with. She was ten times more. She had depth—scars he didn't understand—regrets she actually felt. He wanted to know them, hear them, soothe them. "Candy Caine?" A phone with a text message was shoved beneath his nose followed by the sweet scent of fresh pear and Dixie's husky voice in his ears. "I've been ordered by your mother to find you and come directly to her house for pecan pie." He rolled the magazine up and tucked it into his back pocket, glancing up at her with a smile. "I curse the damn day I taught her how to text. There's no hiding from her, ever." Dixie's giggle, light and easy, drifted to his ears on the soft breeze as the shadows of the trees in the square played across her face. She hiked the strap of her purse over her shoulder and tucked it under her arm. "Well, it is homemade pecan pie. Seriously, what's more text-worthy?" He rose, tearing his eyes from the swell of the tops of her breasts, just barely skimming her sweater's opening. "You ready? Or do you have more men left to charm the pants off?" She tossed her empty dinner carton in a trash can as they began to walk. "Don't be ridiculous, do you really believe I left a crowd of men uncharmed and with pants on? Impossible," she said on a flirty giggle. The bruises Louella had left on her eye were finally beginning to fade, but it still made him wince to see it. "How's the eye?" "Still in my head." "Phew. Good thing. One-eyed Southern belles are a hard sell these days, I hear." She let him slip his hand under her arm as they crossed the street, the temptation to place it at her waist stronger than ever. She cocked her head in his direction and raised an eyebrow. "Then it's a good thing I'm not for sale. Sadly, my reputation precedes me." Caine heard the teasing lilt to her voice, but that underlying hint of seriousness, the one that continually self-deprecated and punished, remained, and it was becoming like a kidney punch every time she used it. "Reputations can change. Or should I say perceptions can?" Why he couldn't just tell her his perception of her changing was a testament to his own damn insecurities. He didn't want to fall into another trap. She gave him a strange look before saying, "Uh-huh. Just like a leopard's spots and a Kardashian's love of publicity. But let's not talk about me. We do that a lot. It's overrated. How about we talk about you?" "Okay, floor's yours." "How's the real-estate business in Miami? Yours in particular." Mostly flailing due to his lack of on-site management for over a month now. "It's been good to me, despite the economy." Dixie shook her finger along with her head. "I don't mean overall. I mean now, as in right now. Because I heard you just the other day talking to someone named Geraldo about escrow and all sorts of big words simple girls like me don't understand. And when I say talking, I mean you were sort of yelling. Which leads me to believe Miami's finest golden boy turned real-estate agent should be back home, managing the empire he built from scratch. You sounded pretty stressed." She was right. Things were slipping back in Miami while he was in Plum Orchard, trying to figure out where the hell to go from here. The realization that he wasn't ready to leave yet was unsettling. That he wasn't ready to leave because Dixie was here left his brain all kinds of shit-wrecked. But he had a feeling there was still much more to discover about her, and he wanted to do that. "The big words are just a front for the real problem, which is that I'm not there to micromanage everything. They'll get over it. I pay them a lot of money to figure it out." Dixie slowed a bit, the rhythm of her heels clicking against the sidewalk changing. She stopped at the cross section where his mother's street met the old dirt road they used to race bikes on. This was the road where he'd threatened to take out Dixie's first boyfriend, Wayne Hicks, for getting her drunk and trying to take advantage of her. Little had he known Dixie was anything but drunk, and no one, least of all poor Wayne Hicks, took advantage of her. The memory, vivid and in living color, made him smile. It was also the road where they'd first made love in the back of his father's old pickup truck. And where he'd proposed to her. Dixie tapped him on the shoulder. "Hey, where'd you go?" Obviously to a place she didn't remember quite the way he did. He spun around on his heel and began walking back toward the intersection. "Sorry. What was the question?" "Do you like living in Miami, Caine?" He stopped again. Did he? He liked the money. He liked the challenge of selling high-end real estate in a less than desirable market. "Do you like Chicago?" She smiled up at him, her dimples deepening in that enticing way she had of using their innocence only to later nail you with her sexy. "I asked first." Caine shrugged, unsure of what she was getting at. "At first, I was in culture shock. It's very unlike Plum Orchard, but it grew on me." He'd forced himself to adjust because no way in hell was he going back to a place where Dixie lived in every nook and cranny. Dixie wiggled a finger into his belt loop and tugged at it before letting go, a familiar gesture from their past when she'd wanted his attention. "But do you like it? Really like it? Do you think of it as home? Lately I'm discovering it's important to like the place you're going to set up shop with your life." When he'd moved to Miami to follow up on a job offer from a college friend's father, it had been the answer to getting the hell out of Plum Orchard and away from the memory of Dixie, but did Florida feel like home? Did he look forward to going back to his ultra swanky town house after a long day? Did it make him feel comfortable and welcomed like his mother's kitchen with its hardwood floors, antiqued white cabinets and woodburning stove did? No. It was just a place to hang his hat—a place to grab some sleep, have the occasional beer, and watch a game on a flat-screen TV he almost never had the time to turn on. "I have a nice town house," he replied, almost defensively. "I bet you do. You deserve a nice town house. Town homes are what make life worth living," she teased. Caine squared his shoulders. "So, Chicago?" Dixie rubbed her arms. "Cold. Brrr." Caine's eyes zeroed in on hers, trying to read where this was leading. "Did you like it?" Had she missed home as much as he was discovering he did? Her eyes darkened momentarily then clouded over to hide whatever it was she was hiding. "It served a purpose. Did I love it? Sometimes, if I'm honest. The shopping was great, the nightlife even better. But mostly, not so much as I got older." And? There was something beyond her getting older and her financial struggles that made Chicago not so likeable, and he wanted to know what. Yet, the way her eyes avoided his told him she wasn't ready to tell him what it was. In fact, her eyes said she especially didn't want to tell him. He kept his next question as casual as possible. "So, you thinking about staying here in the PO?" "Well, of course I am. Who's going to run Call Girls when I beat you senseless and win this whole thing?" she asked on a laugh, sticking her hands in the pockets of her jeans as her feet began to move again. "Tsk-tsk," he chided with amusement. "That has yet to be determined, Mistress Taboo. We still have a couple of weeks left. Don't pack your stilettos just yet." She slowed as they neared the white rose-covered front gate to his mother's. "Can I ask you something?" "You can always ask." "If you win will you stay here, Caine? Move back?" Would he? It was damn good to be home, to see his mother more often than holidays. It was good to walk down the center of town and be greeted with smiles that were familiar. It was good to share a beer or two and chicken wings with some of his old high school buddies at Cooters while they watched the game. He'd missed the hell out of Landon the second he'd left this earth. Yet he'd thought it would be painful to return home and remember his friend as vividly as he did Dixie. Instead, he found it comforting to be surrounded by the things both he and Landon had once loved. Was he ready to admit that? Not yet. So he avoided the answer to her question much the way he was internally avoiding it, too. "I don't need to be here to run things, Dixie." "Like you're running them in Miami from here?" Her skepticism was apparent. Ouch. "Hey, give a guy a break. Landon's will and his little game were a surprise. I wasn't ready for it, so I didn't make the proper arrangements. But remember, Cat's here. She can run things until I get everything sorted out and make decisions." Dixie pushed her way through the gate topped with roses, stopping at the hammock his father liked to nap on. "But what about the girls?" "What about them?" The sun had become a setting ball of fiery orange behind her head, matching the flash of her eyes. "Please don't be such a man right now just to avoid any hint of female drama. You don't really think Plum Orchard's going to just let this go, do you? All this sin they claim has been cast upon them by having Call Girls right under their noses? Landon wasn't one to take pressure from people lightly, Caine. He responded in the way he always did, by doing what he damned well pleased because he had the money to do it. Which is a great attitude to have if you're dead and don't have to live with the consequences. He left that for us to do." "Why does that require me to physically be here?" Her eyes went wide. "Did you miss the knock-down drag-out Louella and I had because she was taunting LaDawn? I think we all know their idea of progress is allowing Rayvonne Purnell to open up a coffee shop that has, and I mean no disrespect, 'That fancy coffee with serving sizes in Eye-talian and full up with whipped cream and purty sprinkles,'" she mimicked the mindset of nearly everyone at the last town council meeting. Caine noted his mother's house was strangely dark, which meant the doors were locked. He dropped down into the hammock, pulling Dixie with him as he considered her very valid point. "Well, to be fair, they did vote in Reyvonne's favor." Dixie turned to face him, slipping her leg underneath her. She shook her head. "That's a far cry from a phone-sex line, Caine. The Mags want to preserve the integrity of small-town America. That doesn't include women like LaDawn who list companionating as a skill on their résumés. The girls need someone here to fight off ornery Nanette and the Mags. They'll eat LaDawn and the others alive if someone isn't here to take up for them." Her fierce response left him full of admiration for her and just a little offended. "You know I'd never let anyone hurt those girls, Dixie. I like them, too." The hard line of her lips softened then curved into a smile before she looked down at her lap, spinning her thumb ring. "I do know that. I do. I'm just being ridiculously overprotective, I guess. But just so we're clear, if you win this I want you to promise me you'll make regular appearances to check on them, Caine. Make sure Louella isn't baking them pies with the paring knife she uses to stab them in the backs." Caine grinned and playfully nudged her to avoid addressing how incredibly sexy and passionate she looked when protecting the women of Call Girls. "Look at you coming to terms with me winning. That's good. Get used to it, pretty lady." She clasped her hands together and reached skyward to stretch, revealing the silky smooth skin of her lightly tanned belly. Her lips curved upward. "Not on your life. I'm just saying it to distract you and stroke your ego before I swoop in like the mean-girl ninja I am and snatch that golden headset of yours right out from under you," she taunted good-naturedly, leaning back on her elbows and exposing the tops of her breasts to his hungry eyes. Would she go back to Chicago if she lost? What would she go back to? "So does that mean you'll leave Plum Orchard if you lose?" "Would you give me a job if I stayed?" she asked, but it wasn't with a challenge in her tone. There was a hint of fear in it—one that cut him off at the knees. Made him want to drag her into his arms and tell her he'd take care of it all. "I need a full-time job, Caine. I have...debts." Damn it, he wasn't willing to feel that way again—not yet. Yet, friend? Landon taunted. Her take-no-prisoners attitude about her financial mess surprised him. She didn't have to pay off the people who'd invested in her restaurant. They were mostly family friends and connections that wouldn't miss the money. Somehow, Dixie had managed to find some principles. His curiosity about what had led to that was eating him alive. Dixie poked at his rib cage, her eyes hopeful when she gazed at him from behind the curtain of her hair. "So would you? Hire me?" "Sure, Dixie. I'll give you a job if I win. You can be Mistress Taboo for as long as you want." She whistled. "Wow. I like this being friends thing. It's made you all warm and squishy." "Do you, Dixie? Do you like us being friends?" The intensity of his question shook him up. He no longer wanted to crush her—at least not in phone sex. Jesus. "It beats wearing my boxing gloves all the time. They're hell on a manicure." Pushing off with his foot, Caine set the hammock to rocking with a chuckle. The motion moved Dixie so close to him he had to fight with his arm to keep from wrapping it around her before she righted herself. "We used to spend a lot of time out here, didn't we?" The pretty tint to her cheeks, the way she avoided eye contact told him she remembered exactly what he was remembering. "We did," she murmured, then gave him a look he pegged for surprise. "You still think about...about that?" "More than I care to admit." Or more than was healthy. Dixie shuddered a breath, a breath he read as touchy territory. "Me, too." But he didn't want to walk on eggshells tonight. He needed to know those memories, that bittersweet chunk of time they'd spent together during their courtship, meant the same things to her they did to him. "Do you remember the time my mom came home, and we were out here, right on this hammock, buck naked like we were the only two people in the world?" Dixie's head fell back on her shoulders, exposing her throat when she laughed out loud. "I remember it vividly. Me hiding in that big bush of rhododendrons while you stumbled through that lame story about how you thought you heard Digger Radcliff's dog rooting around in the garbage cans." She laughed again, carefree and soft. "It would have worked, too, except for your damn pink bra, glowing under the moonlight, screaming, 'Hey, Jo-Lynne! Dixie and Caine just racked up some hammock miles!'" Dixie began to laugh so hard she had to put a hand to her stomach. "Well, there's that—and the fact that it was still stuck around your ankle." Caine began to laugh, too, making the hammock shake until somehow, they were only inches away from each other. Seeing her so untroubled, so like she used to be before her life had taken this abrupt turn, changed everything about what was supposed to be a piece of pecan pie and coffee with his mother. "Do you remember that bra, Dixie?" He couldn't stop himself from asking. Her breathing hitched when he leaned over her, brushing her hair from her eyes. She nodded, and Caine thought he caught her swallowing nervously. "I think you said it was your favorite." "Yeah," he muttered, transfixed by the swell of her peach-colored lips parted just enough to reveal her tongue. "Yeah. It was." Dixie's chest rose and fell in choppy breaths, pushing against his. Her full breasts swelled with each breath she took. He used his knuckles to caress the underside of them, loving the fullness of them, aching to tear the tiny pearl buttons of her sweater off with his teeth before burying himself between them. Her fingers automatically went to his hair, gripping a fistful of it as his tongue wisped over her bottom lip, nibbling it before pulling at it, relishing the plump flesh. "Caine..." she whispered as the tip of her tongue caught his. "Dixie?" he asked, before dragging his index finger between her breasts. "We have to..." She moaned at the pop of the buttons on her sweater. "Talk. I need to..." She gasped when he pressed his open mouth to the column of her neck. "Ask you—and...tell you something." He chuckled, cupping her breast, still encased in a lacy confection of interfering bra. "After." Gripping his hair tighter, she dragged his head upward, forcing him to look at her. What he saw was desire, hot and sweet, but there was something else—something gritty. "Please," she almost begged. Her tone was so urgent, so primal and hoarse Caine paused, looking directly into her eyes, alarmed. "Anything." Dixie's eyes shone bright under the full moon. "Did you sleep with Louella the night our engagement ended? Look at me when you answer me, Caine. Look at me." Where the hell had that come from? Was she serious? Yet, the almost desperate question, the way she demanded he look at her when he answered, said she meant it. Caine looked down at this woman he was fighting tooth and nail not to fall in love with all over again, waiting with dread for his answer. He pinned her with his gaze, his hand encircled her wrist, squeezing it. "No, Dixie. Never." Everything stopped when he spoke those words. There was nothing but Dixie, staring up at him as if she was seeing him for the first time, his blood pumping in his veins, and the crushing thump of his pulse. But she didn't leave him any time to think about it before she wrapped her arms around his neck and pressed her mouth to his. She drove her tongue into his mouth, her lips covering his in a hungry kiss, lifting her legs and wrapping them around his waist. Caine reacted, grinding his cock into the V of her soft thighs, pulling away and clenching his teeth when she tore her mouth from his to put her hands on his chest and roll him from her body. She straddled him right there on the hammock, while crickets chirped and the breeze blew through her tousled hair. His breath caught in his chest, seeing her like this, sitting on top of him, her bare shoulder exposed from his anxious hands tugging at her sweater. The rise and fall of her chest. Her waist, narrowed and flaring out to accentuate the swell of her full hips. Her lips, now red and swollen from their kiss. "Wait," he said huskily, unwilling to lose this moment—this small window of opportunity when their inexorable need for each other wasn't clouded by angry words or revenge. He grabbed hold of her wrists, imprisoning them. "Let me look—touch." Dixie clenched her fingers into fists, resting them on his abdomen, her breathing heavy, her eyes colliding with his. Caine walked his fingers up over her shoulder and along her neck until he reached her mouth. He pressed his index finger to her lips. "I want you, Dixie. Need you. Now." She shivered at his insistent words. "I need you, too, Caine," she breathed before reaching for his belt buckle. Dixie's eyes captured his for only a moment before she pulled the belt open and unzipped his jeans. He throbbed against the tight cotton of his boxer-briefs for agonizing seconds, and then her hands were around him, stroking him as she slid downward. Her hands tugged at the material at his waist, yanking his jeans down until they were around his ankles. The cool air washed over his skin. Caine's boots dug into the soft earth to keep them anchored and to prevent himself from losing his mind when Dixie's hot breath made contact with his bare cock. His hiss of pleasure wheezed from his lungs at her lips, wrapping around him and taking him into her mouth with a slow swipe of her tongue. His hips bucked upward as she swirled it around the heated length of him, dragging along the sensitive skin with raspy precision. His hands went to her head, palming the back of it, following the movement of her up-and-down motion. "Christ, Dixie. More. I need more." He ground out the demand just seconds before her passes intensified. White-hot need raged in his veins, his cock swelled in the wet cavern of her mouth, and his balls drew up tight against his body in preparation. Caine's hand found hers gripped firmly around him. He covered it, increasing the pressure, clenching his teeth as their fingers entwined. The tips of his fingers drifted into the wet heat of her mouth, and Dixie nipped at them, pushing them aside. He drove upward, heedless to anything but the wet heat of her tongue, and the soft noises she made as she devoured the length of him. He reared upward one last time when she used her tongue to tease the sensitive spot just beneath the head of his cock before he pulled back, and her hands took the place of her mouth. Caine lost control when she whispered, "Come, Caine. Come for me." He exploded in a brilliant flash of color, and Dixie's soft hair splayed out over his abdomen. His lungs screamed out a rush of air, and his cock pulsed in her hand with hot release. He reached for her shoulders, pulling her upward until her forehead rested on his chin, fighting for breath. Her arms slipped under his shoulders the familiar way they always had after she made love to him and his cradled her, fusing them together. The easy sway of the hammock rocked them, while Caine struggled to find the words that would express what was happening to him. Both of them jumped when the floodlight attached to his mother's porch bathed them in the glare of light. Jo-Lynne's voice seeped from behind the screened windows. "Attention Caine Donovan and Dixie Davis! If I find one wayward bra in my backyard, someone's in for a lickin'! Now both of you make yourselves fit to sit at my kitchen table and join me inside for some pecan pie. Fully clothed, please!" Dixie muffled a giggle against his shirt, her shoulders crumbling in the effort. "Caught," he said on a laugh. She nodded with a snort. "Like a fly on sticky paper." Before they succumbed to the decadence of his mother's pecan pie, he needed to know—to understand—what had brought about the question of Louella. Or more importantly, who had planted that notion in her pretty head. Her urgency left him restless to settle whatever had her so troubled. He cupped the back of her head, kissing the top of it. "So hey, that talk? After pecan pie and coffee? Jo-Lynne's not exactly known for her patience." She sat up, the warmth of her body replaced with the cool air of nothing but space between them. The carefree look of moments ago was replaced with a troubled pair of eyes, but then Dixie smiled, leaning in to press a quick kiss to his lips. "After pecan pie. And there are tissues in my purse." Dixie squirmed off the hammock, toeing the jeans around his ankles with a grin full of mischief. She began to back up into a light trot toward the house and yelled, "Race ya to the kitchen, Donovan. Last one in gets no pie!" Caine pushed off the hammock with his knees and dived for the shadow of the big maple in his mother's backyard, grunting when he tripped on the tangle of his jeans and hit the ground hard. Reaching for Dixie's purse, he made good use of the tissues and pulled his jeans back up. Rolling to a sitting position, he glanced up at the warm glow in the kitchen, shining through the screen porch windows. Dixie was pouring coffee with one hand while the fingers on her other folded napkins, and his mother sliced her famous pecan pie. They were smiling and chatting, their mouths moving in time with their bobbing heads, making him smile along with them. Caine's gut tightened. It was exactly like the visual he'd created in his mind a long time ago when he'd thought about their future together. Exactly. Eighteen Em and Dixie strolled arm in arm across the square toward the brightly lit gazebo where the annual Founder's Day parade was wrapping up and the slide show of Plum Orchard's founding fathers would begin. The early-evening air was sweet with the scent of cotton candy and hot buttered popcorn. Children played in the grassy area in front of the gazebo, small American flags in hand. Replicas of pilgrim hats and Puritan bonnets tossed on blankets as far as the eye could see made Dixie smile. This was what her childhood had been filled with: town events concocted specifically to inspire gatherings full of family, food and Plum Orchard residential pride. Plum Orchard celebrated their Founder's Day in the tradition of Thanksgiving—only without the turkey and dressing. Her father had always told her it was exactly like when the pilgrims hopped off the Mayflower at Plymouth. Except no one accused the original settlers of Plum Orchard of stealin' anything, he would joke with a chuckle. Em plucked at the sweater covering Dixie's arm as they strolled past a local dressed as a pilgrim, making balloon animals for the children. "So, I hear there was some funny business in the backyard of one Miss Jo-Lynne Donovan yesterday. I thought I was your person. You're supposed to tell your person about all funny business." Dixie stopped in her tracks, rocking back on her heels. "Is there nothing sacred in this town?" "Oh, now stop bein' offended. Digger was in Madge's, and he casually remarked he'd seen you and Caine swingin' on the hammock in Jo-Lynne's backyard when he let Dewie out for his nightly run. I just assumed funny business occurred because it always does whenever you two are anywhere near each other." Dixie breathed a sigh of relief. The last thing she needed was everyone in town abuzz with her private matters. She was still trying to process Caine's denial. Process the fact that she'd been beating herself up, agonizing over her lack of self-esteem and willpower, only to discover nothing had happened between Caine and Louella. Her gut told her Caine wasn't lying. "So did you talk to him?" Em pressed, stopping in front of a table brimming with punch and delectable desserts made by the locals. "Sort of. I think I cleared one thing up. Hey, as my person, can I tell you something? You don't have to get involved if you don't want to. Just say the word, and I'll hush." Em's face was relaxed and open. "Tell away." "Caine said he didn't sleep with Louella." Em's red-glossed lips thinned. She tucked a strand of her hair back into the neat bun at the back of her head and clucked her tongue. "Told you I didn't believe it. Not for a daggone second. The impression I got was that Louella was just salvaging what was left of her pride. As if to say the smoke from Caine's love haze for you had finally cleared, and he realized she was the better woman. But I never totally believed it because I saw that man's face after you hightailed it outta there, Dixie. Once the dust settled, he was hurtin'. We could all see it even though he tried to hide it while he handled the aftermath." Caine's pain, voiced out loud, was like a knife in her back. Em's words left her breathless. "So you asked him about it?" Dixie gulped in some air. "I did, and he said never." No, Dixie. Never. Those words had haunted her throughout last night while she'd waited for Caine to come back to the big house so they could talk. Finally, around dawn, she'd given up and fallen asleep, but before she had, she'd had an epiphany. The agony she'd suffered just thinking something had gone on between Louella and Caine had been true, was nothing less than she deserved. She'd deserved to have it chip away at her heart bit by ugly bit. "Hah! I knew it," Em chirped, slapping her hand against her thigh. "That Louella's a lyin' snake! I think we all knew it. We were all just too afraid to call her a phony to her face. But not anymore. No, ma'am. I'm gonna march right up to her and—" Dixie grabbed at her arm. "No, Emmaline!" Em's gasp of shock rang out. "No? No? What is this word no? Maybe I'll even beat it out of her with a microphone." Dixie took a deep breath and closed her eyes. When she reopened them, she shot Em a pleading look. "I'm going to ask you to let it go. It's enough that Louella knows she lied. When she goes to bed at night, she's not right with herself. That's plenty of guilt. Trust me." Em gave her a sour look. "You know, Miss Dixie, there's a time for letting things go, and there's a time for loadin' up your gun for bear. Louella made you miserable for over a month now, and you want to just forget it ever happened?" "It's nothing I didn't do to her, Em," she whispered harshly, letting her eyes skim the crowd that had begun to gather for the Founder's Day speech. "So I spent a month fretting over something that didn't happen? That's nothing compared to what I did to her in high school." Em's sigh of exasperation sliced through the air, raspy and crisp. She put her hands in the wide pockets of her ruffled skirt and rocked back on her heels. "You were in high school, for heaven's sake! Not well over the age of consent, still carrying a childhood grudge. You made a stupid bet with Louella. Yes, it was childish and for no other reason than to show her you could best her, but it was most certainly not like claiming you'd made the hokey-pokey with her man. You've done nothin' but good since you stepped back on Plum Orchard soil, and still, you beat yourself up. The Mother Teresa act ends now, Dixie. You stink at pulling it off anyway. Mother Teresa would never have red hair with gaudy nails to match. And if you won't let me take that viper Louella to task, at the very least, promise me you'll stop this ridiculous need to accept whatever's handed to you as a bizarre form of just deserts!" Dixie broke the intense moment by pinching Em's cheeks with a grin. "Does this mean you like me now, Em? Like really, really like me? Because I like you," she teased. Em burst into a fit of giggles. "It means I like you better 'n I did a month and a half ago, but no more than that. Hear me?" Dixie rolled her eyes. "If I can't have you, I don't want nobody, baby," she sang, doing her version of John Travolta's infamous Stayin' Alive move by jabbing her finger up and down. But Em wasn't impressed. She folded her arms over her chest and gave her an eyeful of admonishment. "Oh, Dixie Davis, how the tides have turned." "Yup." She nodded, dancing around her in a circle. "You wanna use it against me at a later date? Dixie Davis Begs Emmaline Amos to Be Her BFF. It'd make a good Plum Orchard Gazette headline." Em bumped butts with her and wiggled, letting out another giggle. "Gloating is a deadly sin." She grabbed Dixie's hand and twirled her. "That's gluttony," Dixie said, ducking under her arm and giggling with her. Em's laughter mingled with Dixie's, but the sudden sharp screech of the microphone in the center of the gazebo made them both turn their heads. "C'mon. It looks like Mayor Hale's gettin' ready to give his umpteenth, long-winded Founder's Day speech. I don't want the boys to miss seeing their daddy's picture up on that big projection screen." Em's soon-to-be ex-husband's family would be honored tonight in the slide show they did every year as a memorial to those who'd made Plum Orchard what it was today. Dixie hoped it would make Em's boys smile again, proud their father—even if he wasn't physically here—had left his mark. After spending a great deal of time with Em's sons in the guesthouse pool and on several very messy dining excursions, Dixie privately worried about Clifton Junior and his angry take on his parents' divorce. He was so sullen and withdrawn, it hurt her to the very core to see him so torn and broken. They made their way across the grass, stepping over spilled cups and popcorn kernels. Dixie's eyes loosely scanned the crowd for Caine while she kept an eye out for Em's mother and the boys. A chubby older hand rose from a quilted blanket in the corner of the square, waving at them. Em's face lit up at the sight of her mother and two sons. Their smiles, so much like Em's, were covered in gooey pink and blue cotton candy. Gareth jumped up from the ground, propelling his stout five-year-old body at Dixie's legs. His arms wrapped around her, and he smiled up at her. "Miss Dixie!" She bent and scooped him up, swinging him in the air before settling him on her hip. "Well, who's this fine-lookin' gentleman?" She lifted his pilgrim hat and tweaked his nose. "It's me, Miss Dixie. Garef," he replied with serious, indignant eyes. The pronunciation of his name melted her on the spot. "Oh, my gravy, it is!" She batted her eyelashes at him. "Why, I hardly recognized you without your swimmies, Gareth. When did you get to be such a man?" Dixie walked her fingers up his stomach and planted a kiss on his sticky cheek. "I'm not no man. I'm five." He held up five sticky fingers to prove it. She gave him a squeeze before setting him back on the blanket in order to dig through her purse to find some wet wipes. Clifton Junior sat quietly, as stoic as she'd ever seen any eight-year-old, on the edge of the quilt, his eyes, the same blue as Em's, taking everything in and sharing none of it. Her heart shuddered in her chest for Clifton. He was angry and hurt over his father's departure, according to Em. He had little to say these days—not even to the pediatric divorce therapist Em had insisted he make weekly visits to. Dixie wiped Gareth up, said her hellos to Em's mother then plunked down beside him, fanning her dress out over her knees and following his eyes to a delicious man with a child around Clifton's age. The man, with hair so dark it gleamed like the feathers on a raven, and features so chiseled you could strike a match on them, threw a baseball to a young girl with fiery red hair. He was gorgeous and graceful, and between throws, he was eyeing up Em as if she was the last woman on earth. Clifton Junior's hungry eyes watched the ball, but he said nothing. He didn't have to. His eyes did all the talking. Damn Clifton Senior for abandoning his sons. Dixie looked off into the distance with him, taking on the same air of indifference he exuded. "Do you like baseball?" His small shoulders shrugged under his SpongeBob sweatshirt with a hearty dose of that indifference. "Sort of." "Really? Not me. I think it's kind of stupid." "Everything's stupid." Dixie nodded. "Yep. Pretty much everything." He almost looked at her, but instead lifted his chin, giving it a defiant edge. "You don't think everything's stupid. You're an adult." "I do so. And adults think stuff's stupid, too. Kids aren't the only ones allowed to hate stupid stuff." Clifton's lips thinned at her response, clearly irritated his efforts to spoil a lovely evening weren't working out. "Being here is stupid. Who cares about Founder's Day anyway? My dad's not even here to see it." Dixie let a bored sigh escape her lips while gazing up at the fading sun. "Yeah. Who wants to sit around and eat cotton candy and popcorn? I mean, seriously. And sparklers? For sissies." "And girls." "Not this girl. No way would I even consider holding a sparkler." "Really?" "Truly." "You're not like most girls." "Nope. I'm a bona fide not-like-most-girls kind of girl. Ask anyone." "You don't make any sense." "Which is probably pretty stupid." He laughed. It was faint, and it wasn't willingly, but it was. "You're weird." Em gasped her disapproval, latching onto Clifton's chin and forcing him to look at her. "Clifton Junior! You apologize this instant to Miss Dixie! I'm sorry, Dixie! I don't know what's come over him as of late." Dixie held up a lazy hand, one that indicated she was neither upset nor offended. "No, no. Clifton and I were just talking about things that are stupid. And being weird is definitely not stupid. Right, Clifton?" She gave him a sidelong glance and winked as if they had a special secret. He smiled back at her. A real smile. One filled with the kind of mischief eight-year-olds should have glued to their sweet faces at all times. "Right, Miss Dixie, and I didn't mean that you were weird in a rude way. I meant it in a good way," he said, before jumping up and running toward the direction of the good-looking man and his daughter. He didn't approach them. Rather he stood on the outskirts of their game of catch, watching. Em leaned into her and patted Dixie's leg with affection. "He smiled. I haven't seen him smile in near three months now. Leave it to you and your charm to turn even the crankiest, most sullen of boys into a puddle of goo." "It just takes time, I guess. He's missin' Clifton Senior. He'll adjust. I know he will. He's of your loins. That means he's a fighter just like you." "My loins..." Em snorted, settling in next to her between a basket of fried chicken and the potato salad. "Speaking of your loins, I was wondering...did they have anything to do with him?" She hitched her jaw in the direction of tall, dark and lickable. "Because I don't think wow covers all—" Dixie made a circular motion with her hand "—that." Em peered into the twilight of the coming evening, halting all movement. Dixie saw her throat work up and down in a nervous gesture, and her hand, always a sure sign she was flustered, go directly to her throat. "My." "Uh-huh. He's worthy of an ohhhh," she cooed. "So is he that Jax you were talking about?" "So what if he is?" Her response was defensive, as if it were unseemly to be caught looking at any other man but Clifton. "Because if it is, it's my duty to tell you, he's been gobbling you up with his eyes since you plunked your cute behind on the ground." "Oh, he has not." "Has so, has so, has so. Watch. In three—two—one." And right on cue, the gorgeous Jax looked up in Em's direction. Their eyes met and held, locking gazes for a brief moment. Even Dixie held her breath at the sizzle between the two of them before he broke the spell by nodding to Em in acknowledgment, then returned to his game of catch. Dixie cleared her throat. "So don't you ever tell me you're not attractive, Emmaline Amos. Because you managed to attract the likes of Mr. Sinfully Sexy on a Stick. Go, you." "I did not," Em blustered, hiding her eyes with the curtain of her hair by unleashing her tight bun and giving her locks a shake. "He was probably looking at you. It's neither here nor there. I'm not on the market. Speaking of markets, where's Caine?" "He texted me to tell me he had some emergency with work in Miami, and he'd be back soon." She was a bagful of vulnerability. She didn't understand where last night left them, if it left them anywhere at all. She was afraid this new Dixie and Caine was nothing more than a long goodbye. Caine hadn't made any commitments to her last night other than he was willing to hear her out, and he hadn't even shown up for that. Caine's words had contradicted themselves. He'd said if he won Call Girls, he could run it from Miami. Yet, when he'd looked up at her, lying beneath her on that hammock, and whispered, I need you, with such intensity, she'd been certain they meant something more than just physically. Or had they? Win or lose Call Girls, he could go right back to his life in Miami and run the company from some beach house while women in colorful bikinis strolled past his lounge chair in the sand under a blazing Florida sky. He still might not trust her the way she needed him to. Worrying that Caine was just holding his breath until she made another horrible mistake was no longer an option. He had to trust her enough to believe she'd begun to make the right choices. How could anyone live like that when you knew you'd surely make more mistakes? Her chest tightened when she realized, she couldn't go on with Caine this way anymore. Wouldn't. Em was dead on. Dixie was all about punishing herself, and being with Caine, making love to him, spending time with him while almost certain she'd lose him all over again, was the biggest form of self-inflicted harm going. To postpone the inevitable, to drag out the end, hurt almost as much as letting go. The very idea of letting Caine go—actually saying the words out loud—was almost more than she could wrap her mind around. Her throat went dry, and her limbs went buttery soft simply thinking about it. Somewhere, in the crazy recesses of her mind, she'd always thought she and Caine would find each other again. Now that she'd grown up and experienced one of the hardest journeys of her life before returning to Plum Orchard, she knew that was all just make-believe. But it had been the old Dixie's way of thinking. The reformed Dixie saw Caine needed more substance in a woman. He needed trust—deep trust. He needed someone who would match his brand of integrity one selfless act at a time. As the world continued around her, as children waved sparklers, as people laughed and chatted, and the high school marching band pounded out a patriotic beat, everything screamed to a screeching halt for her. She would do the right thing and stop torturing herself by ending whatever this was with Caine. She'd apologize to him once and for all, and then she'd walk away—for good. Even if everything inside her rebelled against the very idea. Her fingers twisted the quilt beneath her to quell the bone-deep ache. Leaning forward, Dixie scrunched her eyes shut and fought for breath when her phone buzzed. Her hand blindly reached for it, stupidly hoping it might be Caine—afraid it might be Caine. Do you know how much I love you, Dixie-Cup? More than I love my own spleen. That's a whole lotta love. Love yourself that much, too, would you, please? She read the text from Landon again, hearing him in her head as if he was sitting right next to her, eating Em's mother's potato salad. She pinched the bridge of her nose. Love yourself, Dixie. Love yourself enough to know when to let go. Upon Landon's words, Dixie knew exactly what she had to do. "Hey! Where'd you go there?" Em asked, smiling down at her, holding her hand out to offer her help up. "The slide show's about to start. Gareth wants to be front and center to see his grandparents and Clifton Senior up on that big screen. You comin'?" Dixie took a ragged breath and accepted Em's hand with a squeeze. "I wouldn't miss it." She held out her hand to Gareth who smiled his chubby grin from behind Em's knees. "Let's go see Daddy, little man." Mayor Hale had begun to read off the prestigious list of the founders of Plum Orchard with their accompanying pictures while Em, Dixie and Gareth plunged into the crowd, pushing their way toward the front to get closer to the gazebo. Sandwiched between a scowling Nanette and Reverend Watson, Dixie swung Gareth up in her arms so he could see the screen. Em tugged on his T-shirt from behind. "Daddy's next!" "Next up," Mayor Hale's voice rumbled into the microphone, "we have the Amos family, five generations strong and still going. One of our original founding fathers..." She forced herself to focus on the pictures projected and Mayor Hale's deep drone rather than dwell on what she planned to tell Caine. "And next up, Clifton Amos Senior!" The crowd politely clapped and Dixie squeezed Gareth to show her excitement for him, bouncing him up and down. "Son of Harlow and Idalee, one-time all-state football champ, six-time 4-H blue ribbon holder, father of two fine young men, Clifton Junior, now a strapping eight years old, and little Gareth, five, and last—but certainly not least—husband to the lovely Emmaline Amos. Join me in a round of applause for such an esteemed member of our little burg, Clifton Amos, everybody!" The projection screen flashed to the next picture, but the hands that had so heartily joined together moments ago, stopped—dead. She heard Em's shrill scream just before the silence. From the corner of her eye, Dixie caught her covering her mouth in order to muffle it. There were gasps from the crowd—piercing and outraged. There were faint whispers—hissed and uncomfortable. Then there was silence, and nothing but Gareth's words when he placed his chubby hands on each of her cheeks and stared down at her with stoic confusion. "Miss Dixie, that ain't my daddy up there. That's a lady." A lady? Oh, sweet fancy Moses, no. Nineteen The silence that met Dixie's ears was deafening. No one moved for what felt like an eternity while she put two and two together. No. Dixie tore her eyes from Gareth's face and looked up at the screen behind Mayor Hale's receding hairline. No. She closed her eyes tight. Please. Please, whoever's in charge, you can have whatever you want from me, if when I open my eyes, this all turns out to be a horrible dream. Her eyes popped open again followed by her mouth. Clifton swam before her line of vision, larger than life. But he wasn't the Clifton of the plaid flannel hunting jacket and John Deere ball cap. He was, according to the small bio beneath his picture, Trixy LeMieux and second runner-up in the Atlanta Miss Cross Dresser 2014 pageant. Dixie's eyes widened as she absorbed Clifton dressed as a woman. He wore a sapphire dress, dazzling in sequins from his toes to the curve-hugging bodice where breasts spilled from the sleeveless frock. Enviable silver stilettos with black, glossy heels, pointed outward in a pose reminiscent of Miss America, from beneath the tapered hemline. He wore heavy makeup, thickly applied to his jaw to cover his stubble line, flirty false eyelashes, silver and pink glitter eye shadow, and bloodred lipstick. The crown on his head sat atop long, thick locks in a gleaming ebony where they fell, tumbling into an enormous bouquet of red roses. Clifton eyed the camera with a half smile as kittenish as any Dixie'd ever been able to produce. Yet there was something in his eyes, something that stuck with her. He looked happy. Happy and alive. As the yelling around her reeling head began to penetrate her slow-motion brain, Dixie clung to Gareth, instinctively spinning around so he wouldn't see any more, making a frantic search for Em. Em stormed toward her, her fists tight at her sides, her feet crashing against the ground. "Give me my child," she seethed between her clenched teeth, her red lips twisted into a sneer. "Em! Oh, God, how can I help?" Who would have done something like this? How had someone found out? Em handed Gareth to her mother who had Clifton Junior by the hand. "Take them where they can't hear me, Mama," she ordered in a tone that held one part ferocious, two parts livid. The three scurried off into the dark, their backs becoming tiny dots until the night swallowed them up. Dixie gripped Em's arm in an urgent gesture. All she needed to do was get her away from the gaping, judgmental eyes of all of Plum Orchard. They'd figure out the rest when they found somewhere more private than the square. "Hold on to me. I'll get you out of here." "The—hell—you—will!" she roared. She yanked her arm from Dixie's grasp and shook her off with a hard shove. The crowd of people around them parted, making a half circle of more astonished eyes. Nanette and Essie hovered on the fringes of the circle, arms hooked together, eyes narrowed in Dixie's direction. Dixie frowned, not understanding. "Em! C'mon. Let's get out of here!" Instead of accepting Dixie's hand, Em circled her, her chest heaving beneath the scooped neckline of her red dress. Her sweet face, always so open and warm, was a closed mask of barely contained anger. Her mouth twisted into a sneer. "I'd rather be flayed alive under the noonday sun and have vinegar poured in my open wounds than ever go anywhere with you again, Dixie Davis!" Startled, Dixie still didn't understand the words coming from her person's mouth. "What?" Em moved in on her, shoving her face at Dixie, her finger pointed and suddenly digging into the spot beneath Dixie's collarbone as if she was hammering a nail. "What?" She mimicked Dixie's surprise in a sticky-sweet impression, giving her a poke so full of rage, it knocked Dixie backward. "Poor, poor persecuted Dixie. Everyone's always picking on sorry little ol' you. Don't you pull that—that—" she jammed her face farther into Dixie's "—bullshit with me! Don't you dare 'what' me like you're all innocence and light! You were only one of two souls alive who knew about Clifton. You don't expect me to believe it was Marybell, do you? Do you?" she screamed. Dixie's heart throbbed in her ears while ice ran in her veins. "Em! No! I swear to you on everything I have, I didn't tell anyone. I made a promise to you—" Em's snort ripped through the strange quiet of the square, cutting her off. "Just like you made a promise to Caine, Dixie? You're a filthy liar and a disgraceful human being. I told you, Dixie. I warned you, if you hurt my boys, you'd pay!" Dixie was too stunned to speak. Her instinct to run, to hide from such a horrible accusation was thwarted only by the fleeting thought that this was what vengeance, the karma she so deserved, was all about. But Em couldn't possibly believe she'd told anyone her secret. "Em, please." She reached for Em's arm only to have it snatched from her vicinity as if she was riddled with plague. Dixie held up her hands and backed away, forcing herself to remain calm. There was an explanation. There had to be an explanation. "Please listen to me. I would never, ever hurt those boys. I would never hurt you. I know I've said things like this before, but I swear to you, I never told a soul. Not a solitary soul. No one talks to me, Em. Everyone treats me like the town pariah. Think about that. Who'd listen to me if I told them anyway?" Dixie's eyes sought Em's, pleading, blinking back tears. "Ohhh, you're good, Dixie. So, so good," she said with dripping sarcasm. "Those big, wide eyes, those crocodile tears. I'm sure you'd love me to believe that, wouldn't you? Stupid Emmaline, always chasin' after the trail of pathetic breadcrumbs you leave in your wake like some lovesick puppy. I'm sure you'd love for me to believe you had nothing to do with this so I could go on bein' your loyal minion. Well, no—more!" she bellowed. Dixie held her ground. She ignored the spiteful eyes of the Mags, and the disgusted eyes of the Senior Mags. She turned her cheek to the onlookers, gawking at her public demise with knowing nods of their heads. One more time, she forced herself to stay calm even as her stomach rebelled. "Em, I'm begging you. Please listen to me. Let's go somewhere where we can sit down and talk about this rationally, and you'll see I'm telling you the truth. I did not tell anyone. No one." "Well, you musta told someone, Dixie! You might not have done that," she pointed at the projection screen, still displaying the damning image of Clifton/Trixie. "But it got out somehow!" Dixie shook her head, fighting her hysteria in order to make Em believe her. "Em, I'm telling you the truth—" "The truth? You wouldn't know the truth if it slapped you in your two faces! Maybe you just wanted back in with those disgusting Mags, and this was your way to get it. A big juicy bit of gossip for you to spread like the vermin you all are." Em whipped around, turning her rant to the crowd, her eyes wild and shining with fury even as tears streamed from her eyes. "And yes!" she yelled, her scream coming out harshly hoarse, her finger stabbing out her targets. "I called the lot of you vermin, always takin' pleasure in someone else's pain. You hear me, Louella Palmer, Annabelle Pruitt, and Lesta-Sue Arnold? I meant all of you with your vicious mouths and backstabbin' ways! Be warned, Dixie, you will pay for hurting my children! I hope you all, and especially you, Dixie Davis, rot in hell!" she spat before spinning around and running toward the curb. Dixie watched as Em careened right into the man who'd watched her while he played catch with his little girl. He righted her with large hands, his eyes under the streetlamp full of concern, before Em let out a sharp cry and tore herself from his reach. Dixie closed her eyes and swallowed so hard, she was sure it echoed around the square. The eyes of Plum Orchard burned holes into her back as sure as if they were holding lit matches to her skin. And still, she couldn't move. Not as their horror over Em's accusations sank in, and the twitter of chatter ignited. Not even as some residents made wide circles around her as if her mere existence would bring them harm. She didn't move an inch, not a muscle. For all the hurt that had resulted from the cruel pranks she'd once perpetrated, she would not run from everyone's chance to finally hold her accountable. She lifted her chin and let them all look, refusing to look away. Let them nod their heads as if to say, No big surprise here. As she stared straight ahead, she almost wished Caine were here. Funny, even though she'd resolved to end things between them she still looked for him, needed his chest to bury her face in. In her head, she heard Landon's voice, Sugarpie, someday, for all the mean things you've done, you're gonna have to pay the piper, and I'm bettin' it won't be pretty. Someday had arrived. * * * She didn't know how long she stood there. The only thing she could see was Em's face, angry and riddled with hurt, and it had immobilized her. The only thing she could hear was Em's stifled cry of shock, Gareth's innocent question, Clifton Junior's tears as he sobbed, and Em's damning words. You're a filthy liar. A disgraceful excuse for a human being! over and over again as though someone had pressed replay. If it had only been minutes, it certainly felt like an eternity judging from her stiff legs and ice-cold hands. Whether she breathed during that time also escaped her, though when she finally exhaled, she almost gagged. The litter in the deserted square came into focus, just as the goose bumps on her arms made her shiver. Her clenched fingers, achy and tight, released. She shook them, looking down to find they were red and splotchy from the cold. Dixie wrapped her arms around her waist and sank to her haunches, taking slow, deep breaths, aching for Em and the boys. How could she make this right so Em would allow her to help her through this? "If it isn't Plum Orchard's favorite snitch." Her head snapped up to see Louella, smiling down at her, ankles crossed, arms folded at her chest and eyes full of smug delight. Dixie buried her face in her arms. "Go away, Louella," she mumbled into her folded arms, not caring that she was obviously here to gloat. Louella toed the grass at Dixie's feet with her royal blue pump. "Somebody had a rough night." Misery prevented Dixie from rising. Everything in her hurt. "Somebody asked you to go away." "What comes around goes around, they say." "Then they enjoyed a little full circle tonight, didn't they?" Louella's laughter slipped from her lips. "I'll say. You really did it this time, Dixie. This was almost as good as your engagement party." Her pride nudged her from the inside. Just because you're paying for all those sins, doesn't mean you have to be Louella's whipping boy. Stand up, Dixie. Look her in the eye, take it like a man, but don't take it sitting down. Dixie pushed off her knees with her hands and rose. "Okay, you've had your due, right? See mean Dixie taken down a notch while everyone points and laughs. But in the process someone was hurt, Louella. Someone who's always kind to everyone. Worse, there were children involved. I can't think of anything more despicable than involving innocent children." Louella's eyes shifted from smug to dark for a fleeting moment before settling on haughty. "Maybe you should have thought about that before you told everyone her secret. Who knew the king of the rednecks liked sparkly pretties? Surprise!" Her anger spiked hard and fast, and it was all Dixie could do not to snake her hand out and flatten her palm against Louella's taut cheek. "Don't speak those words to me, Louella. I won't defend myself to the likes of you, but I won't have you speak about Em or Clifton that way either. Em's the most decent human being I know. How dare you poke at her when she's hurting so badly." "You mean just like you used to do?" "Is that enough? Or do you want blood?" Dixie held out her wrists as an offering. Louella made a face and rolled her eyes. "No. I just want you gone. You're bad news, Dixie. Everybody knows it. So why don't you go on back to Chicago, and take your bevy of trashy women with you." Suddenly, she was bone-weary. Exhausted from either having her guard up all the time, or trying to live down her bad reputation. "Look, Louella, let's just call a truce, okay? I'm asking you this because, though there were plenty of bad times, we had some good ones, too." Louella put a finger to her chin as if in thought. "You mean like the time you knew I was crazy about Caine and you went out with him anyway? That was super good." Dixie's sigh was ragged. "Okay, it's time for some reality here. You've let this eat you up for too long, and you've lost perspective. First, I wasn't in that bet alone. Second, you couldn't have known you loved Caine any more than I did. You'd been on one date with him. The rest was just hero-worship from afar, a high school crush. How do you know you would have fallen in love with him? Maybe you would have hated each other." "Or maybe we wouldn't have," she said angrily. "You didn't exactly give him the chance to find out before you were all over him like some kind of fungus. You knew how I felt about him, and you chased after him anyway. So we'll never know, will we?" Someone needed to give Louella the cold splash of reality she so deserved. She was so eaten up by her petty jealousy and anger she was headed straight for where Dixie was right now. "Obviously, he didn't want to find out, Louella. He didn't ask for another date. I'm not saying that to be cruel. I'm just saying it because it's the truth. But you won anyway, didn't you? You won big. Remember the microphone suspiciously placed by your phone at our engagement party? Remember the chaos?" The fire that broke out when Dixie had chased after Caine and knocked down all those beautiful candles. The pounding thunder and rain that had come out of nowhere like some foreboding sign. God, she'd never forget the words she'd said. I win, you lose, Louella Palmer! That'll teach you to bet Dixie Davis when it comes to a marriage proposal. Wish Caine and I lots of little Donovans. "You righted all the wrongs. We broke up. I can't figure why you're still so angry about it almost ten years later. You won in the end." "I did that to right a potential wrong, Dixie. Marrying you was wrong. No man should marry a woman who was just in it to do a victory lap. You didn't deserve Caine then, and you don't deserve him now." More truth. But she still had an ounce left of reason in her, and she was going to spend it on Louella because she had nothing to lose. "But I don't have him now. So if you still want him so badly, why don't you go get him? You got him mere minutes after I left our engagement party, right? Isn't that what you told Em and the rest of the Mags?" Louella's mouth opened to construct a plausible denial, but Dixie held up a hand to stop her. "Let's lay all our cards on the table. You wanted me to believe you'd slept with Caine. Please don't insult me by denying it. You wouldn't have told the Mags, and Em specifically, if you didn't. It's a weapon to hurt me, but I'm here to tell you, whether you did or not, I'm over it. Caine was a free man, and you were free, too. Isn't the knowledge that it would hurt me enough payback for what I did? And if it isn't, what is? What will ever be enough?" Dixie got the impression Louella didn't know the answer to that question any more than she did. "You're a sad excuse for a human being, Dixie. You turn my stomach." She didn't have the energy left to fight with Louella when the very thought of what Em was going through was on the table. "Yes, yes, yes, Louella. I was a horrible person. Said it a hundred times since I've been back. The point is I'm trying to be better, but you're taking up where I left off. Do you want to end up like me? Like this? Didn't you just bear witness to what happens when you're an absolute bitch? It bites you in the ass." Louella's face had disbelief written all over it. "When did this change come about? I'd love to know. We'd all love to know, in fact. Care to share what made coldhearted snake Dixie turn into Dixie-Do-Gooder?" No. There'd be no sharing something that was none of her business. It was personal and a deep pain she'd live with always. Ignoring the question, Dixie appealed to Louella from a different angle. "Here's the score. The girls and I aren't hurting anyone. We're just trying to make a living. Why does that make you see red? Is it because I'm involved in it? Are you just out to make me miserable simply because I exist? Because I have some news for you, I plan to exist for a long time." Louella's silence gave Dixie hope she was at least reaching her. She stuck her hand out to Louella, an olive branch of fingers attached to a wrist. "So can we just let this grudge go, please? People who had nothing to do with what happened between us are getting hurt. LaDawn and Marybell have never uttered an unkind word to you, so why involve them? To hurt me? There's nothing you have left in your bag of tricks that can hurt me, Louella. I'm all hurt out. Everyone knows everythin' there is to know about my jaded past. All the sharp thorns left in my backside have scarred over." More silence as Dixie watched the wheels turn inside Louella's head. She hoped that meant Louella was considering her proposition. "So if you stop, I'll stop," Dixie continued. "Swear it. And I'll never steal another man from you for as long as I live. My hand to God," she joked, hoping to lighten their conversation. "Deal?" Louella looked at Dixie's hand as if it was smeared in cow dung, turning her nose up at it. She leaned in close so Dixie wouldn't miss a venomous word. "I won't stop until you're gone from this town, Dixie. You and your bunch of ex-hooker friends are a blight on Plum Orchard's good name. No one wanted you to come back to begin with. After what you did to poor Em tonight, I'd bet they want you here even less." Louella spun on her heel, picking her way across the square, leaving behind a haunting tune of pending doom playing in Dixie's head. * * * Sanjeev's quiet presence made her turn around to acknowledge him before returning to packing her bags. Mona and Lisa stretched their bulky bodies, whimpering their joy at his presence. "So you're running away?" "You heard?" "Who hasn't, Dixie? The Maldives heard. They called, by the way. Something about an offer for a place to hide?" Dixie snorted. "Don't you start on me, Sanjeev. I'm not running away or hiding because I did something wrong. I would never, ever betray Em's trust. Are you kidding me? What little trust I had was like earning a seat on the space shuttle to Mars. But I can't seem to convince anyone of that." She'd texted Em five times since she'd walked back from the square to nothing but profound silence. And it hurt. But it had proven something to her. No matter what the situation, no one would ever say, "Not a snowball's chance in hell would Dixie Davis ever do that." She'd always be the scapegoat. While that was more than fair, it was no way to live. Sanjeev placed a comforting hand on her shoulder. "You've convinced me, Dixie. Landon always said you'd come around. He was right. I believe." She scooped up a pile of her underwear and dumped them into the suitcase. "I'm a real beacon of hope for mean girls everywhere." "Does this mean you'll forfeit the contest, too, Dixie?" Yes. It meant she was forfeiting everything that had to do with Plum Orchard. Caine, Em, the Mags, and Call Girls. For a little while, she'd convinced herself you really could come home again. She'd basked in the familiar, fallen in love all over again with her small town, despite the small minds and gossip. She'd found a way to ignore the comments and whispers—until tonight. "Yes. I'm forfeiting. I'm only causing trouble for the girls anyway, Sanjeev. Louella and every last one of the Mags want them gone. Me in the mix makes her want them doubly gone. If I'm out, maybe she'll ease up a little and let them live their lives in peace." "And if she doesn't?" "Poison some of that fabulous curry you make and take it to her house for dinner, for me, would you?" Sanjeev laughed, tightening his hand on her shoulder and forcing her to turn around. "Dixie?" "Sanjeev?" His somber eyes, full of peace and understanding, gazed into hers. "It's always darkest before the dawn." Her clipped laughter dripped sarcasm. "It doesn't mean my days aren't always going to be dark just because the sun comes up. I think I've proven that. You can have buckets full of sunshine and still have a really crappy day." He tilted his dark head, his deep brown eyes searching hers. "I think what it really means is things have to be intolerable before they're tolerable." She cupped his cheek and stroked the smooth, enviably perfect skin with her thumb. "Look at you, all learning the ways of the American. And after all this time. I'm so proud." But Sanjeev frowned and shook his head. "Who'll see to Em and the boys? Who'll protect her from that wretch Louella and her evil Magnolia posse if not you?" Dixie turned away and threw a batch of socks on top of her underwear. "I can't protect someone who would rather see my entrails wrapped around a tree and tied in a bow, Sanjeev." "It isn't like you haven't stood up to things like this before. It never stopped you in the past." "In the past I had no shame. Nowadays, I have so much I could open a shame bank and share my endless supply with the shameless of the world." "I wish you'd reconsider, Dixie. You're meant to be here." Dropping one of her scarves into the pile, she twisted her fingers into it to keep from screaming her frustration. "I wish you weren't the only one that believed me. But you're an island, my friend." "You do know that your past will always make you have to prove yourself in the present, don't you?" "Is that your way of telling me whenever something horrible happens, I'm always going to be the most likely suspect?" "I suppose it is." "Well, thanks for clearing that up. If I wasn't aware of that before tonight, I'm pretty clear now." "You're being saucy." "And just a little mouthy," she admitted. "I could do it as long as the gossip and sly comments were just about me. I can handle it because I invented it. But now it involves Em and her children. I can't allow that. I won't." "Did you speak to Emmaline?" "She's not talking to me. I want to help, Sanjeev. Can you even imagine how horrible this will be for her and the boys? She was so afraid everyone would find out. She knew exactly how everyone would react, and she was right." "That you're more concerned for Emmaline and the boys than your own persecution surely proves you would never share such a sensitive issue, doesn't it?" "Not when you've done the things I've done, Sanjeev. Wasn't it you just a minute ago who said I'd always be suspect? Though, I wish someone would look at the evidence a little closer. What could I possibly gain from exposing something like that? Or from gossiping about it? When I did some of the things I did, I did them to gain something. Like a boyfriend who had a car. Or a seat on the Miss Cherokee Rose float." "Who do you think would do something so distasteful? Are you certain no one else knew?" Sanjeev prodded. Dixie bobbed her head. "The only other person who knew was Marybell. She was there when Em told us, and I can promise you, she'd never betray Em." Sanjeev's face grew skeptical. "Of that you're sure?" Yes. She knew Marybell would never hurt anyone, let alone Em. "I'm positive, Sanjeev. I wouldn't even insult her by asking her a question like that. So how did someone all the way here in Plum Orchard find out about what Clifton's been doing in Atlanta? Besides, Marybell wouldn't think twice about someone cross-dressing. She's a phone-sex operator, for gravy's sake. It's no big deal to her. She loves Em. Everyone at Call Girls does." She loved Em, damn it. Sanjeev clapped his hands together as if they'd made some revelation. "Then we've narrowed our suspects, yes? It was clearly someone who is mean of spirit. I can name at least one woman. Cough—Louella Palmer—cough," he joked, running his words together. Defeat crushed her just thinking about trying to prove it. If Louella had done this, she wouldn't leave a trail of evidence that led back to her. She wasn't that stupid. "Listen, Matlock, we're not narrowing anything. I almost don't want to know who'd do something so cruel because it would be all I could do not to bring the old Dixie back for some good old-fashioned stoning. And what would Louella gain by doing something so evil?" Sanjeev, usually so serene and unruffled, sighed in aggravation. He gripped her arm, making her stop what she was doing. "She'd hurt you, Dixie, of course! It's clear she'll use any avenue to get to you. You don't really think she opposes Call Girls, do you? She opposes you being here because of Call Girls, which is why she attacked LaDawn and Marybell. She saw your friendship with Emmaline blossoming. If she makes you miserable enough to give her exactly what she wants, which is your swift departure back to Chicago, she'll use whoever and whatever it takes to do so. How can you not see that?" Dixie shook her head in disbelief. "But this involved Em's children, Sanjeev. That's unbelievably cruel, even for Louella." "Bah," he groused, throwing an impatient hand up. "She eats nails for breakfast and kicks walkers out from under unsuspecting senior citizens. Louella's blinded by her jealousy of you. As you well know, that ugly emotion can produce uncharacteristic behavior." That emotion she was painfully aware of. "Nobody knows the green-eyed monster like I do. In fact, I think I slept with him once." Sanjeev wasn't biting at her joke. He was, however, becoming exasperated with her. The tic in his jaw pulsed, a rare occurrence for him to become ruffled. "Then the answer is clear. You have to stay in order to slay the monster." "I can't slay something if my heroine is uncooperative. Em's never going to forgive me. And if what you say about Louella turns out to be true, then I'm going back to Chicago where no one knows me enough to hate my guts quite like that—or hurt anyone else who's even remotely involved with me." "Ah, to lick your wounds," he baited with a sly smile. She clenched her teeth together. "Yes." "And to poverty." "That, too." And to work at McDonald's until she paid off her debts. "And Caine?" Right. And Caine. Dixie's heart sank like a rock, landing in her stomach and settling there. "What about Caine?" she asked nonchalantly. "You'll leave without saying goodbye to him? He'll be distressed to see he didn't have the chance to crow his phone-sex victory." "I haven't seen him since last night. When the smoke clears, I'm sure he'll resurface. Give him my best." Her offhand attitude helped hide the searing pain in her heart. "How unlike him to disappear when there's a damsel in distress." "Well, you're forgetting who the damsel is. I'd disappear if I could, if I were him, too." "Still, very curious..." Dixie breathed deeply, forcing her next words out of her mouth. "It's better we don't see each other anyway." "Really then? I wouldn't have thought that was how you felt by the way the two of you have seen so much of each other recently." Her stomach twisted into a tight knot. She closed her eyes to wash away Caine's face—to keep the memory of their last moments together at bay. "That's all part of our dysfunction. We can't stay away from each other. But it's time to end the cycle. I won't ever be the kind of woman Caine needs, and I definitely can't live up to his high expectations. There's no breathing room for me to screw up—or be accused of screwing up. If I'm always going to be suspect, I'll spend every waking moment proving to him and everyone else I'm innocent if something goes wrong. That's no way to have a relationship, and you know it, Sanjeev. It's done, and that's that." "Then there's nothing left to say but goodbye, is there?" Dixie dropped her makeup bag and turned around when she heard how hoarse Sanjeev's voice had become. He held out his arms, and she went to him, squeezing her eyes shut to keep from shedding more tears. "I'll miss you so much, Sanjeev. Thank you—for everything. The wonderful meals, the company, but most of all, for your friendship." "If I can't talk you into staying, will you at least remember this one thing?" "More twisted clichés and metaphors to suit my pathetic existence?" "Mona likes the cheesy beef kibble, but particular Lisa will only eat the chicken." Dixie laughed into his shoulder and nodded, squeezing him extra hard so she had his soothing warmth to carry her back to Chicago. Sanjeev pulled away first, tilting her chin upward with one finger, his eyes glossy. "Until next we meet, Dixie Davis." "Until next time," she whispered, biting the inside of her cheek to fight the tremble of her lips. And then he was gone, and Dixie was left alone with nothing but the tick of the clock on her dresser. Right back where she'd begun. Twenty Caine dropped into the ugly red chair in his Miami condo living room and made a face at the chair the overpriced interior designer Landon had sent to redecorate when he'd bought the place had chosen. "It's still damn ugly, and it's still uncomfortable, and I don't give a shit if it's facing the ocean," he complained to his phone after reading another of Landon's texts. Are you lonesome tonight? Is your heart filled with pain? Will you come back again? Tell me, Caine, are you lonesome tonight? It was as if the bastard knew what he was doing—how he felt at any given moment. And yes, he was lonesome. Lonesome for home, lonesome for family, lonesome for Dixie and he'd only been gone a day. There. All out in the open now. Now, what to do about it. How to go about it. How to fix this for good. How to trust again. How to explain everything to Dixie. He closed his eyes and took a deep breath—when he opened them again, he saw nothing that made him feel as if any of this was even his. Everything in his condo had been picked out by someone else, carefully placed by a stranger's hand, and he hated most of it. And he never realized how much he hated it until just this moment. How much he'd dreaded getting on that plane to come back to it. He wanted to go home and stay home—for good. With Dixie. If she'd have him after what he'd done. Maybe she'd be angry. Maybe she'd be really angry. He'd been hard on her. Too hard. Too unforgiving. An asshole. He'd tested, he'd pushed, he'd taunted—and still, she'd damn well soldiered on. He wanted this Dixie. Maybe more than he'd ever wanted her. But maybe she won't want you? Maybe not, but he wasn't going out without a fight. His phone chirped, leaving him leery. One more taunting text message from the grave...his eyes fell to the phone, concern instantly replacing his irritability. "Sanjeev? Is everything okay?" "No, Caine Donovan, nothing is okay." As he listened to Sanjeev tell him what happened at the Founder's Day picnic, his blood boiled over. "Son of a bitch," he growled into the phone. "Something must be done, Caine." "I'll handle it. Thanks, buddy." Sanjeev's sigh of relief only added to his anxiety. Sanjeev rarely stuck his nose into anything. When he did, Landon always said that was when you should sit up and take notice. Caine sat up, his fingers skipping through a quick Google search. Tapping the number that came up, he added it to his phone and dialed it while his teeth clenched tight. "Louella? Caine Donovan. We need to talk." * * * "I'm sorry to drag you in here on your last night in Plum Orchard, Dixie, but Marybell's down with the flu, and you know how busy Saturday nights are around here," Cat apologized. "You were the only option. Sheree's out of town, and LaDawn can only handle so many calls. I'll divert the calls for the girls to you. Just do your best." Dixie's plan had been to leave as fast as she could throw her bag and Mona and Lisa into her car, and drive until she was too tired to drive anymore, until Cat called and asked if she'd take Marybell's shift. "I'm happy to help," she offered dully, adjusting the box for her personal belongings. "No you're not. You want to get in your car and drive the hell out of here as fast as your beat-up old car can go." "Like I'm on fire," she tried to joke, but it sounded bleak and depressing. Cat pulled her into a hug, rubbing her back. "I wish you'd reconsider. You were beating Caine by at least twenty-five percent. You could've had it all if that kept up." Dixie breathed deeply, her stomach a jumble of butterflies, all fighting to find their way out of her intestines. "If all means Em hating my guts and the rest of Plum Orchard giving me the stink eye, I'd rather not have it all. I'll settle for just some." "Not a single one of us believe you were responsible, Dixie. Not one of us. Em'll come around." Dixie forced herself to smile. She would not be pathetic. If she was going, she was going with her head held high. "Not even LaDawn? Look at me, winning friends and influencing people." Cat held her away from her. "LaDawn was your biggest supporter. She offered to find out who did this to Em and drop them off at her old place of business in Atlanta." "Aw, LaDawn and me sittin' in a tree. Surely her offer to have someone beaten to death for me is a sign of her undying devotion," she teased, gripping Cat's hand. Despite her preconceived notions about what the women of Call Girls would be like, she'd fallen in love with them. Had come to respect their grit and determination to survive. "I'm really going to miss you guys." "You don't gotta miss us if you don't go nowhere, Dixie. Closed mouths don't get fed, as my mama, rest her soul, used to say! Cain't win a contest if you ain't playin'!" LaDawn shouted from her office. Dixie couldn't help but laugh before Cat sobered and said, "You're only givin' that awful Louella what she wants, Dixie. You know that, right?" "Maybe so, but if she's responsible for what happened to Em, what's the right thing to do, Cat? Stay here so she can cause more damage just so I can say I didn't let Louella Palmer beat me?" Cat's expression was grim. "I see your point, but—" Dixie shook her head, moving toward the hallway leading to her office. "No buts. Em's children's suffering is absolutely not an option. Now, no more of this sad talk. Send some of those LaDawn calls my way, so I can leave y'all proud." She padded down the hallway. As she passed LaDawn's office, she taunted good-naturedly, "Look out, LaDawn! Mistress Taboo's in the house, and she's gonna turn your floggers inside out!" LaDawn's cackle filled her ears, making Dixie's heart clench. "You go on, Vanilla. Do me proud now—show 'em you da man!" * * * Headset in place, Dixie gathered her personal belongings from her desk, taking a last glimpse of the picture of her and Landon. She traced his beaming smile with her fingertip, and wondered what he'd have to say about this mess. "Well, now look, would you? A fine mess you've made, pal. I hope you're happy up there, you big pain in my ass," she said fondly. The realization that Landon had done this—the challenge between her and Caine for his company, the text messages—all of it—had been about bringing her home. Sanjeev's words, You were meant to be here said with such conviction had been the final clue. He knew what Landon's motives were. He had all along. Even in death, Landon had been trying to save her. He knew coming home, confronting her past, tying up all of the ugly loose ends she'd left dangling was what she'd need to finally move forward. She might not have known she was ready to come home, though. So he'd invented a reason that would keep her here longer than just attending his funeral. He'd known familiar places and treasured faces would comfort her at a time when her entire world was upside down. Someday, she'd like to know how he'd known what was really going on with her life in Chicago. He just hadn't counted on the unforeseeable. Her earpiece rang again, making her sit up straighter. Taking over for LaDawn meant she'd better be on her toes. It took her mind off leaving, off the possibility that she'd never see Caine again. "This is Mistress Taboo. Are you worthy?" "Mistress Taboo, it's real fine to hear your voice." Walker. His deep, rumbly voice made her heart begin that excited thrum. Then it took a nosedive. In all her mad preparation to leave, she hadn't given thought to what she'd say to the list of clients she'd managed to accrue—or if she'd say anything at all. But Walker probably wouldn't be too disappointed by her hanging up her phone-sex hat. He didn't really call for advice. Why did Walker call? "Walker—it's nice to hear your voice, too." "You don't sound like yourself tonight, Mistress Taboo. Why's that?" How uncanny he should sense that when she was so focused on sounding normal. "I think I'm catching a cold." It was as good of an excuse as any. "My mama always used to say, some milk boiled with onions and pepper in it'll cure all your ills." Where had she heard that before? "Ugh. I think I'll stick to over-the-counter stuff." "So how've you been, Mistress Taboo? What's new in your world since we last talked?" "Well, I do have a little bit of news, and you're the first client I'm sharing it with." "I'm all sorts of honored." "I'm leaving Call Girls." Her heart squeezed at the admission. Walker's pause was long before he cleared his throat and asked, "Better prospects somewhere else?" Sure. Some might call the fast food industry a better prospect. "I don't know if you'd call it better, but it's better all round, better for me." Better for Em and her boys. She would miss Em so much. "For you? You've got me frettin' now, Mistress Taboo. Does it have to do with what we talked about the last time?" There was no twirling her hair or flushed cheeks over his supposed concern this time, just a dead weight in the pit of her stomach. "How is it we're always talking about me?" "Because I'd bet my last dime you're prettier. Prettier wins the spotlight. Plus, if you're leaving Call Girls, who's it hurtin'? We won't talk to each other after tonight." His soothing voice somehow managed to lull her anxiety. He was right. What difference would it make if she told him the truth? She relented, her reluctance to share ebbing away. "My past has come back to haunt me in ways I not only deserve, but it's also come back to haunt others, too. It's affected someone who's..." She stumbled. "She's very important to me. If I go back to Chicago, I'm hoping it'll all die down, and she'll be able to have some peace." "So this isn't about the former fiancé you talked about? I'm confused." It would always be about Caine, wouldn't it? "It's about him, and my friend, one who was reluctant to befriend me after the pranks I played on her in high school. It's about everyone, I guess," she said with stark honesty. "I guess the point here is I come from a place where everybody knows everybody, a small town that can be very judgmental and set in its ways. The things I did here, the people I hurt, well...they don't forget. And I don't blame them. But a recent incident reminded me of something. I'll always be the usual suspect for everything that goes wrong. No matter how I go about redeeming myself, the first bad thing that happens, everyone's going to look to me." "Do you want to share this latest incident?" he asked tentatively with what sounded like genuine concern. Dixie shook her head at Walker's "no pressure here" inquiry even as she cringed with the memory of last night. "Not on your life. I will tell you, it made the Rapture look like a church social. Suffice it to say, it was ugly, and it devastated someone I really love in front of a bunch of people. It hurt members of her family, too. That's the worst part of this." What would it be like for Clifton Junior and Gareth at school come Monday? Children could be so incredibly cruel. She couldn't bear the idea that everyone would be laughing and teasing them. Especially as fine a line as Clifton Junior was walking. "Any idea who might have done this?" Dixie tried to pinpoint something, anything from her conversation with Louella that was even a small hint she'd been involved, but she couldn't think of anything other than the boys and Em. "There's only one person I can think of. And if it was her, this person, she would have done it to hurt me. I can't live with that. I won't allow it." "Still don't see what that has to do with your ex-fiancé." "Basically, it's just another nail in my coffin. When he hears about what happened, I'll be the first person he suspects talked out of turn and spilled my friend's secret, because I'm the only other person alive who knew about it." The more she said that out loud, the harder the time she was having processing who could've found out about Clifton. Certainly no one from Plum Orchard frequented a place where they held a cross-dressing pageant. Or did they? "So he's pretty judgmental." She leaned back in the chair, a visual of Caine's handsome face, smiling up at her, floating before her eyes. "No," she whispered into the phone, the true ache of longing strong and harsh. "He's not judgmental. I used to think he was, but now I realize, he just lives his life to a pretty high standard. He's an amazing person. The kind of person who expects everyone else to be as honest and decent as he is. And he should expect that, but it's damn hard to live up to when you know you're bound to make more mistakes. Perfect and I are about as far apart as they get." "Do you really think he expects perfection from you? That's rather unrealistic." "No. Maybe I'm not saying this right. I just mean that he deserves a woman who's as amazing as he is. I think if something like this happens again, he'll always suspect me first—even if he doesn't say it out loud. Who wouldn't? My past, our past, will always get in the way of him trusting me fully. Is that any way for two people to live? Him always waiting for the hammer to drop? Me always unsure if he completely trusts me. He deserves better, so much better. But I deserve better, too." As she said the words, Dixie nodded her head with conviction. She did deserve that. She'd earned it. "So clean-slate mentality?" Wouldn't that be a relief? She shrugged her shoulders, pushing the mouse on her desk back and forth. "I guess I'd like a clean slate. I'd like to start over with people who know me as I am now, who aren't tainted by the things I've done to hurt them." "So answer me this, why did you do all those things to people?" She dropped her head to her hand. "I've learned a lot about myself since I've been back, and one of those things was sort of coming to terms with my relationship with my mother. She wasn't an easy woman to please. She's...all about appearances and how much money she has in her bank account. So I stopped trying to please her and went out of my way to do everything in my power to displease her. I guess I rebelled. Big. It's certainly no excuse, but I think I manipulated and lied and forced people to give me what I needed because she didn't." "That's pretty honest. So what brought about this huge change in your life?" Dixie closed her eyes and swallowed. She'd never confessed this to anyone. Not even Landon—because it was a pain that might have dulled, but it would never really leave. It would always taint her soul. It would always ache when she least expected it. "What do you have to lose by sharing, Mistress Taboo? If this is our last phone call, let's go out with a bang, huh?" he coaxed. "Someone did to me what I'd done to countless others." There. She'd said it. She hated it, but she'd done it. "Like?" "Used me and discarded me." The raspy breath Walker took almost hurt her ear. "How?" he said, his question tight and clipped, striking Dixie as an odd reaction to someone he hardly knew. "It's so cliché," she half joked. But wasn't that always the way life paid you back—in the way of a cliché you should've seen coming from a mile off? "Tell me anyway. Pretend I'm someone else if it makes you feel comfortable." Now that the floodgates were open, there was no stopping her. No stopping the guilt and anguish that ate her up from the moment she woke, until the moment she put her head on her pillow. She hid it with jokes, but it was always there. It was the truth she'd wanted to share with Caine. So he understood she'd not just changed—she'd had a complete soul makeover. From the deepest part of her, her core had experienced a shift. There'd been plenty of dark during that shift, but then there'd been light. And it was that light she looked for whenever she began to falter. "When I left home and went back to Chicago after my engagement ended, I was angry and bitter, sort of in an 'I'll show you' head-on disaster way. It's stupid, I know, but I acted out as if the person I was hoping to hurt with my behavior would actually see it or hear about it. I wanted him to think I'd gone right on living like he was just a bump in the road." "The ex-fiancé again." "Yes. He was my target. Anyway, I behaved badly. God. So badly." She closed her eyes to stave off the memories of how badly. "I stayed out all night. Drank too much, ignored my duties at my restaurant. You know the drill." "I'm familiar with it." "I became involved with a man, a rich man. It started out as me using him much in the way I did everyone who had something I wanted. He had a nice car, a nice credit card. All the stuff mean girls like me live for. So we became involved...." "Did you love him?" His question sounded as if he'd asked it with a locked jaw. Had she loved Mason? When she compared what she felt for Caine to how she'd felt about Mason, no. "No, it wasn't like that. After a while, I even liked his companionship more than I liked his credit card. But I wouldn't call what we shared love exactly. Though, arrogant ass that I was, I was sure he loved me. I was wrong, but at the time, I thought I held all the cards the way I always had." "So what happened?" Her fingers trembled, clutching the edge of her desk until her knuckles were white. Just say it, Dixie. Say it. Own it. The silence between them crackled with an expectant hum. "Mistress Taboo?" "I got pregnant." This time, the pause Walker made was painfully long. "You have a child?" he rasped into the phone. Dixie closed her eyes, her throat threatening to close up. "No." The word shot from her lips like a ball from a cannon. "No. I lost the baby. I miscarried three months into my pregnancy." God, she didn't know if she could do this. It was a mistake to share something so personal. The hurt returned like a hammer, pounding nails into her heart. "Damn. I'm sorry. So damn sorry." His remorse sounded so genuine that she found herself reassuring him. "Don't be. It was almost two and a half years ago now." But sometimes, like now, it still felt like yesterday. "If you say you deserved it because of your past, that it's some kind of karma for all your wrongdoin', I just might have to hunt you down and give you a good what-for. Don't say that, Mistress Taboo. Just don't." Walker's insistent tone, his demand she not blame herself might have been cause for alarm had he been any other caller. Yet, Dixie only felt at ease. "I wouldn't go that far. I did deserve what happened in the aftermath." "What could you have possibly done to deserve fallout after a miscarriage?" "Well, for starters, the person I was involved with was married." She thought she heard him hiss until he asked, "Did you know?" "Not until I was two months pregnant and I finally told him." Thinking about Mason and that conversation now made her shudder. His suspicious disbelief. His cavalier dismissal of her tears. His final cruel words. "So let me guess," Walker said, harsh sarcasm riddling his words. "He offered to help get rid of it." Her disgust for Mason's solution to the problem came out in the way of a repulsed snort. "Among other things. Not that it wasn't a viable option, mind you. It was how he went about offering to dispose of his inconvenience. His callous disregard for how I felt about it. I'd planned to break it off with him anyway. I was foolish enough to believe I could raise the baby on my own, and in being honest with myself, I didn't love him. I didn't want to spend the rest of my life with him. But I did hope he'd want to be a father." "But he couldn't because he was married." The disgust in his voice rang true to Dixie's ears. "Right, and he made it clear he wanted to stay that way." "So he was just using you as a fling?" That no longer made her angry. Nowadays, it just made her sad that he'd go right on hurting his wife the way he had countless others before her. "He sure was. That I couldn't see the signs, that I never had a clue he was married, still blows me away. No one had ever done something like that to me. But Mason had me fooled for almost a year." "And?" "I got into a car accident," she said, almost choking on the words. A horrible, metal-screeching, tire-gnashing collision that was totally her fault. "And lost the baby," Walker responded quietly. "But not a scratch on me..." she almost sobbed. Saying those words years later still held the same amount of crushing disbelief they'd held the first time she'd said it to a roomful of doctors and nurses. "It was almost as if it never happened. I lived, but the baby—my baby died." When Walker spoke again, his voice was gruff. "I don't know what to say." Now that she'd spoken the words that had haunted her to a total stranger, she couldn't seem to stop spilling her guts. "When I woke up in the hospital and realized how much I'd wanted the baby, I also realized something else—something more important. This was what it felt like to be humiliated—used." "That son of a bitch." "But I also realized I had not a single friend to turn to for help with my recuperation. There was no one to talk to about the gut-wrenching ache in my empty stomach with the baby gone, or the huge hole in my heart. There was just me, and an agonizing pain that stole my will to get out of bed each day." "You really had no one?" Dixie's bark of laughter into the quiet room was sarcastic. "After the things I'd done? Who do you call when the person who's always done the kickin' is suddenly the kicked? The only real friend I've ever had was overseas in the Baltic, or on safari, or something, and I couldn't call my mother. It would have infuriated her to find out I was pregnant with a married man's child. The shame alone would have made her turn her back on me. Though," she reflected derisively, "she probably wouldn't have been surprised." Walker swallowed into the phone, his voice husky when he asked, "So what happened next?" A smile rimmed her lips. Hope happened, hope and a hand to guide her to the path of self-forgiveness. "Mrs. Kowalski." "And she was?" She reached for her cell phone and clicked on her gallery of pictures, scrolling to Agnes's. The one of her holding up a bag of fresh peaches she'd scored at the farmer's market in order to make Dixie a peach pie so she'd have something from home to comfort her. Her smiling face, round and wrinkled, made Dixie grin. "Who was Mrs. Kowalski?" "My neighbor from downstairs in my apartment building. She'd seen the accident. It was raining. I was in a rush, and I pulled out in front of a big pickup truck. Anyway, she came to the hospital every day. Brought me homemade cookies, muffins, all sorts of goodies to entice me to eat. When I came home, she'd knitted me a blanket and had a freezer full of casseroles and meals at the ready so I wouldn't have to cook. We became friends—good friends. Over many games of checkers and Parcheesi and lemon Bundt cake, she taught me what it was to give selflessly. She didn't have to nurse me back to health. She had a husband and two granddaughters. Her life was already pretty full. But she did it because she was kind, and loving, and she didn't judge me—not even after I told her everything about my horrid past." "Acceptance," he murmured gently. "That's what healed you." Dixie nodded, setting her phone down and promising herself she'd print out Agnes's pictures and frame them when she got back to Chicago. "That and something Mrs. Kowalski said to me one day that stuck like glue. When I let go of what I am, I become what I might be." "Lao Tzu," Walker provided with a low rumble. She smiled at his wisdom. "You know it?" "I do." That conversation with Agnes still warmed her. "I wanted to be a better person for a baby that was never going to exist. I know that sounds utterly ridiculous, but it's what helped me move forward after my miscarriage, stop blaming myself." "I can't imagine losing a child, let alone thinking you were responsible for that loss." His quiet words washed over her like a warm balm. They rang deep with truth, honest and pure. "But you really weren't, Mistress Taboo. If it didn't happen that night, it might have happened on another. I believe there's a reason for everything, but I don't believe the reason is you were a horrible bitch." Dixie closed her eyes and took a deep breath. "Thank you for that, Walker." "So I'm guessing by this point your restaurant had failed?" "It was mostly in ruins. I tried to save it at the eleventh hour, but it didn't work out. But Mrs. Kowalski's words reminded me, I didn't have to keep going in that direction. So I changed direction and decided no matter what, I'd pay off my debts to the people who took a foolish chance on a slacker like me. And that's why I'm at Call Girls." "And Mrs. Kowalski? Is she still in your life?" Dixie's breathing hitched again, her smile sad. "She died of a heart attack about a year after she saved me from myself. I'll never forget her, and I'll never be able to repay her." "Maybe the idea is to pay it forward to someone who needs saving like you did?" "Maybe it is," she murmured. "So, I'm just gonna keep right on bein' nosy about this, Mistress Taboo. Why haven't you told your ex-fiancé about this? That these are the things that shaped you into who you are today? Wouldn't he understand?" Shame. More shame than any one person had room for. "I wanted to. I was actually going to the other night, but he had to leave town, and now, after what's happened with my friend, there's no point. He'll come back to hear about this mess, and there'll be no goin' back." "So what?" Her eyes widened. "So what? Are you kidding me? Who wants to be persecuted over and over?" Walker barked a rebuttal. "Who wants to have someone make decisions for them they have no right to make?" Dixie's spine stiffened. "Huh?" "You're deciding this man won't believe you at all before he has the chance to decide for himself. Maybe he'll look at you and say, 'Mistress Taboo, I believe. And no one could convince me otherwise.' It sounds like it's because you don't want to face the final piper. If you're so into movin' forward, then wouldn't it only be fair to tell him that? Is he such a jackass he wouldn't at least listen?" "No. It's not that. He'd...listen. It's just that—" "Then what can one more shot at it hurt? The worst that can happen is he says forget it, and you do what you originally intended. Go back to Chicago." She sat in quiet contemplation. "Do you want to run away again, Mistress Taboo?" he taunted. "Didn't you say you did that once when you went to Chicago?" "Ye...yes." "Then quit doing it again, Mistress Taboo." A fire began to simmer in her belly. Maybe Walker was right. Maybe she should just confront Caine. Tell him about Mason and the baby, and how much she needed him to trust she was a different person. Maybe she should search high and low until she found out who'd done this awful thing to Em? Maybe... "Mistress Taboo?" "Yes?" she asked, out of breath and excited. Walker chuckled. "I hear your wheels turnin'." "Maybe you're right. Maybe I should do just that," she said with a definitive nod, her long-lost courage poking its head out. Yeah. She didn't have to settle. She didn't have to be poor Dixie. If she was going to leave this town for good, by hell, she was going down in a blaze of truthful glory! The thought left her breathless and determined. "Thank you, Walker. Thank you." She heard him smile into the phone. "You're most welcome, Mistress Taboo. So I guess this is goodbye, huh?" Dixie grinned. "I'm not sure now that you've pumped me up like I'm headed for a round with Mike Tyson. Maybe I'll stick around to fight another day, huh?" "Boo-yah. Boo-yah." Boo-yah, indeed. "So maybe I'll talk to you soon, Walker, but if I don't, thank you. You're..." "The fire under your butt." She laughed. "Sizzling hot." "Goodbye, Mistress Taboo. It's always been a pleasure." "Until another time, Walker, goodbye," she breathed into the phone, her throat clogged, her chest tight. Clicking off the earpiece, she let her hand linger on her ear as though she could keep him close for just a little longer. Whoever Walker was, wherever he came from, he'd been exactly what she needed tonight. She sat back in her chair, entwining her fingers behind her head and closing her eyes. It was time to hatch mission "Make Everyone Listen." "I'll be your Jewel of the Nile, baby," in Michael Douglas's voice echoed through the wall to her office, catching her attention. She jolted forward on her chair. Caine was back? Maybe he'd taken a red-eye? Dixie popped out of her chair, the burning embers Walker had stoked under her backside cheering her on. By hook or by crook, it was time to set the record straight. Dixie threw open the door to her office and rounded the corner to Caine's office door. She pressed her ear to it and heard. "C'mon, baby, take a voyage on the Starship Enterprise," in Jean-Luc's voice. He was here. Her heart began to thrum, and a nervous smile flitted across her lips. It was now or never. She hesitated, her stomach a flutter of butterfly wings. What if... Maybe you should let him decide, Mistress Taboo.Walker's words came back in a rush. Dixie used the side of her fist to bang on the door. "Caine! Open up—it's me. We need to talk!" There it was again. "I love the way you look tonight," Frank Sinatra's voice sang. Wow—this client sure liked variety. "Caine! Open up," she yelled, banging the wood again. "Dixie! What are you doing?" Cat asked, her face worried. "Caine's not here. He's in Miami until tomorrow, honey." Dixie grabbed Cat and pulled her close to the door. "Then he has a ghost or a twin in there. Listen." "C'mon, baby. Take a voyage on the Starship Enterprise," Jean-Luc repeated. Cat's mouth fell open just as Dixie reached for the doorknob. It was locked. Cat dug around in her jeans and pulled out a shiny brass key with a conspiratorial smile, her hands steady as she drove it into the lock. The door popped open, and they both peered inside. To an empty office and a phone hooked up to a charger. The phone buzzed bright and blue each time it spoke—recorded messages. "I'll be your hunk-a-hunk-a burnin' love," Elvis said. Dixie's eyes narrowed. Cat's did, too. Caine had been cheating? Maybe even all along? Ohhh! Oh, the dirty, low-down, let's-be-friends, son of a horse's ass! No disrespect to Miss Jo-Lynne, of course. Dixie scooped up the phone and eyed it. All the tuning him out so she wouldn't have to suffer the indignity of hearing him talk dirty to a string of women, and he'd never been talking to anyone at all. They were the same phrases she heard over and over just before she put her headset on and listened to some loud Waylon Jennings until she had a call. Oh. Oh, Caine Donovan. What have you done? Dixie's eyes narrowed again, but her lungs worked right as rain when she yelped, "Caine Donovan, when I'm through with you, you're gonna wish for a slow, painful death!" The ultimate challenge, the one where she plucked each luscious strand of hair from his head as she poked him with a cattle prod had begun. May the odds be ever in your favor, Candy Caine! Twenty-One "Oh, Candy Caine... Wake up, sugahh plummm," Dixie cooed with a thick drawl in Caine's ear, moving her fingers along his sumptuously hard belly until she skimmed the top of his navy blue boxer-briefs. The reaction she received made her smile that much wider. It was the evening after her discovery Caine had been cheating, and to Dixie's delight, he'd arrived home in the late afternoon, telling Sanjeev he needed a nap. His arms, warm and strong, sculpted and hard, automatically reached for her, but his eyes remained closed. "Is that you, Mistress Taboo?" he asked, deep and sexy-sleepy. "Uh-huh," she whispered, letting her tongue flit over the shell of his ear. Caine moaned low and hot against her cheek, kneading her back with his hands. "To what do I owe this visit? Body snatchers, right? You're Dixie's doppelganger." Dixie giggled in her best breathy Marilyn Monroe. "Now you're just talkin' silly." Her hand roamed over the width of his bicep as she nipped her way down his neck, purring as she went. "Possession? A demon possessed you?" "No possession." "Too much plum wine?" "Why I never," she said, pretending to be offended. "Oh, you have so," he accused, though his voice was light. "Okay, guilty. But I'm definitely not under the influence right now. Unless you count your intoxicating deliciousness." Caine stilled his roving hands. "My what?" Dixie lifted her head, letting her eyes meet his confused gaze. She fluttered her eyelashes. "You heard me." She returned her lips to the heat of his skin, taking pride in his hiss of pleasure. "Question?" he squawked. "Ask," she demanded, enjoying the tension in his muscles as her fingers skimmed the crisp hair on his belly just below the surface of the white band of his underwear. "What's this about?" Letting the length of her hair tease his stomach, something she knew drove him wild, Dixie rolled her head to see if he was looking at her. "You really wanna know what this is about, Candy Caine?" "I do, Mistress Taboo." Just the way she'd planned, she popped up, hopping off his bed. "Here's what this is all about!" she bellowed like some demented warrior, dumping an entire pitcher's worth of ice-cold water on his head. Caine reared up off the bed with a roar, water spraying everywhere. "What the hell?" Dixie bolted for the door with a scream, Caine's Call Girls cell phone in her hand. She waved it above her head as she made a break for her bedroom. "Cheater, cheater, toe jam eater!" she singsonged, slipping into her room and turning to slam the door on his lying face. When she'd hatched her payback, she hadn't counted on Caine still being as quick as he'd ever been. His dripping wet fingers grasped the edge of the heavy door and gave it a good shove, sending her backward to the frantic barks of Mona and Lisa. She stumbled, but caught herself by stabilizing her feet and thanking her Zumba DVD for giving her the balance of a dancer. Dixie held up the phone between them as if it was some sort of super shield. "You low-down, lying, underhanded, self-righteous cheater! Ohhhhh," she sneered up at him. "All that garbage about playing fair and bein' friends all while you were cheating!" Caine, his tanned skin glistening from the water like some Adonis, went all humble and pleading, surprising her. "Okay, first, I'm not a cheater, and if you'll just let me explain, you'll know why I did it." But Dixie threw her hand up, phone in place. "Don't you dare come any closer, cheater! I worked the phones legitimately while you were cheating! I'd bet you never answered a single call! All that Dixie bashing, all that takin' me to task over and over for being underhanded all those years ago, and you were doing the same thing. Racking all those fake numbers up for calls that never even existed. How did you do that, Candy Caine? One of your computer friends from Miami help you out, maybe? Like maybe the same person who designed your trashy website?" Cat had rechecked Caine's numbers, and they were solid. But if he hadn't been talking to anyone, how had he managed to finagle any numbers at all? His chest, glorious and dripping wet, rose and fell. "Dixie, if you'll just hear me out." "You should be ashamed of yourself!" She grabbed a picture of Landon from her nightstand. "Landon's ashamed of you, aren't you, Landon?" Dixie thrust the picture of Landon into Caine's face. "Tell him, Landon." She held the picture back up to her ear with a smirk. "What's that you say? Caine's just as smarmy as I am?" "Dixie," he warned in that oh-so-reasonable tone of his, moving closer, the material of his boxer-briefs sticking to him in all the right places. "If you'll just hear what I have to say, I can clear—" Dixie hopped up on her bed, making Mona and Lisa bark excitedly, cutting Caine off. "Don't you dare come near me, Donovan! Get out of my bedroom right now, or I'll start throwing things and have Sanjeev charge you for them," she threatened, reaching for a very expensive vase with her toe. "Dixie," he said once more, inching closer, leaving behind a pool of water in his wake. "Get—out!" she hollered, snatching the vase up and hurling it at his head with the speed of a torpedo. Caine ducked just in time before the vase crashed against the wall, splintering into a million gold and green pieces. Sanjeev flew around the corner as one of her perfume bottles sailed across the room. Dixie laughed out loud when it cracked and the room filled with the scent of pears. "Get out, Caine!" "Dixie Davis!" Sanjeev bellowed, freezing all movement. Even Mona and Lisa stopped hopping around the bed. "The two of you will cease this instant!" "But!" They both yelled together in protest. Sanjeev threw a hand up, his normally calm eyes flashing all sorts of threats. "Enough! I will not have you desecrate the beauty of my housekeeping with your foolishness. Caine? You will leave this room now. You will go back to yours, and you will mop up every last drop of water on my meticulously maintained carpets!" Caine frowned, but his expression was tame. "But she started it." "Pay very close attention to me, Caine Donovan. I don't care who started this. You will do as I've told you, or you'll suffer the consequences!" Caine's eyes grew petulant. He crossed his arms over his chest in a defiant gesture. "What're you gonna do? Take away my dessert?" "Yes!" Sanjeev whisper-yelled. "Damn," Caine muttered, backing toward the door. Sanjeev's gaze swung to her, all hot and fiery. "And you, Mistress Taboo. You will clean up every last splinter of glass or never shall another of my curries pass your lips. Do we understand each other?" Dixie pouted in response, straightening her sweater and smoothing it over her jeans. "Oh, no, young lady. I'm not Landon. Your sullen act and sultry-pouty lips do not pull the wool over my eyes. Dustpan and broom, nowwwww!" He jabbed his finger toward her bathroom. Dixie dropped on her butt to the bed, bouncing as she fell to the mattress. She tried one last sad face only to be greeted by Sanjeev's surprisingly stern eyes, before slinking off the edge with a huff. Sanjeev turned his back to whistle to the dogs. "Mona, Lisa, come! I won't allow you to dwell with heathens!" While his back was turned, Dixie stuck her tongue out at a contrite Caine, jamming her fingers into her ears and wiggling them. Caine's lips thinned. Sanjeev whipped around as if he had eyes in the back of his head. "Dixie Davis! The bathroom. Now. March!" he ordered. "And not another word from either of you. Not. One." Caine left first, stomping back to his bedroom and slamming the door like a ten-year-old. The moment he left, Dixie collapsed, muffling a fit of giggles. She'd won—fair and square. No tricks, no loopholes, just her and a phone. "This—" Sanjeev muttered, eyes ablaze "—is amusing how, Dixie?" Dixie hopped off the bed and hugged him hard. "I'm sorry I made a mess of your floors, but it was for a good cause." "Define a cause worthy enough to sully my housekeeping reputation?" Dixie gave him another quick hug. "Soon. All shall be revealed soon." Sanjeev gave her a quick squeeze of her shoulder before putting his stern face back in place. "But not before you clean this up!" Dixie waved him off with a chuckle. "Like it never happened. Swear." As Sanjeev took his leave, his face perplexed, Dixie smiled to herself, celebrating her victory on the inside. It was official. She'd won Call Girls fair and square, and that meant she could pay back her investors. But before she decided if she'd run it from Plum Orchard or Chicago, there was one more thing left to do, and it would depend on Em. Two things left to do if you counted Candy Caine. * * * "Why am I here again, Cat?" Dixie groused. She needed to set phase two of her plan in motion, and it didn't involve her operating the phones tonight. "I told you, Dixie. Whatever Marybell had, LaDawn has, too, and now that Caine's out of the picture because he's a dirty, rotten, cheating scoundrel, we need your help. If Call Girls is gonna be yours, you gotta help in the bad times as well as the good." "Okay," she conceded. "Let me get my headset and get settled." She moved toward the hall entryway leading to the offices. Cat spun around in her chair. "Hey! Did I tell ya how glad I am you're stayin', Dixie?" Dixie smiled at her, beaming when she remembered Cat's response to that information at Hank's office. They'd both gone and tattled on Caine together first thing this morning while Caine slept after a red-eye in from Miami, proving he'd been cheating. Caine had found a way to rack up fake phone calls to his operator line, and fake money to the Call Girls kitty, but there was absolutely no way he'd been taking real calls like she had. Oddly, he hadn't inflated his numbers, though he'd still cheated because, according to Hank, his clients weren't genuine Call Girls callers. Oh, technicalities, how I love thee, she'd yelped in triumph. All of Dixie's clients were real, and when Hank had ruled as such, Cat had jumped up and down over the fact that Dixie was considering staying in Plum Orchard. Dixie sent Cat a warm smile. "I think you screamed your approval once or twice." Cat rushed up behind her and gave her a hug. "Well, lemme say it again. I'm happy you're the new boss." "Happier than if Caine was your boss?" she teased, tugging at Cat's hair. "Now, don't go makin' me pick. I like Caine, too. I know there's an altruistic reason he did what he did. We just don't know it yet." "Oh, fine. Keep the truth to yourself if it makes you feel better. I'm off to answer those calls." She gave Cat a quick hug and headed to her office. Her office. The words made her beam with pride. She'd done it, and she'd done it without taking advantage, twisting the rules—well, not a lot—and hard work. She hoped somewhere up there, Agnes and Landon were proud. When Dixie entered her workspace, there were pink and purple balloons attached to her chair and a banner on the wall behind her desk that read, You Are Worthy, Mistress Taboo! Dixie grinned, but only briefly before her headset was buzzing. She grabbed it off her desk and placed it over her head, pushing balloons out of her way to sit down. "This is Mistress Taboo, are you worthy?" "I hope I am." "Walker!" she squealed in delight. "I'm so glad you called again." "I'm assuming, because I was able to reach you, you're staying at Call Girls?" She raised a fist upward. "You assume correctly. Boy, what a difference a day makes," she said, giddy and excited about the future for the first time in a long time. "You were so right." "So you talked to your ex-fiancé, then?" "Not quite, but I did revert back to my mean-girl ways—only momentarily. Promise." She was still patting herself on the back, rejuvenated by the sight of Caine, soaking wet in all his hard ab-ed, delicious half nakedness, dripping wet and in shock. She almost heard Walker's smile turn to disappointment. "But I thought those days were over?" "Well, so did I, until I caught him cheating." "Cheating?" Walker rasped as though he were defending Caine. "He sure doesn't sound like the kind of guy who'd cheat. Not after the way you described him. Where'd Mr. Honorable go?" "Well, I didn't think so either, but here we are." "So he had another woman?" "No! It's one of those long stories I'm infamous for, but here's the gist of it." Dixie briefly explained Landon, and the challenge to win Call Girls, and how they'd come to find out Caine hadn't ever taken any calls at all. Walker whistled into the phone. "Wow. He's a crafty guy. Whaddaya suppose his motive for that was?" Dixie paused for the thousandth time, wondering the same thing. "I don't know. It's really strange. He didn't inflate his call numbers—which is what I would have done back in the day if I'd had access to something like that. So it wasn't like he was trying to beat me that way. It makes no sense. He knew if he cheated he'd have to forfeit the company." "Do you want to find out why he did it? I might be able to help." Dixie stopped all motion. How could Walker help her find out what Caine had been up to? A rush of alarmed goose bumps skittered along her spine, but it didn't stop her from saying, "More than I want to keep a kidney." "Can you meet me by the pool outside the guesthouse?" The pool? Outside? Now there weren't just alarms going off in her head, she was hearing DEFCON bells and whistles. "My pool? In my town?" "Right here in Plum Orchard," he said low and sweet. Dixie looked around at her office with suspicion, her breaths coming in choppy puffs. How did Walker know about Plum Orchard? Oh, God, what had she done by confiding in him? He'd tracked where her calls were coming from, and now he was stalking her. Damn technology. She forced herself to slow down. "Okay, what's going on? This isn't funny." "Just come to the pool," he reassured in that lazy Southern accent. "And keep the phone to your ear while you walk over." Dixie froze, her hands like ice. She shook her head. "No. If this is some kind of joke—not funny." "Don't be afraid, Dixie." Panic screamed in her veins. He knew her name. Oh, Dixie Davis, of all the foolhardy, stupid, ill-informed things you've done, talking to a man who calls for sex on a phone has to be right up there with a too-stupid-to-live horror-movie heroine! "It's okay, Dixie. I promise," he coaxed, silky and smooth. Dixie hesitated only a moment before she began making a break for the reception area where she'd left Cat, like, at any moment, Walker was going to pop out of the bedroom closet door, gleaming knife in hand. She pushed out of her door and found all three women, Marybell, Cat, and LaDawn, waiting on her with expectant faces. Her mouth fell open then snapped shut as her mind raced forward. "I—I thought you had the flu, LaDawn? Hold on. What's going on here?" "Don't you have a phone call right now, Mistress Taboo?" LaDawn reminded, smiling wide. "Remember what my mama said, closed mouth don't get no food. You need to eat. Keep talkin', honey." Cat shooed her forward toward the door leading out to the pool as Marybell grabbed her hand with an excited giggle and virtually dragged her through the entryway door. Dixie gripped the phone to her ear, even as her feet struggled to keep up with Marybell's. Her fear was replaced with bouts of panic. "Walker! I want you to tell me what's going on right now. This isn't funny, you hear? Not laughing!" But Walker was laughing. Right in her ear. "Keep walking, Dixie. Trust, would you? There has to be trust in a relationship, right? Isn't that what you said?" As they entered the pool area, Dixie stopped dead, refusing to move another inch. She pulled her hand from Marybell's grip and turned on the women. "You three better tell me what's going on right now! I'm your boss now, you know, and if you thought for one fine second—" "Dixie?" Walker interrupted her rant. "What?" she hissed into her earpiece. "Look up, honey," he insisted. Oh, no she would not. In fact, she was going to close her eyes tighter. "I will not." LaDawn gripped her chin and forced it upward. "Look up, boss, or I'm gonna pry your eyes open with my fingers like I'm pryin' open a can of sardines." Her eyes propped open just as she was mentally putting on her boxing gloves to take on LaDawn. And then her jaw dropped open to the sounds of the girls cheering, and Caine wiggling his fingers at her. Always Caine. Standing on the far side of the pool amidst the twinkling lights, as handsome as ever, a phone to his ear, he held out his hand to her. His eyes were tender, and his smile was wide. Dixie's heart responded with a clench. LaDawn, Cat and Marybell squealed in delight, each giving her a quick hug before nudging her toward Caine who took her hand and pulled her close to his side. Just to his left stood Em, holding Gareth in one arm and Clifton Junior's hand. Her face wreathed in a warm smile directed at Dixie. Her confused expression was greeted with laughter and a cheerful vibe from everyone she didn't understand. And there was cake just off to Caine's left, a beautiful confection of tier after tier of sparkling white iced cake with the words, You Are Worthy. But forget the cake. Caine was Walker? She'd kill him. Twenty-Two Gareth leaned forward and placed both of his hands on either side of Dixie's cheeks. "My daddy's a lady just like you, Dixie," he informed her with a chubby smile. "But you're prettier," he exclaimed, making everyone laugh. Despite her shock, Dixie couldn't resist reaching up and squeezing his wrists before he was swept away, and Em was reaching for her, her words tear-filled, her face full of anguish and worry. "I'm so sorry. As long as I live, Dixie Davis, I will never, ever doubt you again. Say you'll forgive me or I'll have to throw myself off the bridge all dramatic-like. That's always messy cleanup for Coon Ryder. You don't want poor old Coon overworked, do you? I need a person, Dixie," she pleaded. "Obviously, I really need a person to stuff a dirty sock in my big trap after my awful behavior the other night. Say you'll be my person again?" Dixie didn't know what to say. No sooner had she grasped the idea that whatever was going on was about her, but Em was taking back what she'd said last night. "But I don't understand...." "You will," Em whispered in her ear, grinning at her. "Right now, all you have to say is you'll be my person again." Dixie squeezed her hand, trying not to choke on her words or her disbelief. "I'll always be your person, Em. For as long as you want me." Caine pulled her back to him, fitting her to his lean side, holding up a microphone to his mouth. Seeing him with the phone in his hand made every fuzzy moment up till now clear. Like a bolt of thunder, Dixie was struck and everything came together. "Wait! Warm milk boiled with an onion and pepper," she whispered up to him. Caine laughed. "Mama always said it was the best cure for a cold. Can't believe I slipped up like that." "You're Walker...." she murmured as all the confessing and cleansing came back to her in humiliating droves. "You?" "I'm Walker, honey. Don't you remember? It's my grandfather's middle name, and the one my mother flat-out refused to give me?" Even though she nodded her head with a chuckle, she tweaked his arm, wrapped tightly around her hard, her senses returning in full force. "You do realize you have to die for toying with me this way, right? You'll pay, Donovan. Maybe not right now because I can't process this totally, but pay you will." He turned Dixie toward him, pulling her close so their lower bodies fastened together. He dropped a kiss on her nose, brushing her hair from her eyes with tender fingers. "Aw, c'mon now, Mistress Taboo. Don't be mad. I had to find a way to talk to you again without all the noise between us. There was just too much of it. We were always at each other's throats. Calling you as Walker was the perfect solution. In a sense, it gave both of us a clean slate. I wanted to get to know this new Dixie. The one who protected Em at Madge's and went a round with Louella. Once I started calling you, I couldn't stop. And even though I knew you'd kill me if you knew, it helped me to understand you in a way I never did before. I get it now, Dixie," he said, husky and low, his blue eyes honing in on her. "I get you." Dixie's lust for the lopping off of Caine's head ebbed, replaced with shock. Her words stuck to the roof of her mouth like thick peanut butter, refusing to cooperate. "Forgive me, honey?" Caine asked, sweet as pie. So many emotions assaulted her. This was Caine asking her for forgiveness. A Caine who now knew all her dirty secrets. A Caine who knew all her secrets and was still standing beside her. But wait... On tiptoe, Dixie rose and pressed a kiss to his lips, savoring the firm silkiness before she let him have it. "Not just yet, Candy Caine. We have to talk privately. But first, explain the rigged phone calls. You were cheating! Can I tell you the flak I've taken for that sort of behavior?" His smile was amused and smug—and perfect—so perfect. "I had a software developer I know hack the system and create calls that never really existed. It looked like calls were coming through, and my numbers reflected that. Then I had him hack Call Girls' bank accounts and dump some money into it so I wouldn't get caught." Her eyes narrowed even if she had to admit it was genius. "Clever you." "Right?" he teased. She poked him in the ribs with a hard jab. "And exactly where were you while you were jeweling everyone's Nile?" He barked a laugh. "You're gonna kill me." "I can only do that once. So make sure you cover everything in your confession." His granite face went sheepish, but his arms tightened around her. "Sleeping. Not at first. At first I did answer a call or two, but that was damn weird. After that, I recorded what you heard through the wall last night, hit play, locked the door, climbed out the window, and went back and watched TV, hot-tubbed, had some pie with Sanjeev. I just forgot to take the damn thing with me when I went to Miami." Dixie gasped her outrage, but it wasn't real—her heart was too full for true outrage. "Sanjeev knew? My kill list grows." "He knew. But it was all for the greater good, honey. Scout's honor." "Explain your definition of the greater good." "I did it so you'd win." Her head fell back on her shoulders when she laughed out loud. "I knew it! I knew you couldn't be dishonest or cheat. I just didn't know what your motives were for rigging the calls." Dixie softened then, cupping Caine's hard jaw with trembling fingers. "But why would you do it? You could have just gone home, Caine and I would've won anyway." He nodded his dark head, bringing her hand to his lips and brushing his mouth over it. "Yep. But then I couldn't have stayed to torture you—or win your favor." Dixie's heart pumped hard against her ribs, her stomach a bundle of knots, but she managed to give him a coy batting of her eyelashes. She didn't bother to harp on the fact that Caine had handed her Call Girls. She didn't balk at the fact that it was a win by default—something she no longer took any pleasure in. Instead, she smiled at the fact that Caine was in it to win her. "Why, Candy Caine, you want to win my favor? Little ol' me? I do declare. Mistress Taboo is listening. The real question is, are you worthy?" she asked in her best Scarlett O'Hara impression. "You'll see, Miss Davis," he teased, dropping a gentle kiss to her lips, leaving her wanting more. So much more. "But first, I need to know something." "Ask me anything." "You wanna give this thing another go? A real one? No secrets. No agendas, just you, me and a phone-sex business?" All the years of missing him, all the angry words that had passed between them, her guilt over that horrible night, every last doubt she'd had about whether Caine would ever be able to trust her the way she needed him to, exploded and disappeared in one word. "Yes," she whispered, throwing her arms around his neck and lifting her lips for the kiss she'd waited for since the night they'd parted ways. The girls cheered as they surrounded them, but Dixie only heard about that later, when everyone retold the story about the night Caine Donovan threw a multimillion-dollar contest in Dixie Davis's favor just to win her heart. She was too busy kissing the life out of him. * * * Dixie caught Caine's gaze from across the pool where he was parked beneath a palm tree, holding a longneck and listening to LaDawn and Marybell tell him how they'd met Landon. She tilted her head toward the road leading away from Landon's and to his mother's house just off the square with a come-hither smile, waggling a finger at him in invitation. Recognizing what she meant, Caine wiggled his eyebrows with a delicious grin and held up two fingers to signify he needed a couple of minutes. The evening was cool, and she only had a T-shirt on. She rubbed her arms and smiled at the pool, abuzz with food and music and faces that actually smiled back at her. Em came up behind her, throwing a sweater over her shoulders. She spun Dixie around to face her. "I'm despicable." Dixie couldn't resist the temptation to tease. "The despicablest." "That I ever in a million years thought you could do something so horrible to us." Em shook her head with disgust. "Well, I'll have to bring you lattes for a million years to make up for it." "And some of that pizza with anchovies," Dixie said on a giggle. "I'll bring you all of the Backstreet Boys, if you'll forget all the horrible things I said." Dixie pursed her lips in mock thought. "Only if it's Nick Carter. I'll settle for nothing less." "Done." Dixie threw her arms around Em's neck and squeezed her tight. "It's okay, Em. Really. I get why you thought it was me. What I don't get is, who was it?" Em swiped at a tear, escaping down her cheek. "Know what? I'm gonna let the hero in all this tell you all about it. If looks could devour a person, from the way Caine looks at you from across that pool, we won't have time to talk anyway." Caine came up behind Dixie, slipping his arms around her waist. "Let's blow this Popsicle stand, huh? We have business to attend to," he murmured in her ear, sending a shiver of anticipation along her spine. She leaned back against him, sheltered, warm, accepted. "We'll talk tomorrow, Em?" Em blew her nose into a tissue and flapped her hand at them. "You two go do unseemly things to each other. I'll be at the offices tomorrow." She blew them a kiss before making her way back through the shrubs and disappearing. Dixie turned in Caine's arms, planting small kisses on his jawline. "I believe we have a date, Candy Caine. Lead the way." "I know the perfect place." He grabbed her hand and began walking in the shadows. Dixie ran behind him, forcing herself to focus on the road rather than his perfect butt in tight jeans. When he finally came to a stop, Dixie had to muffle a giggle. "You remembered," she said with a breathy sigh. He planted a quick kiss on her lips, rubbing his nose against hers. "I'll always remember. Now hurry it up, Dixie Davis. I need you naked, but I know you're gonna make me wait until you have all the answers to your questions just like a typical female," he teased. Dixie cocked her head at him, planting her hands on her hips. "A typical female, huh? I'll have you know, I'm anything but typical. Race ya to our spot!" she shouted before she took off running toward the dirt road. Caine was hot on her heels, his footsteps thundering behind her until he caught up, looping an arm around her waist and carrying her the rest of the way to their favorite tree. Dixie's laughter rang out when he plunked her down on a warm plaid blanket where glasses and a bottle of wine sat. "You had this all figured out, didn't you? Presumptuous much?" Caine dropped down beside her, raising himself up on one arm while slipping the button of her jeans open. "I sure as hell am," he teased, rimming the outline of her mouth with his tongue. Her nipples beaded, tight and hungry, just as her fingers threaded through his hair. "But wait," she moaned. "You promised me answers." His sigh was ragged, but his fingers didn't stop moving. Pulling up her shirt, he dragged it over her head. "Right, answers," he repeated, unhooking the clasp on her bra with nimble fingers and cupping her breast. "What am I answering?" "What about Miami?" Please don't go back. "Well, now, you don't think I'm going to go back to Miami when there's this hot, rich, totally awesome Southern belle right here in Plum Orchard? Why work when I can lounge by the pool and watch you work?" Dixie laughed, snuggling closer to his broad chest. "I'm being serious." "I'm giving up the real-estate business. If this trip taught me one of many things, it was that I miss the hell out of my mama's pecan pie. Oh, and you. And PO and life here in general. That's why I was in Miami—to close up shop. So is there anything else? Or can we get to the business of tearing each other's clothes off. If I'm not inside you soon, it's gonna get ugly." Dixie almost forgot everything but the firm warmth of his hand, his thumb as it caressed her nipple. "Who took those pictures of Clifton? How did they get mixed in at the Founder's Day celebration?" Caine's hands stilled for a moment, almost eliciting a whimper from her. "Louella Palmer hired an investigator from Atlanta to take them." Dixie sat straight up, astonished. "But how did she know? How did you find out it was her?" "Sanjeev called me in Miami to tell me you were leaving and why, and when I heard the whole story, I knew it was her." "You really believed I had nothing to do with it?" Caine cupped her jaw, bringing her in for a brief kiss. "Don't look so surprised, Dixie. Of course, I didn't believe you had anything to do with it. I saw how you defended Em at Madge's. I was there when you took out Louella, remember? I gotta confess, all those talks with Walker, all that honesty, stung, and it made me see what a judgmental prick I'd been. All this time you were beating yourself up, and I was still angry with you. Angry about what happened. Angry about wanting you even after what happened." Dixie let her hands thread through his hair, loving the silky feel of it. "I get it." "That has to stop, Dixie," he grated. "Stop thinking it's okay for anyone to treat you badly because of something that happened a decade ago. It's not your due, and if I damn well see it happen, whoever it is, is going to wish they'd stayed the hell in bed." His overprotective stance left her breathless. Dixie gazed at him intently, loving him so completely. "I think Walker hashed that all out enough for one lifetime, don't you?" Caine's face went from hard to soft in seconds. "The baby. Jesus, Dixie. If I'd known..." She shook her head, the hot swell of tears pricking her eyelids. "Please, please don't," she whispered up at him, caressing his cheek. "It's done now. I just have to let it heal the rest of the way, okay?" "But I need you to know I see what an ass I was being." "Oh, I've seen," she teased, reaching for the buttons on his shirt. "Just tell me how you figured out it was Louella. How could she have possibly found something like that out when only three people here in Plum Orchard knew?" Caine gave her a smile of satisfaction. "I did what every good golden boy does, I plied her with wine over a late lunch today, cornered her and got it out of her. She overheard you and Em talking about it one night outside of Call Girls when you were by the pool, according to her." Dixie fell back on the warmth of the blanket, pulling Caine with her, but she couldn't hide her surprise. "You made Louella Palmer confess? I didn't think Jack Bauer could get her to confess." "Oh, you'd be surprised at my interrogation tactics. I know things about Louella she wouldn't like getting out. I hated to resort to it, but I was willing to do whatever it took to prove you were innocent. To prove to Em you were innocent. Next, we work on damage control so Em and the boys don't suffer. I'll do the best I can to keep the gossip mill from percolating." That wasn't enough for Dixie. No good would come from the entire town seeing that picture of Clifton. "But it's Plum Orchard. The gossip mill is set on percolate." "You know what, sweetheart, you don't give everyone in Plum Orchard a fair shake because you've been on their bad side for too long. When someone's struggling, they don't just gossip about it, they rally together, bake a cake and gossip about it while the batter sets. Sure, plenty of them are small-minded, especially the Mags, senior and junior, but those are just the loudest hens in the henhouse." "And they rule the roost. I'm worried, Caine." Dixie shook her head, incredibly sad for Louella. "She went too far. This involved children, Caine." "If there's fallout, we'll help Em and the boys. Together. Now, can we get back to what's important here?" "What's that?" she purred, pushing his shirt out of the way to allow her hands free rein over his pecs. "You. Naked," he muttered, closing his mouth over hers. She tore her lips from his. "Hold that thought for one second, Golden Boy." "Woman, if you aren't naked in ten seconds and counting, I won't be held accountable for the lack of foreplay." "Just one more thing. And it's really, really important," she said, bracketing his face with her hands. "Nine...eight..." "Thank you," she murmured, her eyes filling with tears for this new beginning. "Thank you for trusting me. You'll never know what that means." "I don't know if I told you this, but that's what someone does when they love you." "I love you, too, Caine." So, so much it might have hurt if it wasn't so sweet with fulfillment. "So quiet time now?" he asked with a smile before he didn't give her the chance to answer. Caine planted his lips on hers, and Dixie responded by rearing up against him, stroking his tongue, reveling in his hot skin. His lips slipped away, moving along the column of her throat until he was skimming her breasts, tonguing her nipples to sharp peaks, pushing at her jeans until they were off, and she was naked. Caine's head lifted, his eyes hungry, devouring her as he slid his fingers between the folds of her flesh, thumbing her clit. "Christ, Dixie, you're so wet and hot." His voice, husky and low, sent a thrill of pleasure through her. His words always stirred something in Dixie she couldn't quite define. Something wicked and untamed. Pulling at his shoulders, she dragged him upward, fumbling with his jeans until he was free of clothing, and his cock was hard against her hip. "I say we skip the foreplay," she managed to squeak out on a whimper. "I can't wait." Caine's response was primal, his growl of pleasure from somewhere deep in his chest. He pushed her legs up high around his waist, and without another word, drove deep within her. Dixie's head thrashed, pressing her hips hard against his until he was embedded in her slick heat. She fought a scream of pleasure at his first thrust, writhing, clutching his back, begging him with her body to make her come. With each thrust, Caine's cock scraped her clit until she was swollen and so needy, tears seeped from the corners of her eyes. Dixie opened her eyes briefly to watch Caine as he thrust into her, his muscled back under the moonlight, her fingers digging into his solid flesh, his body calling to hers to be one with his—forever. At that very moment, she understood what it was to love someone so completely—so deeply—she felt it in her pores, her bones, her heart, her mind. "Come, baby," he said, brushing his lips over hers, his words ragged and raw. "Come hard." With those words, she tensed around him, moving faster beneath him until she exploded with a sharply sweet burn of release. Her head reeled with the pleasure roaring through her veins. Her pride swelled at his hot release. And then there was the heave of chests, crashing together to pump air back into their lungs, and Caine was enveloping her in his arms, crushing her against him, pulling her closer, burying his face in her neck. Her soul was full again, heavy with the piece of her she'd been missing all these years. Full and finally complete. * * * Sanjeev awaited them at the wide front door of the big house, Mona and Lisa at his feet. "I see we've thoroughly debauched one another?" Dixie's face turned red, and she buried it in Caine's shoulder, running her guilty fingers through her mussed hair. "You missed a leaf," Caine remarked, winking at Sanjeev before he picked it out of her hair and let it flutter to the ground. "Well, now that you've taken care of that," Sanjeev said on a smirk, "we have at least an hour before you absolutely must continue your mad race to procreation. This leaves us time for the final bit of business to tend to. Follow me, sex-starved lovebirds." Sanjeev began making his way down the marble-floored hallway to Landon's study. Dixie had avoided this room almost the entire time she'd been here. Everything about it screamed Landon at the top of its lungs. From the large desk made of cherrywood, to the wall with dozens of pictures of all of them together, smiling, laughing, loving each other, to the teak giraffe figurines and African masks. She inhaled, savoring the smell of his cologne, closing her eyes and remembering his warm hugs, his deep voice full of excitement about his next adventure. Sanjeev motioned to the chocolate-colored leather couch. "Sit. I have a message from Landon." Neither Caine nor Dixie spoke. She couldn't if she'd tried anyway. Being so near Landon's things brought back a sharp pain—one that reminded her he wasn't here to see who she'd become—how far she'd gone—and that she'd finally landed right back in Caine's arms. Where she belonged. "What's going on, Sanjeev?" Caine asked, sitting next to Dixie on the sofa and drawing her close to him. Sanjeev located the remote for Landon's flat screen, covering almost an entire wall in front of them, and pointed it at the Blu-ray player. "Watch." The DVD flickered on, making Dixie gasp at the image of Landon in his hospital bed. His face was drawn and pale, his lips chapped, but the twinkle in his eyes was as bright as it had ever been—the purest of souls reflected in his kind handsome face. Dixie stifled a sob, her eyes riveted to the TV. Transfixed by the tubes hanging from his arm, the bottles of medication to the right of his bed, the oxygen mask lying beside him. God, I miss you so much, friend. When Landon spoke, his voice was thready and weak, but Dixie's heart clutched it tight to her chest. "So, by now you've either gotten those fool heads of yours on straight, or you've killed each other. I'm hopin' for the former, but I've prepared for the latter. By now, you also know, I set you both up, because how much pinin' can a BFF take from the two of you?" he said on a chuckle. Clearing his throat after a blast of oxygen, he continued. "Ten years is long enough for you both to come to your senses, and if it isn't, then you shouldn't be allowed a second longer of precious time on Earth when it could be used by someone who wants to really live. "Call Girls was an opportunity I knew both of you couldn't resist. I know you, my Dixie-Cup. I knew you were in dire financial straits with all those investors. I also knew you were determined not to ask for help. But I couldn't just leave money to ya, honeybee. First, you'da been mad, and your pride would have ruined everything. Second, you woulda just taken your mound of cash and gone right back where you don't belong. You belong in Plum Orchard, Dixie, not Chicago. I'm bettin' it didn't feel like it at first with all the anger and resentment some feel, but I'm also bettin' you showed everybody what you're really made of these days. It was time to heal up your wounds after you lost your little one." A sob escaped Dixie's lips. That Landon had known, and never said a word.... Caine kissed the top of her head, pulling her hand into his to hold it tight. "I know you're clutchin' your pearls in shock right now. I know all about your miscarriage, Dixie-Cup. I don't have all this money for nothin'. You hurt me bad when I found out about it. But I was off on some trip I can't remember now, unreachable, so I'm choosin' to believe you just couldn't find me instead of believin' it was a pain so deep you couldn't even tell me. But you did find Agnes, right? I left a whole bunch o' money to that Agnes's grandkids—a nice college fund for takin' such good care of you. "Point is you needed to come home and be surrounded by the people who would eventually love the new Dixie. By the people you love despite bein' a bunch of gossipy old biddies. You just have to look hard. I put Em in charge because not only did I grow to love her in this last month or so, but also because she's pretty dang honest. And I knew she'd keep y'all on the straight and narrow. "Most of all, Dixie, you needed Caine, and he needed you. In fact, I'm not sure who needed who more at this point. Caine was lost, too, you know. No social life to speak of, work, work, work. And still, he was always thinkin' about you. He'd never say it. For a while your name was strictly forbidden until I reminded him that my life was a part of yours, and if he couldn't hear about our friendship, he wasn't just a sissy, he was gonna be one less friend. Same way I did you. "So what's a BFF to do but die a dramatic death, hatch up a devious plan while he's doin' it, and leave behind a will that forced you two to do what you do best—hand-to-hand combat. Then I set up texts to taunt you both—spur you both into some action, let you know I was thinkin' about you." Dixie and Caine looked at each other with wonder. "You got them, too?" she asked. Caine nodded with a grin and a shake of his head. "I laughed and laughed when I thought about what your faces would look like when you found out I owned a phone-sex company. Then I laughed some more when I thought about what you'd look like when you both heard the rules of the game. I laughed harder when Sanjeev suggested we make the rule about you both stayin' in the big house together while you competed. "But I didn't laugh when I considered what you'd get out of this if you'd just listen to your hearts. Dixie, honey. You. Are. Worthy. Of love and trust and all the good things that come with livin' a clean life. "I wanted you to have each other forever. That part didn't make me laugh. I cried. I cried because I wouldn't be there to see it. I cried because I wouldn't finally have the two people I love most in the world together again with me—just like it used to be. I'm sure sorry I'm missin' that, y'all. "But wherever I am, I want you two blind asshats to know this—I love you both more than I loved some of my very own family members. "So I hope you jackasses found each other. I hope it was a hard road, too. You owe me for ten years of my life I lost listenin' to your whinin'. I don't know which of you I hope won, because if this went the way I laid bets with Sanjeev, who won won't matter. You're now a couple, and you're gonna have that big weddin' you were supposed to have ten years ago to seal the deal, and Call Girls is rightfully both of yours—the big house, too. "Hang on to what you have. Live. Really, really live. Be happy. Laugh. Smile. Argue. Make up. But do it. Don't let the past keep you from anything. "And last, know that I love you more than my spleen, Dixie-Cup. I'd do anything for you. This time, I did the thing I knew was right for you. Caine. Caine is right for you, and you're right for him. "Go make babies. Call one of 'em Landon, okay? "As for me, well, I'll see ya when I see ya. I love you both, even from all the way up here." The DVD clicked off, and Dixie buried her face in Caine's chest, tears streaming from her eyes, leaving the silence of Landon gone forever, weighing heavy in the room. Caine cleared his throat, the muscles clenched and straining. "You've had this the whole time, Sanjeev?" Sanjeev's eyes met Caine's, earnest and soft. "I have. I had two, actually. The other, the one where you two don't figure this whole thing out shall be burned promptly. On that note, I did as the man who rescued me so many years ago asked. One hundred lifetimes could never repay that debt—or do justice to the love and gratitude I felt for him." Dixie let a breath escape her lips, shuddering on its way out. She wiped the tears from her eyes and rose, breaking her silence by pointing a finger at Sanjeev. "You are the master at keeping secrets, pal. I bow to you." Sanjeev gave them a small smile. "I've also mastered the art of texting. I'm rather proud of that." "It was you?" Caine asked. "I was only the messenger. Landon was the author. He wrote them all out before he passed and left them for me to send to you both." Dixie shook her head and grinned. "Was there anything he didn't think of? Bet he's feeling pretty pleased with himself right now." Sanjeev nodded with another smile. "Oh, he was pleased long before now. I was laughing at you. Silly, lovesick fools. Oh, the moping and carrying on from the two of you is enough to make even the happiest of people seek a painful death." They all laughed at that as Caine rose, too, sticking his hand out at Sanjeev to shake it and slap him on the back. "So, I guess this means you'll stay on here at the big house? Keep running things for us?" "Surely you jest if you think I'd leave all of this and Mona and Lisa in the hands of the two of you?" he said on a chuckle. "The big house would fall to ashes, and Mona and Lisa would starve for all the attention you'd give them while you're sequestered in that bedroom up there. Now we're off, aren't we, precious beasts?" he said, looking down at Mona and Lisa who sat at perfect attention. "Come, furry friends, there's celebratory kibble to be had!" Sanjeev left them, the dogs following his every step in a rustle of toenails against the marble floor. Caine held his arms open, and Dixie went to them, slipping her hands under his biceps and curling into him. "I miss him so much. It aches, you know?" Caine nodded against the top of her head. "I do know. I miss him every day." At that very second, both of their phones chirped an incoming text. They each fumbled in their pockets to pull their phones out, sharing a gasp of laughter. "Sanjeev! Knock it off!" Caine yelled toward the kitchen. "Last one, I promise!" Sanjeev yelled back, making them both laugh. Caine held up his phone just as Dixie did so they could read each other's message. Have you started making those babies yet? Your ovaries will dry up if you don't make with the haste, Dixie-Cup, and Caine, your sperm will turn to dust if you both don't get a move on. The big bed awaits you... Love you both more than Bobby Flay. Always, my friends, Landon Caine took Dixie in his arms, nuzzling her cheek with his lips, stirring that familiar ache deep inside her. "So how do you feel about making babies?" She gave him a flirtatious smile. "Oh, I dunno. Do you think you're worthy?" Caine shot her a smile before planting a hard kiss on her lips, one that made her moan in anticipation. Then he let her go and began to back away, pushing open the double doors to Landon's study and taking off up the stairs, "Argh, matey! Last one to the big bed be unworthy!" he said in Captain Jack Sparrow's voice. Dixie chased after him, racing up the wide staircase, giggling like a schoolgirl. Whatever would happen, it would happen with Caine. She stopped short of the bedroom's door and looked up at the ceiling, closing her eyes and summoning a picture of the last time she'd seen Landon—dapper, healthy, plowing through life and telling her all about it over her favorite wine. "I love you more than I love my own spleen, Landon Wells. Thank you, friend, thank you," she whispered, blowing a kiss into the palm of her hand and shooting it upward. Placing her fingers on the partially opened door, Dixie walked right through, a smile on her face—and this time, she was going to keep walking—toward Caine. Toward the love she'd waited a lifetime to find her way back to. Toward whatever their kind of forever meant. * * * * * We hope you enjoyed this Harlequin ebook. Connect with us on Harlequin.com for info on our new releases, access to exclusive offers, free online reads and much more! Other ways to keep in touch: Harlequin.com/newsletters Facebook.com/HarlequinBooks Twitter.com/HarlequinBooks HarlequinBlog.com ISBN-13: 9781460331170 TALK DIRTY TO ME Copyright © 2014 by Dakota Cassidy All rights reserved. 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 publisher, Harlequin Enterprises Limited, 225 Duncan Mill Road, Don Mills, Ontario, Canada M3B 3K9. This is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental. This edition published by arrangement with Harlequin Books S.A. ® and ™ are trademarks of the publisher. Trademarks indicated with ® are registered are registered in the United States Patent and Trademark Office, the Canadian Intellectual Property Office and in other countries. www.Harlequin.com	{ "redpajama_set_name": "RedPajamaBook" }	5,140
\section{Supplementary Information:} \subsection{Soft-core particles -- in the absence of external potential} The phase behavior and dynamics of repulsively interacting colloidal suspensions have been extensively studied in literature~\cite{Broughton1982a,Falck2004,Lahtinen2001}. The two-dimensional fluid is known to undergo a freezing transition into a triangular lattice solid, arguably via a hexatic phase, with increasing density~\cite{Kosterlitz1973,Nelson1979,Young1979,Sengupta2000}. Without going into the intricacies of identifying the hexatic~\cite{Broughton1982a,Sengupta2000}, to determine the fluid-solid transition point we calculate the solid- order parameter $\|S(G_2)\|$ of the soft-disk system, in the absence of any external driving, as a function of the density of the system, depicted in \fref{fig:opfreesystem}(a). At phase transition, $\rho^\ast = 1.01$, $\|S(G_2)\|$ shows a discontinuous increase with density (Fig.\ref{fig:opfreesystem}($a$) shows MD simulation result of 4096 particles). At this point the order-parameter jumps increase to a value $\|S(G_2)\|=0.31$. {In all our simulations, even in the presence of ratchet-driving, we identify a phase transition to a solid phase whenever $\|S(G_2)\|$ crosses the value $0.31$.} Note that this value is close to the phase transition criterion of $\|S(G_2)\|=0.35$ used in recent literature~\cite{Mondal2012}. As depicted in the inset of Fig.\ref{fig:opfreesystem}($a$), the pressure versus density behavior shows non-monotonicity signifying a phase-coexistence of a solid with density $\rho_s =1.01$ and liquid with $\rho_l=0.99$~\cite{Broughton1982a}. In the overdamped regime, the internal relaxation is due to diffusive motion of particles. The density dependence of diffusivity $D(\rho)$ in soft-disk particles has been previously studied by Lahtinen {\it et. al} \cite{Falck2004,Lahtinen2001} and exhibits a linear dependence on $\rho$. Using our MD simulations, we studied the same to obtain the linear density dependence $D=D_0(1-\rho/\rho_c)$, where $D_0$ is the diffusivity of a noninteracting system, and the fitted value $\rho_c = 1.04 $ (Fig.\ref{fig:opfreesystem}($b$)). \begin{figure}[h] \hspace{-0.7cm} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.82]{mean_op_dens_free_sys}}; \node at (60pt, -32pt) {{\large \it (a)}}; \end{tikzpicture} \hspace{0.25cm} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.75]{diffusion_coeff_free_system}}; \node at (60pt, -32pt) {{\large \it (b)}}; \end{tikzpicture} \caption{($a$)~Plot of order parameter against density for a free system. The discontinuity of the order parameter occurs at a density near $\rho^\ast =1.01$. The inset shows pressure versus density behavior and is close to Ref.~\cite{Broughton1982a}. ($b$)~Plot of the diffusion coefficient of the free system, averaged over both directions, against the density of the suspension. The solid line is a linear fit of $D=D_0(1-\rho/\rho_c)$ to the data with $\rho_c = 1.04$.} \label{fig:opfreesystem} \end{figure} \subsection{Soft-core particles -- under stochastic ratcheting} In order to study the effect of stochastic ratcheting, the system was evolved under the potential $\beta U_{\rm ext}(y,t)=\beta V_0(t) \left[\sin\left( 2 \pi y/\lambda \right) +\alpha \sin\left(4\pi y/\lambda \right) \right]$, with $\beta V_0(t)$ switching between 0 and $1$, and asymmetry parameter $\alpha=0.2$. The switching is done stochastically with a rate $f$. During our MD simulation, at each time step the value of the external potential $V(t)$ is switched between 0 and 1 with a probability $f\delta t$, and left unaltered with a probability $(1-f\delta t)$, where $\delta t$ is the size of integration time-step. We have always waited for the system to reach steady state before collecting data presented in all our analysis. In this section, we first focus on systems driven by a ratchet of fixed periodicity $\lambda=1 \sigma$. The driving potential is not commensurate with the density, unlike the system discussed in the main text. The impact on the non-interacting system itself is different, as $\lambda$ is no more a function of density. The integrated directed current $\langle j_y \rangle$ exhibits resonance at a fixed value of the frequency of $f_0 = 3.5 $, independent of densities (see Fig.~\ref{fig:flux}~($a$)). The ballistic time-scale $\tau_b$ to travel a potential-valley of length $\lambda$ follows the kinematic relation $\lambda\sim (U_0/\lambda) \tau_b^2$ leading to $\tau_b \sim \lambda/\sqrt{U_0}$, independent of density as $\lambda$ itself is a constant. As a result the resonance frequency is also constant. If, instead, an interacting system is driven by the same ratcheting potential with $\lambda=1\sigma$, the resonance frequency remains approximately constant at lower densities ($\rho \lesssim 0.5$) indicating a free-particle like ballistic motion. However, at higher densities the resonance frequency drops { approximately linearly with density (inset of Fig.~\ref{fig:flux}~($c$))}. This indicates a cross-over from ballistic to diffusive transport with diffusive time-scale $\tau_D \sim \lambda^2/D_0(1-\rho/\rho_c)$, and the corresponding resonance frequency $\sim (1-\rho/\rho_c)$. This behavior should be contrasted against the $f \sim \rho(1-\rho/\rho_c)$ behavior for {\em commensurately } ratcheting soft- disks at high densities (Fig.3 of main text). Note that the reasoning used here is similar to that we applied for interacting particles driven by a commensurate ratchet (See Eq.s (2)--(4) of main text). \begin{figure}[t] \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{fluxplot_ni_low_dens}}; \node at (10pt, 70pt) {{\large \it (a)}}; \end{tikzpicture} \hspace{1.5cm} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{fluxplot_intincmns_low_dens}}; \node at (10pt, 70pt) {{\large \it (b)}}; \end{tikzpicture} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{fluxplot_intincmns_high_dens1}}; \node at (10pt, 70pt) {{\large \it (c)}}; \end{tikzpicture} \hspace{1.5cm} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.52]{op_plot_supp}}; \node at (10pt, 70pt) {{\large \it (d)}}; \end{tikzpicture} \hspace{0.5cm} \caption{(Color Online) ($a$)~Plot of the averaged directed current for a non-interacting system driven by a periodic potential with periodicity $1\sigma$, for densities $\rho =0.1$ ({\Large $\circ$}), $0.2$ ({\large $\square$}), $0.3$ ($\bdiamond$) and $0.6$ ($\triangle$). The resonance in the measured flux occurs at a fixed frequency $f_0 = 3.5 $. ($b$)~Plot of the measured flux for an interacting system driven by a periodic potential with peroidicity $1\sigma$, for densities $\rho =0.1$ ({\color{myblue} \large $\circ$}), $0.2$ ({\color{mypurple} $\square$}), $0.3$ ({\color{myokker} $\bdiamond$}) and $0.6$ ({\color{myblue} $\triangle$}). The solid lines are fit to the data using the functional form of Eq.~(2) in the main text. ($c$)~Plot of the measured flux for an interacting system driven by a periodic potential with peroidicity $1\sigma$, for densities $\rho =0.8$ ({\color{myblue} \Large $\bullet$}), $0.92$ ({\color{mypurple} $\blacksquare$}) and $0.98$ ({\color{myokker} $\fdiamond$}). The solid lines are fit to the data using the functional form of Eq.~(2) in the main text. The inset depicts the dependence of the resonance frequency on the density of the suspension. The solid line is a linear fit of $f_0 \sim (1-\rho/\rho_c)$ to the data with $\rho_c\approx 1.13$. ($d$)~Plot of the measured order parameter for the reciprocal lattice vectors $\mathbf{G}_2$ (empty symbols) and $\mathbf{G}_1$ (inset, filled symbols) for densities $\rho = 0.94$~({\color{myblue} \Large $\bullet$, $\circ$}) and $0.96$~({\color{mypurple}\footnotesize $\blacksquare$,$\square$}).} \label{fig:flux} \end{figure} It is known that, in case of laser induced freezing (LIF), the soft-core system of particles undergo a modulated liquid to solid transition near $\rho=0.95$ and a periodic commensurate potential of strength $\beta U_0=1$~\cite{Chaudhuri2006}. As we argued in the main text, at very high frequencies of weakly asymmetric commensurate ratcheting, one expects to recover fluid-solid transitions similar to LIF. In Fig.~\ref{fig:flux}~($d$) we present the dependence on the ratcheting frequency $f$ of the solid order parameter $S(\mathbf{G}_2)$ (defined in main text) at densities $\rho =0.94$ and $0.96$. This shows that at high enough frequencies $f \gg 10$, although the effective time-integrated periodic potential strength $\beta U_0 <1$, the system at $\rho=0.96$ already shows freezing, a reminiscent of LIF. \begin{figure}[t] \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{diffusion_coeff_y_low_dens_freq_intcmns}}; \node at (10pt, 70pt) {{\large \it (a)}}; \end{tikzpicture} \hspace{1.5cm} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{diffusion_coeff_y_high_dens_freq_intcmns}}; \node at (10pt, 70pt) {{\large \it (b)}}; \end{tikzpicture} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{diffusion_coeff_x_low_dens_freq_intcmns}}; \node at (10pt, 70pt) {{\large \it (c)}}; \end{tikzpicture} \hspace{1.5cm} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{diffusion_coeff_x_high_dens_freq_intcmns}}; \node at (10pt, 70pt) {{\large \it (d)}}; \end{tikzpicture} \caption{Plot of the measured diffusion coefficient,normalized by the diffusion coefficient of a free system $D_0(\rho)$, for an interacting system driven by a periodic and a commensurate potenial, along (in ($a$) and ($b$)) and perpendicular (in ($c$) and ($d$)) to the direction of the external drive , as a function of frequency for densities of the suspension $\rho =0.2$ ({\color{myblue} {\Large $\circ$}}), $0.5$ ({\color{mypurple} $\square$}), $0.8$ ({\color{myokker} {$\bdiamond$}}), $0.92$ ({\color{myblue} {\Large $\bullet$}}), $0.94$ ({\color{mypurple} $\blacksquare$}) and $0.96$ ({\color{myokker} {$\fdiamond$}}). The solid lines are guide to the eye. } \label{fig:diff_coeff} \end{figure} \begin{figure}[t] \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{diffusion_coeff_y_3dplot}}; \node at (10pt, 70pt) {{\large \it (a)}}; \end{tikzpicture} \hspace{1.5cm} \begin{tikzpicture} \node[above,right] (img) at (0,0) {\includegraphics[scale=0.7]{diffusion_coeff_x_3dplot}}; \node at (10pt, 70pt) {{\large \it (b)}}; \end{tikzpicture} \caption{Plot of the measured diffusion coefficient,normalized by the diffusion coefficient of a free system $D_0(\rho)$, for an interacting system driven by a periodic and a commensurate potential, along ($a$) and perpendicular ($b$) to the direction of the external drive, as a function of frequency and density. } \label{fig:diff_coeff_3d} \end{figure} Finally, we consider the asymmetric diffusivity of the soft-core particle system, driven by a ratchet with periodicity commensurate with the density. We obtain the density and ratcheting-frequency dependence of the diffusivity. In Fig.~\ref{fig:diff_coeff}, we show the frequency dependence of diffusion coefficients $D_y$, $D_x$ along and perpendicular to the direction of ratcheting drive, respectively. We observe that $D_y < D_x$, since to diffuse in the direction of driving, particles have to climb potential barriers. As with the directed current, we also observe a resonance in the diffusion coefficient when the density is relatively small (see \fref{fig:diff_coeff}(a) and (c)). The properties of $D_{x,y}(f,\rho)$ is further presented as a surface plot in Fig.~\ref{fig:diff_coeff_3d}, in the low density regime where the resonance structure is seen. The magnitude of both $D_x$ and $D_y$ increases with density, the effect being more pronounced for $D_x$ and persists for densities lower than $\approx 0.3$ (see Fig.~\ref{fig:diff_coeff_3d}), beyond which the magnitude of the diffusion constants steadily decreases. At higher densities, the resonance structure is lost, the diffusivities remain approximately constant at low frequencies, decreasing at frequencies $f \gtrsim1$ (Fig.s~\ref{fig:diff_coeff}($b$) and (d)).	{ "redpajama_set_name": "RedPajamaArXiv" }	8,789
Come and get them everybody! Choose and pick off these wonderful looking flowers to bring your spirits to nature! Photos are now available at https://stevenanthony123456.smugmug.com/Flowers-and-Plants/ You can also purchase my first photo book at http://www.blurb.com/b/7135384-here-comes-the-daffodils-series for $6.98 in PDF format. Enjoy!	{ "redpajama_set_name": "RedPajamaC4" }	1,877
{"url":"https:\/\/luisrguzmanjr.wordpress.com\/tag\/composite-numbers\/","text":"You are currently browsing the tag archive for the \u2018composite numbers\u2019 tag.\n\nProofs from THE BOOK by Martin Aigner and G\u00fcnter Ziegler begins by giving six proofs of the infinity of primes. We will go over the third proof. Before we go over this proof, lets cover some background.\n\n\u2014 1. Fermat Numbers \u2014\n\nFermat numbers are defined by\n\n$\\displaystyle F_{n}=2^{2^{n}}+1,$\n\nso that ${F_{0}=3,F_{1}=5,F_{2}=17,F_{3}=257,}$ and ${F_{4}=65537}$. They are of great interest in many ways: for example, it was proved by Gauss that, if ${F_{n}}$, is a prime ${p}$, then a regular polygon of ${p}$ sides can be inscribed in a circle by Euclidean methods. The property of the Fermat numbers which is relevant here is\n\nTheorem 1 No two Fermat numbers have a common divisor greater than 1.\n\nWe will prove this theorem later.\n\nThe first four Fermat numbers are prime, and Fermat conjectured that all were prime. Euler, however, found in 1732 that\n\n$\\displaystyle F_{5}=2^{2^{5}}+1=641\\cdot6700417$\n\nis composite.\n\nIn 1880 Landry proved that\n\n$\\displaystyle F_{6}=2^{2^{6}}+1=274177\\cdot67280421310721.$\n\nIt is currently known that ${F_{n}}$, is composite for ${5 \\leq n \\leq 32}$. Factoring Fermat numbers is extremely difficult as a result of their large size. ${F_{12}}$ has ${5}$ known factors with ${C1187}$ remaining (where ${C}$ denotes a composite number with $n$ digits). ${F_{13}}$ has ${4}$ known factors with ${C2391}$ remaining. ${F_{14}}$ has no known factors but is composite. There are currently four Fermat numbers that are known to be composite, but for which no single factor is known: ${F_{14},F_{20},F_{22},}$ and ${F_{14}}$. In all the other cases proved to be composite a factor is known. No prime ${F_{n}}$ has been found beyond ${F_{4}}$, and it seems unlikely that any more will be found using current computational methods and hardware.\n\n\u2014 2. Infinitude of Primes Theorem \u2014\n\nWe are now ready to prove Euclid\u2019s Second Theorem, also called the Infinitude of Primes Theorem using the third proof in Proofs from THE BOOK.\n\nTheorem 2 (Euclid\u2019s Second Theorem) The number of primes is infinite.\n\nProof: Next let us look at the Fermat numbers ${F_{n}=2^{2^{n}}+1}$ for ${n=0,1,2,\\cdots}$. We will show that any two Fermat numbers are relatively prime (Theorem 1); hence there must be infinitely many primes. To this end, we verify the recursion\n\n$\\displaystyle \\prod_{k=0}^{n-1}F_{k}=F_{n}-2\\quad(n\\geq1),$\n\nfrom which our assertion follows immediately. Indeed, if ${m}$ is a divisor of, say, ${F_{k}}$ and ${F_{n}}$ ${(k, then ${m}$ divides ${2}$, and hence ${m=1}$ or ${2}$. But ${m=2}$ is impossible since all Fermat numbers are odd.\n\nTo prove the recursion we use induction on ${n}$. For ${n=1}$ we have ${F_{0}=3}$ and ${F_{1}-2=5-2=3}$. With induction we now conclude\n\n$\\displaystyle \\begin{array}{rcl} \\prod_{k=0}^{n} F_{k} & = & \\left(\\prod_{k=0}^{n-1}F_{k}\\right)F_{n}\\\\ & = & (F_{n}-2)F_{n}\\\\ & = & (2^{2^{n}}-1)(2^{2^{n}}+1)\\\\ & = & 2^{2^{n+1}}-1\\\\ & = & F_{n+1}-2. \\Box \\end{array}$\n\nI have re-posted this to test my changes to the latex2wp.py and terrystyle.py programs, compiled with Python Software Foundation\u2019s Python 2.7.2 64 bit version, to add support for LyX 1.6.9. The code change incorporates some additional theorem-like environments, macros, font styles, and the numbering has been change so that the different theorem-like types each have a separate counter (e.g., theorem 1, theorem 2, lemma 1, proposition 1, theorem 3, lemma 2, \u2026, as opposed to theorem 1, theorem 2, lemma 3, proposition 4, \u2026). Furthermore, I have provided more background information which will benefit those without a background in abstract algebra.\n\nProofs from THE BOOK by Martin Aigner and G\u00fcnter Ziegler begins by giving six proofs of the infinity of primes. The first proof is what they call \"the oldest Book Proof\" attributed to Euclid. Before we go over this proof, lets cover some background. Read the rest of this entry \u00bb\n\nProofs from THE BOOK by Martin Aigner and G\u00fcnter Ziegler begins by giving six proofs of the infinity of primes. The first proof is what they call \u201cthe oldest Book Proof\u201d attributed to Euclid. Before we go over this proof, lets cover some background. Read the rest of this entry \u00bb","date":"2020-08-14 14:04:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 40, \"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\": 0.831356942653656, \"perplexity\": 446.4979237308184}, \"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-2020-34\/segments\/1596439739328.66\/warc\/CC-MAIN-20200814130401-20200814160401-00052.warc.gz\"}"}	null	null
{"url":"https:\/\/effort.academickids.com\/encyclopedia\/index.php\/Dimensional_analysis","text":"# Dimensional analysis\n\nDimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the plausibility of derived equations. Only like dimensioned quantites may be added, subtracted, compared, or equated. When unlike dimensioned quantities appear opposite of the \"+\" or \"-\" or \"=\" sign, that physical equation is not plausible which might prompt one to correct errors before proceeding to use it. When like dimensioned quantities or unlike dimensioned quantites are multiplied or divided, their dimensions are likewise multiplied or divided. When dimensioned quantites are raised to a power or a power root, the same is done to the dimensions attached to those quantities.\n\nThe dimensions of a physical quantity is associated with symbols, such as M, L, T which represent mass, length and time, each raised to rational powers. For instance, the dimension of the physical variable, speed, is distance\/time (L\/T) and the dimension of a force is mass\u00d7distance\/time\u00b2 or ML\/T\u00b2. In mechanics, every dimension can be expressed in terms of distance (which physicists often call \"length\"), time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another other set of dimensions. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents quantity of electric charge.\n\nThe units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but a length always has a dimension of L whether it is measured in meters, feet, inches, miles or micrometres. Dimensional symbols, such as L, form a group: there is an identity, 1; there is an inverse to L, which is 1\/L, and L raised to any rational power p is a member of the group, having an inverse of 1\/L raised to the power p. There are conversion factors between units; for example one meter is equal to 39.37 inches, but a meter and an inch are both associated with the same symbol, L.\n\nIn the most primitive form, dimensional analysis may be used to check the correctness of physical equations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. Moreover, the two sides of any equation must have the same dimensions. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat cannot be added. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is 0.477.\n\nIt should be noted that very different physical quantities may have the same dimensions: work and torque, for example, both have the same dimensions, M L2T-2. An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct! However, torque multiplied by an angular twist measured in (dimensionless) radians is work or energy. (The radian is the mathematically natural measure of an angle and is the ratio of arc of a circle swept by such an angle divided by the radius of the circle. That ratio of like dimensioned quantities, length over length, is dimensionless.)\n\nThe value of a dimensionful physical quantity is written as the product of a unit within the dimension and a dimensionless numerical factor. Strictly, when like dimensioned quantities are added or subtracted or compared, these dimensioned quantities must be expressed in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, conceptually, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 meter added to 1 foot is a length, but it would not be correct to add 1 to 1 to get the result. A conversion factor, which is a ratio of like dimensioned quantities and is equal to the dimensionless unity:\n\n[itex] 1 \\ \\operatorname{ft} = 0.3048 \\ \\operatorname{m} \\ [itex] is identical to saying [itex] 1 = \\frac{0.3048 \\ \\operatorname{m}}{1 \\ \\operatorname{ft}} [itex]\n\nThe factor [itex] 0.3048 \\frac{\\operatorname{m}}{\\operatorname{ft}} [itex] is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantites of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1 used to convert units, is used to convert quanties to the same units so that their numrical values can be added or subtracted.\n\n [itex] 1 \\operatorname{m} + 1 \\operatorname{ft} [itex] [itex]= 1 \\operatorname{m} + 1 \\operatorname{ft} \\times 0.3048 \\frac{\\operatorname{m}}{\\operatorname{ft}} [itex] [itex]=1 \\operatorname{m} + 1 \\operatorname{ft} \\!\\!\\!\\! \/ \\times 0.3048 \\frac{\\operatorname{m}}{\\operatorname{ft} \\!\\!\\!\\! \/} [itex] [itex]=1 \\operatorname{m} + 0.3048 \\operatorname{m} [itex] [itex]=1.3048 \\operatorname{m} \\ [itex]\n\nOnly in this manner, it is meaningful to speak of adding like dimensioned quantities of differing units, although to do so mathematically, all units must be the same. It is not meaningful, either physically or mathematically, to speak of adding unlike dimensioned physical quantities such as adding length (say, in meters) to mass (perhaps in kilograms).\n\nThe Buckingham \u03c0-theorem forms the basis of the central tool of dimensional analysis. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n-m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown.\n\n## A worked example\n\nA typical application of dimensional analysis occurs in fluid dynamics. If a moving fluid meets an object, it exerts a force on the object, according to a complicated (and not completely understood) law. We might suppose that the variables involved under some conditions to be the speed, density and viscosity of the fluid, the size of the body (expressed in terms of its frontal area [itex]A[itex]), and the drag force. Using the algorithm of the \u03c0-theorem, one can reduce these five variables to two dimensionless parameters: the drag coefficient and the Reynolds number.\n\nAlternatively, one can derive the dimensionless parameters via direct manipulation of the underlying differential equations.\n\nThat this is so becomes obvious when the drag force [itex]F[itex] is expressed as part of a function of the other variables in the problem:\n\n[itex]\n\nf(F,u,A,\\rho,\\nu)=0. \\! [itex] This rather odd form of expression is used because it does not assume a one-one relationship. Here, [itex]f[itex] is some function (as yet unknown) that takes five arguments. We note that the right hand side is zero in any system of units; so it should be possible to express the relationship described by [itex]f[itex] in terms of only dimensionless groups.\n\nThere are many ways of combining the five arguments of [itex]f[itex] to form dimensionless groups, but the Buckingham Pi theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by\n\n[itex]\n\nRe=\\frac{u\\sqrt{A}}{\\nu} [itex]\n\nand the drag coefficient, given by\n\n[itex]\n\nC_D=\\frac{F}{\\rho Au^2}. [itex]\n\nThus the original law involving a function of five variables may be replaced by one involving only two:\n\n[itex]\n\nf\\left(\\frac{F}{\\rho Au^2},\\frac{u\\sqrt{A}}{\\nu}\\right)=0. [itex] where [itex]f[itex] is some function of two arguments. The original law is then reduced to a law involving only these two numbers.\n\nBecause the only unknown in the above equation is [itex]F[itex], it is possible to express it as\n\n[itex]\n\n\\frac{F}{\\rho Au^2}=f\\left(\\frac{u\\sqrt{A}}{\\nu}\\right) [itex]\n\nor\n\n[itex]\n\nF=\\rho Au^2f(Re). \\! [itex]\n\nThus the force is simply [itex]\\rho Au^2[itex] times some (as yet unknown) function of the Reynolds number: a considerably simpler system than the original five-argument function given above.\n\nDimensional analysis thus makes a very complex problem (trying to determine the behaviour of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.\n\nThe analysis also gives other information for free, so to speak. We know that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project.\n\nTo empirically determine the Reynolds number dependence, instead of experimenting on huge bodies with fast flowing fluids (such as real-size airplanes in wind-tunnels), one may just as well experiment on small models with slow flowing, more viscous fluids, because these two systems are similar.\n\n## References\n\n\u2022 Barenblatt, G. I., \"Scaling, Self-Similarity, and Intermediate Asymptotics\", Cambridge University Press, 1996\n\u2022 Bridgman, P. W., \"Dimensional Analysis\", Yale University Press, 1937\n\u2022 Langhaar, H. L., \"Dimensional Analysis and Theory of Models\", Wiley, 1951\n\u2022 Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949, 42(6)\n\u2022 Porter, \"The Method of Dimensions\", Methuen, 1933\n\u2022 Boucher and Alves, Dimensionless Numbers, Chem. Eng. Progress, 1960, 55, pp.55-64\n\u2022 Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345\n\u2022 Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140, 167-177\n\u2022 Rayleigh, Lord, The Principle of Similitude, Nature 1915, 95, pp. 66-68\n\u2022 Silberberg, I. H. and McKetta J. J., Jr., Learning How to Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6), p.101; (7), p. 129\n\u2022 Van Driest, E. R., On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34, March\n\u2022 Perry, J. H. et al., \"Standard System of Nomenclature for Chemical Engineering Unit Operations\", Trans. Am. Inst. Chem. Engrs., 1944, 40, 251\n\u2022 Moody, L. F., \"Friction Factors for Pipe Flow\", Trans. Am. Soc. Mech. Engrs., 1944, 66, 671\n\n\u2022 Art and Cultures\n\u2022 Countries of the World\u00a0(http:\/\/www.academickids.com\/encyclopedia\/index.php\/Countries)\n\u2022 Space and Astronomy","date":"2021-06-24 11:46:19","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\": 0.8317407965660095, \"perplexity\": 1185.391538487254}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488553635.87\/warc\/CC-MAIN-20210624110458-20210624140458-00182.warc.gz\"}"}	null	null
I had a discussion with a couple of the girls on the workshop yesterday about how being diagnosed is bittersweet. On the one hand you are dealt with something you wouldn't wish on even your worse enemy and on the other, it makes you take stock of your life and makes you want to make the most out of it. But the best thing is, it's only September so there are plenty other things I could do! If I'm honest, I have no idea what will happen before the end of the year but I'm excited to find out. Like I said before, having cancer is bittersweet, it is only now that I am really enjoying life and having the guts to do all the things I want to. So my advise to all of you reading this is if there is something you want to do, just do it! Don't be afraid, don't think you are being selfish, just get on with it! The only person you are letting down is yourself and if like me, you'll be excited about what it will lead to next.	{ "redpajama_set_name": "RedPajamaC4" }	5,816
Том-Йелте Слагтер (; род. , Гронинген, Нидерланды) — голландский профессиональный шоссейный велогонщик, выступающий за команду мирового тура «». Чемпион Нидерландов 2010 года среди андеров в групповой гонке. Достижения 2010 Чемпионат Нидерландов U23 1-й Групповая гонка 1-й Этап 2 Круг Арденн 4-й Тур де л'Авенир 7-й Льеж — Бастонь — Льеж U23 8-й Вламсе Пейл 2012 5-й Гран-при Квебека 6-й Тур Омана 2013 1-й Тур Даун Андер 1-й Молодёжная классификация 1-й Этап 3 Тур Альберты 1-й Горная классификация 7-й Гран-при Квебека 9-й Тур де л'Айн 2014 Париж — Ницца 1-й Этапы 4 & 7 2-й Гран-при Мигеля Индурайна 6-й Флеш Валлонь 6-й Льеж — Бастонь — Льеж 2015 3-й Тур Альберты 1-й Этапы 3 & 4 4-й Гран-при Квебека 9-й Флеш Валлонь 9-й Гран-при Мигеля Индурайна 10-й Гран-при Монреаля 2016 1-й Этап 1 Тур дю От-Вар 4-й Три варезенские долины 8-й Классика Сан-Себастьяна 9-й Стрела Брабанта 2017 Тур Австрии 1-й Этап 2 3-й Гран-при Монреаля 4-й Тур Альберты 6-й Гран-при Квебека 2018 3-й Тур Даун Андер Статистика выступлений Гранд-туры Ссылки Слагтер, Том-Йелте на FirstCycling Велогонщики Нидерландов	{ "redpajama_set_name": "RedPajamaWikipedia" }	9,910
OMG! Facebook's Creepy Captcha Test Is Forcing Users To Upload Selfies The social network giant Facebook which is created by the Mark Zuckerberg has now begun to experiment with a new method of identity verification, which consists of analyzing a photograph of the face or it will ask for selfies of the user in question. This is how the new Facebook CAPTCHA will work Facebook, the popular social network created by Mark Zuckerberg, has begun to experiment with a new method of identity verification, which consists of analyzing a photograph of the face of the user in question. Obviously, the social network giant Facebook has become a platform that exceeds what we can call a "traditional" social network. The millions of users with whom account not only use it to share content, but also to inform themselves and develop initiatives, among other things. That is why ensuring the identity of users has become one of the most important tasks by developers, who have to deal with false profiles every day or the questioned false news advertised by the entire social platform. While the social network giant Facebook provides several identification systems, such as two-step authentication or login through the profile photo, now plans to begin testing a new way to identify the identity of a profile. According to the reports, the social network giant Facebook's new identity verification system would be a user's profile picture or you can say that the system will ask you for your selfie so that the system can accurately identify the face of the individual. It would be an alternative method to the classic CAPTCHA that exists in the systems. https://twitter.com/flexlibris/status/935635282564734977 In effect, the new identification system would be, essentially, an automated process that would analyze the image and verify that it is a single photo, not repeated and not extracted from any network. The system is both automatic and manual, so it is activated whenever suspicious activity is detected in the account, as confirmed by Facebook to Wired. Regarding the availability of the verification process, some users claim that the system is producing some failures and access under this modality is taking some time, so it is expected that the developers will improve the process until the moment in which it will be made available to all users of the social network. So, what do you think about this new security measure? Simply share your views and thoughts in the comment section below.	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	1,372
Welcome to Change Your Mindset (formerly known as Improv Is No Joke) where it is all about believing that strong communication skills are the best way in delivering your technical accounting knowledge and growing your business. The way of building stronger communication skills is by embracing the principles of applied improvisation. Your host is Peter Margaritis, CPA a.k.a. The Accidental Accountant, will interview financial professionals and business leaders to find their secret in building stronger relationships with their clients, customers, associates, and peers, all the while, growing their businesses. Jennifer Elder works with financial leaders to become more strategic, stay ahead of the competition, and be more successful. As a consultant and keynote speaker, Jennifer is known for being energetic and enthusiastic, and she has the natural talent for taking complicated topics and making them simple, practical, and immediately implementable. CPA Practice Advisor named Jennifer one of the Top 25 Women in Accounting in 2018. The American Institute of Certified Public Accountants and the Maryland Association of Certified Public Accountants named her a Woman to Watch in 2015. She has been awarded Outstanding Educator by the American Institute of CPAs five times. And in 2018, Jennifer earned the designation of Certified Speaking Professional, making her one of only 10 people worldwide who hold both the CPA and CSP designations (but maybe I can push that number from 10 to 11 in 2019). Future-ready CFOs, and future-ready accountants in general, have to think about what's going on in the world and how it might affect your organization, which means you want to be looking at trends. What are the trends in the world at large? What are the trends in your industry? In the accounting and finance world right now, the trend everybody's looking at and talking about is technology. How are things like artificial intelligence going to change everything? When Jennifer teaches classes on the skills of the future and presentation skills, she breaks it down into five words you need to focus on to really add value to your organization: what, so what, and now what. Now what, then, is the action that's going to help your client move forward. Change Your Mindset is now being distributed on C-Suite Radio. You can find Change Your Mindset and many other outstanding business podcasts on C-Suite Radio by going to www.c-suiteradio.com.	{ "redpajama_set_name": "RedPajamaC4" }	8,993
\section{Introduction} \label{intro} An \emph{$m$-cover} of the Hermitian surface $\mathrm{H}(3,q^2)$ is a set $\mathcal{S}$ of lines of $\mathrm{H}(3,q^2)$ such that each point of $\mathrm{H}(3,q^2)$ lies on precisely $m$ elements of $\mathcal{S}$. An $m$-cover is sometimes also called a \emph{regular system of order $m$}. It was shown by Beniamino Segre (1965) \cite{Segre:1965aa} that $\mathrm{H}(3,q^2)$ does not have spreads, that is $1$-covers, for every prime power $q$. Moreover, Segre determined that the only (nontrivial) $m$-covers of $\mathrm{H}(3,q^2)$, $q$ odd, have $m=\tfrac{q+1}{2}$ \cite{Segre:1965aa}. He called these $m$-covers \textit{hemisystems} as they constitute half of the lines on every point. Hemisystems of $\mathrm{H}(3,q^2)$ have piqued the interest of finite geometers and algebraic graph theorists alike due to their links to extremal configurations in combinatorics, such as strongly regular graphs \cite{Cameron:1978aa}, partial quadrangles \cite{Cameron:1975aa} and association schemes \cite{Dam:2013aa}. In 2011, Penttila and Williford defined the concept of a \textit{relative hemisystem} on $\mathrm{H}(3,q^2)$, for $q$ a power of 2 \cite{penttila2011new}. In this context, we consider points and lines that are disjoint or \textit{external} to a symplectic subgeometry $\mathrm{W}(3,q)$. In particular, a relative hemisystem of $\mathrm{H}(3,q^2)$ is a set $\mathcal{R}$ of external lines with respect to $\mathrm{W}(3,q)$ such that each external point lies on $\tfrac{q}{2}$ elements of $\mathcal{R}$. The interest in relative hemisystems stems from the fact that they give rise to primitive $Q$-polynomial association schemes that cannot be constructed from distance regular graphs (see \cite[Theorem 3]{penttila2011new}). These particular sorts of association schemes were considered rare before the introduction of relative hemisystems, and the first infinite family of relative hemisystems gave rise to the first infinite family of such association schemes. In this paper, we generalise the concept of a relative hemisystem naturally to a \textit{relative $m$-cover}. Let $\Gamma$ be a generalised quadrangle with a subquadrangle $\Gamma'$. A relative $m$-cover of $\Gamma$ is a set $\mathcal{R}$ of external lines with respect to $\Gamma'$ such that each external point lies on $m$ elements of $\mathcal{R}$. By considering relative $m$-covers, we are able to prove an analogue of Segre's result that initiated the study of hemisystems. In particular, we prove that every nontrivial relative $m$-cover of a certain generalised quadrangle $\Gamma$ with respect to a \emph{doubly subtended} subquadrangle $\Gamma'$ is a relative hemisystem; which includes the case that $\Gamma$ is $\mathrm{H}(3,q^2)$ and $\Gamma'$ is $\mathrm{W}(3,q)$. The definition of a doubly subtended subquadrangle appears in the next section in the context of subtended spreads of lines. \begin{theorem} \label{bigthm} Let $\Gamma$ be a generalised quadrangle of order $(q^2,q)$ and suppose it has a doubly subtended subquadrangle $\Gamma'$ of order $(q,q)$. Let $\mathcal{R}$ is a nontrivial relative $m$-cover of $\Gamma$ with respect to $\Gamma'$. Then: \begin{enumerate}[(a)] \item $q$ is even and $\mathcal{R}$ is a relative hemisystem, that is, $m=\frac{q}{2}$. \item If $\sigma$ is an involutory automorphism fixing $\Gamma'$ point-wise, then the image of $\mathcal{R}$ under $\sigma$ is its complement. \end{enumerate} \end{theorem} Thas \cite{Thas:1998aa} showed that when $q$ is even, the only generalised quadrangle $\Gamma$ of order $(q^2,q)$ with a doubly subtended subquadrangle $\Gamma'$ of order $(q,q)$ is when $\Gamma=\mathrm{H}(3,q^2)$ and $\Gamma'=\mathrm{W}(3,q)$. For $q$ odd, the only known examples are $\mathrm{H}(3,q^2)$ (with doubly subtended quadrangle $\mathrm{W}(3,q)$) and the \emph{Kantor-Knuth} flock generalised quadrangles. (For \emph{dual translation generalised quadrangles}, K. Thas \cite{Thas:2007aa} showed that these two families of examples are precisely the only generalised quadrangles of order $(q^2,q)$ with doubly subtended subquadrangles). \newpage \begin{corollary}\leavevmode \begin{enumerate} \item[\textrm{(a)}] A nontrivial relative $m$-cover of $\mathrm{H}(3,q^2)$ with respect to a symplectic subgeometry $\mathrm{W}(3,q)$ is a relative hemisystem. \item[\textrm{(b)}] A Kantor-Knuth flock generalised quadrangle of order $(q^2,q)$, $q$ odd, does not have a nontrivial relative $m$-cover with respect to a double subtended subquadrangle. \end{enumerate} \end{corollary} We leave as an open problem whether or not the first part of Theorem \ref{bigthm} holds generally for relative $m$-covers of nonclassical generalised quadrangles of order $(q^2,q)$. \section{Preliminaries} \label{prelims} The central object of study in this paper is the Hermitian surface $\mathrm{H}(3,q^2)$, whose points and lines form a generalised quadrangle, and the language and theory of generalised quadrangles will be beneficial. Furthermore, some of our preliminary results on relative $m$-covers hold in the more general context of generalised quadrangles, which may be of interest. A \textit{generalised quadrangle of order $(s,t)$} is a point-line incidence structure satisfying the following axioms. \begin{itemize} \item Any two points are incident with at most one line. \item Every point is incident with $t+1$ lines. \item Every line is incident with $s+1$ points. \item (GQ-axiom) For any point $P$ and line $\ell$ that are not incident, there is a unique point $P'$ on $\ell$ that is collinear with $P$. \end{itemize} Given a generalised quadrangle $\Gamma$, we say that $\Gamma'$ is a \emph{subquadrangle} of $\Gamma$ if it is a generalised quadrangle, and if the sets of points and lines of $\Gamma'$ are proper subsets of the points and lines of $\Gamma$. The \textit{dual} of a generalised quadrangle of order $(s,t)$ is obtained by swapping the point and line sets while maintaining incidence. The dual is also a generalised quadrangle, and it is of order $(t,s)$. For additional background on generalised quadrangles, see \cite{Payne:2009aa}. In this paper we are interested in generalised quadrangles $\Gamma$ of order $(q^2,q)$, for some $q$ a power of two, and their subquadrangles $\Gamma'$ of order $(q,q)$. The classical generalised quadrangles of order $(q^2,q)$ are the Hermitian spaces $\mathrm{H}(3,q^2)$, obtained by taking the totally isotropic subspaces of $\mathrm{PG}(3,q^2)$ with respect to a Hermitian form. The symplectic space $\mathrm{W}(3,q)$, which is obtained by taking the points of $\mathrm{PG}(3,q)$ together with the totally isotropic lines with respect to an alternating form, is a generalised quadrangle of order $(q,q)$ and can be embedded as a subgeometry (and indeed a \textit{subquadrangle}) of $\mathrm{H}(3,q^2)$ (see \cite[\S 3.5(a)]{Payne:2009aa}). We call points and lines that lie in $\mathrm{H}(3,q^2)$ but not in $\mathrm{W}(3,q)$ \textit{external points} and \textit{external lines} respectively. We denote the set of external points by $\mathcal{P}_E$ and the set of external lines by $\mathcal{L}_E$. A simple calculation shows that \begin{equation}\label{basiccount} \|\mathcal{P}_E\| = (q^2+1)(q^3-q)\quad \text{and} \quad \|\mathcal{L}_E\| = q^2(q^2-1). \end{equation} We call a subset $\mathcal{R}$ of external lines a \textit{relative $m$-cover} of $\Gamma$ with respect to $\Gamma'$ if every external point is incident with exactly $m$ lines of $\mathcal{R}$. There is always a relative $0$-cover and a relative $q$-cover of $\Gamma$ with respect to $\Gamma'$, obtained by taking none or all of the external lines. We will call these two cases \textit{trivial}. \begin{lemma} \label{RmCsize} Suppose that $\mathcal{R}$ is a relative $m$-cover of $\Gamma$ with respect to a subquadrangle $\Gamma'$. Then $\|\mathcal{R}\| = m(q^3-q)$. \end{lemma} \begin{proof} We double count pairs $(X,\ell)$, where $X$ is an external point and $\ell$ is a line of the relative $m$-cover incident with $X$. By Equation \eqref{basiccount}, we have $(q^2+1)(q^3-q)\cdot m = \|\mathcal{R}\|(q^2+1)$, and so $\|\mathcal{R}\| = m(q^3-q)$. \end{proof} A \textit{spread} of a generalised quadrangle is a set $S$ of lines such that each point is incident with precisely one element of $S$. Given an external line $\ell$, it can be seen that the lines concurrent with $\ell$ and meeting a subquadrangle $\Gamma'$ in $q+1$ points form a spread $\mathcal{S}_\ell$ of $\Gamma'$. We say that $\mathcal{S}_\ell$ is the spread \textit{subtended} by $\ell$. Furthermore, a spread of $\Gamma'$ is \textit{doubly subtended} if it is subtended by two external lines $\ell$ and $\bar{\ell}$, and we then say that $\ell$ and $\bar{\ell}$ are \textit{antipodes}. We say that $\Gamma'$ is \textit{doubly subtended} if every subtended spread of $\Gamma'$ is doubly subtended. Generalised quadrangles of order $(q^2,q)$ with doubly subtended subquadrangles of order $(q,q)$ were first considered by Brown \cite{brown1997generalized}, albeit in the context of the dual. Brown \cite[Corollary 2.2]{brown1997generalized} showed that the size of the intersection of two subtended spreads $\mathcal{S}_\ell$ and $\mathcal{S}_n$ is one of the following: \begin{center} \begin{tabular}{ll} \toprule $q^2+1$ & if $\mathcal{S}_\ell=\mathcal{S}_n$,\\ $1$ & if $\mathcal{S}_\ell\ne \mathcal{S}_n$ and $n$ is concurrent with either $\ell$ or $\bar{\ell}$,\\ $q+1$ & if $\mathcal{S}_\ell\ne \mathcal{S}_n$ and $n$ is not concurrent with $\ell$ nor $\bar{\ell}$.\\ \bottomrule \end{tabular} \end{center} He further showed that a generalised quadrangle of order $(q^2,q)$ has a doubly subtended subquadrangle if and only if there is an involutory automorphism that fixes the subquadrangle pointwise. This involutory automorphism swaps antipodes $\ell$ and $\bar{\ell}$. \begin{lemma} \label{barlemma} Let $\ell$ be an external line. Then a line meeting $\ell$ is concurrent with $\bar{\ell}$ if and only if it also meets $\Gamma'$. \end{lemma} \begin{proof} By definition, $\ell$ and $\bar{\ell}$ subtend the same spread of $\Gamma'$, which means that every line concurrent with $\ell$ that meets $\Gamma'$ must also be concurrent with $\bar{\ell}$. As for the `if' implication, suppose $k$ is a line concurrent with both $\ell$ and $\bar{\ell}$. Then the point that is the intersection of $\ell$ and $k$ must have one line on it meeting $\Gamma'$, and, by the previous argument, that line must be concurrent with $\bar{\ell}$. To prevent the existence of a triangle, $k$ must meet $\Gamma'$. \end{proof} \begin{lemma} \label{extlinesint} Let $\ell$ and $n$ be two nonconcurrent external lines in a generalised quadrangle $\Gamma$ of order $(q^2,q)$ containing a doubly subtended subquadrangle $\Gamma'$ of order $(q,q)$. Then the number of external lines concurrent with both $\ell$ and $n$ is: \begin{enumerate}[(a)] \item $0$ if $n = \bar{\ell}$, \item $q-2$ if $n$ is concurrent with $\ell$, \item $q^2-q$ if $n$ is not concurrent with either $\ell$ or $\bar{\ell}$, or \item $q^2$ if $n$ is concurrent with $\bar{\ell}$. \end{enumerate} \end{lemma} \begin{proof} Suppose $n$ is concurrent with $\ell$ and that $P$ is their point of intersection. To prevent the existence of a triangle, the only lines that can be concurrent with both $\ell$ and $n$ (but not equal to either) are the $q-2$ remaining external lines that are incident with $P$. Now suppose $n$ is not concurrent with or equal to $\ell$. By the `GQ-axiom', $\ell$ and $n$ are concurrent with $q^2+1$ lines together in $\Gamma$. We must determine how many of these are external lines. The lines concurrent with both $\ell$ and $n$ and meeting $\Gamma'$ are exactly the lines in the intersection of the spreads subtended by $\ell$ and $n$. Therefore, if $\ell$ and $n$ subtend the same spread (that is, $n= \bar{\ell}$), then there must be zero external lines concurrent with both. If the intersection of the spreads subtended by $\ell$ and $n$ has size $q+1$ (that is, $n$ is concurrent with neither $\ell$ or $\bar{\ell}$), then $\ell$ and $n$ are concurrent with $q^2-q$ external lines together. Finally, if $n$ is concurrent with $\bar{\ell}$, which means that size of the intersection of their subtended spreads is equal to one, then $\ell$ and $n$ must be concurrent with $q^2$ external lines together. \end{proof} For an overview of the theory of association schemes, see \cite{Lint:2001aa}. Penttila and Williford \cite{penttila2011new} showed that there is an association scheme on the external points of a generalised quadrangle of order $(q,q^2)$ with a doubly subtended subquadrangle of order $(q,q)$ which arises from considering the intersection of subtended ovoids. Since we are interested in $\mathrm{H}(3,q^2)$, we state the association scheme in terms of the dual generalised quadrangle, which means we now have an association scheme on external lines. The original statement of the following result stipulated that $q$ is a power of $2$. However, it can be readily deduced from its proof that the result does not need this requirement. \begin{theorem}[Penttila and Williford {\cite[Theorem 1]{penttila2011new}}] \label{extlinesAS} Let $\Gamma$ be a generalised quadrangle of order $(q^2,q)$ containing a doubly subtended generalised quadrangle $\Gamma'$ of order $(q,q)$. Let $\mathcal{L}_E$ be the set of external lines of $\Gamma$ relative to $\Gamma'$. Then the following set of relations on $\mathcal{L}_E$, together with the equality relation $\Lambda_0$, form a cometric association scheme on $\mathcal{L}_E$. \begin{center} \begin{tabular}{ll} \toprule \textsc{Relation} & \textsc{Description}\\ \midrule $(\ell,n)\in \Lambda_1$ &$\ell$ and $n$ are not concurrent and $n$ is concurrent with $\bar{\ell}$.\\ $(\ell,n)\in \Lambda_2$ & $\ell$ and $n$ are not concurrent and $n$ is not concurrent with $\bar{\ell}$.\\ $(\ell,n)\in \Lambda_3$ & $\ell$ and $n$ are concurrent.\\ $(\ell,n)\in \Lambda_4$ & $n = \bar{\ell}$. \\ \bottomrule \end{tabular} \end{center} \end{theorem} The unique basis of minimal idempotents $\{E_i \mid 0 \leqslant i \leqslant 4\}$ of the associated Bose--Mesner algebra $\mathcal{A}$ can be constructed once we know the dual eigenmatrix $Q$ of the association scheme which we give below (see also \cite[p.~505]{penttila2011new}). Each of these minimal idempotents projects onto a simultaneous eigenspace of $\mathcal{A}$, and together these give a decomposition $\mathbb{C}^{\mathcal{L}_E} = V_0 \perp \dots \perp V_4$ of the vector space over $\mathbb{C}$ with the set of external lines $\mathcal{L}_E$ as a basis. Note that $V_0$ is generated by the all-ones vector $j$. \begin{equation} \label{Q} Q = \begin{bmatrix} 1 & \frac{q(q-1)^2}{2} & \frac{(q-2)(q+1)(q^2+1)}{2} & \frac{q(q-1)(q^2+1)}{2} & \frac{q(q^2+1)}{2} \\[1em] 1 & \frac{q(q-1)}{2}& \frac{(q-2)(q+1)}{2} & \frac{-q(q-1)}{2} & \frac{-q(q-1)}{2}\\[1em] 1 & 0 & -(q+1) & 0 & q \\[1em] 1 & \frac{-q(q-1)}{2}& \frac{(q-2)(q+1)}{2} & \frac{q(q-1)}{2} & \frac{-q(q-1)}{2}\\[1em] 1 & \frac{-q(q-1)^2}{2} & \frac{(q-2)(q+1)(q^2+1)}{2} & \frac{-q(q-1)(q^2+1)}{2} & \frac{q(q^2+1)}{2} \\ \end{bmatrix}. \end{equation} \section{Proof of Main Theorem} Let $\Gamma= \mathrm{H}(3,q^2)$, for some prime-power $q$, and let $\Gamma' = \mathrm{W}(3,q)$. Suppose $\mathcal{S}$ is a set of lines of $\Gamma$. Define $\mathcal{S}^{\perp_E}$ to be the set of external lines that are concurrent with every member of $\mathcal{S}$. We denote the characteristic vector of $\mathcal{S}$ in $\mathcal{L}_E$ by $\chi_\mathcal{S}$. Let $[P]$ denote the set of lines incident with a particular point $P$, with $\chi_{[P]}$ being its characteristic vector. If $\mathcal{R}$ is a relative $m$-cover then the scalar product $\chi_{[P]} \cdot \chi_{\mathcal{R}}$ is equal to $m$, by the definition of a relative $m$-cover, and by Lemma \ref{RmCsize}. Therefore, we have: \[ \big( (q^3-q) \chi_{[P]} - j\big) \cdot \chi_{\mathcal{R}} = m(q^3-q) - m(q^3-q) = 0. \] Also, recalling the association scheme on external lines given in Theorem \ref{extlinesAS}, and the associated adjacency and minimal idempotent matrices, we have \begin{align} \left( (q^3-q) \chi_{[P]} - j\right) E_0 & = \frac{1}{q^2(q^2-1)}\left( (q^3-q) \chi_{[P]} - j\right) J \\ & = \frac{(q^3-q)}{q^2(q^2-1)}\ (qj) - \frac{q^2(q^2-1)}{q^2(q^2-1)}j\\ & = 0. \end{align} By \eqref{Q}, we can express each of the other $E_i$ matrices in terms of adjacency matrices as $E_i=\frac{1}{\|\mathcal{L}_E\|}\sum_{j=0}^{4}Q_{ji}A_j$, where $Q_{ji}$ is the $(j,i)$-entry of $Q$, or in more detail: {\small \begin{align} \numberthis\label{E1} E_1 &= \frac{1}{\|\mathcal{L}_E\|} \left(\frac{q(q-1)^2}{2} \mathrm{I} + \frac{q(q-1)}{2} A_1 - \frac{q(q-1)}{2} A_3 - \frac{q(q-1)^2}{2} A_4\right), \\ E_2 &= \frac{1}{\|\mathcal{L}_E\|} \left(\frac{(q-2)(q+1)(q^2+1)}{2} \mathrm{I} + \frac{(q-2)(q+1)}{2} A_1 - (q+1)A_2 \right.\\ &\qquad \left. +\, \frac{(q-2)(q+1)}{2} A_3 + \frac{(q-2)(q+1)(q^2+1)}{2} A_4 \right),\numberthis\label{E2} \\ E_3 &= \frac{1}{\|\mathcal{L}_E\|} \left(\frac{q(q-1)(q^2+1)}{2} \mathrm{I} - \frac{q(q-1)}{2} A_1 + \frac{q(q-1)}{2} A_3 - \frac{q(q-1)(q^2+1)}{2} A_4\right),\numberthis\label{E3} \\ E_4&= \frac{1}{\|\mathcal{L}_E\|} \left(\frac{q(q^2+1)}{2} \mathrm{I} - \frac{q(q-1)}{2} A_1 + qA_2- \frac{q(q-1)}{2} A_3 + \frac{q(q^2+1)}{2} A_4\right). \numberthis\label{E4} \end{align}} Let $\Gamma$ be a generalised quadrangle of order $(q^2,q)$ with a doubly subtended subquadrangle $\Gamma'$ of order $(q,q)$. Let $\sigma$ be the involutory automorphism of $\Gamma$ fixing $\Gamma'$ pointwise. The following result shows that the \emph{dual degree set} of $\chi_{[P]}$, where $P$ is an external point, is $\{1\}$. \begin{proposition} \label{chipinspan} Let $P$ be an external point. Then $\chi_{[P]} E_1 = 0$ and $\chi_{[P]} E_i \neq 0$ for $i \in \{2,3,4\}$. Hence, $\chi_{[P]} \in V_0\perp V_2\perp V_3\perp V_4$. \end{proposition} \begin{proof} Let $\bar{P} = P^\sigma$ and define $W$ be the set of external lines concurrent with the line on $P$ meeting the doubly subtended quadrangle $\Gamma'$, but not incident with $P$ or $\bar{P}$. Since $\sigma$ is an automorphism, $P$ is incident with $\ell $ if and only if $ \bar{P}$ is incident with $\bar{\ell}$. Now, $\chi_{[{P}]} \mathrm{I} = \chi_{[{P}]}$ trivially, and $\chi_{[{P}]} A_4 = \chi_{[\bar{P}]}$ because the $i$th position in the resulting vector will be 1 if and only if the image of the corresponding external line under $\sigma$ is incident with $P$. Let us now calculate $ \chi_{[P]} A_3$. This is equivalent to counting how many external lines on $P$ are concurrent with each of the other external lines. If $\ell$ is an external line incident with $P$, then the other $q-1$ external lines on $P$ are concurrent with it and so the corresponding positions in $ \chi_{[P]} A_3$ will have value $q-1$. If $\ell$ is an external line not incident with $P$, then, by the `GQ-axiom', either $\ell$ is concurrent with an external line on $P$, in which case the corresponding value in $ \chi_{[P]} A_3$ will be 1, or $\ell$ is concurrent with the line on $P$ meeting the subquadrangle $\Gamma'$, and so contributes 0 to $ \chi_{[P]} A_3$. Summarising, we have $$\chi_{[P]} A_3 = (q-1)\chi_{[P]} + (j-\chi_{[P]} - \chi_W-\chi_{[\bar{P}]}) = j + (q-2)\chi_{[P]} - \chi_W-\chi_{[\bar{P}]}.$$ We now calculate $ \chi_{[P]} A_1$. This is equivalent to counting how many external lines on $P$ are concurrent with the image of each of the other external lines under $\sigma$. Since $\sigma$ preserves incidence, if $\ell$ is a line incident with $P$, then an external line $\bar{n}$ is concurrent with $\ell$ if and only if $\bar{\ell}$ is concurrent with $n$. Therefore, we may equivalently count how many lines incident with $\bar{P}$ are concurrent with each external line $n$, which is the same as calculating $\chi_{[\bar{P}]} A_3$. Therefore, $$\chi_{[P]} A_1 = (q-1)\chi_{[\bar{P}]} + (j-\chi_{[P]}-\chi_W-\chi_{[\bar{P}]}) = (q-2)\chi_{[\bar{P}]} + j -\chi_W - \chi_{[P]}.$$ We will now calculate $\chi_{[P]} A_2$, using the fact that the sum of the adjacency matrices is the all-ones matrix $J$: \begin{align} \chi_{[P]} A_2 &= \chi_{[P]} J - \left( \chi_{[P]} I + \chi_{[P]} A_1 + \chi_{[P]} A_3 + \chi_{[P]} A_4 \right)\\ & = qj - \Big( \chi_{[P]} + (q-2)\chi_{[\bar{P}]} + j -\chi_W - \chi_{[P]} + j \\ & \qquad + (q-2)\chi_{[P]} - \chi_W-\chi_{[\bar{P}]}+ \chi_{[\bar{P}]} \Big)\\ & = (q-2)j - (q-2)(\chi_{[\bar{P}]} + \chi_{[P]}) + 2 \chi_W. \end{align} With respect to the set $\{j,\chi_{[P]},\chi_W,\chi_{[\bar{P}]}\} $ we can summarise our calculations as follows: \begin{align} \left(\chi_{[P]} A_i \right) &= (\chi_{[P]},j,\chi_W,\chi_{[\bar{P}]}) \begin{bmatrix} 1 & -1 & -(q-2) & q-2 & 0 \\ 0 & 1 & q-2 & 1 & 0 \\ 0 & -1 & 2 & -1 & 0 \\ 0 & q-2 & -(q-2) & -1 & 1 \\ \end{bmatrix}. \end{align Therefore, {\footnotesize \begin{align} \frac{1}{\|\mathcal{L}_E\|}( \chi_{[P]} A_i )Q&=\frac{1}{q^2(q^2-1)}(\chi_{[P]},j,\chi_W,\chi_{[\bar{P}]}) \begin{bmatrix} 1 & -1 & -(q-2) & q-2 & 0 \\ 0 & 1 & q-2 & 1 & 0 \\ 0 & -1 & 2 & -1 & 0 \\ 0 & q-2 & -(q-2) & -1 & 1 \\ \end{bmatrix}Q\\ &=\frac{(\chi_{[P]},j,\chi_W,\chi_{[\bar{P}]})}{q^2(q^2-1)} \begin{bmatrix} 0 & 0 & \tfrac{1}{2}q(q-2)(q+1)^2 & \tfrac{1}{2}q^2(q^2-1) & q(q+1)\\ q & 0 & 0 & 0 & -q \\ 0 & 0 & -q(q+1) & 0 & q(q+1) \\ 0 & 0 & \tfrac{1}{2}q(q-2)(q+1)^2 & -\tfrac{1}{2} q^2(q^2-1) & q(q+1)\\ \end{bmatrix} \end{align} } Note that only the second column of the matrix above is zero, and so $\chi_{[P]} E_1 = 0$ and $\chi_{[P]} E_i \neq 0$ for $i \in \{2,3,4\}$. \end{proof} \begin{theorem} \label{chipsspan} Let $\chi_{[P]}$ be the characteristic vector of the set of external lines incident with an external point $P$. Then the set of vectors $\{ \chi_{[P]} \mid P \in \mathcal{P}_E\}$ is a spanning set of $V_0\perp V_2 \perp V_3 \perp V_4$. \end{theorem} \begin{proof} By Proposition \ref{chipinspan}, $\chi_{[P]} \in V_0\perp V_2 \perp V_3 \perp V_4 $. Let $A$ be the matrix whose rows are the $\chi_{[P]}$ vectors. To prove that $\{ \chi_{[P]} \mid P \in \mathcal{P}_E\}$ spans $V_0\perp V_2 \perp V_3 \perp V_4 $, it is sufficient to show that the rank of $A$ is equal to $\mathrm{dim}(V_0\perp V_2 \perp V_3 \perp V_4) = \|\mathcal{L}_E\| - \mathrm{dim}(V_1)$. Consider the matrix $M=A^TA$. The rows of $M$ correspond to counting how many points incident with a particular external line are incident with each of the other external lines. The diagonal entries of $M$ are each equal to $q^2+1$, the entries whose row and column correspond to concurrent lines are equal to 1, and the entries are zero otherwise. In particular, the rows of $M$ are of the form $(q^2+1)\chi_\ell + \chi_{\{\ell\}^{\perp_E}}$ for some external line $\ell$. Denote the row of $M$ corresponding to the external line $\ell$ by $M_\ell$, and recall that the size of $\{\ell\}^{\perp_E}$ is $(q-1)(q^2+1)$, since each of the points of $\ell$ is incident with $q-1$ external lines not equal to $\ell$. Note that $\mathrm{rank}(M) \leqslant \mathrm{rank}(A)$, since $M$ is constructed by taking linear combinations of the rows of $A$. Also notice that $M$ has constant column sum equal to $q(q^2+1)$, and so the all-ones vector $j$ lies in the row space of $M$. We first show that $\chi_\ell + \chi_{\bar{\ell}}$ is in the row space of $M$. Define the set \[ U_{\ell} := \{ M_\ell\} \cup \{ M_{\ell'} \mid \ell' \text{ is concurrent with } \ell \}. \] Then the sum of the vectors in $U_\ell$ is \begin{align} &(q^2+1)\chi_\ell + \chi_{\{\ell\}^{\perp_E}} + (q^2+1)\chi_{\{\ell\}^{\perp_E}} + (q-1)(q^2+1)\chi_\ell \\ & \quad + (q-2)\chi_{\{\ell\}^{\perp_E}} + q^2\chi_{\{\bar{\ell}\}^{\perp_E}} + (q^2-q)\chi_{\{\ell, \bar{\ell} \}^{\perp_2}}, \end{align} where $\chi_{\{\ell, \bar{\ell} \}^{\perp_2}} $ is the set of lines that are neither concurrent with nor equal to $\ell$ or $\bar{\ell}$. The first two terms originate from $M_\ell$. The third term arises because $M_{\ell'}$ contributes $(q^2+1)\chi_{\ell'}$ for every $\ell' \in \{\ell\}^{\perp_E}$. Every line of $\{\ell\}^{\perp_E}$ is concurrent with $\ell$ by definition, and this is how the fourth term arises. The fifth term comes about because every element of $\{\ell\}^{\perp_E}$ is also concurrent with the $q-2$ elements of $\{\ell\}^{\perp_E}$ that lie on the same point of intersection with $\ell$ (and no other elements of $\{\ell\}^{\perp_E}$). The sixth and seventh terms arise because for every external line $n$ not equal to $\ell$ or $\bar{\ell}$ and not concurrent with $\ell$, Lemma \ref{extlinesint} implies that the size of $\left\{ \ell, n \right\}^{\perp_E}$ is $q^2$ if $n$ is concurrent with $\bar{\ell}$, and $q^2-q$ otherwise. Simplifying slightly, we write the sum as \[ q(q^2+1)\chi_\ell+ (q^2+q)\chi_{\ell^{\perp_E}} + q^2\chi_{\bar{\ell}^{\perp_E}} + (q^2-q)\chi_{\{\ell, \bar{\ell} \}^{\perp_2}} . \] Consider the sum of all of the vectors in $U_{\ell} \cup U_{\bar{\ell}}$. After some simplification, we arrive at the expression \begin{equation} \label{eq1} q(q^2+1)(\chi_\ell+ \chi_{\bar{\ell}}) + (2q^2+q)(\chi_{\ell^{\perp_E}}+ \chi_{\bar{\ell}^{\perp_E}}) + (2q^2-2q)\chi_{\{\ell, \bar{\ell} \}^{\perp_2}}. \end{equation} Recall that $j$ is in the row space of $M$. We calculate the difference between \eqref{eq1} and $(2q^2-2q)j$ and arrive at the following expression: \[ q(q^2-2q+3)(\chi_\ell+ \chi_{\bar{\ell}}) + 3q(\chi_{\ell^{\perp_E}}+ \chi_{\bar{\ell}^{\perp_E}}). \] We now subtract $3q(M_\ell+M_{\bar{\ell}})$: \begin{align} &q(q^2-2q+3)(\chi_\ell+ \chi_{\bar{\ell}}) + 3q(\chi_{\ell^{\perp_E}}+ \chi_{\bar{\ell}^{\perp_E}}) - 3q(M_\ell+M_{\bar{\ell}}) \\ &\quad =q(q^2-2q+3)(\chi_\ell+ \chi_{\bar{\ell}}) - 3q(q^2+1)(\chi_\ell+ \chi_{\bar{\ell}})\\ & \quad = -2q^2(q+1)(\chi_\ell+ \chi_{\bar{\ell}}). \end{align} Therefore, $\chi_\ell+ \chi_{\bar{\ell}}$ is in the row space of $M$. The $\chi_\ell$ vectors form a spanning set for the vector space over $\mathbb{C}^{\mathcal{L}_E}$ by definition. Therefore, the set $\{\chi_\ell E_i \mid \ell \in \mathcal{L}_E\}$ forms a basis for $V_i$ for $0 \leqslant i \leqslant 4$. Hence, to show that $V_i$ lies in the row space of $M$, it is sufficient to show that $\chi_\ell E_i$ does, for any choice of external line $\ell$. We first calculate the $\chi_\ell E_i$, based on their expressions as the linear combinations of adjacency matrices above (see Equations \eqref{E1}, \eqref{E2}, \eqref{E3}, \eqref{E4}): \begin{align} \chi_\ell E_0 &= \frac{1}{q^2(q^2-1)}j, \\ \chi_\ell E_1 &= \frac{1}{2q(q+1)}((q-1)(\chi_\ell - \chi_{\bar{\ell}}) +\chi_{\bar{\ell}^{\perp_E}} - \chi_{\ell^{\perp_E}}),\\ \chi_\ell E_2 &= \frac{1}{2q^2(q-1)}(-2j+ q((q-1)^2\chi_\ell + \chi_{\ell^{\perp_E}} + (q-1)^2\chi_{\bar{\ell}} + \chi_{\bar{\ell}^{\perp_E}})),\\ \chi_\ell E_3 &= \frac{1}{2q(q+1)}((q^2+1)\chi_\ell - (q^2+1) \chi_{\bar{\ell}} -\chi_{\bar{\ell}^{\perp_E}} + \chi_{\ell^{\perp_E}}),\\ \chi_\ell E_4 &= \frac{1}{2q(q^2-1)}(2j+(q+1)((q-1)(\chi_\ell +\chi_{\bar{\ell}}) - \chi_{\bar{\ell}^{\perp_E}} - \chi_{\ell^{\perp_E}})). \end{align} Recall that the rows $M_\ell$ of $M$ are of the form $(q^2+1)\chi_\ell + \chi_{\{\ell\}^{\perp_E}}$ for each external line $\ell$. Then $ \chi_\ell E_3$ is in the row space of $M$, since it is a linear combination of the $M_\ell - M_{\bar{\ell}}$ vectors. Now consider $\chi_\ell E_2$. Since $j$ is in the row space of $M$, we only need to show that $(q-1)^2\chi_\ell + \chi_{\ell^{\perp_E}} + (q-1)^2\chi_{\bar{\ell}} + \chi_{\bar{\ell}^{\perp_E}}$ is also in the row space of $M$. We can rewrite this vector as \[ ((q^2+1)\chi_\ell + \chi_{\ell^{\perp_E}} ) + ((q^2+1)\chi_{\bar{\ell}} + \chi_{\bar{\ell}^{\perp_E}} ) + ((q-1)^2-(q^2+1))(\chi_\ell+\chi_{\bar{\ell}}). \] This vector lies in the row space of $M$ since the first two terms are $M_\ell$ and $M_{\bar{\ell}}$, and $\chi_\ell+ \chi_{\bar{\ell}}$ also lies in the row space of $M$. Finally, consider $\chi_\ell E_4$. Using the same reasoning as in the last case, it is sufficient to show that $(q-1)(\chi_\ell +\chi_{\bar{\ell}}) - \chi_{\bar{\ell}^{\perp_E}} - \chi_{\ell^{\perp_E}}$ is in the row space of $M$. We can rewrite this expression as \[ -((q^2+1)\chi_\ell + \chi_{\ell^{\perp_E}} ) - ((q^2+1)\chi_{\bar{\ell}} + \chi_{\bar{\ell}^{\perp_E}} ) + ((q-1)+(q^2+1))(\chi_\ell+\chi_{\bar{\ell}}), \] which is in the rowspace of $M$. Therefore, $V_0 \perp V_2 \perp V_3 \perp V_4$ is a subspace of the row space of $M$. This implies that $M$, and therefore $A$, has rank at least the dimension of $V_0 \perp V_2 \perp V_3 \perp V_4$. Hence $A$ has rank equal to the dimension of $V_0 \perp V_2 \perp V_3 \perp V_4$ and its rows (that is, the set $\{ \chi_{[P]} \mid P \in \mathcal{P}_E\}$) form a spanning set of $V_0 \perp V_2 \perp V_3 \perp V_4$. \end{proof} \begin{corollary} \label{spancoroll} The set of vectors $\{(q^3-q)\chi_{[P]} - j \mid P \in \mathcal{P}_E \}$ forms a spanning set of $V_2 \perp V_3 \perp V_4$. \end{corollary} \begin{proof} By Theorem \ref{chipsspan}, the set $\{ \chi_{[P]} \mid P \in \mathcal{P}_E\}$ forms a spanning set of $V_0 \perp V_2 \perp V_3 \perp V_4$. Now, $((q^3-q)\chi_{[P]} - j)E_0 = \frac{1}{q^2(q^2-1)}\left((q^3-q)q - q^2(q^2-1)\right) = 0$, so $(q^3-q)\chi_{[P]} - j \in V_2\perp V_3 \perp V_4$ for every external point $P$. Therefore, the set $\{(q^3-q)\chi_{[P]} - j \mid P \in \mathcal{P}_E \}$ forms a spanning set of $V_2 \perp V_3 \perp V_4$. \end{proof} \begin{corollary} \label{SinV0V1} Let $\mathcal{R}$ be a relative $m$-cover of a generalised quadrangle $\Gamma$ of order $(q^2,q)$ relative to a doubly subtended subquadrangle $\Gamma'$ of order $(q,q)$. Then $\chi_\mathcal{R}$ lies in $V_0 \perp V_1$. \end{corollary} \begin{proof} By Corollary \ref{spancoroll}, it is sufficient to show that $\chi_\mathcal{R} \cdot ((q^3-q)\chi_{[P]} - j ) = 0$ for all $P \in \mathcal{P}_E$. By Lemma \ref{RmCsize}, we have \[ \chi_\mathcal{R} \cdot ((q^3-q)\chi_{[P]} - j ) = m(q^3-q) - m(q^3-q) = 0. \] \end{proof} We are now ready to prove Theorem \ref{bigthm}. \begin{proof}[of Theorem \ref{bigthm}] Let $\mathcal{R}^\sigma$ denote the image of $\mathcal{R}$ under $\sigma$. We first calculate the product of $\chi_\mathcal{R}$ with each of the adjacency matrices. First, $\chi_\mathcal{R}A_4 = \chi_{\mathcal{R}^\sigma}$ since there will be a 1 in the $j$th position of $\chi_\mathcal{R}A_4 $ if and only if the image of the corresponding line under $\sigma$ is contained in $\mathcal{R}$, and 0 otherwise. Let us now calculate $\chi_\mathcal{R}A_3$. This is equivalent to counting how many elements of $\mathcal{R}$ are concurrent with each external line. If $\ell$ is an external line in $\mathcal{R}$, then it will be concurrent with $(m-1)(q^2+1)$ lines of $\mathcal{R}$ (that is, $m-1$ for each point on the line). If $\ell$ is not in $\mathcal{R}$, then there will be $m(q^2+1)$ lines of $\mathcal{R}$ concurrent with $\ell$. Summarising, \[ \chi_\mathcal{R}A_3 = (q^2+1)(m-1)\chi_\mathcal{R} + m(q^2+1)(j-\chi_\mathcal{R}) = m(q^2+1)j-(q^2+1)\chi_\mathcal{R}. \] We now calculate $\chi_\mathcal{R}A_1$. This is equivalent to counting the number of elements of $\mathcal{R}$ that are concurrent with the image $\bar{\ell}$ of each external line $\ell$ under $\sigma$. Since $\sigma$ preserves incidence, if $n$ is a line in $\mathcal{R}$, then $n$ is concurrent with $\bar{\ell}$ if and only if $\bar{n}$ is concurrent with $\ell$. Therefore, we may equivalently count how many lines in $\mathcal{R}^\sigma$ are concurrent with each external line $\ell$, which is the same as calculating $\chi_{\mathcal{R}^\sigma} A_3$. Using the calculation we just completed, we have \[ \chi_\mathcal{R} A_1 = \chi_{\mathcal{R}^\sigma} A_3 = m(q^2+1)j-(q^2+1)\chi_{\mathcal{R}^\sigma}. \] Now, it remains to calculate $\chi_\mathcal{R}A_2$. We will make use of the fact that the sum of all of the adjacency matrices is equal to $J$. Therefore, \begin{align} \chi_\mathcal{R}A_2 & = \chi_\mathcal{R}J - \left( \chi_\mathcal{R}I + \chi_\mathcal{R} A_1 + \chi_\mathcal{R} A_3 + \chi_\mathcal{R} A_4 \right)\\ & = m(q^3-q)j- \big(\chi_\mathcal{R} + m(q^2+1)j-(q^2+1)\chi_{\mathcal{R}^\sigma} \\ & \qquad + m(q^2+1)j-(q^2+1)\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma}\big)\\ & = m(q^3-2q^2-q-2)j+q^2(\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma}). \end{align} We now make use of Equations \eqref{E2}, \eqref{E3} and \eqref{E4} to compute the projections of $\chi_\mathcal{R}$ into $V_2$, $V_3$ and $V_4$. They are as follows: \begin{align} \chi_\mathcal{R} E_2 & = \frac{1}{q^2(q^2-1)} \left(2mq(q+1)j - q^2(q+1)(\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma})\right),\\ \chi_\mathcal{R} E_3 &= 0, \\ \chi_\mathcal{R}E_4 &= \frac{1}{q^2(q^2-1)}\left(\frac{q^2(q+1)^2}{2}(\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma}) -mq(q+1)^2j \right). \end{align} By Corollary \ref{SinV0V1}, $\chi_\mathcal{R} \in V_0 \perp V_1$ and therefore $\chi_\mathcal{R} E_i=0$ for $i \in \{2,3,4\}$. We only need to consider $i=2$: here we see that $2m j = q(\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma})$. Therefore, $\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma}$ is the constant vector with value $2m/q$, and since the values of $\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma}$ lie in $\{0,1,2\}$, it follows immediately that $q$ is even, $m=q/2$, and $\chi_\mathcal{R} + \chi_{\mathcal{R}^\sigma}=j$. That is, $\mathcal{R}$ is a relative hemisystem and $\mathcal{R}^\sigma$ is the complement of $\mathcal{R}$. \end{proof} \subsection*{Acknowledgements} The authors are indebted to Michael Giudici for many discussions on the material in this work. The first author acknowledges the support of the Australian Research Council Future Fellowship FT120100036. The second author acknowledges the support of a Hackett Postgraduate Scholarship. The authors also thank Daniel Horsley for his mention of an elementary result during his talk at Combinatorics 2016, which was instrumental in the proof of the main result of this paper.	{ "redpajama_set_name": "RedPajamaArXiv" }	8,343
#include <utexas_planning/planners/vi/estimator.h> namespace utexas_planning { namespace vi { Estimator::~Estimator() {} } /* vi / } / utexas_planning */	{ "redpajama_set_name": "RedPajamaGithub" }	447
{"url":"https:\/\/paperswithcode.com\/method\/non","text":"Deep Tabular Learning\n\n# Network On Network\n\nIntroduced by Luo et al. in Network On Network for Tabular Data Classification in Real-world Applications\n\nNetwork On Network (NON) is practical tabular data classification model based on deep neural network to provide accurate predictions. Various deep methods have been proposed and promising progress has been made. However, most of them use operations like neural network and factorization machines to fuse the embeddings of different features directly, and linearly combine the outputs of those operations to get the final prediction. As a result, the intra-field information and the non-linear interactions between those operations (e.g. neural network and factorization machines) are ignored. Intra-field information is the information that features inside each field belong to the same field. NON is proposed to take full advantage of intra-field information and non-linear interactions. It consists of three components: field-wise network at the bottom to capture the intra-field information, across field network in the middle to choose suitable operations data-drivenly, and operation fusion network on the top to fuse outputs of the chosen operations deeply\n\n#### Papers\n\nPaper Code Results Date Stars","date":"2022-10-01 21:07:05","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\": 0.9208659529685974, \"perplexity\": 1394.3353542858526}, \"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-2022-40\/segments\/1664030336921.76\/warc\/CC-MAIN-20221001195125-20221001225125-00113.warc.gz\"}"}	null	null
Many people scour the internet looking for ways to save money and, unfortunately, end up learning money saving myths. Many of these myths seem realistic. However, they are counterproductive. In this article, we discuss the most common heating myths that you may have always believed to be true.	{ "redpajama_set_name": "RedPajamaC4" }	4,518
Wheels, Tires, Lift Kit, and Push Bar. 1Not available with special financing, lease and some other offers. See dealer for details. 2Delivery and Handling fee. See dealer for details 3Dealer added accessories. Photos may not reflect accessories currently on the vehicle. Please see dealer for details. 4Not available with special financing, lease and some other offers. See dealer for details. 5Excludes base models. Monthly payment is $16.67 for every $1,000 you finance. Example down payment: 7.9%. Must finance with GM Financial. Some customers may not qualify. Not available with lease and some other offers. Take new retail delivery by 4/30/2019. See dealer for details. Price includes standard Incentives and Dealer Handling. Please call 720-506-3013 to schedule your test drive today! Price includes dealer added accessories. Prices do not include tax, tags and title fees. All Elway Prices include non-qualifying manufacturer rebates/incentives and are subject to change at any time. While every reasonable effort is made to ensure the accuracy of this information we are not responsible for any errors or omissions contained on these pages. Please verify any information with the sales manager. EPA are estimates only. D&H is included in the Total Selling Price. MPG estimates on this website are EPA estimates; your actual mileage may vary. For used vehicles, MPG estimates are EPA estimates for the vehicle when it was new. The EPA periodically modifies its MPG calculation methodology; all MPG estimates are based on the methodology in effect when the vehicles were new (please see the "Fuel Economy" portion of the EPA's website for details, including a MPG recalculation tool). The features and options listed are for the new 2018 Chevrolet Colorado and may not apply to this specific vehicle. Tax, title, license (unless itemized above) are extra. Not available with special finance, lease and some other offers.	{ "redpajama_set_name": "RedPajamaC4" }	8,132
Q: scrapy - condition on allowed_domains I would like to know if it's possible, in python, to make a condition on a link like this : if my_link is allowed: I tried : allowed_domains = ['exemple.com'] if mylink.exemple.com in allowed_domains: something So my link is allowed but not written in allowed_domains... How could I do please? A: You can look at how OffsiteMiddleware is implemented (link), specifically should_follow method. You might reuse the logic from there.	{ "redpajama_set_name": "RedPajamaStackExchange" }	4,453

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.