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Send Flowers for Frederick Frederick J. Dougherty On Monday, December 21, 2020, Frederick J. Dougherty passed away peacefully in his home at the age of 90 in the company of his children. Fred shared 63 years of marriage with his wife Joan who recently passed away. Fred is survived by four children Charlotte Inman of Middleton, Frederick Dougherty of Middleton, Trudy Kiley of Amesbury and Rachel Crosscup of Middleton. Nine grandchildren, 11 great-grandchildren, and numerous nieces and nephews. The couple was predeceased by a daughter Laura. Fred was raised in Lynn, Massachusetts-the youngest of 7 children. He was a well-known, popular presence in the town of Middleton where he resided for over 50 years. Fred was honorably discharged after faithfully performing his duties in the United States Army during the Korean War where he served in Japan. Using the machinist skills and training he received in the military Fred went on to realize a life-long career as an accomplished machinist. He climbed the ranks at the Polaroid Corporation from machinist to research development and ultimately mechanical engineering. He retired from Polaroid as an engineering supervisor in 1980 and then opened FJ Dougherty Machine, Inc. Fred enjoyed listening to Baseball on the radio. And, together, with his wife Joan he never missed a televised Bruins game. "Good, Better, Best. Never Take a Rest. Until the good is better and the better is best". At Fred's request, there will be no services. Donations in his memory may be made to The Middleton Council on Aging, P.O. Box 855, Middleton, MA 01949. Assisting the family is the Mackey Funeral Home. To share a condolence, kindly visit www.mackeyfuneralhome.com. To send flowers to Frederick's family, please visit our floral store. You can still show your support by sending flowers directly to the family Middleton Council on Aging PO BOX 855, Middleton MA 01949	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	9,363
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Pyrus asiae-mediae är en rosväxtart som först beskrevs av Mikhail Grigoríevič Popov, och fick sitt nu gällande namn av Maleev. Pyrus asiae-mediae ingår i släktet päronsläktet, och familjen rosväxter. Inga underarter finns listade i Catalogue of Life. Källor Päronsläktet asiae-mediae	{ "redpajama_set_name": "RedPajamaWikipedia" }	7,161
((volatile unsigned long )0x40058000) #define GPIO_PORTA_AHB_DATA_R (((volatile unsigned long )0x400583FC)) #define GPIO_PORTA_AHB_DIR_R (((volatile unsigned long )0x40058400)) #define GPIO_PORTA_AHB_IS_R (((volatile unsigned long )0x40058404)) #define GPIO_PORTA_AHB_IBE_R (((volatile unsigned long )0x40058408)) #define GPIO_PORTA_AHB_IEV_R (((volatile unsigned long )0x4005840C)) #define GPIO_PORTA_AHB_IM_R (((volatile unsigned long )0x40058410)) #define GPIO_PORTA_AHB_RIS_R (((volatile unsigned long )0x40058414)) #define GPIO_PORTA_AHB_MIS_R (((volatile unsigned long )0x40058418)) #define GPIO_PORTA_AHB_ICR_R (((volatile unsigned long )0x4005841C)) #define GPIO_PORTA_AHB_AFSEL_R (((volatile unsigned long )0x40058420)) #define GPIO_PORTA_AHB_DR2R_R (((volatile unsigned long )0x40058500)) #define GPIO_PORTA_AHB_DR4R_R (((volatile unsigned long )0x40058504)) #define GPIO_PORTA_AHB_DR8R_R (((volatile unsigned long )0x40058508)) #define GPIO_PORTA_AHB_ODR_R (((volatile unsigned long )0x4005850C)) #define GPIO_PORTA_AHB_PUR_R (((volatile unsigned long )0x40058510)) #define GPIO_PORTA_AHB_PDR_R (((volatile unsigned long )0x40058514)) #define GPIO_PORTA_AHB_SLR_R (((volatile unsigned long )0x40058518)) #define GPIO_PORTA_AHB_DEN_R (((volatile unsigned long )0x4005851C)) #define GPIO_PORTA_AHB_LOCK_R (((volatile unsigned long )0x40058520)) #define GPIO_PORTA_AHB_CR_R (((volatile unsigned long )0x40058524)) #define GPIO_PORTA_AHB_AMSEL_R (((volatile unsigned long )0x40058528)) #define GPIO_PORTA_AHB_PCTL_R (((volatile unsigned long )0x4005852C)) #define GPIO_PORTA_AHB_ADCCTL_R (((volatile unsigned long )0x40058530)) #define GPIO_PORTA_AHB_DMACTL_R (((volatile unsigned long )0x40058534)) #define GPIO_PORTA_AHB_SI_R (((volatile unsigned long )0x40058538)) //************************************************************************** // // GPIO registers (PORTB AHB) // //*************************************************************************** #define GPIO_PORTB_AHB_DATA_BITS_R \ ((volatile unsigned long )0x40059000) #define GPIO_PORTB_AHB_DATA_R (((volatile unsigned long )0x400593FC)) #define GPIO_PORTB_AHB_DIR_R (((volatile unsigned long )0x40059400)) #define GPIO_PORTB_AHB_IS_R (((volatile unsigned long )0x40059404)) #define GPIO_PORTB_AHB_IBE_R (((volatile unsigned long )0x40059408)) #define GPIO_PORTB_AHB_IEV_R (((volatile unsigned long )0x4005940C)) #define GPIO_PORTB_AHB_IM_R (((volatile unsigned long )0x40059410)) #define GPIO_PORTB_AHB_RIS_R (((volatile unsigned long )0x40059414)) #define GPIO_PORTB_AHB_MIS_R (((volatile unsigned long )0x40059418)) #define GPIO_PORTB_AHB_ICR_R (((volatile unsigned long )0x4005941C)) #define GPIO_PORTB_AHB_AFSEL_R (((volatile unsigned long )0x40059420)) #define GPIO_PORTB_AHB_DR2R_R (((volatile unsigned long )0x40059500)) #define GPIO_PORTB_AHB_DR4R_R (((volatile unsigned long )0x40059504)) #define GPIO_PORTB_AHB_DR8R_R (((volatile unsigned long )0x40059508)) #define GPIO_PORTB_AHB_ODR_R (((volatile unsigned long )0x4005950C)) #define GPIO_PORTB_AHB_PUR_R (((volatile unsigned long )0x40059510)) #define GPIO_PORTB_AHB_PDR_R (((volatile unsigned long )0x40059514)) #define GPIO_PORTB_AHB_SLR_R (((volatile unsigned long )0x40059518)) #define GPIO_PORTB_AHB_DEN_R (((volatile unsigned long )0x4005951C)) #define GPIO_PORTB_AHB_LOCK_R (((volatile unsigned long )0x40059520)) #define GPIO_PORTB_AHB_CR_R (((volatile unsigned long )0x40059524)) #define GPIO_PORTB_AHB_AMSEL_R (((volatile unsigned long )0x40059528)) #define GPIO_PORTB_AHB_PCTL_R (((volatile unsigned long )0x4005952C)) #define GPIO_PORTB_AHB_ADCCTL_R (((volatile unsigned long )0x40059530)) #define GPIO_PORTB_AHB_DMACTL_R (((volatile unsigned long )0x40059534)) #define GPIO_PORTB_AHB_SI_R (((volatile unsigned long )0x40059538)) //************************************************************************** // // GPIO registers (PORTC AHB) // //*************************************************************************** #define GPIO_PORTC_AHB_DATA_BITS_R \ ((volatile unsigned long )0x4005A000) #define GPIO_PORTC_AHB_DATA_R (((volatile unsigned long )0x4005A3FC)) #define GPIO_PORTC_AHB_DIR_R (((volatile unsigned long )0x4005A400)) #define GPIO_PORTC_AHB_IS_R (((volatile unsigned long )0x4005A404)) #define GPIO_PORTC_AHB_IBE_R (((volatile unsigned long )0x4005A408)) #define GPIO_PORTC_AHB_IEV_R (((volatile unsigned long )0x4005A40C)) #define GPIO_PORTC_AHB_IM_R (((volatile unsigned long )0x4005A410)) #define GPIO_PORTC_AHB_RIS_R (((volatile unsigned long )0x4005A414)) #define GPIO_PORTC_AHB_MIS_R (((volatile unsigned long )0x4005A418)) #define GPIO_PORTC_AHB_ICR_R (((volatile unsigned long )0x4005A41C)) #define GPIO_PORTC_AHB_AFSEL_R (((volatile unsigned long )0x4005A420)) #define GPIO_PORTC_AHB_DR2R_R (((volatile unsigned long )0x4005A500)) #define GPIO_PORTC_AHB_DR4R_R (((volatile unsigned long )0x4005A504)) #define GPIO_PORTC_AHB_DR8R_R (((volatile unsigned long )0x4005A508)) #define GPIO_PORTC_AHB_ODR_R (((volatile unsigned long )0x4005A50C)) #define GPIO_PORTC_AHB_PUR_R (((volatile unsigned long )0x4005A510)) #define GPIO_PORTC_AHB_PDR_R (((volatile unsigned long )0x4005A514)) #define GPIO_PORTC_AHB_SLR_R (((volatile unsigned long )0x4005A518)) #define GPIO_PORTC_AHB_DEN_R (((volatile unsigned long )0x4005A51C)) #define GPIO_PORTC_AHB_LOCK_R (((volatile unsigned long )0x4005A520)) #define GPIO_PORTC_AHB_CR_R (((volatile unsigned long )0x4005A524)) #define GPIO_PORTC_AHB_AMSEL_R (((volatile unsigned long )0x4005A528)) #define GPIO_PORTC_AHB_PCTL_R (((volatile unsigned long )0x4005A52C)) #define GPIO_PORTC_AHB_ADCCTL_R (((volatile unsigned long )0x4005A530)) #define GPIO_PORTC_AHB_DMACTL_R (((volatile unsigned long )0x4005A534)) #define GPIO_PORTC_AHB_SI_R (((volatile unsigned long )0x4005A538)) //************************************************************************** // // GPIO registers (PORTD AHB) // //*************************************************************************** #define GPIO_PORTD_AHB_DATA_BITS_R \ ((volatile unsigned long )0x4005B000) #define GPIO_PORTD_AHB_DATA_R (((volatile unsigned long )0x4005B3FC)) #define GPIO_PORTD_AHB_DIR_R (((volatile unsigned long )0x4005B400)) #define GPIO_PORTD_AHB_IS_R (((volatile unsigned long )0x4005B404)) #define GPIO_PORTD_AHB_IBE_R (((volatile unsigned long )0x4005B408)) #define GPIO_PORTD_AHB_IEV_R (((volatile unsigned long )0x4005B40C)) #define GPIO_PORTD_AHB_IM_R (((volatile unsigned long )0x4005B410)) #define GPIO_PORTD_AHB_RIS_R (((volatile unsigned long )0x4005B414)) #define GPIO_PORTD_AHB_MIS_R (((volatile unsigned long )0x4005B418)) #define GPIO_PORTD_AHB_ICR_R (((volatile unsigned long )0x4005B41C)) #define GPIO_PORTD_AHB_AFSEL_R (((volatile unsigned long )0x4005B420)) #define GPIO_PORTD_AHB_DR2R_R (((volatile unsigned long )0x4005B500)) #define GPIO_PORTD_AHB_DR4R_R (((volatile unsigned long )0x4005B504)) #define GPIO_PORTD_AHB_DR8R_R (((volatile unsigned long )0x4005B508)) #define GPIO_PORTD_AHB_ODR_R (((volatile unsigned long )0x4005B50C)) #define GPIO_PORTD_AHB_PUR_R (((volatile unsigned long )0x4005B510)) #define GPIO_PORTD_AHB_PDR_R (((volatile unsigned long )0x4005B514)) #define GPIO_PORTD_AHB_SLR_R (((volatile unsigned long )0x4005B518)) #define GPIO_PORTD_AHB_DEN_R (((volatile unsigned long )0x4005B51C)) #define GPIO_PORTD_AHB_LOCK_R (((volatile unsigned long )0x4005B520)) #define GPIO_PORTD_AHB_CR_R (((volatile unsigned long )0x4005B524)) #define GPIO_PORTD_AHB_AMSEL_R (((volatile unsigned long )0x4005B528)) #define GPIO_PORTD_AHB_PCTL_R (((volatile unsigned long )0x4005B52C)) #define GPIO_PORTD_AHB_ADCCTL_R (((volatile unsigned long )0x4005B530)) #define GPIO_PORTD_AHB_DMACTL_R (((volatile unsigned long )0x4005B534)) #define GPIO_PORTD_AHB_SI_R (((volatile unsigned long )0x4005B538)) //************************************************************************** // // GPIO registers (PORTE AHB) // //*************************************************************************** #define GPIO_PORTE_AHB_DATA_BITS_R \ ((volatile unsigned long )0x4005C000) #define GPIO_PORTE_AHB_DATA_R (((volatile unsigned long )0x4005C3FC)) #define GPIO_PORTE_AHB_DIR_R (((volatile unsigned long )0x4005C400)) #define GPIO_PORTE_AHB_IS_R (((volatile unsigned long )0x4005C404)) #define GPIO_PORTE_AHB_IBE_R (((volatile unsigned long )0x4005C408)) #define GPIO_PORTE_AHB_IEV_R (((volatile unsigned long )0x4005C40C)) #define GPIO_PORTE_AHB_IM_R (((volatile unsigned long )0x4005C410)) #define GPIO_PORTE_AHB_RIS_R (((volatile unsigned long )0x4005C414)) #define GPIO_PORTE_AHB_MIS_R (((volatile unsigned long )0x4005C418)) #define GPIO_PORTE_AHB_ICR_R (((volatile unsigned long )0x4005C41C)) #define GPIO_PORTE_AHB_AFSEL_R (((volatile unsigned long )0x4005C420)) #define GPIO_PORTE_AHB_DR2R_R (((volatile unsigned long )0x4005C500)) #define GPIO_PORTE_AHB_DR4R_R (((volatile unsigned long )0x4005C504)) #define GPIO_PORTE_AHB_DR8R_R (((volatile unsigned long )0x4005C508)) #define GPIO_PORTE_AHB_ODR_R (((volatile unsigned long )0x4005C50C)) #define GPIO_PORTE_AHB_PUR_R (((volatile unsigned long )0x4005C510)) #define GPIO_PORTE_AHB_PDR_R (((volatile unsigned long )0x4005C514)) #define GPIO_PORTE_AHB_SLR_R (((volatile unsigned long )0x4005C518)) #define GPIO_PORTE_AHB_DEN_R (((volatile unsigned long )0x4005C51C)) #define GPIO_PORTE_AHB_LOCK_R (((volatile unsigned long )0x4005C520)) #define GPIO_PORTE_AHB_CR_R (((volatile unsigned long )0x4005C524)) #define GPIO_PORTE_AHB_AMSEL_R (((volatile unsigned long )0x4005C528)) #define GPIO_PORTE_AHB_PCTL_R (((volatile unsigned long )0x4005C52C)) #define GPIO_PORTE_AHB_ADCCTL_R (((volatile unsigned long )0x4005C530)) #define GPIO_PORTE_AHB_DMACTL_R (((volatile unsigned long )0x4005C534)) #define GPIO_PORTE_AHB_SI_R (((volatile unsigned long )0x4005C538)) //************************************************************************** // // GPIO registers (PORTF AHB) // //*************************************************************************** #define GPIO_PORTF_AHB_DATA_BITS_R \ ((volatile unsigned long )0x4005D000) #define GPIO_PORTF_AHB_DATA_R (((volatile unsigned long )0x4005D3FC)) #define GPIO_PORTF_AHB_DIR_R (((volatile unsigned long )0x4005D400)) #define GPIO_PORTF_AHB_IS_R (((volatile unsigned long )0x4005D404)) #define GPIO_PORTF_AHB_IBE_R (((volatile unsigned long )0x4005D408)) #define GPIO_PORTF_AHB_IEV_R (((volatile unsigned long )0x4005D40C)) #define GPIO_PORTF_AHB_IM_R (((volatile unsigned long )0x4005D410)) #define GPIO_PORTF_AHB_RIS_R (((volatile unsigned long )0x4005D414)) #define GPIO_PORTF_AHB_MIS_R (((volatile unsigned long )0x4005D418)) #define GPIO_PORTF_AHB_ICR_R (((volatile unsigned long )0x4005D41C)) #define GPIO_PORTF_AHB_AFSEL_R (((volatile unsigned long )0x4005D420)) #define GPIO_PORTF_AHB_DR2R_R (((volatile unsigned long )0x4005D500)) #define GPIO_PORTF_AHB_DR4R_R (((volatile unsigned long )0x4005D504)) #define GPIO_PORTF_AHB_DR8R_R (((volatile unsigned long )0x4005D508)) #define GPIO_PORTF_AHB_ODR_R (((volatile unsigned long )0x4005D50C)) #define GPIO_PORTF_AHB_PUR_R (((volatile unsigned long )0x4005D510)) #define GPIO_PORTF_AHB_PDR_R (((volatile unsigned long )0x4005D514)) #define GPIO_PORTF_AHB_SLR_R (((volatile unsigned long )0x4005D518)) #define GPIO_PORTF_AHB_DEN_R (((volatile unsigned long )0x4005D51C)) #define GPIO_PORTF_AHB_LOCK_R (((volatile unsigned long )0x4005D520)) #define GPIO_PORTF_AHB_CR_R (((volatile unsigned long )0x4005D524)) #define GPIO_PORTF_AHB_AMSEL_R (((volatile unsigned long )0x4005D528)) #define GPIO_PORTF_AHB_PCTL_R (((volatile unsigned long )0x4005D52C)) #define GPIO_PORTF_AHB_ADCCTL_R (((volatile unsigned long )0x4005D530)) #define GPIO_PORTF_AHB_DMACTL_R (((volatile unsigned long )0x4005D534)) #define GPIO_PORTF_AHB_SI_R (((volatile unsigned long )0x4005D538)) //************************************************************************** // // EEPROM registers (EEPROM) // //*************************************************************************** #define EEPROM_EESIZE_R (((volatile unsigned long )0x400AF000)) #define EEPROM_EEBLOCK_R (((volatile unsigned long )0x400AF004)) #define EEPROM_EEOFFSET_R (((volatile unsigned long )0x400AF008)) #define EEPROM_EERDWR_R (((volatile unsigned long )0x400AF010)) #define EEPROM_EERDWRINC_R (((volatile unsigned long )0x400AF014)) #define EEPROM_EEDONE_R (((volatile unsigned long )0x400AF018)) #define EEPROM_EESUPP_R (((volatile unsigned long )0x400AF01C)) #define EEPROM_EEUNLOCK_R (((volatile unsigned long )0x400AF020)) #define EEPROM_EEPROT_R (((volatile unsigned long )0x400AF030)) #define EEPROM_EEPASS0_R (((volatile unsigned long )0x400AF034)) #define EEPROM_EEPASS1_R (((volatile unsigned long )0x400AF038)) #define EEPROM_EEPASS2_R (((volatile unsigned long )0x400AF03C)) #define EEPROM_EEINT_R (((volatile unsigned long )0x400AF040)) #define EEPROM_EEHIDE_R (((volatile unsigned long )0x400AF050)) #define EEPROM_EEDBGME_R (((volatile unsigned long )0x400AF080)) #define EEPROM_PP_R (((volatile unsigned long )0x400AFFC0)) //*************************************************************************** // // System Exception Module registers (SYSEXC) // //*************************************************************************** #define SYSEXC_RIS_R (((volatile unsigned long )0x400F9000)) #define SYSEXC_IM_R (((volatile unsigned long )0x400F9004)) #define SYSEXC_MIS_R (((volatile unsigned long )0x400F9008)) #define SYSEXC_IC_R (((volatile unsigned long )0x400F900C)) //*************************************************************************** // // Hibernation module registers (HIB) // //*************************************************************************** #define HIB_RTCC_R (((volatile unsigned long )0x400FC000)) #define HIB_RTCM0_R (((volatile unsigned long )0x400FC004)) #define HIB_RTCLD_R (((volatile unsigned long )0x400FC00C)) #define HIB_CTL_R (((volatile unsigned long )0x400FC010)) #define HIB_IM_R (((volatile unsigned long )0x400FC014)) #define HIB_RIS_R (((volatile unsigned long )0x400FC018)) #define HIB_MIS_R (((volatile unsigned long )0x400FC01C)) #define HIB_IC_R (((volatile unsigned long )0x400FC020)) #define HIB_RTCT_R (((volatile unsigned long )0x400FC024)) #define HIB_RTCSS_R (((volatile unsigned long )0x400FC028)) #define HIB_DATA_R (((volatile unsigned long )0x400FC030)) //*************************************************************************** // // FLASH registers (FLASH CTRL) // //*************************************************************************** #define FLASH_FMA_R (((volatile unsigned long )0x400FD000)) #define FLASH_FMD_R (((volatile unsigned long )0x400FD004)) #define FLASH_FMC_R (((volatile unsigned long )0x400FD008)) #define FLASH_FCRIS_R (((volatile unsigned long )0x400FD00C)) #define FLASH_FCIM_R (((volatile unsigned long )0x400FD010)) #define FLASH_FCMISC_R (((volatile unsigned long )0x400FD014)) #define FLASH_FMC2_R (((volatile unsigned long )0x400FD020)) #define FLASH_FWBVAL_R (((volatile unsigned long )0x400FD030)) #define FLASH_FWBN_R (((volatile unsigned long )0x400FD100)) #define FLASH_FSIZE_R (((volatile unsigned long )0x400FDFC0)) #define FLASH_SSIZE_R (((volatile unsigned long )0x400FDFC4)) #define FLASH_ROMSWMAP_R (((volatile unsigned long )0x400FDFCC)) #define FLASH_RMCTL_R (((volatile unsigned long )0x400FE0F0)) #define FLASH_BOOTCFG_R (((volatile unsigned long )0x400FE1D0)) #define FLASH_USERREG0_R (((volatile unsigned long )0x400FE1E0)) #define FLASH_USERREG1_R (((volatile unsigned long )0x400FE1E4)) #define FLASH_USERREG2_R (((volatile unsigned long )0x400FE1E8)) #define FLASH_USERREG3_R (((volatile unsigned long )0x400FE1EC)) #define FLASH_FMPRE0_R (((volatile unsigned long )0x400FE200)) #define FLASH_FMPRE1_R (((volatile unsigned long )0x400FE204)) #define FLASH_FMPRE2_R (((volatile unsigned long )0x400FE208)) #define FLASH_FMPRE3_R (((volatile unsigned long )0x400FE20C)) #define FLASH_FMPPE0_R (((volatile unsigned long )0x400FE400)) #define FLASH_FMPPE1_R (((volatile unsigned long )0x400FE404)) #define FLASH_FMPPE2_R (((volatile unsigned long )0x400FE408)) #define FLASH_FMPPE3_R (((volatile unsigned long )0x400FE40C)) //*************************************************************************** // // System Control registers (SYSCTL) // //*************************************************************************** #define SYSCTL_DID0_R (((volatile unsigned long )0x400FE000)) #define SYSCTL_DID1_R (((volatile unsigned long )0x400FE004)) #define SYSCTL_DC0_R (((volatile unsigned long )0x400FE008)) #define SYSCTL_DC1_R (((volatile unsigned long )0x400FE010)) #define SYSCTL_DC2_R (((volatile unsigned long )0x400FE014)) #define SYSCTL_DC3_R (((volatile unsigned long )0x400FE018)) #define SYSCTL_DC4_R (((volatile unsigned long )0x400FE01C)) #define SYSCTL_DC5_R (((volatile unsigned long )0x400FE020)) #define SYSCTL_DC6_R (((volatile unsigned long )0x400FE024)) #define SYSCTL_DC7_R (((volatile unsigned long )0x400FE028)) #define SYSCTL_DC8_R (((volatile unsigned long )0x400FE02C)) #define SYSCTL_PBORCTL_R (((volatile unsigned long )0x400FE030)) #define SYSCTL_SRCR0_R (((volatile unsigned long )0x400FE040)) #define SYSCTL_SRCR1_R (((volatile unsigned long )0x400FE044)) #define SYSCTL_SRCR2_R (((volatile unsigned long )0x400FE048)) #define SYSCTL_RIS_R (((volatile unsigned long )0x400FE050)) #define SYSCTL_IMC_R (((volatile unsigned long )0x400FE054)) #define SYSCTL_MISC_R (((volatile unsigned long )0x400FE058)) #define SYSCTL_RESC_R (((volatile unsigned long )0x400FE05C)) #define SYSCTL_RCC_R (((volatile unsigned long )0x400FE060)) #define SYSCTL_GPIOHBCTL_R (((volatile unsigned long )0x400FE06C)) #define SYSCTL_RCC2_R (((volatile unsigned long )0x400FE070)) #define SYSCTL_MOSCCTL_R (((volatile unsigned long )0x400FE07C)) #define SYSCTL_RCGC0_R (((volatile unsigned long )0x400FE100)) #define SYSCTL_RCGC1_R (((volatile unsigned long )0x400FE104)) #define SYSCTL_RCGC2_R (((volatile unsigned long )0x400FE108)) #define SYSCTL_SCGC0_R (((volatile unsigned long )0x400FE110)) #define SYSCTL_SCGC1_R (((volatile unsigned long )0x400FE114)) #define SYSCTL_SCGC2_R (((volatile unsigned long )0x400FE118)) #define SYSCTL_DCGC0_R (((volatile unsigned long )0x400FE120)) #define SYSCTL_DCGC1_R (((volatile unsigned long )0x400FE124)) #define SYSCTL_DCGC2_R (((volatile unsigned long )0x400FE128)) #define SYSCTL_DSLPCLKCFG_R (((volatile unsigned long )0x400FE144)) #define SYSCTL_SYSPROP_R (((volatile unsigned long )0x400FE14C)) #define SYSCTL_PIOSCCAL_R (((volatile unsigned long )0x400FE150)) #define SYSCTL_PIOSCSTAT_R (((volatile unsigned long )0x400FE154)) #define SYSCTL_PLLFREQ0_R (((volatile unsigned long )0x400FE160)) #define SYSCTL_PLLFREQ1_R (((volatile unsigned long )0x400FE164)) #define SYSCTL_PLLSTAT_R (((volatile unsigned long )0x400FE168)) #define SYSCTL_DC9_R (((volatile unsigned long )0x400FE190)) #define SYSCTL_NVMSTAT_R (((volatile unsigned long )0x400FE1A0)) #define SYSCTL_PPWD_R (((volatile unsigned long )0x400FE300)) #define SYSCTL_PPTIMER_R (((volatile unsigned long )0x400FE304)) #define SYSCTL_PPGPIO_R (((volatile unsigned long )0x400FE308)) #define SYSCTL_PPDMA_R (((volatile unsigned long )0x400FE30C)) #define SYSCTL_PPHIB_R (((volatile unsigned long )0x400FE314)) #define SYSCTL_PPUART_R (((volatile unsigned long )0x400FE318)) #define SYSCTL_PPSSI_R (((volatile unsigned long )0x400FE31C)) #define SYSCTL_PPI2C_R (((volatile unsigned long )0x400FE320)) #define SYSCTL_PPUSB_R (((volatile unsigned long )0x400FE328)) #define SYSCTL_PPCAN_R (((volatile unsigned long )0x400FE334)) #define SYSCTL_PPADC_R (((volatile unsigned long )0x400FE338)) #define SYSCTL_PPACMP_R (((volatile unsigned long )0x400FE33C)) #define SYSCTL_PPPWM_R (((volatile unsigned long )0x400FE340)) #define SYSCTL_PPQEI_R (((volatile unsigned long )0x400FE344)) #define SYSCTL_PPEEPROM_R (((volatile unsigned long )0x400FE358)) #define SYSCTL_PPWTIMER_R (((volatile unsigned long )0x400FE35C)) #define SYSCTL_SRWD_R (((volatile unsigned long )0x400FE500)) #define SYSCTL_SRTIMER_R (((volatile unsigned long )0x400FE504)) #define SYSCTL_SRGPIO_R (((volatile unsigned long )0x400FE508)) #define SYSCTL_SRDMA_R (((volatile unsigned long )0x400FE50C)) #define SYSCTL_SRHIB_R (((volatile unsigned long )0x400FE514)) #define SYSCTL_SRUART_R (((volatile unsigned long )0x400FE518)) #define SYSCTL_SRSSI_R (((volatile unsigned long )0x400FE51C)) #define SYSCTL_SRI2C_R (((volatile unsigned long )0x400FE520)) #define SYSCTL_SRUSB_R (((volatile unsigned long )0x400FE528)) #define SYSCTL_SRCAN_R (((volatile unsigned long )0x400FE534)) #define SYSCTL_SRADC_R (((volatile unsigned long )0x400FE538)) #define SYSCTL_SRACMP_R (((volatile unsigned long )0x400FE53C)) #define SYSCTL_SREEPROM_R (((volatile unsigned long )0x400FE558)) #define SYSCTL_SRWTIMER_R (((volatile unsigned long )0x400FE55C)) #define SYSCTL_RCGCWD_R (((volatile unsigned long )0x400FE600)) #define SYSCTL_RCGCTIMER_R (((volatile unsigned long )0x400FE604)) #define SYSCTL_RCGCGPIO_R (((volatile unsigned long )0x400FE608)) #define SYSCTL_RCGCDMA_R (((volatile unsigned long )0x400FE60C)) #define SYSCTL_RCGCHIB_R (((volatile unsigned long )0x400FE614)) #define SYSCTL_RCGCUART_R (((volatile unsigned long )0x400FE618)) #define SYSCTL_RCGCSSI_R (((volatile unsigned long )0x400FE61C)) #define SYSCTL_RCGCI2C_R (((volatile unsigned long )0x400FE620)) #define SYSCTL_RCGCUSB_R (((volatile unsigned long )0x400FE628)) #define SYSCTL_RCGCCAN_R (((volatile unsigned long )0x400FE634)) #define SYSCTL_RCGCADC_R (((volatile unsigned long )0x400FE638)) #define SYSCTL_RCGCACMP_R (((volatile unsigned long )0x400FE63C)) #define SYSCTL_RCGCEEPROM_R (((volatile unsigned long )0x400FE658)) #define SYSCTL_RCGCWTIMER_R (((volatile unsigned long )0x400FE65C)) #define SYSCTL_SCGCWD_R (((volatile unsigned long )0x400FE700)) #define SYSCTL_SCGCTIMER_R (((volatile unsigned long )0x400FE704)) #define SYSCTL_SCGCGPIO_R (((volatile unsigned long )0x400FE708)) #define SYSCTL_SCGCDMA_R (((volatile unsigned long )0x400FE70C)) #define SYSCTL_SCGCHIB_R (((volatile unsigned long )0x400FE714)) #define SYSCTL_SCGCUART_R (((volatile unsigned long )0x400FE718)) #define SYSCTL_SCGCSSI_R (((volatile unsigned long )0x400FE71C)) #define SYSCTL_SCGCI2C_R (((volatile unsigned long )0x400FE720)) #define SYSCTL_SCGCUSB_R (((volatile unsigned long )0x400FE728)) #define SYSCTL_SCGCCAN_R (((volatile unsigned long )0x400FE734)) #define SYSCTL_SCGCADC_R (((volatile unsigned long )0x400FE738)) #define SYSCTL_SCGCACMP_R (((volatile unsigned long )0x400FE73C)) #define SYSCTL_SCGCEEPROM_R (((volatile unsigned long )0x400FE758)) #define SYSCTL_SCGCWTIMER_R (((volatile unsigned long )0x400FE75C)) #define SYSCTL_DCGCWD_R (((volatile unsigned long )0x400FE800)) #define SYSCTL_DCGCTIMER_R (((volatile unsigned long )0x400FE804)) #define SYSCTL_DCGCGPIO_R (((volatile unsigned long )0x400FE808)) #define SYSCTL_DCGCDMA_R (((volatile unsigned long )0x400FE80C)) #define SYSCTL_DCGCHIB_R (((volatile unsigned long )0x400FE814)) #define SYSCTL_DCGCUART_R (((volatile unsigned long )0x400FE818)) #define SYSCTL_DCGCSSI_R (((volatile unsigned long )0x400FE81C)) #define SYSCTL_DCGCI2C_R (((volatile unsigned long )0x400FE820)) #define SYSCTL_DCGCUSB_R (((volatile unsigned long )0x400FE828)) #define SYSCTL_DCGCCAN_R (((volatile unsigned long )0x400FE834)) #define SYSCTL_DCGCADC_R (((volatile unsigned long )0x400FE838)) #define SYSCTL_DCGCACMP_R (((volatile unsigned long )0x400FE83C)) #define SYSCTL_DCGCEEPROM_R (((volatile unsigned long )0x400FE858)) #define SYSCTL_DCGCWTIMER_R (((volatile unsigned long )0x400FE85C)) #define SYSCTL_PCWD_R (((volatile unsigned long )0x400FE900)) #define SYSCTL_PCTIMER_R (((volatile unsigned long )0x400FE904)) #define SYSCTL_PCGPIO_R (((volatile unsigned long )0x400FE908)) #define SYSCTL_PCDMA_R (((volatile unsigned long )0x400FE90C)) #define SYSCTL_PCHIB_R (((volatile unsigned long )0x400FE914)) #define SYSCTL_PCUART_R (((volatile unsigned long )0x400FE918)) #define SYSCTL_PCSSI_R (((volatile unsigned long )0x400FE91C)) #define SYSCTL_PCI2C_R (((volatile unsigned long )0x400FE920)) #define SYSCTL_PCUSB_R (((volatile unsigned long )0x400FE928)) #define SYSCTL_PCCAN_R (((volatile unsigned long )0x400FE934)) #define SYSCTL_PCADC_R (((volatile unsigned long )0x400FE938)) #define SYSCTL_PCACMP_R (((volatile unsigned long )0x400FE93C)) #define SYSCTL_PCEEPROM_R (((volatile unsigned long )0x400FE958)) #define SYSCTL_PCWTIMER_R (((volatile unsigned long )0x400FE95C)) #define SYSCTL_PRWD_R (((volatile unsigned long )0x400FEA00)) #define SYSCTL_PRTIMER_R (((volatile unsigned long )0x400FEA04)) #define SYSCTL_PRGPIO_R (((volatile unsigned long )0x400FEA08)) #define SYSCTL_PRDMA_R (((volatile unsigned long )0x400FEA0C)) #define SYSCTL_PRHIB_R (((volatile unsigned long )0x400FEA14)) #define SYSCTL_PRUART_R (((volatile unsigned long )0x400FEA18)) #define SYSCTL_PRSSI_R (((volatile unsigned long )0x400FEA1C)) #define SYSCTL_PRI2C_R (((volatile unsigned long )0x400FEA20)) #define SYSCTL_PRUSB_R (((volatile unsigned long )0x400FEA28)) #define SYSCTL_PRCAN_R (((volatile unsigned long )0x400FEA34)) #define SYSCTL_PRADC_R (((volatile unsigned long )0x400FEA38)) #define SYSCTL_PRACMP_R (((volatile unsigned long )0x400FEA3C)) #define SYSCTL_PREEPROM_R (((volatile unsigned long )0x400FEA58)) #define SYSCTL_PRWTIMER_R (((volatile unsigned long )0x400FEA5C)) //*************************************************************************** // // Micro Direct Memory Access registers (UDMA) // //*************************************************************************** #define UDMA_STAT_R (((volatile unsigned long )0x400FF000)) #define UDMA_CFG_R (((volatile unsigned long )0x400FF004)) #define UDMA_CTLBASE_R (((volatile unsigned long )0x400FF008)) #define UDMA_ALTBASE_R (((volatile unsigned long )0x400FF00C)) #define UDMA_WAITSTAT_R (((volatile unsigned long )0x400FF010)) #define UDMA_SWREQ_R (((volatile unsigned long )0x400FF014)) #define UDMA_USEBURSTSET_R (((volatile unsigned long )0x400FF018)) #define UDMA_USEBURSTCLR_R (((volatile unsigned long )0x400FF01C)) #define UDMA_REQMASKSET_R (((volatile unsigned long )0x400FF020)) #define UDMA_REQMASKCLR_R (((volatile unsigned long )0x400FF024)) #define UDMA_ENASET_R (((volatile unsigned long )0x400FF028)) #define UDMA_ENACLR_R (((volatile unsigned long )0x400FF02C)) #define UDMA_ALTSET_R (((volatile unsigned long )0x400FF030)) #define UDMA_ALTCLR_R (((volatile unsigned long )0x400FF034)) #define UDMA_PRIOSET_R (((volatile unsigned long )0x400FF038)) #define UDMA_PRIOCLR_R (((volatile unsigned long )0x400FF03C)) #define UDMA_ERRCLR_R (((volatile unsigned long )0x400FF04C)) #define UDMA_CHASGN_R (((volatile unsigned long )0x400FF500)) #define UDMA_CHIS_R (((volatile unsigned long )0x400FF504)) #define UDMA_CHMAP0_R (((volatile unsigned long )0x400FF510)) #define UDMA_CHMAP1_R (((volatile unsigned long )0x400FF514)) #define UDMA_CHMAP2_R (((volatile unsigned long )0x400FF518)) #define UDMA_CHMAP3_R (((volatile unsigned long )0x400FF51C)) //*************************************************************************** // // Micro Direct Memory Access (uDMA) offsets (UDMA) // //************************************************************************* #define UDMA_SRCENDP 0x00000000 // DMA Channel Source Address End // Pointer #define UDMA_DSTENDP 0x00000004 // DMA Channel Destination Address // End Pointer #define UDMA_CHCTL 0x00000008 // DMA Channel Control Word //************************************************************************* // // NVIC registers (NVIC) // //*************************************************************************** #define NVIC_INT_TYPE_R (((volatile unsigned long )0xE000E004)) #define NVIC_ACTLR_R (((volatile unsigned long )0xE000E008)) #define NVIC_ST_CTRL_R (((volatile unsigned long )0xE000E010)) #define NVIC_ST_RELOAD_R (((volatile unsigned long )0xE000E014)) #define NVIC_ST_CURRENT_R (((volatile unsigned long )0xE000E018)) #define NVIC_ST_CAL_R (((volatile unsigned long )0xE000E01C)) #define NVIC_EN0_R (((volatile unsigned long )0xE000E100)) #define NVIC_EN1_R (((volatile unsigned long )0xE000E104)) #define NVIC_EN2_R (((volatile unsigned long )0xE000E108)) #define NVIC_EN3_R (((volatile unsigned long )0xE000E10C)) #define NVIC_EN4_R (((volatile unsigned long )0xE000E110)) #define NVIC_DIS0_R (((volatile unsigned long )0xE000E180)) #define NVIC_DIS1_R (((volatile unsigned long )0xE000E184)) #define NVIC_DIS2_R (((volatile unsigned long )0xE000E188)) #define NVIC_DIS3_R (((volatile unsigned long )0xE000E18C)) #define NVIC_DIS4_R (((volatile unsigned long )0xE000E190)) #define NVIC_PEND0_R (((volatile unsigned long )0xE000E200)) #define NVIC_PEND1_R (((volatile unsigned long )0xE000E204)) #define NVIC_PEND2_R (((volatile unsigned long )0xE000E208)) #define NVIC_PEND3_R (((volatile unsigned long )0xE000E20C)) #define NVIC_PEND4_R (((volatile unsigned long )0xE000E210)) #define NVIC_UNPEND0_R (((volatile unsigned long )0xE000E280)) #define NVIC_UNPEND1_R (((volatile unsigned long )0xE000E284)) #define NVIC_UNPEND2_R (((volatile unsigned long )0xE000E288)) #define NVIC_UNPEND3_R (((volatile unsigned long )0xE000E28C)) #define NVIC_UNPEND4_R (((volatile unsigned long )0xE000E290)) #define NVIC_ACTIVE0_R (((volatile unsigned long )0xE000E300)) #define NVIC_ACTIVE1_R (((volatile unsigned long )0xE000E304)) #define NVIC_ACTIVE2_R (((volatile unsigned long )0xE000E308)) #define NVIC_ACTIVE3_R (((volatile unsigned long )0xE000E30C)) #define NVIC_ACTIVE4_R (((volatile unsigned long )0xE000E310)) #define NVIC_PRI0_R (((volatile unsigned long )0xE000E400)) #define NVIC_PRI1_R (((volatile unsigned long )0xE000E404)) #define NVIC_PRI2_R (((volatile unsigned long )0xE000E408)) #define NVIC_PRI3_R (((volatile unsigned long )0xE000E40C)) #define NVIC_PRI4_R (((volatile unsigned long )0xE000E410)) #define NVIC_PRI5_R (((volatile unsigned long )0xE000E414)) #define NVIC_PRI6_R (((volatile unsigned long )0xE000E418)) #define NVIC_PRI7_R (((volatile unsigned long )0xE000E41C)) #define NVIC_PRI8_R (((volatile unsigned long )0xE000E420)) #define NVIC_PRI9_R (((volatile unsigned long )0xE000E424)) #define NVIC_PRI10_R (((volatile unsigned long )0xE000E428)) #define NVIC_PRI11_R (((volatile unsigned long )0xE000E42C)) #define NVIC_PRI12_R (((volatile unsigned long )0xE000E430)) #define NVIC_PRI13_R (((volatile unsigned long )0xE000E434)) #define NVIC_PRI14_R (((volatile unsigned long )0xE000E438)) #define NVIC_PRI15_R (((volatile unsigned long )0xE000E43C)) #define NVIC_PRI16_R (((volatile unsigned long )0xE000E440)) #define NVIC_PRI17_R (((volatile unsigned long )0xE000E444)) #define NVIC_PRI18_R (((volatile unsigned long )0xE000E448)) #define NVIC_PRI19_R (((volatile unsigned long )0xE000E44C)) #define NVIC_PRI20_R (((volatile unsigned long )0xE000E450)) #define NVIC_PRI21_R (((volatile unsigned long )0xE000E454)) #define NVIC_PRI22_R (((volatile unsigned long )0xE000E458)) #define NVIC_PRI23_R (((volatile unsigned long )0xE000E45C)) #define NVIC_PRI24_R (((volatile unsigned long )0xE000E460)) #define NVIC_PRI25_R (((volatile unsigned long )0xE000E464)) #define NVIC_PRI26_R (((volatile unsigned long )0xE000E468)) #define NVIC_PRI27_R (((volatile unsigned long )0xE000E46C)) #define NVIC_PRI28_R (((volatile unsigned long )0xE000E470)) #define NVIC_PRI29_R (((volatile unsigned long )0xE000E474)) #define NVIC_PRI30_R (((volatile unsigned long )0xE000E478)) #define NVIC_PRI31_R (((volatile unsigned long )0xE000E47C)) #define NVIC_PRI32_R (((volatile unsigned long )0xE000E480)) #define NVIC_PRI33_R (((volatile unsigned long )0xE000E484)) #define NVIC_PRI34_R (((volatile unsigned long )0xE000E488)) #define NVIC_CPUID_R (((volatile unsigned long )0xE000ED00)) #define NVIC_INT_CTRL_R (((volatile unsigned long )0xE000ED04)) #define NVIC_VTABLE_R (((volatile unsigned long )0xE000ED08)) #define NVIC_APINT_R (((volatile unsigned long )0xE000ED0C)) #define NVIC_SYS_CTRL_R (((volatile unsigned long )0xE000ED10)) #define NVIC_CFG_CTRL_R (((volatile unsigned long )0xE000ED14)) #define NVIC_SYS_PRI1_R (((volatile unsigned long )0xE000ED18)) #define NVIC_SYS_PRI2_R (((volatile unsigned long )0xE000ED1C)) #define NVIC_SYS_PRI3_R (((volatile unsigned long )0xE000ED20)) #define NVIC_SYS_HND_CTRL_R (((volatile unsigned long )0xE000ED24)) #define NVIC_FAULT_STAT_R (((volatile unsigned long )0xE000ED28)) #define NVIC_HFAULT_STAT_R (((volatile unsigned long )0xE000ED2C)) #define NVIC_DEBUG_STAT_R (((volatile unsigned long )0xE000ED30)) #define NVIC_MM_ADDR_R (((volatile unsigned long )0xE000ED34)) #define NVIC_FAULT_ADDR_R (((volatile unsigned long )0xE000ED38)) #define NVIC_CPAC_R (((volatile unsigned long )0xE000ED88)) #define NVIC_MPU_TYPE_R (((volatile unsigned long )0xE000ED90)) #define NVIC_MPU_CTRL_R (((volatile unsigned long )0xE000ED94)) #define NVIC_MPU_NUMBER_R (((volatile unsigned long )0xE000ED98)) #define NVIC_MPU_BASE_R (((volatile unsigned long )0xE000ED9C)) #define NVIC_MPU_ATTR_R (((volatile unsigned long )0xE000EDA0)) #define NVIC_MPU_BASE1_R (((volatile unsigned long )0xE000EDA4)) #define NVIC_MPU_ATTR1_R (((volatile unsigned long )0xE000EDA8)) #define NVIC_MPU_BASE2_R (((volatile unsigned long )0xE000EDAC)) #define NVIC_MPU_ATTR2_R (((volatile unsigned long )0xE000EDB0)) #define NVIC_MPU_BASE3_R (((volatile unsigned long )0xE000EDB4)) #define NVIC_MPU_ATTR3_R (((volatile unsigned long )0xE000EDB8)) #define NVIC_DBG_CTRL_R (((volatile unsigned long )0xE000EDF0)) #define NVIC_DBG_XFER_R (((volatile unsigned long )0xE000EDF4)) #define NVIC_DBG_DATA_R (((volatile unsigned long )0xE000EDF8)) #define NVIC_DBG_INT_R (((volatile unsigned long )0xE000EDFC)) #define NVIC_SW_TRIG_R (((volatile unsigned long )0xE000EF00)) #define NVIC_FPCC_R (((volatile unsigned long )0xE000EF34)) #define NVIC_FPCA_R (((volatile unsigned long )0xE000EF38)) #define NVIC_FPDSC_R (((volatile unsigned long )0xE000EF3C)) //*************************************************************************** // // The following are defines for the bit fields in the WDT_O_LOAD register. // //************************************************************************* #define WDT_LOAD_M 0xFFFFFFFF // Watchdog Load Value #define WDT_LOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the WDT_O_VALUE register. // //************************************************************************* #define WDT_VALUE_M 0xFFFFFFFF // Watchdog Value #define WDT_VALUE_S 0 //************************************************************************* // // The following are defines for the bit fields in the WDT_O_CTL register. // //************************************************************************* #define WDT_CTL_WRC 0x80000000 // Write Complete #define WDT_CTL_INTTYPE 0x00000004 // Watchdog Interrupt Type #define WDT_CTL_RESEN 0x00000002 // Watchdog Reset Enable #define WDT_CTL_INTEN 0x00000001 // Watchdog Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the WDT_O_ICR register. // //************************************************************************* #define WDT_ICR_M 0xFFFFFFFF // Watchdog Interrupt Clear #define WDT_ICR_S 0 //************************************************************************* // // The following are defines for the bit fields in the WDT_O_RIS register. // //************************************************************************* #define WDT_RIS_WDTRIS 0x00000001 // Watchdog Raw Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the WDT_O_MIS register. // //************************************************************************* #define WDT_MIS_WDTMIS 0x00000001 // Watchdog Masked Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the WDT_O_TEST register. // //************************************************************************* #define WDT_TEST_STALL 0x00000100 // Watchdog Stall Enable //************************************************************************* // // The following are defines for the bit fields in the WDT_O_LOCK register. // //************************************************************************* #define WDT_LOCK_M 0xFFFFFFFF // Watchdog Lock #define WDT_LOCK_UNLOCKED 0x00000000 // Unlocked #define WDT_LOCK_LOCKED 0x00000001 // Locked //************************************************************************* // // The following are defines for the bit fields in the GPIO_O_IM register. // //************************************************************************* #define GPIO_IM_GPIO_M 0x000000FF // GPIO Interrupt Mask Enable #define GPIO_IM_GPIO_S 0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_O_RIS register. // //************************************************************************* #define GPIO_RIS_GPIO_M 0x000000FF // GPIO Interrupt Raw Status #define GPIO_RIS_GPIO_S 0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_O_MIS register. // //************************************************************************* #define GPIO_MIS_GPIO_M 0x000000FF // GPIO Masked Interrupt Status #define GPIO_MIS_GPIO_S 0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_O_ICR register. // //************************************************************************* #define GPIO_ICR_GPIO_M 0x000000FF // GPIO Interrupt Clear #define GPIO_ICR_GPIO_S 0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_O_LOCK register. // //************************************************************************* #define GPIO_LOCK_M 0xFFFFFFFF // GPIO Lock #define GPIO_LOCK_UNLOCKED 0x00000000 // The GPIOCR register is unlocked // and may be modified #define GPIO_LOCK_LOCKED 0x00000001 // The GPIOCR register is locked // and may not be modified #define GPIO_LOCK_KEY 0x4C4F434B // Unlocks the GPIO_CR register //************************************************************************* // // The following are defines for the bit fields in the GPIO_O_SI register. // //************************************************************************* #define GPIO_SI_SUM 0x00000001 // Summary Interrupt //************************************************************************* // // The following are defines for the bit fields in the GPIO_PCTL register for // port A. // //************************************************************************* #define GPIO_PCTL_PA7_M 0xF0000000 // PA7 mask #define GPIO_PCTL_PA7_I2C1SDA 0x30000000 // I2C1SDA on PA7 #define GPIO_PCTL_PA6_M 0x0F000000 // PA6 mask #define GPIO_PCTL_PA6_I2C1SCL 0x03000000 // I2C1SCL on PA6 #define GPIO_PCTL_PA5_M 0x00F00000 // PA5 mask #define GPIO_PCTL_PA5_SSI0TX 0x00200000 // SSI0TX on PA5 #define GPIO_PCTL_PA4_M 0x000F0000 // PA4 mask #define GPIO_PCTL_PA4_SSI0RX 0x00020000 // SSI0RX on PA4 #define GPIO_PCTL_PA3_M 0x0000F000 // PA3 mask #define GPIO_PCTL_PA3_SSI0FSS 0x00002000 // SSI0FSS on PA3 #define GPIO_PCTL_PA2_M 0x00000F00 // PA2 mask #define GPIO_PCTL_PA2_SSI0CLK 0x00000200 // SSI0CLK on PA2 #define GPIO_PCTL_PA1_M 0x000000F0 // PA1 mask #define GPIO_PCTL_PA1_U0TX 0x00000010 // U0TX on PA1 #define GPIO_PCTL_PA0_M 0x0000000F // PA0 mask #define GPIO_PCTL_PA0_U0RX 0x00000001 // U0RX on PA0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_PCTL register for // port B. // //************************************************************************* #define GPIO_PCTL_PB7_M 0xF0000000 // PB7 mask #define GPIO_PCTL_PB7_SSI2TX 0x20000000 // SSI2TX on PB7 #define GPIO_PCTL_PB7_T0CCP1 0x70000000 // T0CCP1 on PB7 #define GPIO_PCTL_PB6_M 0x0F000000 // PB6 mask #define GPIO_PCTL_PB6_SSI2RX 0x02000000 // SSI2RX on PB6 #define GPIO_PCTL_PB6_T0CCP0 0x07000000 // T0CCP0 on PB6 #define GPIO_PCTL_PB5_M 0x00F00000 // PB5 mask #define GPIO_PCTL_PB5_SSI2FSS 0x00200000 // SSI2FSS on PB5 #define GPIO_PCTL_PB5_T1CCP1 0x00700000 // T1CCP1 on PB5 #define GPIO_PCTL_PB5_CAN0TX 0x00800000 // CAN0TX on PB5 #define GPIO_PCTL_PB4_M 0x000F0000 // PB4 mask #define GPIO_PCTL_PB4_SSI2CLK 0x00020000 // SSI2CLK on PB4 #define GPIO_PCTL_PB4_T1CCP0 0x00070000 // T1CCP0 on PB4 #define GPIO_PCTL_PB4_CAN0RX 0x00080000 // CAN0RX on PB4 #define GPIO_PCTL_PB3_M 0x0000F000 // PB3 mask #define GPIO_PCTL_PB3_I2C0SDA 0x00003000 // I2C0SDA on PB3 #define GPIO_PCTL_PB3_T3CCP1 0x00007000 // T3CCP1 on PB3 #define GPIO_PCTL_PB2_M 0x00000F00 // PB2 mask #define GPIO_PCTL_PB2_I2C0SCL 0x00000300 // I2C0SCL on PB2 #define GPIO_PCTL_PB2_T3CCP0 0x00000700 // T3CCP0 on PB2 #define GPIO_PCTL_PB1_M 0x000000F0 // PB1 mask #define GPIO_PCTL_PB1_U1TX 0x00000010 // U1TX on PB1 #define GPIO_PCTL_PB1_T2CCP1 0x00000070 // T2CCP1 on PB1 #define GPIO_PCTL_PB0_M 0x0000000F // PB0 mask #define GPIO_PCTL_PB0_U1RX 0x00000001 // U1RX on PB0 #define GPIO_PCTL_PB0_T2CCP0 0x00000007 // T2CCP0 on PB0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_PCTL register for // port C. // //************************************************************************* #define GPIO_PCTL_PC7_M 0xF0000000 // PC7 mask #define GPIO_PCTL_PC7_U3TX 0x10000000 // U3TX on PC7 #define GPIO_PCTL_PC7_WT1CCP1 0x70000000 // WT1CCP1 on PC7 #define GPIO_PCTL_PC6_M 0x0F000000 // PC6 mask #define GPIO_PCTL_PC6_U3RX 0x01000000 // U3RX on PC6 #define GPIO_PCTL_PC6_WT1CCP0 0x07000000 // WT1CCP0 on PC6 #define GPIO_PCTL_PC5_M 0x00F00000 // PC5 mask #define GPIO_PCTL_PC5_U4TX 0x00100000 // U4TX on PC5 #define GPIO_PCTL_PC5_U1TX 0x00200000 // U1TX on PC5 #define GPIO_PCTL_PC5_WT0CCP1 0x00700000 // WT0CCP1 on PC5 #define GPIO_PCTL_PC5_U1CTS 0x00800000 // U1CTS on PC5 #define GPIO_PCTL_PC4_M 0x000F0000 // PC4 mask #define GPIO_PCTL_PC4_U4RX 0x00010000 // U4RX on PC4 #define GPIO_PCTL_PC4_U1RX 0x00020000 // U1RX on PC4 #define GPIO_PCTL_PC4_WT0CCP0 0x00070000 // WT0CCP0 on PC4 #define GPIO_PCTL_PC4_U1RTS 0x00080000 // U1RTS on PC4 #define GPIO_PCTL_PC3_M 0x0000F000 // PC3 mask #define GPIO_PCTL_PC3_TDO 0x00001000 // TDO on PC3 #define GPIO_PCTL_PC3_T5CCP1 0x00007000 // T5CCP1 on PC3 #define GPIO_PCTL_PC2_M 0x00000F00 // PC2 mask #define GPIO_PCTL_PC2_TDI 0x00000100 // TDI on PC2 #define GPIO_PCTL_PC2_T5CCP0 0x00000700 // T5CCP0 on PC2 #define GPIO_PCTL_PC1_M 0x000000F0 // PC1 mask #define GPIO_PCTL_PC1_TMS 0x00000010 // TMS on PC1 #define GPIO_PCTL_PC1_T4CCP1 0x00000070 // T4CCP1 on PC1 #define GPIO_PCTL_PC0_M 0x0000000F // PC0 mask #define GPIO_PCTL_PC0_TCK 0x00000001 // TCK on PC0 #define GPIO_PCTL_PC0_T4CCP0 0x00000007 // T4CCP0 on PC0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_PCTL register for // port D. // //************************************************************************* #define GPIO_PCTL_PD7_M 0xF0000000 // PD7 mask #define GPIO_PCTL_PD7_U2TX 0x10000000 // U2TX on PD7 #define GPIO_PCTL_PD7_WT5CCP1 0x70000000 // WT5CCP1 on PD7 #define GPIO_PCTL_PD7_NMI 0x80000000 // NMI on PD7 #define GPIO_PCTL_PD6_M 0x0F000000 // PD6 mask #define GPIO_PCTL_PD6_U2RX 0x01000000 // U2RX on PD6 #define GPIO_PCTL_PD6_WT5CCP0 0x07000000 // WT5CCP0 on PD6 #define GPIO_PCTL_PD5_M 0x00F00000 // PD5 mask #define GPIO_PCTL_PD5_U6TX 0x00100000 // U6TX on PD5 #define GPIO_PCTL_PD5_WT4CCP1 0x00700000 // WT4CCP1 on PD5 #define GPIO_PCTL_PD4_M 0x000F0000 // PD4 mask #define GPIO_PCTL_PD4_U6RX 0x00010000 // U6RX on PD4 #define GPIO_PCTL_PD4_WT4CCP0 0x00070000 // WT4CCP0 on PD4 #define GPIO_PCTL_PD3_M 0x0000F000 // PD3 mask #define GPIO_PCTL_PD3_SSI3TX 0x00001000 // SSI3TX on PD3 #define GPIO_PCTL_PD3_SSI1TX 0x00002000 // SSI1TX on PD3 #define GPIO_PCTL_PD3_WT3CCP1 0x00007000 // WT3CCP1 on PD3 #define GPIO_PCTL_PD2_M 0x00000F00 // PD2 mask #define GPIO_PCTL_PD2_SSI3RX 0x00000100 // SSI3RX on PD2 #define GPIO_PCTL_PD2_SSI1RX 0x00000200 // SSI1RX on PD2 #define GPIO_PCTL_PD2_WT3CCP0 0x00000700 // WT3CCP0 on PD2 #define GPIO_PCTL_PD1_M 0x000000F0 // PD1 mask #define GPIO_PCTL_PD1_SSI3FSS 0x00000010 // SSI3FSS on PD1 #define GPIO_PCTL_PD1_SSI1FSS 0x00000020 // SSI1FSS on PD1 #define GPIO_PCTL_PD1_I2C3SDA 0x00000030 // I2C3SDA on PD1 #define GPIO_PCTL_PD1_WT2CCP1 0x00000070 // WT2CCP1 on PD1 #define GPIO_PCTL_PD0_M 0x0000000F // PD0 mask #define GPIO_PCTL_PD0_SSI3CLK 0x00000001 // SSI3CLK on PD0 #define GPIO_PCTL_PD0_SSI1CLK 0x00000002 // SSI1CLK on PD0 #define GPIO_PCTL_PD0_I2C3SCL 0x00000003 // I2C3SCL on PD0 #define GPIO_PCTL_PD0_WT2CCP0 0x00000007 // WT2CCP0 on PD0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_PCTL register for // port E. // //************************************************************************* #define GPIO_PCTL_PE5_M 0x00F00000 // PE5 mask #define GPIO_PCTL_PE5_U5TX 0x00100000 // U5TX on PE5 #define GPIO_PCTL_PE5_I2C2SDA 0x00300000 // I2C2SDA on PE5 #define GPIO_PCTL_PE5_CAN0TX 0x00800000 // CAN0TX on PE5 #define GPIO_PCTL_PE4_M 0x000F0000 // PE4 mask #define GPIO_PCTL_PE4_U5RX 0x00010000 // U5RX on PE4 #define GPIO_PCTL_PE4_I2C2SCL 0x00030000 // I2C2SCL on PE4 #define GPIO_PCTL_PE4_CAN0RX 0x00080000 // CAN0RX on PE4 #define GPIO_PCTL_PE1_M 0x000000F0 // PE1 mask #define GPIO_PCTL_PE1_U7TX 0x00000010 // U7TX on PE1 #define GPIO_PCTL_PE0_M 0x0000000F // PE0 mask #define GPIO_PCTL_PE0_U7RX 0x00000001 // U7RX on PE0 //************************************************************************* // // The following are defines for the bit fields in the GPIO_PCTL register for // port F. // //************************************************************************* #define GPIO_PCTL_PF4_M 0x000F0000 // PF4 mask #define GPIO_PCTL_PF4_T2CCP0 0x00070000 // T2CCP0 on PF4 #define GPIO_PCTL_PF3_M 0x0000F000 // PF3 mask #define GPIO_PCTL_PF3_SSI1FSS 0x00002000 // SSI1FSS on PF3 #define GPIO_PCTL_PF3_CAN0TX 0x00003000 // CAN0TX on PF3 #define GPIO_PCTL_PF3_T1CCP1 0x00007000 // T1CCP1 on PF3 #define GPIO_PCTL_PF3_TRCLK 0x0000E000 // TRCLK on PF3 #define GPIO_PCTL_PF2_M 0x00000F00 // PF2 mask #define GPIO_PCTL_PF2_SSI1CLK 0x00000200 // SSI1CLK on PF2 #define GPIO_PCTL_PF2_T1CCP0 0x00000700 // T1CCP0 on PF2 #define GPIO_PCTL_PF2_TRD0 0x00000E00 // TRD0 on PF2 #define GPIO_PCTL_PF1_M 0x000000F0 // PF1 mask #define GPIO_PCTL_PF1_U1CTS 0x00000010 // U1CTS on PF1 #define GPIO_PCTL_PF1_SSI1TX 0x00000020 // SSI1TX on PF1 #define GPIO_PCTL_PF1_T0CCP1 0x00000070 // T0CCP1 on PF1 #define GPIO_PCTL_PF1_C1O 0x00000090 // C1O on PF1 #define GPIO_PCTL_PF1_TRD1 0x000000E0 // TRD1 on PF1 #define GPIO_PCTL_PF0_M 0x0000000F // PF0 mask #define GPIO_PCTL_PF0_U1RTS 0x00000001 // U1RTS on PF0 #define GPIO_PCTL_PF0_SSI1RX 0x00000002 // SSI1RX on PF0 #define GPIO_PCTL_PF0_CAN0RX 0x00000003 // CAN0RX on PF0 #define GPIO_PCTL_PF0_T0CCP0 0x00000007 // T0CCP0 on PF0 #define GPIO_PCTL_PF0_NMI 0x00000008 // NMI on PF0 #define GPIO_PCTL_PF0_C0O 0x00000009 // C0O on PF0 #define GPIO_PCTL_PF0_TRD2 0x0000000E // TRD2 on PF0 //************************************************************************* // // The following are defines for the bit fields in the SSI_O_CR0 register. // //************************************************************************* #define SSI_CR0_SCR_M 0x0000FF00 // SSI Serial Clock Rate #define SSI_CR0_SPH 0x00000080 // SSI Serial Clock Phase #define SSI_CR0_SPO 0x00000040 // SSI Serial Clock Polarity #define SSI_CR0_FRF_M 0x00000030 // SSI Frame Format Select #define SSI_CR0_FRF_MOTO 0x00000000 // Freescale SPI Frame Format #define SSI_CR0_FRF_TI 0x00000010 // Texas Instruments Synchronous // Serial Frame Format #define SSI_CR0_FRF_NMW 0x00000020 // MICROWIRE Frame Format #define SSI_CR0_DSS_M 0x0000000F // SSI Data Size Select #define SSI_CR0_DSS_4 0x00000003 // 4-bit data #define SSI_CR0_DSS_5 0x00000004 // 5-bit data #define SSI_CR0_DSS_6 0x00000005 // 6-bit data #define SSI_CR0_DSS_7 0x00000006 // 7-bit data #define SSI_CR0_DSS_8 0x00000007 // 8-bit data #define SSI_CR0_DSS_9 0x00000008 // 9-bit data #define SSI_CR0_DSS_10 0x00000009 // 10-bit data #define SSI_CR0_DSS_11 0x0000000A // 11-bit data #define SSI_CR0_DSS_12 0x0000000B // 12-bit data #define SSI_CR0_DSS_13 0x0000000C // 13-bit data #define SSI_CR0_DSS_14 0x0000000D // 14-bit data #define SSI_CR0_DSS_15 0x0000000E // 15-bit data #define SSI_CR0_DSS_16 0x0000000F // 16-bit data #define SSI_CR0_SCR_S 8 //************************************************************************* // // The following are defines for the bit fields in the SSI_O_CR1 register. // //************************************************************************* #define SSI_CR1_EOT 0x00000010 // End of Transmission #define SSI_CR1_SOD 0x00000008 // SSI Slave Mode Output Disable #define SSI_CR1_MS 0x00000004 // SSI Master/Slave Select #define SSI_CR1_SSE 0x00000002 // SSI Synchronous Serial Port // Enable #define SSI_CR1_LBM 0x00000001 // SSI Loopback Mode //************************************************************************* // // The following are defines for the bit fields in the SSI_O_DR register. // //************************************************************************* #define SSI_DR_DATA_M 0x0000FFFF // SSI Receive/Transmit Data #define SSI_DR_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the SSI_O_SR register. // //************************************************************************* #define SSI_SR_BSY 0x00000010 // SSI Busy Bit #define SSI_SR_RFF 0x00000008 // SSI Receive FIFO Full #define SSI_SR_RNE 0x00000004 // SSI Receive FIFO Not Empty #define SSI_SR_TNF 0x00000002 // SSI Transmit FIFO Not Full #define SSI_SR_TFE 0x00000001 // SSI Transmit FIFO Empty //************************************************************************* // // The following are defines for the bit fields in the SSI_O_CPSR register. // //************************************************************************* #define SSI_CPSR_CPSDVSR_M 0x000000FF // SSI Clock Prescale Divisor #define SSI_CPSR_CPSDVSR_S 0 //************************************************************************* // // The following are defines for the bit fields in the SSI_O_IM register. // //************************************************************************* #define SSI_IM_TXIM 0x00000008 // SSI Transmit FIFO Interrupt Mask #define SSI_IM_RXIM 0x00000004 // SSI Receive FIFO Interrupt Mask #define SSI_IM_RTIM 0x00000002 // SSI Receive Time-Out Interrupt // Mask #define SSI_IM_RORIM 0x00000001 // SSI Receive Overrun Interrupt // Mask //************************************************************************* // // The following are defines for the bit fields in the SSI_O_RIS register. // //************************************************************************* #define SSI_RIS_TXRIS 0x00000008 // SSI Transmit FIFO Raw Interrupt // Status #define SSI_RIS_RXRIS 0x00000004 // SSI Receive FIFO Raw Interrupt // Status #define SSI_RIS_RTRIS 0x00000002 // SSI Receive Time-Out Raw // Interrupt Status #define SSI_RIS_RORRIS 0x00000001 // SSI Receive Overrun Raw // Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the SSI_O_MIS register. // //************************************************************************* #define SSI_MIS_TXMIS 0x00000008 // SSI Transmit FIFO Masked // Interrupt Status #define SSI_MIS_RXMIS 0x00000004 // SSI Receive FIFO Masked // Interrupt Status #define SSI_MIS_RTMIS 0x00000002 // SSI Receive Time-Out Masked // Interrupt Status #define SSI_MIS_RORMIS 0x00000001 // SSI Receive Overrun Masked // Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the SSI_O_ICR register. // //************************************************************************* #define SSI_ICR_RTIC 0x00000002 // SSI Receive Time-Out Interrupt // Clear #define SSI_ICR_RORIC 0x00000001 // SSI Receive Overrun Interrupt // Clear //************************************************************************* // // The following are defines for the bit fields in the SSI_O_DMACTL register. // //************************************************************************* #define SSI_DMACTL_TXDMAE 0x00000002 // Transmit DMA Enable #define SSI_DMACTL_RXDMAE 0x00000001 // Receive DMA Enable //************************************************************************* // // The following are defines for the bit fields in the SSI_O_CC register. // //************************************************************************* #define SSI_CC_CS_M 0x0000000F // SSI Baud Clock Source #define SSI_CC_CS_SYSPLL 0x00000000 // Either the system clock (if the // PLL bypass is in effect) or the // PLL output (default) #define SSI_CC_CS_PIOSC 0x00000005 // PIOSC //************************************************************************* // // The following are defines for the bit fields in the UART_O_DR register. // //************************************************************************* #define UART_DR_OE 0x00000800 // UART Overrun Error #define UART_DR_BE 0x00000400 // UART Break Error #define UART_DR_PE 0x00000200 // UART Parity Error #define UART_DR_FE 0x00000100 // UART Framing Error #define UART_DR_DATA_M 0x000000FF // Data Transmitted or Received #define UART_DR_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_RSR register. // //************************************************************************* #define UART_RSR_OE 0x00000008 // UART Overrun Error #define UART_RSR_BE 0x00000004 // UART Break Error #define UART_RSR_PE 0x00000002 // UART Parity Error #define UART_RSR_FE 0x00000001 // UART Framing Error //************************************************************************* // // The following are defines for the bit fields in the UART_O_ECR register. // //************************************************************************* #define UART_ECR_DATA_M 0x000000FF // Error Clear #define UART_ECR_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_FR register. // //************************************************************************* #define UART_FR_TXFE 0x00000080 // UART Transmit FIFO Empty #define UART_FR_RXFF 0x00000040 // UART Receive FIFO Full #define UART_FR_TXFF 0x00000020 // UART Transmit FIFO Full #define UART_FR_RXFE 0x00000010 // UART Receive FIFO Empty #define UART_FR_BUSY 0x00000008 // UART Busy #define UART_FR_CTS 0x00000001 // Clear To Send //************************************************************************* // // The following are defines for the bit fields in the UART_O_ILPR register. // //************************************************************************* #define UART_ILPR_ILPDVSR_M 0x000000FF // IrDA Low-Power Divisor #define UART_ILPR_ILPDVSR_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_IBRD register. // //************************************************************************* #define UART_IBRD_DIVINT_M 0x0000FFFF // Integer Baud-Rate Divisor #define UART_IBRD_DIVINT_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_FBRD register. // //************************************************************************* #define UART_FBRD_DIVFRAC_M 0x0000003F // Fractional Baud-Rate Divisor #define UART_FBRD_DIVFRAC_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_LCRH register. // //************************************************************************* #define UART_LCRH_SPS 0x00000080 // UART Stick Parity Select #define UART_LCRH_WLEN_M 0x00000060 // UART Word Length #define UART_LCRH_WLEN_5 0x00000000 // 5 bits (default) #define UART_LCRH_WLEN_6 0x00000020 // 6 bits #define UART_LCRH_WLEN_7 0x00000040 // 7 bits #define UART_LCRH_WLEN_8 0x00000060 // 8 bits #define UART_LCRH_FEN 0x00000010 // UART Enable FIFOs #define UART_LCRH_STP2 0x00000008 // UART Two Stop Bits Select #define UART_LCRH_EPS 0x00000004 // UART Even Parity Select #define UART_LCRH_PEN 0x00000002 // UART Parity Enable #define UART_LCRH_BRK 0x00000001 // UART Send Break //************************************************************************* // // The following are defines for the bit fields in the UART_O_CTL register. // //************************************************************************* #define UART_CTL_RXE 0x00000200 // UART Receive Enable #define UART_CTL_TXE 0x00000100 // UART Transmit Enable #define UART_CTL_LBE 0x00000080 // UART Loop Back Enable #define UART_CTL_LIN 0x00000040 // LIN Mode Enable #define UART_CTL_HSE 0x00000020 // High-Speed Enable #define UART_CTL_EOT 0x00000010 // End of Transmission #define UART_CTL_SMART 0x00000008 // ISO 7816 Smart Card Support #define UART_CTL_SIRLP 0x00000004 // UART SIR Low-Power Mode #define UART_CTL_SIREN 0x00000002 // UART SIR Enable #define UART_CTL_UARTEN 0x00000001 // UART Enable //************************************************************************* // // The following are defines for the bit fields in the UART_O_IFLS register. // //************************************************************************* #define UART_IFLS_RX_M 0x00000038 // UART Receive Interrupt FIFO // Level Select #define UART_IFLS_RX1_8 0x00000000 // RX FIFO >= 1/8 full #define UART_IFLS_RX2_8 0x00000008 // RX FIFO >= 1/4 full #define UART_IFLS_RX4_8 0x00000010 // RX FIFO >= 1/2 full (default) #define UART_IFLS_RX6_8 0x00000018 // RX FIFO >= 3/4 full #define UART_IFLS_RX7_8 0x00000020 // RX FIFO >= 7/8 full #define UART_IFLS_TX_M 0x00000007 // UART Transmit Interrupt FIFO // Level Select #define UART_IFLS_TX1_8 0x00000000 // TX FIFO <= 1/8 full #define UART_IFLS_TX2_8 0x00000001 // TX FIFO <= 1/4 full #define UART_IFLS_TX4_8 0x00000002 // TX FIFO <= 1/2 full (default) #define UART_IFLS_TX6_8 0x00000003 // TX FIFO <= 3/4 full #define UART_IFLS_TX7_8 0x00000004 // TX FIFO <= 7/8 full //************************************************************************* // // The following are defines for the bit fields in the UART_O_IM register. // //************************************************************************* #define UART_IM_LME5IM 0x00008000 // LIN Mode Edge 5 Interrupt Mask #define UART_IM_LME1IM 0x00004000 // LIN Mode Edge 1 Interrupt Mask #define UART_IM_LMSBIM 0x00002000 // LIN Mode Sync Break Interrupt // Mask #define UART_IM_9BITIM 0x00001000 // 9-Bit Mode Interrupt Mask #define UART_IM_OEIM 0x00000400 // UART Overrun Error Interrupt // Mask #define UART_IM_BEIM 0x00000200 // UART Break Error Interrupt Mask #define UART_IM_PEIM 0x00000100 // UART Parity Error Interrupt Mask #define UART_IM_FEIM 0x00000080 // UART Framing Error Interrupt // Mask #define UART_IM_RTIM 0x00000040 // UART Receive Time-Out Interrupt // Mask #define UART_IM_TXIM 0x00000020 // UART Transmit Interrupt Mask #define UART_IM_RXIM 0x00000010 // UART Receive Interrupt Mask #define UART_IM_CTSMIM 0x00000002 // UART Clear to Send Modem // Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the UART_O_RIS register. // //************************************************************************* #define UART_RIS_LME5RIS 0x00008000 // LIN Mode Edge 5 Raw Interrupt // Status #define UART_RIS_LME1RIS 0x00004000 // LIN Mode Edge 1 Raw Interrupt // Status #define UART_RIS_LMSBRIS 0x00002000 // LIN Mode Sync Break Raw // Interrupt Status #define UART_RIS_9BITRIS 0x00001000 // 9-Bit Mode Raw Interrupt Status #define UART_RIS_OERIS 0x00000400 // UART Overrun Error Raw Interrupt // Status #define UART_RIS_BERIS 0x00000200 // UART Break Error Raw Interrupt // Status #define UART_RIS_PERIS 0x00000100 // UART Parity Error Raw Interrupt // Status #define UART_RIS_FERIS 0x00000080 // UART Framing Error Raw Interrupt // Status #define UART_RIS_RTRIS 0x00000040 // UART Receive Time-Out Raw // Interrupt Status #define UART_RIS_TXRIS 0x00000020 // UART Transmit Raw Interrupt // Status #define UART_RIS_RXRIS 0x00000010 // UART Receive Raw Interrupt // Status #define UART_RIS_CTSRIS 0x00000002 // UART Clear to Send Modem Raw // Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the UART_O_MIS register. // //************************************************************************* #define UART_MIS_LME5MIS 0x00008000 // LIN Mode Edge 5 Masked Interrupt // Status #define UART_MIS_LME1MIS 0x00004000 // LIN Mode Edge 1 Masked Interrupt // Status #define UART_MIS_LMSBMIS 0x00002000 // LIN Mode Sync Break Masked // Interrupt Status #define UART_MIS_9BITMIS 0x00001000 // 9-Bit Mode Masked Interrupt // Status #define UART_MIS_OEMIS 0x00000400 // UART Overrun Error Masked // Interrupt Status #define UART_MIS_BEMIS 0x00000200 // UART Break Error Masked // Interrupt Status #define UART_MIS_PEMIS 0x00000100 // UART Parity Error Masked // Interrupt Status #define UART_MIS_FEMIS 0x00000080 // UART Framing Error Masked // Interrupt Status #define UART_MIS_RTMIS 0x00000040 // UART Receive Time-Out Masked // Interrupt Status #define UART_MIS_TXMIS 0x00000020 // UART Transmit Masked Interrupt // Status #define UART_MIS_RXMIS 0x00000010 // UART Receive Masked Interrupt // Status #define UART_MIS_CTSMIS 0x00000002 // UART Clear to Send Modem Masked // Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the UART_O_ICR register. // //************************************************************************* #define UART_ICR_LME5IC 0x00008000 // LIN Mode Edge 5 Interrupt Clear #define UART_ICR_LME1IC 0x00004000 // LIN Mode Edge 1 Interrupt Clear #define UART_ICR_LMSBIC 0x00002000 // LIN Mode Sync Break Interrupt // Clear #define UART_ICR_9BITIC 0x00001000 // 9-Bit Mode Interrupt Clear #define UART_ICR_OEIC 0x00000400 // Overrun Error Interrupt Clear #define UART_ICR_BEIC 0x00000200 // Break Error Interrupt Clear #define UART_ICR_PEIC 0x00000100 // Parity Error Interrupt Clear #define UART_ICR_FEIC 0x00000080 // Framing Error Interrupt Clear #define UART_ICR_RTIC 0x00000040 // Receive Time-Out Interrupt Clear #define UART_ICR_TXIC 0x00000020 // Transmit Interrupt Clear #define UART_ICR_RXIC 0x00000010 // Receive Interrupt Clear #define UART_ICR_CTSMIC 0x00000002 // UART Clear to Send Modem // Interrupt Clear //************************************************************************* // // The following are defines for the bit fields in the UART_O_DMACTL register. // //************************************************************************* #define UART_DMACTL_DMAERR 0x00000004 // DMA on Error #define UART_DMACTL_TXDMAE 0x00000002 // Transmit DMA Enable #define UART_DMACTL_RXDMAE 0x00000001 // Receive DMA Enable //************************************************************************* // // The following are defines for the bit fields in the UART_O_LCTL register. // //************************************************************************* #define UART_LCTL_BLEN_M 0x00000030 // Sync Break Length #define UART_LCTL_BLEN_13T 0x00000000 // Sync break length is 13T bits // (default) #define UART_LCTL_BLEN_14T 0x00000010 // Sync break length is 14T bits #define UART_LCTL_BLEN_15T 0x00000020 // Sync break length is 15T bits #define UART_LCTL_BLEN_16T 0x00000030 // Sync break length is 16T bits #define UART_LCTL_MASTER 0x00000001 // LIN Master Enable //************************************************************************* // // The following are defines for the bit fields in the UART_O_LSS register. // //************************************************************************* #define UART_LSS_TSS_M 0x0000FFFF // Timer Snap Shot #define UART_LSS_TSS_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_LTIM register. // //************************************************************************* #define UART_LTIM_TIMER_M 0x0000FFFF // Timer Value #define UART_LTIM_TIMER_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_9BITADDR // register. // //************************************************************************* #define UART_9BITADDR_9BITEN 0x00008000 // Enable 9-Bit Mode #define UART_9BITADDR_ADDR_M 0x000000FF // Self Address for 9-Bit Mode #define UART_9BITADDR_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_9BITAMASK // register. // //************************************************************************* #define UART_9BITAMASK_MASK_M 0x000000FF // Self Address Mask for 9-Bit Mode #define UART_9BITAMASK_MASK_S 0 //************************************************************************* // // The following are defines for the bit fields in the UART_O_PP register. // //************************************************************************* #define UART_PP_NB 0x00000002 // 9-Bit Support #define UART_PP_SC 0x00000001 // Smart Card Support //************************************************************************* // // The following are defines for the bit fields in the UART_O_CC register. // //************************************************************************* #define UART_CC_CS_M 0x0000000F // UART Baud Clock Source #define UART_CC_CS_SYSCLK 0x00000000 // The system clock (default) #define UART_CC_CS_PIOSC 0x00000005 // PIOSC //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MSA register. // //************************************************************************* #define I2C_MSA_SA_M 0x000000FE // I2C Slave Address #define I2C_MSA_RS 0x00000001 // Receive not send #define I2C_MSA_SA_S 1 //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SOAR register. // //************************************************************************* #define I2C_SOAR_OAR_M 0x0000007F // I2C Slave Own Address #define I2C_SOAR_OAR_S 0 //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SCSR register. // //************************************************************************* #define I2C_SCSR_OAR2SEL 0x00000008 // OAR2 Address Matched #define I2C_SCSR_FBR 0x00000004 // First Byte Received #define I2C_SCSR_TREQ 0x00000002 // Transmit Request #define I2C_SCSR_DA 0x00000001 // Device Active #define I2C_SCSR_RREQ 0x00000001 // Receive Request //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MCS register. // //************************************************************************* #define I2C_MCS_CLKTO 0x00000080 // Clock Timeout Error #define I2C_MCS_BUSBSY 0x00000040 // Bus Busy #define I2C_MCS_IDLE 0x00000020 // I2C Idle #define I2C_MCS_ARBLST 0x00000010 // Arbitration Lost #define I2C_MCS_HS 0x00000010 // High-Speed Enable #define I2C_MCS_ACK 0x00000008 // Data Acknowledge Enable #define I2C_MCS_DATACK 0x00000008 // Acknowledge Data #define I2C_MCS_ADRACK 0x00000004 // Acknowledge Address #define I2C_MCS_STOP 0x00000004 // Generate STOP #define I2C_MCS_ERROR 0x00000002 // Error #define I2C_MCS_START 0x00000002 // Generate START #define I2C_MCS_RUN 0x00000001 // I2C Master Enable #define I2C_MCS_BUSY 0x00000001 // I2C Busy //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SDR register. // //************************************************************************* #define I2C_SDR_DATA_M 0x000000FF // Data for Transfer #define I2C_SDR_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MDR register. // //************************************************************************* #define I2C_MDR_DATA_M 0x000000FF // Data Transferred #define I2C_MDR_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MTPR register. // //************************************************************************* #define I2C_MTPR_HS 0x00000080 // High-Speed Enable #define I2C_MTPR_TPR_M 0x0000007F // SCL Clock Period #define I2C_MTPR_TPR_S 0 //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SIMR register. // //************************************************************************* #define I2C_SIMR_STOPIM 0x00000004 // Stop Condition Interrupt Mask #define I2C_SIMR_STARTIM 0x00000002 // Start Condition Interrupt Mask #define I2C_SIMR_DATAIM 0x00000001 // Data Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SRIS register. // //************************************************************************* #define I2C_SRIS_STOPRIS 0x00000004 // Stop Condition Raw Interrupt // Status #define I2C_SRIS_STARTRIS 0x00000002 // Start Condition Raw Interrupt // Status #define I2C_SRIS_DATARIS 0x00000001 // Data Raw Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MIMR register. // //************************************************************************* #define I2C_MIMR_CLKIM 0x00000002 // Clock Timeout Interrupt Mask #define I2C_MIMR_IM 0x00000001 // Master Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MRIS register. // //************************************************************************* #define I2C_MRIS_CLKRIS 0x00000002 // Clock Timeout Raw Interrupt // Status #define I2C_MRIS_RIS 0x00000001 // Master Raw Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SMIS register. // //************************************************************************* #define I2C_SMIS_STOPMIS 0x00000004 // Stop Condition Masked Interrupt // Status #define I2C_SMIS_STARTMIS 0x00000002 // Start Condition Masked Interrupt // Status #define I2C_SMIS_DATAMIS 0x00000001 // Data Masked Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SICR register. // //************************************************************************* #define I2C_SICR_STOPIC 0x00000004 // Stop Condition Interrupt Clear #define I2C_SICR_STARTIC 0x00000002 // Start Condition Interrupt Clear #define I2C_SICR_DATAIC 0x00000001 // Data Interrupt Clear //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MMIS register. // //************************************************************************* #define I2C_MMIS_CLKMIS 0x00000002 // Clock Timeout Masked Interrupt // Status #define I2C_MMIS_MIS 0x00000001 // Masked Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MICR register. // //************************************************************************* #define I2C_MICR_CLKIC 0x00000002 // Clock Timeout Interrupt Clear #define I2C_MICR_IC 0x00000001 // Master Interrupt Clear //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SOAR2 register. // //************************************************************************* #define I2C_SOAR2_OAR2EN 0x00000080 // I2C Slave Own Address 2 Enable #define I2C_SOAR2_OAR2_M 0x0000007F // I2C Slave Own Address 2 #define I2C_SOAR2_OAR2_S 0 //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MCR register. // //************************************************************************* #define I2C_MCR_SFE 0x00000020 // I2C Slave Function Enable #define I2C_MCR_MFE 0x00000010 // I2C Master Function Enable #define I2C_MCR_LPBK 0x00000001 // I2C Loopback //************************************************************************* // // The following are defines for the bit fields in the I2C_O_SACKCTL register. // //************************************************************************* #define I2C_SACKCTL_ACKOVAL 0x00000002 // I2C Slave ACK Override Value #define I2C_SACKCTL_ACKOEN 0x00000001 // I2C Slave ACK Override Enable //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MCLKOCNT register. // //************************************************************************* #define I2C_MCLKOCNT_CNTL_M 0x000000FF // I2C Master Count #define I2C_MCLKOCNT_CNTL_S 0 //************************************************************************* // // The following are defines for the bit fields in the I2C_O_MBMON register. // //************************************************************************* #define I2C_MBMON_SDA 0x00000002 // I2C SDA Status #define I2C_MBMON_SCL 0x00000001 // I2C SCL Status //************************************************************************* // // The following are defines for the bit fields in the I2C_O_PP register. // //************************************************************************* #define I2C_PP_HS 0x00000001 // High-Speed Capable //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_CFG register. // //************************************************************************* #define TIMER_CFG_M 0x00000007 // GPTM Configuration #define TIMER_CFG_32_BIT_TIMER 0x00000000 // 32-bit timer configuration #define TIMER_CFG_32_BIT_RTC 0x00000001 // 32-bit real-time clock (RTC) // counter configuration #define TIMER_CFG_16_BIT 0x00000004 // 16-bit timer configuration. The // function is controlled by bits // 1:0 of GPTMTAMR and GPTMTBMR //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAMR register. // //************************************************************************* #define TIMER_TAMR_TAPLO 0x00000800 // GPTM Timer A PWM Legacy // Operation #define TIMER_TAMR_TAMRSU 0x00000400 // GPTM Timer A Match Register // Update #define TIMER_TAMR_TAPWMIE 0x00000200 // GPTM Timer A PWM Interrupt // Enable #define TIMER_TAMR_TAILD 0x00000100 // GPTM Timer A Interval Load Write #define TIMER_TAMR_TASNAPS 0x00000080 // GPTM Timer A Snap-Shot Mode #define TIMER_TAMR_TAWOT 0x00000040 // GPTM Timer A Wait-on-Trigger #define TIMER_TAMR_TAMIE 0x00000020 // GPTM Timer A Match Interrupt // Enable #define TIMER_TAMR_TACDIR 0x00000010 // GPTM Timer A Count Direction #define TIMER_TAMR_TAAMS 0x00000008 // GPTM Timer A Alternate Mode // Select #define TIMER_TAMR_TACMR 0x00000004 // GPTM Timer A Capture Mode #define TIMER_TAMR_TAMR_M 0x00000003 // GPTM Timer A Mode #define TIMER_TAMR_TAMR_1_SHOT 0x00000001 // One-Shot Timer mode #define TIMER_TAMR_TAMR_PERIOD 0x00000002 // Periodic Timer mode #define TIMER_TAMR_TAMR_CAP 0x00000003 // Capture mode //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBMR register. // //************************************************************************* #define TIMER_TBMR_TBPLO 0x00000800 // GPTM Timer B PWM Legacy // Operation #define TIMER_TBMR_TBMRSU 0x00000400 // GPTM Timer B Match Register // Update #define TIMER_TBMR_TBPWMIE 0x00000200 // GPTM Timer B PWM Interrupt // Enable #define TIMER_TBMR_TBILD 0x00000100 // GPTM Timer B Interval Load Write #define TIMER_TBMR_TBSNAPS 0x00000080 // GPTM Timer B Snap-Shot Mode #define TIMER_TBMR_TBWOT 0x00000040 // GPTM Timer B Wait-on-Trigger #define TIMER_TBMR_TBMIE 0x00000020 // GPTM Timer B Match Interrupt // Enable #define TIMER_TBMR_TBCDIR 0x00000010 // GPTM Timer B Count Direction #define TIMER_TBMR_TBAMS 0x00000008 // GPTM Timer B Alternate Mode // Select #define TIMER_TBMR_TBCMR 0x00000004 // GPTM Timer B Capture Mode #define TIMER_TBMR_TBMR_M 0x00000003 // GPTM Timer B Mode #define TIMER_TBMR_TBMR_1_SHOT 0x00000001 // One-Shot Timer mode #define TIMER_TBMR_TBMR_PERIOD 0x00000002 // Periodic Timer mode #define TIMER_TBMR_TBMR_CAP 0x00000003 // Capture mode //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_CTL register. // //************************************************************************* #define TIMER_CTL_TBPWML 0x00004000 // GPTM Timer B PWM Output Level #define TIMER_CTL_TBOTE 0x00002000 // GPTM Timer B Output Trigger // Enable #define TIMER_CTL_TBEVENT_M 0x00000C00 // GPTM Timer B Event Mode #define TIMER_CTL_TBEVENT_POS 0x00000000 // Positive edge #define TIMER_CTL_TBEVENT_NEG 0x00000400 // Negative edge #define TIMER_CTL_TBEVENT_BOTH 0x00000C00 // Both edges #define TIMER_CTL_TBSTALL 0x00000200 // GPTM Timer B Stall Enable #define TIMER_CTL_TBEN 0x00000100 // GPTM Timer B Enable #define TIMER_CTL_TAPWML 0x00000040 // GPTM Timer A PWM Output Level #define TIMER_CTL_TAOTE 0x00000020 // GPTM Timer A Output Trigger // Enable #define TIMER_CTL_RTCEN 0x00000010 // GPTM RTC Stall Enable #define TIMER_CTL_TAEVENT_M 0x0000000C // GPTM Timer A Event Mode #define TIMER_CTL_TAEVENT_POS 0x00000000 // Positive edge #define TIMER_CTL_TAEVENT_NEG 0x00000004 // Negative edge #define TIMER_CTL_TAEVENT_BOTH 0x0000000C // Both edges #define TIMER_CTL_TASTALL 0x00000002 // GPTM Timer A Stall Enable #define TIMER_CTL_TAEN 0x00000001 // GPTM Timer A Enable //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_SYNC register. // //************************************************************************* #define TIMER_SYNC_SYNCWT5_M 0x00C00000 // Synchronize GPTM 32/64-Bit Timer // 5 #define TIMER_SYNC_SYNCWT5_NONE 0x00000000 // GPTM 32/64-Bit Timer 5 is not // affected #define TIMER_SYNC_SYNCWT5_TA 0x00400000 // A timeout event for Timer A of // GPTM 32/64-Bit Timer 5 is // triggered #define TIMER_SYNC_SYNCWT5_TB 0x00800000 // A timeout event for Timer B of // GPTM 32/64-Bit Timer 5 is // triggered #define TIMER_SYNC_SYNCWT5_TATB 0x00C00000 // A timeout event for both Timer A // and Timer B of GPTM 32/64-Bit // Timer 5 is triggered #define TIMER_SYNC_SYNCWT4_M 0x00300000 // Synchronize GPTM 32/64-Bit Timer // 4 #define TIMER_SYNC_SYNCWT4_NONE 0x00000000 // GPTM 32/64-Bit Timer 4 is not // affected #define TIMER_SYNC_SYNCWT4_TA 0x00100000 // A timeout event for Timer A of // GPTM 32/64-Bit Timer 4 is // triggered #define TIMER_SYNC_SYNCWT4_TB 0x00200000 // A timeout event for Timer B of // GPTM 32/64-Bit Timer 4 is // triggered #define TIMER_SYNC_SYNCWT4_TATB 0x00300000 // A timeout event for both Timer A // and Timer B of GPTM 32/64-Bit // Timer 4 is triggered #define TIMER_SYNC_SYNCWT3_M 0x000C0000 // Synchronize GPTM 32/64-Bit Timer // 3 #define TIMER_SYNC_SYNCWT3_NONE 0x00000000 // GPTM 32/64-Bit Timer 3 is not // affected #define TIMER_SYNC_SYNCWT3_TA 0x00040000 // A timeout event for Timer A of // GPTM 32/64-Bit Timer 3 is // triggered #define TIMER_SYNC_SYNCWT3_TB 0x00080000 // A timeout event for Timer B of // GPTM 32/64-Bit Timer 3 is // triggered #define TIMER_SYNC_SYNCWT3_TATB 0x000C0000 // A timeout event for both Timer A // and Timer B of GPTM 32/64-Bit // Timer 3 is triggered #define TIMER_SYNC_SYNCWT2_M 0x00030000 // Synchronize GPTM 32/64-Bit Timer // 2 #define TIMER_SYNC_SYNCWT2_NONE 0x00000000 // GPTM 32/64-Bit Timer 2 is not // affected #define TIMER_SYNC_SYNCWT2_TA 0x00010000 // A timeout event for Timer A of // GPTM 32/64-Bit Timer 2 is // triggered #define TIMER_SYNC_SYNCWT2_TB 0x00020000 // A timeout event for Timer B of // GPTM 32/64-Bit Timer 2 is // triggered #define TIMER_SYNC_SYNCWT2_TATB 0x00030000 // A timeout event for both Timer A // and Timer B of GPTM 32/64-Bit // Timer 2 is triggered #define TIMER_SYNC_SYNCWT1_M 0x0000C000 // Synchronize GPTM 32/64-Bit Timer // 1 #define TIMER_SYNC_SYNCWT1_NONE 0x00000000 // GPTM 32/64-Bit Timer 1 is not // affected #define TIMER_SYNC_SYNCWT1_TA 0x00004000 // A timeout event for Timer A of // GPTM 32/64-Bit Timer 1 is // triggered #define TIMER_SYNC_SYNCWT1_TB 0x00008000 // A timeout event for Timer B of // GPTM 32/64-Bit Timer 1 is // triggered #define TIMER_SYNC_SYNCWT1_TATB 0x0000C000 // A timeout event for both Timer A // and Timer B of GPTM 32/64-Bit // Timer 1 is triggered #define TIMER_SYNC_SYNCWT0_M 0x00003000 // Synchronize GPTM 32/64-Bit Timer // 0 #define TIMER_SYNC_SYNCWT0_NONE 0x00000000 // GPTM 32/64-Bit Timer 0 is not // affected #define TIMER_SYNC_SYNCWT0_TA 0x00001000 // A timeout event for Timer A of // GPTM 32/64-Bit Timer 0 is // triggered #define TIMER_SYNC_SYNCWT0_TB 0x00002000 // A timeout event for Timer B of // GPTM 32/64-Bit Timer 0 is // triggered #define TIMER_SYNC_SYNCWT0_TATB 0x00003000 // A timeout event for both Timer A // and Timer B of GPTM 32/64-Bit // Timer 0 is triggered #define TIMER_SYNC_SYNCT5_M 0x00000C00 // Synchronize GPTM 16/32-Bit Timer // 5 #define TIMER_SYNC_SYNCT5_NONE 0x00000000 // GPTM 16/32-Bit Timer 5 is not // affected #define TIMER_SYNC_SYNCT5_TA 0x00000400 // A timeout event for Timer A of // GPTM 16/32-Bit Timer 5 is // triggered #define TIMER_SYNC_SYNCT5_TB 0x00000800 // A timeout event for Timer B of // GPTM 16/32-Bit Timer 5 is // triggered #define TIMER_SYNC_SYNCT5_TATB 0x00000C00 // A timeout event for both Timer A // and Timer B of GPTM 16/32-Bit // Timer 5 is triggered #define TIMER_SYNC_SYNCT4_M 0x00000300 // Synchronize GPTM 16/32-Bit Timer // 4 #define TIMER_SYNC_SYNCT4_NONE 0x00000000 // GPTM 16/32-Bit Timer 4 is not // affected #define TIMER_SYNC_SYNCT4_TA 0x00000100 // A timeout event for Timer A of // GPTM 16/32-Bit Timer 4 is // triggered #define TIMER_SYNC_SYNCT4_TB 0x00000200 // A timeout event for Timer B of // GPTM 16/32-Bit Timer 4 is // triggered #define TIMER_SYNC_SYNCT4_TATB 0x00000300 // A timeout event for both Timer A // and Timer B of GPTM 16/32-Bit // Timer 4 is triggered #define TIMER_SYNC_SYNCT3_M 0x000000C0 // Synchronize GPTM 16/32-Bit Timer // 3 #define TIMER_SYNC_SYNCT3_NONE 0x00000000 // GPTM 16/32-Bit Timer 3 is not // affected #define TIMER_SYNC_SYNCT3_TA 0x00000040 // A timeout event for Timer A of // GPTM 16/32-Bit Timer 3 is // triggered #define TIMER_SYNC_SYNCT3_TB 0x00000080 // A timeout event for Timer B of // GPTM 16/32-Bit Timer 3 is // triggered #define TIMER_SYNC_SYNCT3_TATB 0x000000C0 // A timeout event for both Timer A // and Timer B of GPTM 16/32-Bit // Timer 3 is triggered #define TIMER_SYNC_SYNCT2_M 0x00000030 // Synchronize GPTM 16/32-Bit Timer // 2 #define TIMER_SYNC_SYNCT2_NONE 0x00000000 // GPTM 16/32-Bit Timer 2 is not // affected #define TIMER_SYNC_SYNCT2_TA 0x00000010 // A timeout event for Timer A of // GPTM 16/32-Bit Timer 2 is // triggered #define TIMER_SYNC_SYNCT2_TB 0x00000020 // A timeout event for Timer B of // GPTM 16/32-Bit Timer 2 is // triggered #define TIMER_SYNC_SYNCT2_TATB 0x00000030 // A timeout event for both Timer A // and Timer B of GPTM 16/32-Bit // Timer 2 is triggered #define TIMER_SYNC_SYNCT1_M 0x0000000C // Synchronize GPTM 16/32-Bit Timer // 1 #define TIMER_SYNC_SYNCT1_NONE 0x00000000 // GPTM 16/32-Bit Timer 1 is not // affected #define TIMER_SYNC_SYNCT1_TA 0x00000004 // A timeout event for Timer A of // GPTM 16/32-Bit Timer 1 is // triggered #define TIMER_SYNC_SYNCT1_TB 0x00000008 // A timeout event for Timer B of // GPTM 16/32-Bit Timer 1 is // triggered #define TIMER_SYNC_SYNCT1_TATB 0x0000000C // A timeout event for both Timer A // and Timer B of GPTM 16/32-Bit // Timer 1 is triggered #define TIMER_SYNC_SYNCT0_M 0x00000003 // Synchronize GPTM 16/32-Bit Timer // 0 #define TIMER_SYNC_SYNCT0_NONE 0x00000000 // GPTM 16/32-Bit Timer 0 is not // affected #define TIMER_SYNC_SYNCT0_TA 0x00000001 // A timeout event for Timer A of // GPTM 16/32-Bit Timer 0 is // triggered #define TIMER_SYNC_SYNCT0_TB 0x00000002 // A timeout event for Timer B of // GPTM 16/32-Bit Timer 0 is // triggered #define TIMER_SYNC_SYNCT0_TATB 0x00000003 // A timeout event for both Timer A // and Timer B of GPTM 16/32-Bit // Timer 0 is triggered //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_IMR register. // //************************************************************************* #define TIMER_IMR_WUEIM 0x00010000 // GPTM Write Update Error // Interrupt Mask #define TIMER_IMR_TBMIM 0x00000800 // GPTM Timer B Match Interrupt // Mask #define TIMER_IMR_CBEIM 0x00000400 // GPTM Timer B Capture Mode Event // Interrupt Mask #define TIMER_IMR_CBMIM 0x00000200 // GPTM Timer B Capture Mode Match // Interrupt Mask #define TIMER_IMR_TBTOIM 0x00000100 // GPTM Timer B Time-Out Interrupt // Mask #define TIMER_IMR_TAMIM 0x00000010 // GPTM Timer A Match Interrupt // Mask #define TIMER_IMR_RTCIM 0x00000008 // GPTM RTC Interrupt Mask #define TIMER_IMR_CAEIM 0x00000004 // GPTM Timer A Capture Mode Event // Interrupt Mask #define TIMER_IMR_CAMIM 0x00000002 // GPTM Timer A Capture Mode Match // Interrupt Mask #define TIMER_IMR_TATOIM 0x00000001 // GPTM Timer A Time-Out Interrupt // Mask //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_RIS register. // //************************************************************************* #define TIMER_RIS_WUERIS 0x00010000 // GPTM Write Update Error Raw // Interrupt #define TIMER_RIS_TBMRIS 0x00000800 // GPTM Timer B Match Raw Interrupt #define TIMER_RIS_CBERIS 0x00000400 // GPTM Timer B Capture Mode Event // Raw Interrupt #define TIMER_RIS_CBMRIS 0x00000200 // GPTM Timer B Capture Mode Match // Raw Interrupt #define TIMER_RIS_TBTORIS 0x00000100 // GPTM Timer B Time-Out Raw // Interrupt #define TIMER_RIS_TAMRIS 0x00000010 // GPTM Timer A Match Raw Interrupt #define TIMER_RIS_RTCRIS 0x00000008 // GPTM RTC Raw Interrupt #define TIMER_RIS_CAERIS 0x00000004 // GPTM Timer A Capture Mode Event // Raw Interrupt #define TIMER_RIS_CAMRIS 0x00000002 // GPTM Timer A Capture Mode Match // Raw Interrupt #define TIMER_RIS_TATORIS 0x00000001 // GPTM Timer A Time-Out Raw // Interrupt //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_MIS register. // //************************************************************************* #define TIMER_MIS_WUEMIS 0x00010000 // GPTM Write Update Error Masked // Interrupt #define TIMER_MIS_TBMMIS 0x00000800 // GPTM Timer B Match Masked // Interrupt #define TIMER_MIS_CBEMIS 0x00000400 // GPTM Timer B Capture Mode Event // Masked Interrupt #define TIMER_MIS_CBMMIS 0x00000200 // GPTM Timer B Capture Mode Match // Masked Interrupt #define TIMER_MIS_TBTOMIS 0x00000100 // GPTM Timer B Time-Out Masked // Interrupt #define TIMER_MIS_TAMMIS 0x00000010 // GPTM Timer A Match Masked // Interrupt #define TIMER_MIS_RTCMIS 0x00000008 // GPTM RTC Masked Interrupt #define TIMER_MIS_CAEMIS 0x00000004 // GPTM Timer A Capture Mode Event // Masked Interrupt #define TIMER_MIS_CAMMIS 0x00000002 // GPTM Timer A Capture Mode Match // Masked Interrupt #define TIMER_MIS_TATOMIS 0x00000001 // GPTM Timer A Time-Out Masked // Interrupt //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_ICR register. // //************************************************************************* #define TIMER_ICR_WUECINT 0x00010000 // 32/64-Bit GPTM Write Update // Error Interrupt Clear #define TIMER_ICR_TBMCINT 0x00000800 // GPTM Timer B Match Interrupt // Clear #define TIMER_ICR_CBECINT 0x00000400 // GPTM Timer B Capture Mode Event // Interrupt Clear #define TIMER_ICR_CBMCINT 0x00000200 // GPTM Timer B Capture Mode Match // Interrupt Clear #define TIMER_ICR_TBTOCINT 0x00000100 // GPTM Timer B Time-Out Interrupt // Clear #define TIMER_ICR_TAMCINT 0x00000010 // GPTM Timer A Match Interrupt // Clear #define TIMER_ICR_RTCCINT 0x00000008 // GPTM RTC Interrupt Clear #define TIMER_ICR_CAECINT 0x00000004 // GPTM Timer A Capture Mode Event // Interrupt Clear #define TIMER_ICR_CAMCINT 0x00000002 // GPTM Timer A Capture Mode Match // Interrupt Clear #define TIMER_ICR_TATOCINT 0x00000001 // GPTM Timer A Time-Out Raw // Interrupt //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAILR register. // //************************************************************************* #define TIMER_TAILR_M 0xFFFFFFFF // GPTM Timer A Interval Load // Register #define TIMER_TAILR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBILR register. // //************************************************************************* #define TIMER_TBILR_M 0xFFFFFFFF // GPTM Timer B Interval Load // Register #define TIMER_TBILR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAMATCHR // register. // //************************************************************************* #define TIMER_TAMATCHR_TAMR_M 0xFFFFFFFF // GPTM Timer A Match Register #define TIMER_TAMATCHR_TAMR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBMATCHR // register. // //************************************************************************* #define TIMER_TBMATCHR_TBMR_M 0xFFFFFFFF // GPTM Timer B Match Register #define TIMER_TBMATCHR_TBMR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAPR register. // //************************************************************************* #define TIMER_TAPR_TAPSRH_M 0x0000FF00 // GPTM Timer A Prescale High Byte #define TIMER_TAPR_TAPSR_M 0x000000FF // GPTM Timer A Prescale #define TIMER_TAPR_TAPSRH_S 8 #define TIMER_TAPR_TAPSR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBPR register. // //************************************************************************* #define TIMER_TBPR_TBPSRH_M 0x0000FF00 // GPTM Timer B Prescale High Byte #define TIMER_TBPR_TBPSR_M 0x000000FF // GPTM Timer B Prescale #define TIMER_TBPR_TBPSRH_S 8 #define TIMER_TBPR_TBPSR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAPMR register. // //************************************************************************* #define TIMER_TAPMR_TAPSMRH_M 0x0000FF00 // GPTM Timer A Prescale Match High // Byte #define TIMER_TAPMR_TAPSMR_M 0x000000FF // GPTM TimerA Prescale Match #define TIMER_TAPMR_TAPSMRH_S 8 #define TIMER_TAPMR_TAPSMR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBPMR register. // //************************************************************************* #define TIMER_TBPMR_TBPSMRH_M 0x0000FF00 // GPTM Timer B Prescale Match High // Byte #define TIMER_TBPMR_TBPSMR_M 0x000000FF // GPTM TimerB Prescale Match #define TIMER_TBPMR_TBPSMRH_S 8 #define TIMER_TBPMR_TBPSMR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAR register. // //************************************************************************* #define TIMER_TAR_M 0xFFFFFFFF // GPTM Timer A Register #define TIMER_TAR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBR register. // //************************************************************************* #define TIMER_TBR_M 0xFFFFFFFF // GPTM Timer B Register #define TIMER_TBR_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAV register. // //************************************************************************* #define TIMER_TAV_M 0xFFFFFFFF // GPTM Timer A Value #define TIMER_TAV_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBV register. // //************************************************************************* #define TIMER_TBV_M 0xFFFFFFFF // GPTM Timer B Value #define TIMER_TBV_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_RTCPD register. // //************************************************************************* #define TIMER_RTCPD_RTCPD_M 0x0000FFFF // RTC Predivide Counter Value #define TIMER_RTCPD_RTCPD_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAPS register. // //************************************************************************* #define TIMER_TAPS_PSS_M 0x0000FFFF // GPTM Timer A Prescaler Snapshot #define TIMER_TAPS_PSS_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBPS register. // //************************************************************************* #define TIMER_TBPS_PSS_M 0x0000FFFF // GPTM Timer A Prescaler Value #define TIMER_TBPS_PSS_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TAPV register. // //************************************************************************* #define TIMER_TAPV_PSV_M 0x0000FFFF // GPTM Timer A Prescaler Value #define TIMER_TAPV_PSV_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_TBPV register. // //************************************************************************* #define TIMER_TBPV_PSV_M 0x0000FFFF // GPTM Timer B Prescaler Value #define TIMER_TBPV_PSV_S 0 //************************************************************************* // // The following are defines for the bit fields in the TIMER_O_PP register. // //************************************************************************* #define TIMER_PP_SIZE_M 0x0000000F // Count Size #define TIMER_PP_SIZE_16 0x00000000 // Timer A and Timer B counters are // 16 bits each with an 8-bit // prescale counter #define TIMER_PP_SIZE_32 0x00000001 // Timer A and Timer B counters are // 32 bits each with a 16-bit // prescale counter //************************************************************************* // // The following are defines for the bit fields in the ADC_O_ACTSS register. // //************************************************************************* #define ADC_ACTSS_ASEN3 0x00000008 // ADC SS3 Enable #define ADC_ACTSS_ASEN2 0x00000004 // ADC SS2 Enable #define ADC_ACTSS_ASEN1 0x00000002 // ADC SS1 Enable #define ADC_ACTSS_ASEN0 0x00000001 // ADC SS0 Enable //************************************************************************* // // The following are defines for the bit fields in the ADC_O_RIS register. // //************************************************************************* #define ADC_RIS_INRDC 0x00010000 // Digital Comparator Raw Interrupt // Status #define ADC_RIS_INR3 0x00000008 // SS3 Raw Interrupt Status #define ADC_RIS_INR2 0x00000004 // SS2 Raw Interrupt Status #define ADC_RIS_INR1 0x00000002 // SS1 Raw Interrupt Status #define ADC_RIS_INR0 0x00000001 // SS0 Raw Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the ADC_O_IM register. // //************************************************************************* #define ADC_IM_DCONSS3 0x00080000 // Digital Comparator Interrupt on // SS3 #define ADC_IM_DCONSS2 0x00040000 // Digital Comparator Interrupt on // SS2 #define ADC_IM_DCONSS1 0x00020000 // Digital Comparator Interrupt on // SS1 #define ADC_IM_DCONSS0 0x00010000 // Digital Comparator Interrupt on // SS0 #define ADC_IM_MASK3 0x00000008 // SS3 Interrupt Mask #define ADC_IM_MASK2 0x00000004 // SS2 Interrupt Mask #define ADC_IM_MASK1 0x00000002 // SS1 Interrupt Mask #define ADC_IM_MASK0 0x00000001 // SS0 Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the ADC_O_ISC register. // //************************************************************************* #define ADC_ISC_DCINSS3 0x00080000 // Digital Comparator Interrupt // Status on SS3 #define ADC_ISC_DCINSS2 0x00040000 // Digital Comparator Interrupt // Status on SS2 #define ADC_ISC_DCINSS1 0x00020000 // Digital Comparator Interrupt // Status on SS1 #define ADC_ISC_DCINSS0 0x00010000 // Digital Comparator Interrupt // Status on SS0 #define ADC_ISC_IN3 0x00000008 // SS3 Interrupt Status and Clear #define ADC_ISC_IN2 0x00000004 // SS2 Interrupt Status and Clear #define ADC_ISC_IN1 0x00000002 // SS1 Interrupt Status and Clear #define ADC_ISC_IN0 0x00000001 // SS0 Interrupt Status and Clear //************************************************************************* // // The following are defines for the bit fields in the ADC_O_OSTAT register. // //************************************************************************* #define ADC_OSTAT_OV3 0x00000008 // SS3 FIFO Overflow #define ADC_OSTAT_OV2 0x00000004 // SS2 FIFO Overflow #define ADC_OSTAT_OV1 0x00000002 // SS1 FIFO Overflow #define ADC_OSTAT_OV0 0x00000001 // SS0 FIFO Overflow //************************************************************************* // // The following are defines for the bit fields in the ADC_O_EMUX register. // //************************************************************************* #define ADC_EMUX_EM3_M 0x0000F000 // SS3 Trigger Select #define ADC_EMUX_EM3_PROCESSOR 0x00000000 // Processor (default) #define ADC_EMUX_EM3_COMP0 0x00001000 // Analog Comparator 0 #define ADC_EMUX_EM3_COMP1 0x00002000 // Analog Comparator 1 #define ADC_EMUX_EM3_EXTERNAL 0x00004000 // External (GPIO PB4) #define ADC_EMUX_EM3_TIMER 0x00005000 // Timer #define ADC_EMUX_EM3_ALWAYS 0x0000F000 // Always (continuously sample) #define ADC_EMUX_EM2_M 0x00000F00 // SS2 Trigger Select #define ADC_EMUX_EM2_PROCESSOR 0x00000000 // Processor (default) #define ADC_EMUX_EM2_COMP0 0x00000100 // Analog Comparator 0 #define ADC_EMUX_EM2_COMP1 0x00000200 // Analog Comparator 1 #define ADC_EMUX_EM2_EXTERNAL 0x00000400 // External (GPIO PB4) #define ADC_EMUX_EM2_TIMER 0x00000500 // Timer #define ADC_EMUX_EM2_ALWAYS 0x00000F00 // Always (continuously sample) #define ADC_EMUX_EM1_M 0x000000F0 // SS1 Trigger Select #define ADC_EMUX_EM1_PROCESSOR 0x00000000 // Processor (default) #define ADC_EMUX_EM1_COMP0 0x00000010 // Analog Comparator 0 #define ADC_EMUX_EM1_COMP1 0x00000020 // Analog Comparator 1 #define ADC_EMUX_EM1_EXTERNAL 0x00000040 // External (GPIO PB4) #define ADC_EMUX_EM1_TIMER 0x00000050 // Timer #define ADC_EMUX_EM1_ALWAYS 0x000000F0 // Always (continuously sample) #define ADC_EMUX_EM0_M 0x0000000F // SS0 Trigger Select #define ADC_EMUX_EM0_PROCESSOR 0x00000000 // Processor (default) #define ADC_EMUX_EM0_COMP0 0x00000001 // Analog Comparator 0 #define ADC_EMUX_EM0_COMP1 0x00000002 // Analog Comparator 1 #define ADC_EMUX_EM0_EXTERNAL 0x00000004 // External (GPIO PB4) #define ADC_EMUX_EM0_TIMER 0x00000005 // Timer #define ADC_EMUX_EM0_ALWAYS 0x0000000F // Always (continuously sample) //************************************************************************* // // The following are defines for the bit fields in the ADC_O_USTAT register. // //************************************************************************* #define ADC_USTAT_UV3 0x00000008 // SS3 FIFO Underflow #define ADC_USTAT_UV2 0x00000004 // SS2 FIFO Underflow #define ADC_USTAT_UV1 0x00000002 // SS1 FIFO Underflow #define ADC_USTAT_UV0 0x00000001 // SS0 FIFO Underflow //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSPRI register. // //************************************************************************* #define ADC_SSPRI_SS3_M 0x00003000 // SS3 Priority #define ADC_SSPRI_SS3_1ST 0x00000000 // First priority #define ADC_SSPRI_SS3_2ND 0x00001000 // Second priority #define ADC_SSPRI_SS3_3RD 0x00002000 // Third priority #define ADC_SSPRI_SS3_4TH 0x00003000 // Fourth priority #define ADC_SSPRI_SS2_M 0x00000300 // SS2 Priority #define ADC_SSPRI_SS2_1ST 0x00000000 // First priority #define ADC_SSPRI_SS2_2ND 0x00000100 // Second priority #define ADC_SSPRI_SS2_3RD 0x00000200 // Third priority #define ADC_SSPRI_SS2_4TH 0x00000300 // Fourth priority #define ADC_SSPRI_SS1_M 0x00000030 // SS1 Priority #define ADC_SSPRI_SS1_1ST 0x00000000 // First priority #define ADC_SSPRI_SS1_2ND 0x00000010 // Second priority #define ADC_SSPRI_SS1_3RD 0x00000020 // Third priority #define ADC_SSPRI_SS1_4TH 0x00000030 // Fourth priority #define ADC_SSPRI_SS0_M 0x00000003 // SS0 Priority #define ADC_SSPRI_SS0_1ST 0x00000000 // First priority #define ADC_SSPRI_SS0_2ND 0x00000001 // Second priority #define ADC_SSPRI_SS0_3RD 0x00000002 // Third priority #define ADC_SSPRI_SS0_4TH 0x00000003 // Fourth priority //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SPC register. // //************************************************************************* #define ADC_SPC_PHASE_M 0x0000000F // Phase Difference #define ADC_SPC_PHASE_0 0x00000000 // ADC sample lags by 0.0 #define ADC_SPC_PHASE_22_5 0x00000001 // ADC sample lags by 22.5 #define ADC_SPC_PHASE_45 0x00000002 // ADC sample lags by 45.0 #define ADC_SPC_PHASE_67_5 0x00000003 // ADC sample lags by 67.5 #define ADC_SPC_PHASE_90 0x00000004 // ADC sample lags by 90.0 #define ADC_SPC_PHASE_112_5 0x00000005 // ADC sample lags by 112.5 #define ADC_SPC_PHASE_135 0x00000006 // ADC sample lags by 135.0 #define ADC_SPC_PHASE_157_5 0x00000007 // ADC sample lags by 157.5 #define ADC_SPC_PHASE_180 0x00000008 // ADC sample lags by 180.0 #define ADC_SPC_PHASE_202_5 0x00000009 // ADC sample lags by 202.5 #define ADC_SPC_PHASE_225 0x0000000A // ADC sample lags by 225.0 #define ADC_SPC_PHASE_247_5 0x0000000B // ADC sample lags by 247.5 #define ADC_SPC_PHASE_270 0x0000000C // ADC sample lags by 270.0 #define ADC_SPC_PHASE_292_5 0x0000000D // ADC sample lags by 292.5 #define ADC_SPC_PHASE_315 0x0000000E // ADC sample lags by 315.0 #define ADC_SPC_PHASE_337_5 0x0000000F // ADC sample lags by 337.5 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_PSSI register. // //************************************************************************* #define ADC_PSSI_GSYNC 0x80000000 // Global Synchronize #define ADC_PSSI_SYNCWAIT 0x08000000 // Synchronize Wait #define ADC_PSSI_SS3 0x00000008 // SS3 Initiate #define ADC_PSSI_SS2 0x00000004 // SS2 Initiate #define ADC_PSSI_SS1 0x00000002 // SS1 Initiate #define ADC_PSSI_SS0 0x00000001 // SS0 Initiate //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SAC register. // //************************************************************************* #define ADC_SAC_AVG_M 0x00000007 // Hardware Averaging Control #define ADC_SAC_AVG_OFF 0x00000000 // No hardware oversampling #define ADC_SAC_AVG_2X 0x00000001 // 2x hardware oversampling #define ADC_SAC_AVG_4X 0x00000002 // 4x hardware oversampling #define ADC_SAC_AVG_8X 0x00000003 // 8x hardware oversampling #define ADC_SAC_AVG_16X 0x00000004 // 16x hardware oversampling #define ADC_SAC_AVG_32X 0x00000005 // 32x hardware oversampling #define ADC_SAC_AVG_64X 0x00000006 // 64x hardware oversampling //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCISC register. // //************************************************************************* #define ADC_DCISC_DCINT7 0x00000080 // Digital Comparator 7 Interrupt // Status and Clear #define ADC_DCISC_DCINT6 0x00000040 // Digital Comparator 6 Interrupt // Status and Clear #define ADC_DCISC_DCINT5 0x00000020 // Digital Comparator 5 Interrupt // Status and Clear #define ADC_DCISC_DCINT4 0x00000010 // Digital Comparator 4 Interrupt // Status and Clear #define ADC_DCISC_DCINT3 0x00000008 // Digital Comparator 3 Interrupt // Status and Clear #define ADC_DCISC_DCINT2 0x00000004 // Digital Comparator 2 Interrupt // Status and Clear #define ADC_DCISC_DCINT1 0x00000002 // Digital Comparator 1 Interrupt // Status and Clear #define ADC_DCISC_DCINT0 0x00000001 // Digital Comparator 0 Interrupt // Status and Clear //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSMUX0 register. // //************************************************************************* #define ADC_SSMUX0_MUX7_M 0xF0000000 // 8th Sample Input Select #define ADC_SSMUX0_MUX6_M 0x0F000000 // 7th Sample Input Select #define ADC_SSMUX0_MUX5_M 0x00F00000 // 6th Sample Input Select #define ADC_SSMUX0_MUX4_M 0x000F0000 // 5th Sample Input Select #define ADC_SSMUX0_MUX3_M 0x0000F000 // 4th Sample Input Select #define ADC_SSMUX0_MUX2_M 0x00000F00 // 3rd Sample Input Select #define ADC_SSMUX0_MUX1_M 0x000000F0 // 2nd Sample Input Select #define ADC_SSMUX0_MUX0_M 0x0000000F // 1st Sample Input Select #define ADC_SSMUX0_MUX7_S 28 #define ADC_SSMUX0_MUX6_S 24 #define ADC_SSMUX0_MUX5_S 20 #define ADC_SSMUX0_MUX4_S 16 #define ADC_SSMUX0_MUX3_S 12 #define ADC_SSMUX0_MUX2_S 8 #define ADC_SSMUX0_MUX1_S 4 #define ADC_SSMUX0_MUX0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSCTL0 register. // //************************************************************************* #define ADC_SSCTL0_TS7 0x80000000 // 8th Sample Temp Sensor Select #define ADC_SSCTL0_IE7 0x40000000 // 8th Sample Interrupt Enable #define ADC_SSCTL0_END7 0x20000000 // 8th Sample is End of Sequence #define ADC_SSCTL0_D7 0x10000000 // 8th Sample Diff Input Select #define ADC_SSCTL0_TS6 0x08000000 // 7th Sample Temp Sensor Select #define ADC_SSCTL0_IE6 0x04000000 // 7th Sample Interrupt Enable #define ADC_SSCTL0_END6 0x02000000 // 7th Sample is End of Sequence #define ADC_SSCTL0_D6 0x01000000 // 7th Sample Diff Input Select #define ADC_SSCTL0_TS5 0x00800000 // 6th Sample Temp Sensor Select #define ADC_SSCTL0_IE5 0x00400000 // 6th Sample Interrupt Enable #define ADC_SSCTL0_END5 0x00200000 // 6th Sample is End of Sequence #define ADC_SSCTL0_D5 0x00100000 // 6th Sample Diff Input Select #define ADC_SSCTL0_TS4 0x00080000 // 5th Sample Temp Sensor Select #define ADC_SSCTL0_IE4 0x00040000 // 5th Sample Interrupt Enable #define ADC_SSCTL0_END4 0x00020000 // 5th Sample is End of Sequence #define ADC_SSCTL0_D4 0x00010000 // 5th Sample Diff Input Select #define ADC_SSCTL0_TS3 0x00008000 // 4th Sample Temp Sensor Select #define ADC_SSCTL0_IE3 0x00004000 // 4th Sample Interrupt Enable #define ADC_SSCTL0_END3 0x00002000 // 4th Sample is End of Sequence #define ADC_SSCTL0_D3 0x00001000 // 4th Sample Diff Input Select #define ADC_SSCTL0_TS2 0x00000800 // 3rd Sample Temp Sensor Select #define ADC_SSCTL0_IE2 0x00000400 // 3rd Sample Interrupt Enable #define ADC_SSCTL0_END2 0x00000200 // 3rd Sample is End of Sequence #define ADC_SSCTL0_D2 0x00000100 // 3rd Sample Diff Input Select #define ADC_SSCTL0_TS1 0x00000080 // 2nd Sample Temp Sensor Select #define ADC_SSCTL0_IE1 0x00000040 // 2nd Sample Interrupt Enable #define ADC_SSCTL0_END1 0x00000020 // 2nd Sample is End of Sequence #define ADC_SSCTL0_D1 0x00000010 // 2nd Sample Diff Input Select #define ADC_SSCTL0_TS0 0x00000008 // 1st Sample Temp Sensor Select #define ADC_SSCTL0_IE0 0x00000004 // 1st Sample Interrupt Enable #define ADC_SSCTL0_END0 0x00000002 // 1st Sample is End of Sequence #define ADC_SSCTL0_D0 0x00000001 // 1st Sample Diff Input Select //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFIFO0 register. // //************************************************************************* #define ADC_SSFIFO0_DATA_M 0x00000FFF // Conversion Result Data #define ADC_SSFIFO0_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFSTAT0 register. // //************************************************************************* #define ADC_SSFSTAT0_FULL 0x00001000 // FIFO Full #define ADC_SSFSTAT0_EMPTY 0x00000100 // FIFO Empty #define ADC_SSFSTAT0_HPTR_M 0x000000F0 // FIFO Head Pointer #define ADC_SSFSTAT0_TPTR_M 0x0000000F // FIFO Tail Pointer #define ADC_SSFSTAT0_HPTR_S 4 #define ADC_SSFSTAT0_TPTR_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSOP0 register. // //************************************************************************* #define ADC_SSOP0_S7DCOP 0x10000000 // Sample 7 Digital Comparator // Operation #define ADC_SSOP0_S6DCOP 0x01000000 // Sample 6 Digital Comparator // Operation #define ADC_SSOP0_S5DCOP 0x00100000 // Sample 5 Digital Comparator // Operation #define ADC_SSOP0_S4DCOP 0x00010000 // Sample 4 Digital Comparator // Operation #define ADC_SSOP0_S3DCOP 0x00001000 // Sample 3 Digital Comparator // Operation #define ADC_SSOP0_S2DCOP 0x00000100 // Sample 2 Digital Comparator // Operation #define ADC_SSOP0_S1DCOP 0x00000010 // Sample 1 Digital Comparator // Operation #define ADC_SSOP0_S0DCOP 0x00000001 // Sample 0 Digital Comparator // Operation //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSDC0 register. // //************************************************************************* #define ADC_SSDC0_S7DCSEL_M 0xF0000000 // Sample 7 Digital Comparator // Select #define ADC_SSDC0_S6DCSEL_M 0x0F000000 // Sample 6 Digital Comparator // Select #define ADC_SSDC0_S5DCSEL_M 0x00F00000 // Sample 5 Digital Comparator // Select #define ADC_SSDC0_S4DCSEL_M 0x000F0000 // Sample 4 Digital Comparator // Select #define ADC_SSDC0_S3DCSEL_M 0x0000F000 // Sample 3 Digital Comparator // Select #define ADC_SSDC0_S2DCSEL_M 0x00000F00 // Sample 2 Digital Comparator // Select #define ADC_SSDC0_S1DCSEL_M 0x000000F0 // Sample 1 Digital Comparator // Select #define ADC_SSDC0_S0DCSEL_M 0x0000000F // Sample 0 Digital Comparator // Select #define ADC_SSDC0_S6DCSEL_S 24 #define ADC_SSDC0_S5DCSEL_S 20 #define ADC_SSDC0_S4DCSEL_S 16 #define ADC_SSDC0_S3DCSEL_S 12 #define ADC_SSDC0_S2DCSEL_S 8 #define ADC_SSDC0_S1DCSEL_S 4 #define ADC_SSDC0_S0DCSEL_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSMUX1 register. // //************************************************************************* #define ADC_SSMUX1_MUX3_M 0x0000F000 // 4th Sample Input Select #define ADC_SSMUX1_MUX2_M 0x00000F00 // 3rd Sample Input Select #define ADC_SSMUX1_MUX1_M 0x000000F0 // 2nd Sample Input Select #define ADC_SSMUX1_MUX0_M 0x0000000F // 1st Sample Input Select #define ADC_SSMUX1_MUX3_S 12 #define ADC_SSMUX1_MUX2_S 8 #define ADC_SSMUX1_MUX1_S 4 #define ADC_SSMUX1_MUX0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSCTL1 register. // //************************************************************************* #define ADC_SSCTL1_TS3 0x00008000 // 4th Sample Temp Sensor Select #define ADC_SSCTL1_IE3 0x00004000 // 4th Sample Interrupt Enable #define ADC_SSCTL1_END3 0x00002000 // 4th Sample is End of Sequence #define ADC_SSCTL1_D3 0x00001000 // 4th Sample Diff Input Select #define ADC_SSCTL1_TS2 0x00000800 // 3rd Sample Temp Sensor Select #define ADC_SSCTL1_IE2 0x00000400 // 3rd Sample Interrupt Enable #define ADC_SSCTL1_END2 0x00000200 // 3rd Sample is End of Sequence #define ADC_SSCTL1_D2 0x00000100 // 3rd Sample Diff Input Select #define ADC_SSCTL1_TS1 0x00000080 // 2nd Sample Temp Sensor Select #define ADC_SSCTL1_IE1 0x00000040 // 2nd Sample Interrupt Enable #define ADC_SSCTL1_END1 0x00000020 // 2nd Sample is End of Sequence #define ADC_SSCTL1_D1 0x00000010 // 2nd Sample Diff Input Select #define ADC_SSCTL1_TS0 0x00000008 // 1st Sample Temp Sensor Select #define ADC_SSCTL1_IE0 0x00000004 // 1st Sample Interrupt Enable #define ADC_SSCTL1_END0 0x00000002 // 1st Sample is End of Sequence #define ADC_SSCTL1_D0 0x00000001 // 1st Sample Diff Input Select //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFIFO1 register. // //************************************************************************* #define ADC_SSFIFO1_DATA_M 0x00000FFF // Conversion Result Data #define ADC_SSFIFO1_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFSTAT1 register. // //************************************************************************* #define ADC_SSFSTAT1_FULL 0x00001000 // FIFO Full #define ADC_SSFSTAT1_EMPTY 0x00000100 // FIFO Empty #define ADC_SSFSTAT1_HPTR_M 0x000000F0 // FIFO Head Pointer #define ADC_SSFSTAT1_TPTR_M 0x0000000F // FIFO Tail Pointer #define ADC_SSFSTAT1_HPTR_S 4 #define ADC_SSFSTAT1_TPTR_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSOP1 register. // //************************************************************************* #define ADC_SSOP1_S3DCOP 0x00001000 // Sample 3 Digital Comparator // Operation #define ADC_SSOP1_S2DCOP 0x00000100 // Sample 2 Digital Comparator // Operation #define ADC_SSOP1_S1DCOP 0x00000010 // Sample 1 Digital Comparator // Operation #define ADC_SSOP1_S0DCOP 0x00000001 // Sample 0 Digital Comparator // Operation //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSDC1 register. // //************************************************************************* #define ADC_SSDC1_S3DCSEL_M 0x0000F000 // Sample 3 Digital Comparator // Select #define ADC_SSDC1_S2DCSEL_M 0x00000F00 // Sample 2 Digital Comparator // Select #define ADC_SSDC1_S1DCSEL_M 0x000000F0 // Sample 1 Digital Comparator // Select #define ADC_SSDC1_S0DCSEL_M 0x0000000F // Sample 0 Digital Comparator // Select #define ADC_SSDC1_S2DCSEL_S 8 #define ADC_SSDC1_S1DCSEL_S 4 #define ADC_SSDC1_S0DCSEL_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSMUX2 register. // //************************************************************************* #define ADC_SSMUX2_MUX3_M 0x0000F000 // 4th Sample Input Select #define ADC_SSMUX2_MUX2_M 0x00000F00 // 3rd Sample Input Select #define ADC_SSMUX2_MUX1_M 0x000000F0 // 2nd Sample Input Select #define ADC_SSMUX2_MUX0_M 0x0000000F // 1st Sample Input Select #define ADC_SSMUX2_MUX3_S 12 #define ADC_SSMUX2_MUX2_S 8 #define ADC_SSMUX2_MUX1_S 4 #define ADC_SSMUX2_MUX0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSCTL2 register. // //************************************************************************* #define ADC_SSCTL2_TS3 0x00008000 // 4th Sample Temp Sensor Select #define ADC_SSCTL2_IE3 0x00004000 // 4th Sample Interrupt Enable #define ADC_SSCTL2_END3 0x00002000 // 4th Sample is End of Sequence #define ADC_SSCTL2_D3 0x00001000 // 4th Sample Diff Input Select #define ADC_SSCTL2_TS2 0x00000800 // 3rd Sample Temp Sensor Select #define ADC_SSCTL2_IE2 0x00000400 // 3rd Sample Interrupt Enable #define ADC_SSCTL2_END2 0x00000200 // 3rd Sample is End of Sequence #define ADC_SSCTL2_D2 0x00000100 // 3rd Sample Diff Input Select #define ADC_SSCTL2_TS1 0x00000080 // 2nd Sample Temp Sensor Select #define ADC_SSCTL2_IE1 0x00000040 // 2nd Sample Interrupt Enable #define ADC_SSCTL2_END1 0x00000020 // 2nd Sample is End of Sequence #define ADC_SSCTL2_D1 0x00000010 // 2nd Sample Diff Input Select #define ADC_SSCTL2_TS0 0x00000008 // 1st Sample Temp Sensor Select #define ADC_SSCTL2_IE0 0x00000004 // 1st Sample Interrupt Enable #define ADC_SSCTL2_END0 0x00000002 // 1st Sample is End of Sequence #define ADC_SSCTL2_D0 0x00000001 // 1st Sample Diff Input Select //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFIFO2 register. // //************************************************************************* #define ADC_SSFIFO2_DATA_M 0x00000FFF // Conversion Result Data #define ADC_SSFIFO2_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFSTAT2 register. // //************************************************************************* #define ADC_SSFSTAT2_FULL 0x00001000 // FIFO Full #define ADC_SSFSTAT2_EMPTY 0x00000100 // FIFO Empty #define ADC_SSFSTAT2_HPTR_M 0x000000F0 // FIFO Head Pointer #define ADC_SSFSTAT2_TPTR_M 0x0000000F // FIFO Tail Pointer #define ADC_SSFSTAT2_HPTR_S 4 #define ADC_SSFSTAT2_TPTR_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSOP2 register. // //************************************************************************* #define ADC_SSOP2_S3DCOP 0x00001000 // Sample 3 Digital Comparator // Operation #define ADC_SSOP2_S2DCOP 0x00000100 // Sample 2 Digital Comparator // Operation #define ADC_SSOP2_S1DCOP 0x00000010 // Sample 1 Digital Comparator // Operation #define ADC_SSOP2_S0DCOP 0x00000001 // Sample 0 Digital Comparator // Operation //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSDC2 register. // //************************************************************************* #define ADC_SSDC2_S3DCSEL_M 0x0000F000 // Sample 3 Digital Comparator // Select #define ADC_SSDC2_S2DCSEL_M 0x00000F00 // Sample 2 Digital Comparator // Select #define ADC_SSDC2_S1DCSEL_M 0x000000F0 // Sample 1 Digital Comparator // Select #define ADC_SSDC2_S0DCSEL_M 0x0000000F // Sample 0 Digital Comparator // Select #define ADC_SSDC2_S2DCSEL_S 8 #define ADC_SSDC2_S1DCSEL_S 4 #define ADC_SSDC2_S0DCSEL_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSMUX3 register. // //************************************************************************* #define ADC_SSMUX3_MUX0_M 0x0000000F // 1st Sample Input Select #define ADC_SSMUX3_MUX0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSCTL3 register. // //************************************************************************* #define ADC_SSCTL3_TS0 0x00000008 // 1st Sample Temp Sensor Select #define ADC_SSCTL3_IE0 0x00000004 // 1st Sample Interrupt Enable #define ADC_SSCTL3_END0 0x00000002 // 1st Sample is End of Sequence #define ADC_SSCTL3_D0 0x00000001 // 1st Sample Diff Input Select //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFIFO3 register. // //************************************************************************* #define ADC_SSFIFO3_DATA_M 0x00000FFF // Conversion Result Data #define ADC_SSFIFO3_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSFSTAT3 register. // //************************************************************************* #define ADC_SSFSTAT3_FULL 0x00001000 // FIFO Full #define ADC_SSFSTAT3_EMPTY 0x00000100 // FIFO Empty #define ADC_SSFSTAT3_HPTR_M 0x000000F0 // FIFO Head Pointer #define ADC_SSFSTAT3_TPTR_M 0x0000000F // FIFO Tail Pointer #define ADC_SSFSTAT3_HPTR_S 4 #define ADC_SSFSTAT3_TPTR_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSOP3 register. // //************************************************************************* #define ADC_SSOP3_S0DCOP 0x00000001 // Sample 0 Digital Comparator // Operation //************************************************************************* // // The following are defines for the bit fields in the ADC_O_SSDC3 register. // //************************************************************************* #define ADC_SSDC3_S0DCSEL_M 0x0000000F // Sample 0 Digital Comparator // Select //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCRIC register. // //************************************************************************* #define ADC_DCRIC_DCTRIG7 0x00800000 // Digital Comparator Trigger 7 #define ADC_DCRIC_DCTRIG6 0x00400000 // Digital Comparator Trigger 6 #define ADC_DCRIC_DCTRIG5 0x00200000 // Digital Comparator Trigger 5 #define ADC_DCRIC_DCTRIG4 0x00100000 // Digital Comparator Trigger 4 #define ADC_DCRIC_DCTRIG3 0x00080000 // Digital Comparator Trigger 3 #define ADC_DCRIC_DCTRIG2 0x00040000 // Digital Comparator Trigger 2 #define ADC_DCRIC_DCTRIG1 0x00020000 // Digital Comparator Trigger 1 #define ADC_DCRIC_DCTRIG0 0x00010000 // Digital Comparator Trigger 0 #define ADC_DCRIC_DCINT7 0x00000080 // Digital Comparator Interrupt 7 #define ADC_DCRIC_DCINT6 0x00000040 // Digital Comparator Interrupt 6 #define ADC_DCRIC_DCINT5 0x00000020 // Digital Comparator Interrupt 5 #define ADC_DCRIC_DCINT4 0x00000010 // Digital Comparator Interrupt 4 #define ADC_DCRIC_DCINT3 0x00000008 // Digital Comparator Interrupt 3 #define ADC_DCRIC_DCINT2 0x00000004 // Digital Comparator Interrupt 2 #define ADC_DCRIC_DCINT1 0x00000002 // Digital Comparator Interrupt 1 #define ADC_DCRIC_DCINT0 0x00000001 // Digital Comparator Interrupt 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL0 register. // //************************************************************************* #define ADC_DCCTL0_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL0_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL0_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL0_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL0_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL0_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL0_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL0_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL0_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL0_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL1 register. // //************************************************************************* #define ADC_DCCTL1_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL1_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL1_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL1_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL1_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL1_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL1_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL1_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL1_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL1_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL2 register. // //************************************************************************* #define ADC_DCCTL2_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL2_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL2_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL2_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL2_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL2_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL2_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL2_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL2_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL2_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL3 register. // //************************************************************************* #define ADC_DCCTL3_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL3_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL3_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL3_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL3_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL3_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL3_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL3_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL3_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL3_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL4 register. // //************************************************************************* #define ADC_DCCTL4_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL4_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL4_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL4_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL4_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL4_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL4_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL4_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL4_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL4_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL5 register. // //************************************************************************* #define ADC_DCCTL5_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL5_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL5_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL5_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL5_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL5_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL5_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL5_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL5_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL5_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL6 register. // //************************************************************************* #define ADC_DCCTL6_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL6_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL6_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL6_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL6_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL6_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL6_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL6_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL6_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL6_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCTL7 register. // //************************************************************************* #define ADC_DCCTL7_CIE 0x00000010 // Comparison Interrupt Enable #define ADC_DCCTL7_CIC_M 0x0000000C // Comparison Interrupt Condition #define ADC_DCCTL7_CIC_LOW 0x00000000 // Low Band #define ADC_DCCTL7_CIC_MID 0x00000004 // Mid Band #define ADC_DCCTL7_CIC_HIGH 0x0000000C // High Band #define ADC_DCCTL7_CIM_M 0x00000003 // Comparison Interrupt Mode #define ADC_DCCTL7_CIM_ALWAYS 0x00000000 // Always #define ADC_DCCTL7_CIM_ONCE 0x00000001 // Once #define ADC_DCCTL7_CIM_HALWAYS 0x00000002 // Hysteresis Always #define ADC_DCCTL7_CIM_HONCE 0x00000003 // Hysteresis Once //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP0 register. // //************************************************************************* #define ADC_DCCMP0_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP0_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP0_COMP1_S 16 #define ADC_DCCMP0_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP1 register. // //************************************************************************* #define ADC_DCCMP1_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP1_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP1_COMP1_S 16 #define ADC_DCCMP1_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP2 register. // //************************************************************************* #define ADC_DCCMP2_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP2_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP2_COMP1_S 16 #define ADC_DCCMP2_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP3 register. // //************************************************************************* #define ADC_DCCMP3_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP3_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP3_COMP1_S 16 #define ADC_DCCMP3_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP4 register. // //************************************************************************* #define ADC_DCCMP4_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP4_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP4_COMP1_S 16 #define ADC_DCCMP4_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP5 register. // //************************************************************************* #define ADC_DCCMP5_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP5_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP5_COMP1_S 16 #define ADC_DCCMP5_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP6 register. // //************************************************************************* #define ADC_DCCMP6_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP6_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP6_COMP1_S 16 #define ADC_DCCMP6_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_DCCMP7 register. // //************************************************************************* #define ADC_DCCMP7_COMP1_M 0x0FFF0000 // Compare 1 #define ADC_DCCMP7_COMP0_M 0x00000FFF // Compare 0 #define ADC_DCCMP7_COMP1_S 16 #define ADC_DCCMP7_COMP0_S 0 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_PP register. // //************************************************************************* #define ADC_PP_TS 0x00800000 // Temperature Sensor #define ADC_PP_RSL_M 0x007C0000 // Resolution #define ADC_PP_TYPE_M 0x00030000 // ADC Architecture #define ADC_PP_TYPE_SAR 0x00000000 // SAR #define ADC_PP_DC_M 0x0000FC00 // Digital Comparator Count #define ADC_PP_CH_M 0x000003F0 // ADC Channel Count #define ADC_PP_MSR_M 0x0000000F // Maximum ADC Sample Rate #define ADC_PP_MSR_125K 0x00000001 // 125 ksps #define ADC_PP_MSR_250K 0x00000003 // 250 ksps #define ADC_PP_MSR_500K 0x00000005 // 500 ksps #define ADC_PP_MSR_1M 0x00000007 // 1 Msps #define ADC_PP_RSL_S 18 #define ADC_PP_DC_S 10 #define ADC_PP_CH_S 4 //************************************************************************* // // The following are defines for the bit fields in the ADC_O_PC register. // //************************************************************************* #define ADC_PC_SR_M 0x0000000F // ADC Sample Rate #define ADC_PC_SR_125K 0x00000001 // 125 ksps #define ADC_PC_SR_250K 0x00000003 // 250 ksps #define ADC_PC_SR_500K 0x00000005 // 500 ksps #define ADC_PC_SR_1M 0x00000007 // 1 Msps //************************************************************************* // // The following are defines for the bit fields in the ADC_O_CC register. // //************************************************************************* #define ADC_CC_CS_M 0x0000000F // ADC Clock Source #define ADC_CC_CS_SYSPLL 0x00000000 // Either the system clock (if the // PLL bypass is in effect) or the // 16 MHz clock derived from PLL / // 25 (default) #define ADC_CC_CS_PIOSC 0x00000001 // PIOSC //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACMIS register. // //************************************************************************* #define COMP_ACMIS_IN1 0x00000002 // Comparator 1 Masked Interrupt // Status #define COMP_ACMIS_IN0 0x00000001 // Comparator 0 Masked Interrupt // Status //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACRIS register. // //************************************************************************* #define COMP_ACRIS_IN1 0x00000002 // Comparator 1 Interrupt Status #define COMP_ACRIS_IN0 0x00000001 // Comparator 0 Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACINTEN register. // //************************************************************************* #define COMP_ACINTEN_IN1 0x00000002 // Comparator 1 Interrupt Enable #define COMP_ACINTEN_IN0 0x00000001 // Comparator 0 Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACREFCTL // register. // //************************************************************************* #define COMP_ACREFCTL_EN 0x00000200 // Resistor Ladder Enable #define COMP_ACREFCTL_RNG 0x00000100 // Resistor Ladder Range #define COMP_ACREFCTL_VREF_M 0x0000000F // Resistor Ladder Voltage Ref #define COMP_ACREFCTL_VREF_S 0 //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACSTAT0 register. // //************************************************************************* #define COMP_ACSTAT0_OVAL 0x00000002 // Comparator Output Value //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACCTL0 register. // //************************************************************************* #define COMP_ACCTL0_TOEN 0x00000800 // Trigger Output Enable #define COMP_ACCTL0_ASRCP_M 0x00000600 // Analog Source Positive #define COMP_ACCTL0_ASRCP_PIN 0x00000000 // Pin value of Cn+ #define COMP_ACCTL0_ASRCP_PIN0 0x00000200 // Pin value of C0+ #define COMP_ACCTL0_ASRCP_REF 0x00000400 // Internal voltage reference #define COMP_ACCTL0_TSLVAL 0x00000080 // Trigger Sense Level Value #define COMP_ACCTL0_TSEN_M 0x00000060 // Trigger Sense #define COMP_ACCTL0_TSEN_LEVEL 0x00000000 // Level sense, see TSLVAL #define COMP_ACCTL0_TSEN_FALL 0x00000020 // Falling edge #define COMP_ACCTL0_TSEN_RISE 0x00000040 // Rising edge #define COMP_ACCTL0_TSEN_BOTH 0x00000060 // Either edge #define COMP_ACCTL0_ISLVAL 0x00000010 // Interrupt Sense Level Value #define COMP_ACCTL0_ISEN_M 0x0000000C // Interrupt Sense #define COMP_ACCTL0_ISEN_LEVEL 0x00000000 // Level sense, see ISLVAL #define COMP_ACCTL0_ISEN_FALL 0x00000004 // Falling edge #define COMP_ACCTL0_ISEN_RISE 0x00000008 // Rising edge #define COMP_ACCTL0_ISEN_BOTH 0x0000000C // Either edge #define COMP_ACCTL0_CINV 0x00000002 // Comparator Output Invert //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACSTAT1 register. // //************************************************************************* #define COMP_ACSTAT1_OVAL 0x00000002 // Comparator Output Value //************************************************************************* // // The following are defines for the bit fields in the COMP_O_ACCTL1 register. // //************************************************************************* #define COMP_ACCTL1_TOEN 0x00000800 // Trigger Output Enable #define COMP_ACCTL1_ASRCP_M 0x00000600 // Analog Source Positive #define COMP_ACCTL1_ASRCP_PIN 0x00000000 // Pin value of Cn+ #define COMP_ACCTL1_ASRCP_PIN0 0x00000200 // Pin value of C0+ #define COMP_ACCTL1_ASRCP_REF 0x00000400 // Internal voltage reference // (VIREF) #define COMP_ACCTL1_TSLVAL 0x00000080 // Trigger Sense Level Value #define COMP_ACCTL1_TSEN_M 0x00000060 // Trigger Sense #define COMP_ACCTL1_TSEN_LEVEL 0x00000000 // Level sense, see TSLVAL #define COMP_ACCTL1_TSEN_FALL 0x00000020 // Falling edge #define COMP_ACCTL1_TSEN_RISE 0x00000040 // Rising edge #define COMP_ACCTL1_TSEN_BOTH 0x00000060 // Either edge #define COMP_ACCTL1_ISLVAL 0x00000010 // Interrupt Sense Level Value #define COMP_ACCTL1_ISEN_M 0x0000000C // Interrupt Sense #define COMP_ACCTL1_ISEN_LEVEL 0x00000000 // Level sense, see ISLVAL #define COMP_ACCTL1_ISEN_FALL 0x00000004 // Falling edge #define COMP_ACCTL1_ISEN_RISE 0x00000008 // Rising edge #define COMP_ACCTL1_ISEN_BOTH 0x0000000C // Either edge #define COMP_ACCTL1_CINV 0x00000002 // Comparator Output Invert //************************************************************************* // // The following are defines for the bit fields in the COMP_O_PP register. // //************************************************************************* #define COMP_PP_C2O 0x00040000 // Comparator Output 2 Present #define COMP_PP_C1O 0x00020000 // Comparator Output 1 Present #define COMP_PP_C0O 0x00010000 // Comparator Output 0 Present #define COMP_PP_CMP2 0x00000004 // Comparator 2 Present #define COMP_PP_CMP1 0x00000002 // Comparator 1 Present #define COMP_PP_CMP0 0x00000001 // Comparator 0 Present //************************************************************************* // // The following are defines for the bit fields in the CAN_O_CTL register. // //************************************************************************* #define CAN_CTL_TEST 0x00000080 // Test Mode Enable #define CAN_CTL_CCE 0x00000040 // Configuration Change Enable #define CAN_CTL_DAR 0x00000020 // Disable Automatic-Retransmission #define CAN_CTL_EIE 0x00000008 // Error Interrupt Enable #define CAN_CTL_SIE 0x00000004 // Status Interrupt Enable #define CAN_CTL_IE 0x00000002 // CAN Interrupt Enable #define CAN_CTL_INIT 0x00000001 // Initialization //************************************************************************* // // The following are defines for the bit fields in the CAN_O_STS register. // //************************************************************************* #define CAN_STS_BOFF 0x00000080 // Bus-Off Status #define CAN_STS_EWARN 0x00000040 // Warning Status #define CAN_STS_EPASS 0x00000020 // Error Passive #define CAN_STS_RXOK 0x00000010 // Received a Message Successfully #define CAN_STS_TXOK 0x00000008 // Transmitted a Message // Successfully #define CAN_STS_LEC_M 0x00000007 // Last Error Code #define CAN_STS_LEC_NONE 0x00000000 // No Error #define CAN_STS_LEC_STUFF 0x00000001 // Stuff Error #define CAN_STS_LEC_FORM 0x00000002 // Format Error #define CAN_STS_LEC_ACK 0x00000003 // ACK Error #define CAN_STS_LEC_BIT1 0x00000004 // Bit 1 Error #define CAN_STS_LEC_BIT0 0x00000005 // Bit 0 Error #define CAN_STS_LEC_CRC 0x00000006 // CRC Error #define CAN_STS_LEC_NOEVENT 0x00000007 // No Event //************************************************************************* // // The following are defines for the bit fields in the CAN_O_ERR register. // //************************************************************************* #define CAN_ERR_RP 0x00008000 // Received Error Passive #define CAN_ERR_REC_M 0x00007F00 // Receive Error Counter #define CAN_ERR_TEC_M 0x000000FF // Transmit Error Counter #define CAN_ERR_REC_S 8 #define CAN_ERR_TEC_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_BIT register. // //************************************************************************* #define CAN_BIT_TSEG2_M 0x00007000 // Time Segment after Sample Point #define CAN_BIT_TSEG1_M 0x00000F00 // Time Segment Before Sample Point #define CAN_BIT_SJW_M 0x000000C0 // (Re)Synchronization Jump Width #define CAN_BIT_BRP_M 0x0000003F // Baud Rate Prescaler #define CAN_BIT_TSEG2_S 12 #define CAN_BIT_TSEG1_S 8 #define CAN_BIT_SJW_S 6 #define CAN_BIT_BRP_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_INT register. // //************************************************************************* #define CAN_INT_INTID_M 0x0000FFFF // Interrupt Identifier #define CAN_INT_INTID_NONE 0x00000000 // No interrupt pending #define CAN_INT_INTID_STATUS 0x00008000 // Status Interrupt //************************************************************************* // // The following are defines for the bit fields in the CAN_O_TST register. // //************************************************************************* #define CAN_TST_RX 0x00000080 // Receive Observation #define CAN_TST_TX_M 0x00000060 // Transmit Control #define CAN_TST_TX_CANCTL 0x00000000 // CAN Module Control #define CAN_TST_TX_SAMPLE 0x00000020 // Sample Point #define CAN_TST_TX_DOMINANT 0x00000040 // Driven Low #define CAN_TST_TX_RECESSIVE 0x00000060 // Driven High #define CAN_TST_LBACK 0x00000010 // Loopback Mode #define CAN_TST_SILENT 0x00000008 // Silent Mode #define CAN_TST_BASIC 0x00000004 // Basic Mode //************************************************************************* // // The following are defines for the bit fields in the CAN_O_BRPE register. // //************************************************************************* #define CAN_BRPE_BRPE_M 0x0000000F // Baud Rate Prescaler Extension #define CAN_BRPE_BRPE_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1CRQ register. // //************************************************************************* #define CAN_IF1CRQ_BUSY 0x00008000 // Busy Flag #define CAN_IF1CRQ_MNUM_M 0x0000003F // Message Number #define CAN_IF1CRQ_MNUM_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1CMSK register. // //************************************************************************* #define CAN_IF1CMSK_WRNRD 0x00000080 // Write, Not Read #define CAN_IF1CMSK_MASK 0x00000040 // Access Mask Bits #define CAN_IF1CMSK_ARB 0x00000020 // Access Arbitration Bits #define CAN_IF1CMSK_CONTROL 0x00000010 // Access Control Bits #define CAN_IF1CMSK_CLRINTPND 0x00000008 // Clear Interrupt Pending Bit #define CAN_IF1CMSK_NEWDAT 0x00000004 // Access New Data #define CAN_IF1CMSK_TXRQST 0x00000004 // Access Transmission Request #define CAN_IF1CMSK_DATAA 0x00000002 // Access Data Byte 0 to 3 #define CAN_IF1CMSK_DATAB 0x00000001 // Access Data Byte 4 to 7 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1MSK1 register. // //************************************************************************* #define CAN_IF1MSK1_IDMSK_M 0x0000FFFF // Identifier Mask #define CAN_IF1MSK1_IDMSK_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1MSK2 register. // //************************************************************************* #define CAN_IF1MSK2_MXTD 0x00008000 // Mask Extended Identifier #define CAN_IF1MSK2_MDIR 0x00004000 // Mask Message Direction #define CAN_IF1MSK2_IDMSK_M 0x00001FFF // Identifier Mask #define CAN_IF1MSK2_IDMSK_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1ARB1 register. // //************************************************************************* #define CAN_IF1ARB1_ID_M 0x0000FFFF // Message Identifier #define CAN_IF1ARB1_ID_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1ARB2 register. // //************************************************************************* #define CAN_IF1ARB2_MSGVAL 0x00008000 // Message Valid #define CAN_IF1ARB2_XTD 0x00004000 // Extended Identifier #define CAN_IF1ARB2_DIR 0x00002000 // Message Direction #define CAN_IF1ARB2_ID_M 0x00001FFF // Message Identifier #define CAN_IF1ARB2_ID_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1MCTL register. // //************************************************************************* #define CAN_IF1MCTL_NEWDAT 0x00008000 // New Data #define CAN_IF1MCTL_MSGLST 0x00004000 // Message Lost #define CAN_IF1MCTL_INTPND 0x00002000 // Interrupt Pending #define CAN_IF1MCTL_UMASK 0x00001000 // Use Acceptance Mask #define CAN_IF1MCTL_TXIE 0x00000800 // Transmit Interrupt Enable #define CAN_IF1MCTL_RXIE 0x00000400 // Receive Interrupt Enable #define CAN_IF1MCTL_RMTEN 0x00000200 // Remote Enable #define CAN_IF1MCTL_TXRQST 0x00000100 // Transmit Request #define CAN_IF1MCTL_EOB 0x00000080 // End of Buffer #define CAN_IF1MCTL_DLC_M 0x0000000F // Data Length Code #define CAN_IF1MCTL_DLC_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1DA1 register. // //************************************************************************* #define CAN_IF1DA1_DATA_M 0x0000FFFF // Data #define CAN_IF1DA1_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1DA2 register. // //************************************************************************* #define CAN_IF1DA2_DATA_M 0x0000FFFF // Data #define CAN_IF1DA2_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1DB1 register. // //************************************************************************* #define CAN_IF1DB1_DATA_M 0x0000FFFF // Data #define CAN_IF1DB1_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF1DB2 register. // //************************************************************************* #define CAN_IF1DB2_DATA_M 0x0000FFFF // Data #define CAN_IF1DB2_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2CRQ register. // //************************************************************************* #define CAN_IF2CRQ_BUSY 0x00008000 // Busy Flag #define CAN_IF2CRQ_MNUM_M 0x0000003F // Message Number #define CAN_IF2CRQ_MNUM_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2CMSK register. // //************************************************************************* #define CAN_IF2CMSK_WRNRD 0x00000080 // Write, Not Read #define CAN_IF2CMSK_MASK 0x00000040 // Access Mask Bits #define CAN_IF2CMSK_ARB 0x00000020 // Access Arbitration Bits #define CAN_IF2CMSK_CONTROL 0x00000010 // Access Control Bits #define CAN_IF2CMSK_CLRINTPND 0x00000008 // Clear Interrupt Pending Bit #define CAN_IF2CMSK_NEWDAT 0x00000004 // Access New Data #define CAN_IF2CMSK_TXRQST 0x00000004 // Access Transmission Request #define CAN_IF2CMSK_DATAA 0x00000002 // Access Data Byte 0 to 3 #define CAN_IF2CMSK_DATAB 0x00000001 // Access Data Byte 4 to 7 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2MSK1 register. // //************************************************************************* #define CAN_IF2MSK1_IDMSK_M 0x0000FFFF // Identifier Mask #define CAN_IF2MSK1_IDMSK_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2MSK2 register. // //************************************************************************* #define CAN_IF2MSK2_MXTD 0x00008000 // Mask Extended Identifier #define CAN_IF2MSK2_MDIR 0x00004000 // Mask Message Direction #define CAN_IF2MSK2_IDMSK_M 0x00001FFF // Identifier Mask #define CAN_IF2MSK2_IDMSK_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2ARB1 register. // //************************************************************************* #define CAN_IF2ARB1_ID_M 0x0000FFFF // Message Identifier #define CAN_IF2ARB1_ID_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2ARB2 register. // //************************************************************************* #define CAN_IF2ARB2_MSGVAL 0x00008000 // Message Valid #define CAN_IF2ARB2_XTD 0x00004000 // Extended Identifier #define CAN_IF2ARB2_DIR 0x00002000 // Message Direction #define CAN_IF2ARB2_ID_M 0x00001FFF // Message Identifier #define CAN_IF2ARB2_ID_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2MCTL register. // //************************************************************************* #define CAN_IF2MCTL_NEWDAT 0x00008000 // New Data #define CAN_IF2MCTL_MSGLST 0x00004000 // Message Lost #define CAN_IF2MCTL_INTPND 0x00002000 // Interrupt Pending #define CAN_IF2MCTL_UMASK 0x00001000 // Use Acceptance Mask #define CAN_IF2MCTL_TXIE 0x00000800 // Transmit Interrupt Enable #define CAN_IF2MCTL_RXIE 0x00000400 // Receive Interrupt Enable #define CAN_IF2MCTL_RMTEN 0x00000200 // Remote Enable #define CAN_IF2MCTL_TXRQST 0x00000100 // Transmit Request #define CAN_IF2MCTL_EOB 0x00000080 // End of Buffer #define CAN_IF2MCTL_DLC_M 0x0000000F // Data Length Code #define CAN_IF2MCTL_DLC_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2DA1 register. // //************************************************************************* #define CAN_IF2DA1_DATA_M 0x0000FFFF // Data #define CAN_IF2DA1_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2DA2 register. // //************************************************************************* #define CAN_IF2DA2_DATA_M 0x0000FFFF // Data #define CAN_IF2DA2_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2DB1 register. // //************************************************************************* #define CAN_IF2DB1_DATA_M 0x0000FFFF // Data #define CAN_IF2DB1_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_IF2DB2 register. // //************************************************************************* #define CAN_IF2DB2_DATA_M 0x0000FFFF // Data #define CAN_IF2DB2_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_TXRQ1 register. // //************************************************************************* #define CAN_TXRQ1_TXRQST_M 0x0000FFFF // Transmission Request Bits #define CAN_TXRQ1_TXRQST_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_TXRQ2 register. // //************************************************************************* #define CAN_TXRQ2_TXRQST_M 0x0000FFFF // Transmission Request Bits #define CAN_TXRQ2_TXRQST_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_NWDA1 register. // //************************************************************************* #define CAN_NWDA1_NEWDAT_M 0x0000FFFF // New Data Bits #define CAN_NWDA1_NEWDAT_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_NWDA2 register. // //************************************************************************* #define CAN_NWDA2_NEWDAT_M 0x0000FFFF // New Data Bits #define CAN_NWDA2_NEWDAT_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_MSG1INT register. // //************************************************************************* #define CAN_MSG1INT_INTPND_M 0x0000FFFF // Interrupt Pending Bits #define CAN_MSG1INT_INTPND_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_MSG2INT register. // //************************************************************************* #define CAN_MSG2INT_INTPND_M 0x0000FFFF // Interrupt Pending Bits #define CAN_MSG2INT_INTPND_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_MSG1VAL register. // //************************************************************************* #define CAN_MSG1VAL_MSGVAL_M 0x0000FFFF // Message Valid Bits #define CAN_MSG1VAL_MSGVAL_S 0 //************************************************************************* // // The following are defines for the bit fields in the CAN_O_MSG2VAL register. // //************************************************************************* #define CAN_MSG2VAL_MSGVAL_M 0x0000FFFF // Message Valid Bits #define CAN_MSG2VAL_MSGVAL_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FADDR register. // //************************************************************************* #define USB_FADDR_M 0x0000007F // Function Address #define USB_FADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_POWER register. // //************************************************************************* #define USB_POWER_ISOUP 0x00000080 // Isochronous Update #define USB_POWER_SOFTCONN 0x00000040 // Soft Connect/Disconnect #define USB_POWER_RESET 0x00000008 // RESET Signaling #define USB_POWER_RESUME 0x00000004 // RESUME Signaling #define USB_POWER_SUSPEND 0x00000002 // SUSPEND Mode #define USB_POWER_PWRDNPHY 0x00000001 // Power Down PHY //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXIS register. // //************************************************************************* #define USB_TXIS_EP7 0x00000080 // TX Endpoint 7 Interrupt #define USB_TXIS_EP6 0x00000040 // TX Endpoint 6 Interrupt #define USB_TXIS_EP5 0x00000020 // TX Endpoint 5 Interrupt #define USB_TXIS_EP4 0x00000010 // TX Endpoint 4 Interrupt #define USB_TXIS_EP3 0x00000008 // TX Endpoint 3 Interrupt #define USB_TXIS_EP2 0x00000004 // TX Endpoint 2 Interrupt #define USB_TXIS_EP1 0x00000002 // TX Endpoint 1 Interrupt #define USB_TXIS_EP0 0x00000001 // TX and RX Endpoint 0 Interrupt //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXIS register. // //************************************************************************* #define USB_RXIS_EP7 0x00000080 // RX Endpoint 7 Interrupt #define USB_RXIS_EP6 0x00000040 // RX Endpoint 6 Interrupt #define USB_RXIS_EP5 0x00000020 // RX Endpoint 5 Interrupt #define USB_RXIS_EP4 0x00000010 // RX Endpoint 4 Interrupt #define USB_RXIS_EP3 0x00000008 // RX Endpoint 3 Interrupt #define USB_RXIS_EP2 0x00000004 // RX Endpoint 2 Interrupt #define USB_RXIS_EP1 0x00000002 // RX Endpoint 1 Interrupt //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXIE register. // //************************************************************************* #define USB_TXIE_EP7 0x00000080 // TX Endpoint 7 Interrupt Enable #define USB_TXIE_EP6 0x00000040 // TX Endpoint 6 Interrupt Enable #define USB_TXIE_EP5 0x00000020 // TX Endpoint 5 Interrupt Enable #define USB_TXIE_EP4 0x00000010 // TX Endpoint 4 Interrupt Enable #define USB_TXIE_EP3 0x00000008 // TX Endpoint 3 Interrupt Enable #define USB_TXIE_EP2 0x00000004 // TX Endpoint 2 Interrupt Enable #define USB_TXIE_EP1 0x00000002 // TX Endpoint 1 Interrupt Enable #define USB_TXIE_EP0 0x00000001 // TX and RX Endpoint 0 Interrupt // Enable //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXIE register. // //************************************************************************* #define USB_RXIE_EP7 0x00000080 // RX Endpoint 7 Interrupt Enable #define USB_RXIE_EP6 0x00000040 // RX Endpoint 6 Interrupt Enable #define USB_RXIE_EP5 0x00000020 // RX Endpoint 5 Interrupt Enable #define USB_RXIE_EP4 0x00000010 // RX Endpoint 4 Interrupt Enable #define USB_RXIE_EP3 0x00000008 // RX Endpoint 3 Interrupt Enable #define USB_RXIE_EP2 0x00000004 // RX Endpoint 2 Interrupt Enable #define USB_RXIE_EP1 0x00000002 // RX Endpoint 1 Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the USB_O_IS register. // //************************************************************************* #define USB_IS_DISCON 0x00000020 // Session Disconnect #define USB_IS_SOF 0x00000008 // Start of Frame #define USB_IS_RESET 0x00000004 // RESET Signaling Detected #define USB_IS_RESUME 0x00000002 // RESUME Signaling Detected #define USB_IS_SUSPEND 0x00000001 // SUSPEND Signaling Detected //************************************************************************* // // The following are defines for the bit fields in the USB_O_IE register. // //************************************************************************* #define USB_IE_DISCON 0x00000020 // Enable Disconnect Interrupt #define USB_IE_SOF 0x00000008 // Enable Start-of-Frame Interrupt #define USB_IE_RESET 0x00000004 // Enable RESET Interrupt #define USB_IE_RESUME 0x00000002 // Enable RESUME Interrupt #define USB_IE_SUSPND 0x00000001 // Enable SUSPEND Interrupt //************************************************************************* // // The following are defines for the bit fields in the USB_O_FRAME register. // //************************************************************************* #define USB_FRAME_M 0x000007FF // Frame Number #define USB_FRAME_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_EPIDX register. // //************************************************************************* #define USB_EPIDX_EPIDX_M 0x0000000F // Endpoint Index #define USB_EPIDX_EPIDX_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TEST register. // //************************************************************************* #define USB_TEST_FIFOACC 0x00000040 // FIFO Access #define USB_TEST_FORCEFS 0x00000020 // Force Full-Speed Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO0 register. // //************************************************************************* #define USB_FIFO0_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO0_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO1 register. // //************************************************************************* #define USB_FIFO1_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO1_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO2 register. // //************************************************************************* #define USB_FIFO2_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO2_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO3 register. // //************************************************************************* #define USB_FIFO3_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO3_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO4 register. // //************************************************************************* #define USB_FIFO4_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO4_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO5 register. // //************************************************************************* #define USB_FIFO5_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO5_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO6 register. // //************************************************************************* #define USB_FIFO6_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO6_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FIFO7 register. // //************************************************************************* #define USB_FIFO7_EPDATA_M 0xFFFFFFFF // Endpoint Data #define USB_FIFO7_EPDATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXFIFOSZ register. // //************************************************************************* #define USB_TXFIFOSZ_DPB 0x00000010 // Double Packet Buffer Support #define USB_TXFIFOSZ_SIZE_M 0x0000000F // Max Packet Size #define USB_TXFIFOSZ_SIZE_8 0x00000000 // 8 #define USB_TXFIFOSZ_SIZE_16 0x00000001 // 16 #define USB_TXFIFOSZ_SIZE_32 0x00000002 // 32 #define USB_TXFIFOSZ_SIZE_64 0x00000003 // 64 #define USB_TXFIFOSZ_SIZE_128 0x00000004 // 128 #define USB_TXFIFOSZ_SIZE_256 0x00000005 // 256 #define USB_TXFIFOSZ_SIZE_512 0x00000006 // 512 #define USB_TXFIFOSZ_SIZE_1024 0x00000007 // 1024 #define USB_TXFIFOSZ_SIZE_2048 0x00000008 // 2048 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXFIFOSZ register. // //************************************************************************* #define USB_RXFIFOSZ_DPB 0x00000010 // Double Packet Buffer Support #define USB_RXFIFOSZ_SIZE_M 0x0000000F // Max Packet Size #define USB_RXFIFOSZ_SIZE_8 0x00000000 // 8 #define USB_RXFIFOSZ_SIZE_16 0x00000001 // 16 #define USB_RXFIFOSZ_SIZE_32 0x00000002 // 32 #define USB_RXFIFOSZ_SIZE_64 0x00000003 // 64 #define USB_RXFIFOSZ_SIZE_128 0x00000004 // 128 #define USB_RXFIFOSZ_SIZE_256 0x00000005 // 256 #define USB_RXFIFOSZ_SIZE_512 0x00000006 // 512 #define USB_RXFIFOSZ_SIZE_1024 0x00000007 // 1024 #define USB_RXFIFOSZ_SIZE_2048 0x00000008 // 2048 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXFIFOADD // register. // //************************************************************************* #define USB_TXFIFOADD_ADDR_M 0x000001FF // Transmit/Receive Start Address #define USB_TXFIFOADD_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXFIFOADD // register. // //************************************************************************* #define USB_RXFIFOADD_ADDR_M 0x000001FF // Transmit/Receive Start Address #define USB_RXFIFOADD_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_CONTIM register. // //************************************************************************* #define USB_CONTIM_WTCON_M 0x000000F0 // Connect Wait #define USB_CONTIM_WTID_M 0x0000000F // Wait ID #define USB_CONTIM_WTCON_S 4 #define USB_CONTIM_WTID_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_FSEOF register. // //************************************************************************* #define USB_FSEOF_FSEOFG_M 0x000000FF // Full-Speed End-of-Frame Gap #define USB_FSEOF_FSEOFG_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_LSEOF register. // //************************************************************************* #define USB_LSEOF_LSEOFG_M 0x000000FF // Low-Speed End-of-Frame Gap #define USB_LSEOF_LSEOFG_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_CSRL0 register. // //************************************************************************* #define USB_CSRL0_SETENDC 0x00000080 // Setup End Clear #define USB_CSRL0_RXRDYC 0x00000040 // RXRDY Clear #define USB_CSRL0_STALL 0x00000020 // Send Stall #define USB_CSRL0_SETEND 0x00000010 // Setup End #define USB_CSRL0_DATAEND 0x00000008 // Data End #define USB_CSRL0_STALLED 0x00000004 // Endpoint Stalled #define USB_CSRL0_TXRDY 0x00000002 // Transmit Packet Ready #define USB_CSRL0_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_CSRH0 register. // //************************************************************************* #define USB_CSRH0_FLUSH 0x00000001 // Flush FIFO //************************************************************************* // // The following are defines for the bit fields in the USB_O_COUNT0 register. // //************************************************************************* #define USB_COUNT0_COUNT_M 0x0000007F // FIFO Count #define USB_COUNT0_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXMAXP1 register. // //************************************************************************* #define USB_TXMAXP1_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_TXMAXP1_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRL1 register. // //************************************************************************* #define USB_TXCSRL1_CLRDT 0x00000040 // Clear Data Toggle #define USB_TXCSRL1_STALLED 0x00000020 // Endpoint Stalled #define USB_TXCSRL1_STALL 0x00000010 // Send STALL #define USB_TXCSRL1_FLUSH 0x00000008 // Flush FIFO #define USB_TXCSRL1_UNDRN 0x00000004 // Underrun #define USB_TXCSRL1_FIFONE 0x00000002 // FIFO Not Empty #define USB_TXCSRL1_TXRDY 0x00000001 // Transmit Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRH1 register. // //************************************************************************* #define USB_TXCSRH1_AUTOSET 0x00000080 // Auto Set #define USB_TXCSRH1_ISO 0x00000040 // Isochronous Transfers #define USB_TXCSRH1_MODE 0x00000020 // Mode #define USB_TXCSRH1_DMAEN 0x00000010 // DMA Request Enable #define USB_TXCSRH1_FDT 0x00000008 // Force Data Toggle #define USB_TXCSRH1_DMAMOD 0x00000004 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXMAXP1 register. // //************************************************************************* #define USB_RXMAXP1_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_RXMAXP1_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRL1 register. // //************************************************************************* #define USB_RXCSRL1_CLRDT 0x00000080 // Clear Data Toggle #define USB_RXCSRL1_STALLED 0x00000040 // Endpoint Stalled #define USB_RXCSRL1_STALL 0x00000020 // Send STALL #define USB_RXCSRL1_FLUSH 0x00000010 // Flush FIFO #define USB_RXCSRL1_DATAERR 0x00000008 // Data Error #define USB_RXCSRL1_OVER 0x00000004 // Overrun #define USB_RXCSRL1_FULL 0x00000002 // FIFO Full #define USB_RXCSRL1_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRH1 register. // //************************************************************************* #define USB_RXCSRH1_AUTOCL 0x00000080 // Auto Clear #define USB_RXCSRH1_ISO 0x00000040 // Isochronous Transfers #define USB_RXCSRH1_DMAEN 0x00000020 // DMA Request Enable #define USB_RXCSRH1_DISNYET 0x00000010 // Disable NYET #define USB_RXCSRH1_PIDERR 0x00000010 // PID Error #define USB_RXCSRH1_DMAMOD 0x00000008 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCOUNT1 register. // //************************************************************************* #define USB_RXCOUNT1_COUNT_M 0x00001FFF // Receive Packet Count #define USB_RXCOUNT1_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXMAXP2 register. // //************************************************************************* #define USB_TXMAXP2_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_TXMAXP2_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRL2 register. // //************************************************************************* #define USB_TXCSRL2_CLRDT 0x00000040 // Clear Data Toggle #define USB_TXCSRL2_STALLED 0x00000020 // Endpoint Stalled #define USB_TXCSRL2_STALL 0x00000010 // Send STALL #define USB_TXCSRL2_FLUSH 0x00000008 // Flush FIFO #define USB_TXCSRL2_UNDRN 0x00000004 // Underrun #define USB_TXCSRL2_FIFONE 0x00000002 // FIFO Not Empty #define USB_TXCSRL2_TXRDY 0x00000001 // Transmit Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRH2 register. // //************************************************************************* #define USB_TXCSRH2_AUTOSET 0x00000080 // Auto Set #define USB_TXCSRH2_ISO 0x00000040 // Isochronous Transfers #define USB_TXCSRH2_MODE 0x00000020 // Mode #define USB_TXCSRH2_DMAEN 0x00000010 // DMA Request Enable #define USB_TXCSRH2_FDT 0x00000008 // Force Data Toggle #define USB_TXCSRH2_DMAMOD 0x00000004 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXMAXP2 register. // //************************************************************************* #define USB_RXMAXP2_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_RXMAXP2_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRL2 register. // //************************************************************************* #define USB_RXCSRL2_CLRDT 0x00000080 // Clear Data Toggle #define USB_RXCSRL2_STALLED 0x00000040 // Endpoint Stalled #define USB_RXCSRL2_STALL 0x00000020 // Send STALL #define USB_RXCSRL2_FLUSH 0x00000010 // Flush FIFO #define USB_RXCSRL2_DATAERR 0x00000008 // Data Error #define USB_RXCSRL2_OVER 0x00000004 // Overrun #define USB_RXCSRL2_FULL 0x00000002 // FIFO Full #define USB_RXCSRL2_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRH2 register. // //************************************************************************* #define USB_RXCSRH2_AUTOCL 0x00000080 // Auto Clear #define USB_RXCSRH2_ISO 0x00000040 // Isochronous Transfers #define USB_RXCSRH2_DMAEN 0x00000020 // DMA Request Enable #define USB_RXCSRH2_DISNYET 0x00000010 // Disable NYET #define USB_RXCSRH2_PIDERR 0x00000010 // PID Error #define USB_RXCSRH2_DMAMOD 0x00000008 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCOUNT2 register. // //************************************************************************* #define USB_RXCOUNT2_COUNT_M 0x00001FFF // Receive Packet Count #define USB_RXCOUNT2_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXMAXP3 register. // //************************************************************************* #define USB_TXMAXP3_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_TXMAXP3_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRL3 register. // //************************************************************************* #define USB_TXCSRL3_CLRDT 0x00000040 // Clear Data Toggle #define USB_TXCSRL3_STALLED 0x00000020 // Endpoint Stalled #define USB_TXCSRL3_STALL 0x00000010 // Send STALL #define USB_TXCSRL3_FLUSH 0x00000008 // Flush FIFO #define USB_TXCSRL3_UNDRN 0x00000004 // Underrun #define USB_TXCSRL3_FIFONE 0x00000002 // FIFO Not Empty #define USB_TXCSRL3_TXRDY 0x00000001 // Transmit Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRH3 register. // //************************************************************************* #define USB_TXCSRH3_AUTOSET 0x00000080 // Auto Set #define USB_TXCSRH3_ISO 0x00000040 // Isochronous Transfers #define USB_TXCSRH3_MODE 0x00000020 // Mode #define USB_TXCSRH3_DMAEN 0x00000010 // DMA Request Enable #define USB_TXCSRH3_FDT 0x00000008 // Force Data Toggle #define USB_TXCSRH3_DMAMOD 0x00000004 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXMAXP3 register. // //************************************************************************* #define USB_RXMAXP3_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_RXMAXP3_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRL3 register. // //************************************************************************* #define USB_RXCSRL3_CLRDT 0x00000080 // Clear Data Toggle #define USB_RXCSRL3_STALLED 0x00000040 // Endpoint Stalled #define USB_RXCSRL3_STALL 0x00000020 // Send STALL #define USB_RXCSRL3_FLUSH 0x00000010 // Flush FIFO #define USB_RXCSRL3_DATAERR 0x00000008 // Data Error #define USB_RXCSRL3_OVER 0x00000004 // Overrun #define USB_RXCSRL3_FULL 0x00000002 // FIFO Full #define USB_RXCSRL3_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRH3 register. // //************************************************************************* #define USB_RXCSRH3_AUTOCL 0x00000080 // Auto Clear #define USB_RXCSRH3_ISO 0x00000040 // Isochronous Transfers #define USB_RXCSRH3_DMAEN 0x00000020 // DMA Request Enable #define USB_RXCSRH3_DISNYET 0x00000010 // Disable NYET #define USB_RXCSRH3_PIDERR 0x00000010 // PID Error #define USB_RXCSRH3_DMAMOD 0x00000008 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCOUNT3 register. // //************************************************************************* #define USB_RXCOUNT3_COUNT_M 0x00001FFF // Receive Packet Count #define USB_RXCOUNT3_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXMAXP4 register. // //************************************************************************* #define USB_TXMAXP4_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_TXMAXP4_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRL4 register. // //************************************************************************* #define USB_TXCSRL4_CLRDT 0x00000040 // Clear Data Toggle #define USB_TXCSRL4_STALLED 0x00000020 // Endpoint Stalled #define USB_TXCSRL4_STALL 0x00000010 // Send STALL #define USB_TXCSRL4_FLUSH 0x00000008 // Flush FIFO #define USB_TXCSRL4_UNDRN 0x00000004 // Underrun #define USB_TXCSRL4_FIFONE 0x00000002 // FIFO Not Empty #define USB_TXCSRL4_TXRDY 0x00000001 // Transmit Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRH4 register. // //************************************************************************* #define USB_TXCSRH4_AUTOSET 0x00000080 // Auto Set #define USB_TXCSRH4_ISO 0x00000040 // Isochronous Transfers #define USB_TXCSRH4_MODE 0x00000020 // Mode #define USB_TXCSRH4_DMAEN 0x00000010 // DMA Request Enable #define USB_TXCSRH4_FDT 0x00000008 // Force Data Toggle #define USB_TXCSRH4_DMAMOD 0x00000004 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXMAXP4 register. // //************************************************************************* #define USB_RXMAXP4_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_RXMAXP4_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRL4 register. // //************************************************************************* #define USB_RXCSRL4_CLRDT 0x00000080 // Clear Data Toggle #define USB_RXCSRL4_STALLED 0x00000040 // Endpoint Stalled #define USB_RXCSRL4_STALL 0x00000020 // Send STALL #define USB_RXCSRL4_FLUSH 0x00000010 // Flush FIFO #define USB_RXCSRL4_DATAERR 0x00000008 // Data Error #define USB_RXCSRL4_OVER 0x00000004 // Overrun #define USB_RXCSRL4_FULL 0x00000002 // FIFO Full #define USB_RXCSRL4_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRH4 register. // //************************************************************************* #define USB_RXCSRH4_AUTOCL 0x00000080 // Auto Clear #define USB_RXCSRH4_ISO 0x00000040 // Isochronous Transfers #define USB_RXCSRH4_DMAEN 0x00000020 // DMA Request Enable #define USB_RXCSRH4_DISNYET 0x00000010 // Disable NYET #define USB_RXCSRH4_PIDERR 0x00000010 // PID Error #define USB_RXCSRH4_DMAMOD 0x00000008 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCOUNT4 register. // //************************************************************************* #define USB_RXCOUNT4_COUNT_M 0x00001FFF // Receive Packet Count #define USB_RXCOUNT4_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXMAXP5 register. // //************************************************************************* #define USB_TXMAXP5_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_TXMAXP5_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRL5 register. // //************************************************************************* #define USB_TXCSRL5_CLRDT 0x00000040 // Clear Data Toggle #define USB_TXCSRL5_STALLED 0x00000020 // Endpoint Stalled #define USB_TXCSRL5_STALL 0x00000010 // Send STALL #define USB_TXCSRL5_FLUSH 0x00000008 // Flush FIFO #define USB_TXCSRL5_UNDRN 0x00000004 // Underrun #define USB_TXCSRL5_FIFONE 0x00000002 // FIFO Not Empty #define USB_TXCSRL5_TXRDY 0x00000001 // Transmit Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRH5 register. // //************************************************************************* #define USB_TXCSRH5_AUTOSET 0x00000080 // Auto Set #define USB_TXCSRH5_ISO 0x00000040 // Isochronous Transfers #define USB_TXCSRH5_MODE 0x00000020 // Mode #define USB_TXCSRH5_DMAEN 0x00000010 // DMA Request Enable #define USB_TXCSRH5_FDT 0x00000008 // Force Data Toggle #define USB_TXCSRH5_DMAMOD 0x00000004 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXMAXP5 register. // //************************************************************************* #define USB_RXMAXP5_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_RXMAXP5_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRL5 register. // //************************************************************************* #define USB_RXCSRL5_CLRDT 0x00000080 // Clear Data Toggle #define USB_RXCSRL5_STALLED 0x00000040 // Endpoint Stalled #define USB_RXCSRL5_STALL 0x00000020 // Send STALL #define USB_RXCSRL5_FLUSH 0x00000010 // Flush FIFO #define USB_RXCSRL5_DATAERR 0x00000008 // Data Error #define USB_RXCSRL5_OVER 0x00000004 // Overrun #define USB_RXCSRL5_FULL 0x00000002 // FIFO Full #define USB_RXCSRL5_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRH5 register. // //************************************************************************* #define USB_RXCSRH5_AUTOCL 0x00000080 // Auto Clear #define USB_RXCSRH5_ISO 0x00000040 // Isochronous Transfers #define USB_RXCSRH5_DMAEN 0x00000020 // DMA Request Enable #define USB_RXCSRH5_DISNYET 0x00000010 // Disable NYET #define USB_RXCSRH5_PIDERR 0x00000010 // PID Error #define USB_RXCSRH5_DMAMOD 0x00000008 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCOUNT5 register. // //************************************************************************* #define USB_RXCOUNT5_COUNT_M 0x00001FFF // Receive Packet Count #define USB_RXCOUNT5_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXMAXP6 register. // //************************************************************************* #define USB_TXMAXP6_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_TXMAXP6_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRL6 register. // //************************************************************************* #define USB_TXCSRL6_CLRDT 0x00000040 // Clear Data Toggle #define USB_TXCSRL6_STALLED 0x00000020 // Endpoint Stalled #define USB_TXCSRL6_STALL 0x00000010 // Send STALL #define USB_TXCSRL6_FLUSH 0x00000008 // Flush FIFO #define USB_TXCSRL6_UNDRN 0x00000004 // Underrun #define USB_TXCSRL6_FIFONE 0x00000002 // FIFO Not Empty #define USB_TXCSRL6_TXRDY 0x00000001 // Transmit Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRH6 register. // //************************************************************************* #define USB_TXCSRH6_AUTOSET 0x00000080 // Auto Set #define USB_TXCSRH6_ISO 0x00000040 // Isochronous Transfers #define USB_TXCSRH6_MODE 0x00000020 // Mode #define USB_TXCSRH6_DMAEN 0x00000010 // DMA Request Enable #define USB_TXCSRH6_FDT 0x00000008 // Force Data Toggle #define USB_TXCSRH6_DMAMOD 0x00000004 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXMAXP6 register. // //************************************************************************* #define USB_RXMAXP6_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_RXMAXP6_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRL6 register. // //************************************************************************* #define USB_RXCSRL6_CLRDT 0x00000080 // Clear Data Toggle #define USB_RXCSRL6_STALLED 0x00000040 // Endpoint Stalled #define USB_RXCSRL6_STALL 0x00000020 // Send STALL #define USB_RXCSRL6_FLUSH 0x00000010 // Flush FIFO #define USB_RXCSRL6_DATAERR 0x00000008 // Data Error #define USB_RXCSRL6_OVER 0x00000004 // Overrun #define USB_RXCSRL6_FULL 0x00000002 // FIFO Full #define USB_RXCSRL6_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRH6 register. // //************************************************************************* #define USB_RXCSRH6_AUTOCL 0x00000080 // Auto Clear #define USB_RXCSRH6_ISO 0x00000040 // Isochronous Transfers #define USB_RXCSRH6_DMAEN 0x00000020 // DMA Request Enable #define USB_RXCSRH6_DISNYET 0x00000010 // Disable NYET #define USB_RXCSRH6_PIDERR 0x00000010 // PID Error #define USB_RXCSRH6_DMAMOD 0x00000008 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCOUNT6 register. // //************************************************************************* #define USB_RXCOUNT6_COUNT_M 0x00001FFF // Receive Packet Count #define USB_RXCOUNT6_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXMAXP7 register. // //************************************************************************* #define USB_TXMAXP7_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_TXMAXP7_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRL7 register. // //************************************************************************* #define USB_TXCSRL7_CLRDT 0x00000040 // Clear Data Toggle #define USB_TXCSRL7_STALLED 0x00000020 // Endpoint Stalled #define USB_TXCSRL7_STALL 0x00000010 // Send STALL #define USB_TXCSRL7_FLUSH 0x00000008 // Flush FIFO #define USB_TXCSRL7_UNDRN 0x00000004 // Underrun #define USB_TXCSRL7_FIFONE 0x00000002 // FIFO Not Empty #define USB_TXCSRL7_TXRDY 0x00000001 // Transmit Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXCSRH7 register. // //************************************************************************* #define USB_TXCSRH7_AUTOSET 0x00000080 // Auto Set #define USB_TXCSRH7_ISO 0x00000040 // Isochronous Transfers #define USB_TXCSRH7_MODE 0x00000020 // Mode #define USB_TXCSRH7_DMAEN 0x00000010 // DMA Request Enable #define USB_TXCSRH7_FDT 0x00000008 // Force Data Toggle #define USB_TXCSRH7_DMAMOD 0x00000004 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXMAXP7 register. // //************************************************************************* #define USB_RXMAXP7_MAXLOAD_M 0x000007FF // Maximum Payload #define USB_RXMAXP7_MAXLOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRL7 register. // //************************************************************************* #define USB_RXCSRL7_CLRDT 0x00000080 // Clear Data Toggle #define USB_RXCSRL7_STALLED 0x00000040 // Endpoint Stalled #define USB_RXCSRL7_STALL 0x00000020 // Send STALL #define USB_RXCSRL7_FLUSH 0x00000010 // Flush FIFO #define USB_RXCSRL7_DATAERR 0x00000008 // Data Error #define USB_RXCSRL7_OVER 0x00000004 // Overrun #define USB_RXCSRL7_FULL 0x00000002 // FIFO Full #define USB_RXCSRL7_RXRDY 0x00000001 // Receive Packet Ready //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCSRH7 register. // //************************************************************************* #define USB_RXCSRH7_AUTOCL 0x00000080 // Auto Clear #define USB_RXCSRH7_ISO 0x00000040 // Isochronous Transfers #define USB_RXCSRH7_DMAEN 0x00000020 // DMA Request Enable #define USB_RXCSRH7_PIDERR 0x00000010 // PID Error #define USB_RXCSRH7_DISNYET 0x00000010 // Disable NYET #define USB_RXCSRH7_DMAMOD 0x00000008 // DMA Request Mode //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXCOUNT7 register. // //************************************************************************* #define USB_RXCOUNT7_COUNT_M 0x00001FFF // Receive Packet Count #define USB_RXCOUNT7_COUNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_RXDPKTBUFDIS // register. // //************************************************************************* #define USB_RXDPKTBUFDIS_EP7 0x00000080 // EP7 RX Double-Packet Buffer // Disable #define USB_RXDPKTBUFDIS_EP6 0x00000040 // EP6 RX Double-Packet Buffer // Disable #define USB_RXDPKTBUFDIS_EP5 0x00000020 // EP5 RX Double-Packet Buffer // Disable #define USB_RXDPKTBUFDIS_EP4 0x00000010 // EP4 RX Double-Packet Buffer // Disable #define USB_RXDPKTBUFDIS_EP3 0x00000008 // EP3 RX Double-Packet Buffer // Disable #define USB_RXDPKTBUFDIS_EP2 0x00000004 // EP2 RX Double-Packet Buffer // Disable #define USB_RXDPKTBUFDIS_EP1 0x00000002 // EP1 RX Double-Packet Buffer // Disable //************************************************************************* // // The following are defines for the bit fields in the USB_O_TXDPKTBUFDIS // register. // //************************************************************************* #define USB_TXDPKTBUFDIS_EP7 0x00000080 // EP7 TX Double-Packet Buffer // Disable #define USB_TXDPKTBUFDIS_EP6 0x00000040 // EP6 TX Double-Packet Buffer // Disable #define USB_TXDPKTBUFDIS_EP5 0x00000020 // EP5 TX Double-Packet Buffer // Disable #define USB_TXDPKTBUFDIS_EP4 0x00000010 // EP4 TX Double-Packet Buffer // Disable #define USB_TXDPKTBUFDIS_EP3 0x00000008 // EP3 TX Double-Packet Buffer // Disable #define USB_TXDPKTBUFDIS_EP2 0x00000004 // EP2 TX Double-Packet Buffer // Disable #define USB_TXDPKTBUFDIS_EP1 0x00000002 // EP1 TX Double-Packet Buffer // Disable //************************************************************************* // // The following are defines for the bit fields in the USB_O_DRRIS register. // //************************************************************************* #define USB_DRRIS_RESUME 0x00000001 // RESUME Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the USB_O_DRIM register. // //************************************************************************* #define USB_DRIM_RESUME 0x00000001 // RESUME Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the USB_O_DRISC register. // //************************************************************************* #define USB_DRISC_RESUME 0x00000001 // RESUME Interrupt Status and // Clear //************************************************************************* // // The following are defines for the bit fields in the USB_O_DMASEL register. // //************************************************************************* #define USB_DMASEL_DMACTX_M 0x00F00000 // DMA C TX Select #define USB_DMASEL_DMACRX_M 0x000F0000 // DMA C RX Select #define USB_DMASEL_DMABTX_M 0x0000F000 // DMA B TX Select #define USB_DMASEL_DMABRX_M 0x00000F00 // DMA B RX Select #define USB_DMASEL_DMAATX_M 0x000000F0 // DMA A TX Select #define USB_DMASEL_DMAARX_M 0x0000000F // DMA A RX Select #define USB_DMASEL_DMACTX_S 20 #define USB_DMASEL_DMACRX_S 16 #define USB_DMASEL_DMABTX_S 12 #define USB_DMASEL_DMABRX_S 8 #define USB_DMASEL_DMAATX_S 4 #define USB_DMASEL_DMAARX_S 0 //************************************************************************* // // The following are defines for the bit fields in the USB_O_PP register. // //************************************************************************* #define USB_PP_ECNT_M 0x0000FF00 // Endpoint Count #define USB_PP_USB_M 0x000000C0 // USB Capability #define USB_PP_USB_DEVICE 0x00000040 // DEVICE #define USB_PP_USB_HOSTDEVICE 0x00000080 // HOST #define USB_PP_USB_OTG 0x000000C0 // OTG #define USB_PP_PHY 0x00000010 // PHY Present #define USB_PP_TYPE_M 0x0000000F // Controller Type #define USB_PP_TYPE_0 0x00000000 // The first-generation USB // controller #define USB_PP_ECNT_S 8 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EESIZE register. // //************************************************************************* #define EEPROM_EESIZE_BLKCNT_M 0x07FF0000 // Number of 16-Word Blocks #define EEPROM_EESIZE_WORDCNT_M 0x0000FFFF // Number of 32-Bit Words #define EEPROM_EESIZE_BLKCNT_S 16 #define EEPROM_EESIZE_WORDCNT_S 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEBLOCK register. // //************************************************************************* #define EEPROM_EEBLOCK_BLOCK_M 0x0000FFFF // Current Block #define EEPROM_EEBLOCK_BLOCK_S 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEOFFSET // register. // //************************************************************************* #define EEPROM_EEOFFSET_OFFSET_M \ 0x0000000F // Current Address Offset #define EEPROM_EEOFFSET_OFFSET_S \ 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EERDWR register. // //************************************************************************* #define EEPROM_EERDWR_VALUE_M 0xFFFFFFFF // EEPROM Read or Write Data #define EEPROM_EERDWR_VALUE_S 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EERDWRINC // register. // //************************************************************************* #define EEPROM_EERDWRINC_VALUE_M \ 0xFFFFFFFF // EEPROM Read or Write Data with // Increment #define EEPROM_EERDWRINC_VALUE_S \ 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEDONE register. // //************************************************************************* #define EEPROM_EEDONE_INVPL 0x00000100 // Invalid Program Voltage Level #define EEPROM_EEDONE_WRBUSY 0x00000020 // Write Busy #define EEPROM_EEDONE_NOPERM 0x00000010 // Write Without Permission #define EEPROM_EEDONE_WKCOPY 0x00000008 // Working on a Copy #define EEPROM_EEDONE_WKERASE 0x00000004 // Working on an Erase #define EEPROM_EEDONE_WORKING 0x00000001 // EEPROM Working //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EESUPP register. // //************************************************************************* #define EEPROM_EESUPP_PRETRY 0x00000008 // Programming Must Be Retried #define EEPROM_EESUPP_ERETRY 0x00000004 // Erase Must Be Retried #define EEPROM_EESUPP_EREQ 0x00000002 // Erase Required #define EEPROM_EESUPP_START 0x00000001 // Start Erase //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEUNLOCK // register. // //************************************************************************* #define EEPROM_EEUNLOCK_UNLOCK_M \ 0xFFFFFFFF // EEPROM Unlock //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEPROT register. // //************************************************************************* #define EEPROM_EEPROT_ACC 0x00000008 // Access Control #define EEPROM_EEPROT_PROT_M 0x00000007 // Protection Control #define EEPROM_EEPROT_PROT_RWNPW \ 0x00000000 // This setting is the default. If // there is no password, the block // is not protected and is readable // and writable #define EEPROM_EEPROT_PROT_RWPW 0x00000001 // If there is a password, the // block is readable or writable // only when unlocked #define EEPROM_EEPROT_PROT_RONPW \ 0x00000002 // If there is no password, the // block is readable, not writable //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEPASS0 register. // //************************************************************************* #define EEPROM_EEPASS0_PASS_M 0xFFFFFFFF // Password #define EEPROM_EEPASS0_PASS_S 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEPASS1 register. // //************************************************************************* #define EEPROM_EEPASS1_PASS_M 0xFFFFFFFF // Password #define EEPROM_EEPASS1_PASS_S 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEPASS2 register. // //************************************************************************* #define EEPROM_EEPASS2_PASS_M 0xFFFFFFFF // Password #define EEPROM_EEPASS2_PASS_S 0 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEINT register. // //************************************************************************* #define EEPROM_EEINT_INT 0x00000001 // Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEHIDE register. // //************************************************************************* #define EEPROM_EEHIDE_HN_M 0xFFFFFFFE // Hide Block //************************************************************************* // // The following are defines for the bit fields in the EEPROM_EEDBGME register. // //************************************************************************* #define EEPROM_EEDBGME_KEY_M 0xFFFF0000 // Erase Key #define EEPROM_EEDBGME_ME 0x00000001 // Mass Erase #define EEPROM_EEDBGME_KEY_S 16 //************************************************************************* // // The following are defines for the bit fields in the EEPROM_PP register. // //************************************************************************* #define EEPROM_PP_SIZE_M 0x0000001F // EEPROM Size #define EEPROM_PP_SIZE_S 0 //************************************************************************* // // The following are defines for the bit fields in the SYSEXC_RIS register. // //************************************************************************* #define SYSEXC_RIS_FPIXCRIS 0x00000020 // Floating-Point Inexact Exception // Raw Interrupt Status #define SYSEXC_RIS_FPOFCRIS 0x00000010 // Floating-Point Overflow // Exception Raw Interrupt Status #define SYSEXC_RIS_FPUFCRIS 0x00000008 // Floating-Point Underflow // Exception Raw Interrupt Status #define SYSEXC_RIS_FPIOCRIS 0x00000004 // Floating-Point Invalid Operation // Raw Interrupt Status #define SYSEXC_RIS_FPDZCRIS 0x00000002 // Floating-Point Divide By 0 // Exception Raw Interrupt Status #define SYSEXC_RIS_FPIDCRIS 0x00000001 // Floating-Point Input Denormal // Exception Raw Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the SYSEXC_IM register. // //************************************************************************* #define SYSEXC_IM_FPIXCIM 0x00000020 // Floating-Point Inexact Exception // Interrupt Mask #define SYSEXC_IM_FPOFCIM 0x00000010 // Floating-Point Overflow // Exception Interrupt Mask #define SYSEXC_IM_FPUFCIM 0x00000008 // Floating-Point Underflow // Exception Interrupt Mask #define SYSEXC_IM_FPIOCIM 0x00000004 // Floating-Point Invalid Operation // Interrupt Mask #define SYSEXC_IM_FPDZCIM 0x00000002 // Floating-Point Divide By 0 // Exception Interrupt Mask #define SYSEXC_IM_FPIDCIM 0x00000001 // Floating-Point Input Denormal // Exception Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the SYSEXC_MIS register. // //************************************************************************* #define SYSEXC_MIS_FPIXCMIS 0x00000020 // Floating-Point Inexact Exception // Masked Interrupt Status #define SYSEXC_MIS_FPOFCMIS 0x00000010 // Floating-Point Overflow // Exception Masked Interrupt // Status #define SYSEXC_MIS_FPUFCMIS 0x00000008 // Floating-Point Underflow // Exception Masked Interrupt // Status #define SYSEXC_MIS_FPIOCMIS 0x00000004 // Floating-Point Invalid Operation // Masked Interrupt Status #define SYSEXC_MIS_FPDZCMIS 0x00000002 // Floating-Point Divide By 0 // Exception Masked Interrupt // Status #define SYSEXC_MIS_FPIDCMIS 0x00000001 // Floating-Point Input Denormal // Exception Masked Interrupt // Status //************************************************************************* // // The following are defines for the bit fields in the SYSEXC_IC register. // //************************************************************************* #define SYSEXC_IC_FPIXCIC 0x00000020 // Floating-Point Inexact Exception // Interrupt Clear #define SYSEXC_IC_FPOFCIC 0x00000010 // Floating-Point Overflow // Exception Interrupt Clear #define SYSEXC_IC_FPUFCIC 0x00000008 // Floating-Point Underflow // Exception Interrupt Clear #define SYSEXC_IC_FPIOCIC 0x00000004 // Floating-Point Invalid Operation // Interrupt Clear #define SYSEXC_IC_FPDZCIC 0x00000002 // Floating-Point Divide By 0 // Exception Interrupt Clear #define SYSEXC_IC_FPIDCIC 0x00000001 // Floating-Point Input Denormal // Exception Interrupt Clear //************************************************************************* // // The following are defines for the bit fields in the HIB_RTCC register. // //************************************************************************* #define HIB_RTCC_M 0xFFFFFFFF // RTC Counter #define HIB_RTCC_S 0 //************************************************************************* // // The following are defines for the bit fields in the HIB_RTCM0 register. // //************************************************************************* #define HIB_RTCM0_M 0xFFFFFFFF // RTC Match 0 #define HIB_RTCM0_S 0 //************************************************************************* // // The following are defines for the bit fields in the HIB_RTCLD register. // //************************************************************************* #define HIB_RTCLD_M 0xFFFFFFFF // RTC Load #define HIB_RTCLD_S 0 //************************************************************************* // // The following are defines for the bit fields in the HIB_CTL register. // //************************************************************************* #define HIB_CTL_WRC 0x80000000 // Write Complete/Capable #define HIB_CTL_OSCHYS 0x00040000 // 32 #define HIB_CTL_OSCDRV 0x00020000 // Oscillator Drive Capability #define HIB_CTL_OSCBYP 0x00010000 // Oscillator Bypass #define HIB_CTL_VBATSEL_M 0x00006000 // Select for Low-Battery // Comparator #define HIB_CTL_VBATSEL_1_9V 0x00000000 // 1.9 Volts #define HIB_CTL_VBATSEL_2_1V 0x00002000 // 2.1 Volts (default) #define HIB_CTL_VBATSEL_2_3V 0x00004000 // 2.3 Volts #define HIB_CTL_VBATSEL_2_5V 0x00006000 // 2.5 Volts #define HIB_CTL_BATCHK 0x00000400 // Check Battery Status #define HIB_CTL_BATWKEN 0x00000200 // Wake on Low Battery #define HIB_CTL_VDD3ON 0x00000100 // VDD Powered #define HIB_CTL_VABORT 0x00000080 // Power Cut Abort Enable #define HIB_CTL_CLK32EN 0x00000040 // Clocking Enable #define HIB_CTL_LOWBATEN 0x00000020 // Low Battery Monitoring Enable #define HIB_CTL_PINWEN 0x00000010 // External WAKE Pin Enable #define HIB_CTL_RTCWEN 0x00000008 // RTC Wake-up Enable #define HIB_CTL_HIBREQ 0x00000002 // Hibernation Request #define HIB_CTL_RTCEN 0x00000001 // RTC Timer Enable //************************************************************************* // // The following are defines for the bit fields in the HIB_IM register. // //************************************************************************* #define HIB_IM_WC 0x00000010 // External Write Complete/Capable // Interrupt Mask #define HIB_IM_EXTW 0x00000008 // External Wake-Up Interrupt Mask #define HIB_IM_LOWBAT 0x00000004 // Low Battery Voltage Interrupt // Mask #define HIB_IM_RTCALT0 0x00000001 // RTC Alert 0 Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the HIB_RIS register. // //************************************************************************* #define HIB_RIS_WC 0x00000010 // Write Complete/Capable Raw // Interrupt Status #define HIB_RIS_EXTW 0x00000008 // External Wake-Up Raw Interrupt // Status #define HIB_RIS_LOWBAT 0x00000004 // Low Battery Voltage Raw // Interrupt Status #define HIB_RIS_RTCALT0 0x00000001 // RTC Alert 0 Raw Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the HIB_MIS register. // //************************************************************************* #define HIB_MIS_WC 0x00000010 // Write Complete/Capable Masked // Interrupt Status #define HIB_MIS_EXTW 0x00000008 // External Wake-Up Masked // Interrupt Status #define HIB_MIS_LOWBAT 0x00000004 // Low Battery Voltage Masked // Interrupt Status #define HIB_MIS_RTCALT0 0x00000001 // RTC Alert 0 Masked Interrupt // Status //************************************************************************* // // The following are defines for the bit fields in the HIB_IC register. // //************************************************************************* #define HIB_IC_WC 0x00000010 // Write Complete/Capable Masked // Interrupt Clear #define HIB_IC_EXTW 0x00000008 // External Wake-Up Masked // Interrupt Clear #define HIB_IC_LOWBAT 0x00000004 // Low Battery Voltage Masked // Interrupt Clear #define HIB_IC_RTCALT0 0x00000001 // RTC Alert0 Masked Interrupt // Clear //************************************************************************* // // The following are defines for the bit fields in the HIB_RTCT register. // //************************************************************************* #define HIB_RTCT_TRIM_M 0x0000FFFF // RTC Trim Value #define HIB_RTCT_TRIM_S 0 //************************************************************************* // // The following are defines for the bit fields in the HIB_RTCSS register. // //************************************************************************* #define HIB_RTCSS_RTCSSM_M 0x7FFF0000 // RTC Sub Seconds Match #define HIB_RTCSS_RTCSSC_M 0x00007FFF // RTC Sub Seconds Count #define HIB_RTCSS_RTCSSM_S 16 #define HIB_RTCSS_RTCSSC_S 0 //************************************************************************* // // The following are defines for the bit fields in the HIB_DATA register. // //************************************************************************* #define HIB_DATA_RTD_M 0xFFFFFFFF // Hibernation Module NV Data #define HIB_DATA_RTD_S 0 //************************************************************************* // // The following are defines for the bit fields in the FLASH_FMA register. // //************************************************************************* #define FLASH_FMA_OFFSET_M 0x0003FFFF // Address Offset #define FLASH_FMA_OFFSET_S 0 //************************************************************************* // // The following are defines for the bit fields in the FLASH_FMD register. // //************************************************************************* #define FLASH_FMD_DATA_M 0xFFFFFFFF // Data Value #define FLASH_FMD_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the FLASH_FMC register. // //************************************************************************* #define FLASH_FMC_WRKEY 0xA4420000 // FLASH write key #define FLASH_FMC_COMT 0x00000008 // Commit Register Value #define FLASH_FMC_MERASE 0x00000004 // Mass Erase Flash Memory #define FLASH_FMC_ERASE 0x00000002 // Erase a Page of Flash Memory #define FLASH_FMC_WRITE 0x00000001 // Write a Word into Flash Memory //************************************************************************* // // The following are defines for the bit fields in the FLASH_FCRIS register. // //************************************************************************* #define FLASH_FCRIS_PROGRIS 0x00002000 // PROGVER Raw Interrupt Status #define FLASH_FCRIS_ERRIS 0x00000800 // ERVER Raw Interrupt Status #define FLASH_FCRIS_INVDRIS 0x00000400 // Invalid Data Raw Interrupt // Status #define FLASH_FCRIS_VOLTRIS 0x00000200 // VOLTSTAT Raw Interrupt Status #define FLASH_FCRIS_ERIS 0x00000004 // EEPROM Raw Interrupt Status #define FLASH_FCRIS_PRIS 0x00000002 // Programming Raw Interrupt Status #define FLASH_FCRIS_ARIS 0x00000001 // Access Raw Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the FLASH_FCIM register. // //************************************************************************* #define FLASH_FCIM_PROGMASK 0x00002000 // PROGVER Interrupt Mask #define FLASH_FCIM_ERMASK 0x00000800 // ERVER Interrupt Mask #define FLASH_FCIM_INVDMASK 0x00000400 // Invalid Data Interrupt Mask #define FLASH_FCIM_VOLTMASK 0x00000200 // VOLT Interrupt Mask #define FLASH_FCIM_EMASK 0x00000004 // EEPROM Interrupt Mask #define FLASH_FCIM_PMASK 0x00000002 // Programming Interrupt Mask #define FLASH_FCIM_AMASK 0x00000001 // Access Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the FLASH_FCMISC register. // //************************************************************************* #define FLASH_FCMISC_PROGMISC 0x00002000 // PROGVER Masked Interrupt Status // and Clear #define FLASH_FCMISC_ERMISC 0x00000800 // ERVER Masked Interrupt Status // and Clear #define FLASH_FCMISC_INVDMISC 0x00000400 // Invalid Data Masked Interrupt // Status and Clear #define FLASH_FCMISC_VOLTMISC 0x00000200 // VOLT Masked Interrupt Status and // Clear #define FLASH_FCMISC_EMISC 0x00000004 // EEPROM Masked Interrupt Status // and Clear #define FLASH_FCMISC_PMISC 0x00000002 // Programming Masked Interrupt // Status and Clear #define FLASH_FCMISC_AMISC 0x00000001 // Access Masked Interrupt Status // and Clear //************************************************************************* // // The following are defines for the bit fields in the FLASH_FMC2 register. // //************************************************************************* #define FLASH_FMC2_WRKEY 0xA4420000 // FLASH write key #define FLASH_FMC2_WRBUF 0x00000001 // Buffered Flash Memory Write //************************************************************************* // // The following are defines for the bit fields in the FLASH_FWBVAL register. // //************************************************************************* #define FLASH_FWBVAL_FWB_M 0xFFFFFFFF // Flash Memory Write Buffer //************************************************************************* // // The following are defines for the bit fields in the FLASH_FWBN register. // //************************************************************************* #define FLASH_FWBN_DATA_M 0xFFFFFFFF // Data //************************************************************************* // // The following are defines for the bit fields in the FLASH_FSIZE register. // //************************************************************************* #define FLASH_FSIZE_SIZE_M 0x0000FFFF // Flash Size #define FLASH_FSIZE_SIZE_8KB 0x00000003 // 8 KB of Flash #define FLASH_FSIZE_SIZE_16KB 0x00000007 // 16 KB of Flash #define FLASH_FSIZE_SIZE_32KB 0x0000000F // 32 KB of Flash #define FLASH_FSIZE_SIZE_64KB 0x0000001F // 64 KB of Flash #define FLASH_FSIZE_SIZE_96KB 0x0000002F // 96 KB of Flash #define FLASH_FSIZE_SIZE_128KB 0x0000003F // 128 KB of Flash #define FLASH_FSIZE_SIZE_192KB 0x0000005F // 192 KB of Flash #define FLASH_FSIZE_SIZE_256KB 0x0000007F // 256 KB of Flash //************************************************************************* // // The following are defines for the bit fields in the FLASH_SSIZE register. // //************************************************************************* #define FLASH_SSIZE_SIZE_M 0x0000FFFF // SRAM Size #define FLASH_SSIZE_SIZE_2KB 0x00000007 // 2 KB of SRAM #define FLASH_SSIZE_SIZE_4KB 0x0000000F // 4 KB of SRAM #define FLASH_SSIZE_SIZE_6KB 0x00000017 // 6 KB of SRAM #define FLASH_SSIZE_SIZE_8KB 0x0000001F // 8 KB of SRAM #define FLASH_SSIZE_SIZE_12KB 0x0000002F // 12 KB of SRAM #define FLASH_SSIZE_SIZE_16KB 0x0000003F // 16 KB of SRAM #define FLASH_SSIZE_SIZE_20KB 0x0000004F // 20 KB of SRAM #define FLASH_SSIZE_SIZE_24KB 0x0000005F // 24 KB of SRAM #define FLASH_SSIZE_SIZE_32KB 0x0000007F // 32 KB of SRAM //************************************************************************* // // The following are defines for the bit fields in the FLASH_ROMSWMAP register. // //************************************************************************* #define FLASH_ROMSWMAP_SAFERTOS 0x00000001 // SafeRTOS Present //************************************************************************* // // The following are defines for the bit fields in the FLASH_RMCTL register. // //************************************************************************* #define FLASH_RMCTL_BA 0x00000001 // Boot Alias //************************************************************************* // // The following are defines for the bit fields in the FLASH_BOOTCFG register. // //************************************************************************* #define FLASH_BOOTCFG_NW 0x80000000 // Not Written #define FLASH_BOOTCFG_PORT_M 0x0000E000 // Boot GPIO Port #define FLASH_BOOTCFG_PORT_A 0x00000000 // Port A #define FLASH_BOOTCFG_PORT_B 0x00002000 // Port B #define FLASH_BOOTCFG_PORT_C 0x00004000 // Port C #define FLASH_BOOTCFG_PORT_D 0x00006000 // Port D #define FLASH_BOOTCFG_PORT_E 0x00008000 // Port E #define FLASH_BOOTCFG_PORT_F 0x0000A000 // Port F #define FLASH_BOOTCFG_PORT_G 0x0000C000 // Port G #define FLASH_BOOTCFG_PORT_H 0x0000E000 // Port H #define FLASH_BOOTCFG_PIN_M 0x00001C00 // Boot GPIO Pin #define FLASH_BOOTCFG_PIN_0 0x00000000 // Pin 0 #define FLASH_BOOTCFG_PIN_1 0x00000400 // Pin 1 #define FLASH_BOOTCFG_PIN_2 0x00000800 // Pin 2 #define FLASH_BOOTCFG_PIN_3 0x00000C00 // Pin 3 #define FLASH_BOOTCFG_PIN_4 0x00001000 // Pin 4 #define FLASH_BOOTCFG_PIN_5 0x00001400 // Pin 5 #define FLASH_BOOTCFG_PIN_6 0x00001800 // Pin 6 #define FLASH_BOOTCFG_PIN_7 0x00001C00 // Pin 7 #define FLASH_BOOTCFG_POL 0x00000200 // Boot GPIO Polarity #define FLASH_BOOTCFG_EN 0x00000100 // Boot GPIO Enable #define FLASH_BOOTCFG_DBG1 0x00000002 // Debug Control 1 #define FLASH_BOOTCFG_DBG0 0x00000001 // Debug Control 0 //************************************************************************* // // The following are defines for the bit fields in the FLASH_USERREG0 register. // //************************************************************************* #define FLASH_USERREG0_DATA_M 0xFFFFFFFF // User Data #define FLASH_USERREG0_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the FLASH_USERREG1 register. // //************************************************************************* #define FLASH_USERREG1_DATA_M 0xFFFFFFFF // User Data #define FLASH_USERREG1_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the FLASH_USERREG2 register. // //************************************************************************* #define FLASH_USERREG2_DATA_M 0xFFFFFFFF // User Data #define FLASH_USERREG2_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the FLASH_USERREG3 register. // //************************************************************************* #define FLASH_USERREG3_DATA_M 0xFFFFFFFF // User Data #define FLASH_USERREG3_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DID0 register. // //************************************************************************* #define SYSCTL_DID0_VER_M 0x70000000 // DID0 Version #define SYSCTL_DID0_VER_1 0x10000000 // Second version of the DID0 // register format #define SYSCTL_DID0_CLASS_M 0x00FF0000 // Device Class #define SYSCTL_DID0_CLASS_BLIZZARD \ 0x00050000 // Stellaris(R) Blizzard-class // microcontrollers #define SYSCTL_DID0_MAJ_M 0x0000FF00 // Major Revision #define SYSCTL_DID0_MAJ_REVA 0x00000000 // Revision A (initial device) #define SYSCTL_DID0_MAJ_REVB 0x00000100 // Revision B (first base layer // revision) #define SYSCTL_DID0_MAJ_REVC 0x00000200 // Revision C (second base layer // revision) #define SYSCTL_DID0_MIN_M 0x000000FF // Minor Revision #define SYSCTL_DID0_MIN_0 0x00000000 // Initial device, or a major // revision update #define SYSCTL_DID0_MIN_1 0x00000001 // First metal layer change #define SYSCTL_DID0_MIN_2 0x00000002 // Second metal layer change //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DID1 register. // //************************************************************************* #define SYSCTL_DID1_VER_M 0xF0000000 // DID1 Version #define SYSCTL_DID1_VER_0 0x00000000 // Initial DID1 register format // definition, indicating a // Stellaris LM3Snnn device #define SYSCTL_DID1_VER_1 0x10000000 // Second version of the DID1 // register format #define SYSCTL_DID1_FAM_M 0x0F000000 // Family #define SYSCTL_DID1_FAM_STELLARIS \ 0x00000000 // Stellaris family of // microcontollers, that is, all // devices with external part // numbers starting with LM3S #define SYSCTL_DID1_PRTNO_M 0x00FF0000 // Part Number #define SYSCTL_DID1_PRTNO_LM4F120H5QR \ 0x00040000 // LM4F120H5QR #define SYSCTL_DID1_PINCNT_M 0x0000E000 // Package Pin Count #define SYSCTL_DID1_PINCNT_28 0x00000000 // 28-pin package #define SYSCTL_DID1_PINCNT_48 0x00002000 // 48-pin package #define SYSCTL_DID1_PINCNT_100 0x00004000 // 100-pin package #define SYSCTL_DID1_PINCNT_64 0x00006000 // 64-pin package #define SYSCTL_DID1_PINCNT_144 0x00008000 // 144-pin package #define SYSCTL_DID1_PINCNT_157 0x0000A000 // 157-pin package #define SYSCTL_DID1_TEMP_M 0x000000E0 // Temperature Range #define SYSCTL_DID1_TEMP_C 0x00000000 // Commercial temperature range (0C // to 70C) #define SYSCTL_DID1_TEMP_I 0x00000020 // Industrial temperature range // (-40C to 85C) #define SYSCTL_DID1_TEMP_E 0x00000040 // Extended temperature range (-40C // to 105C) #define SYSCTL_DID1_PKG_M 0x00000018 // Package Type #define SYSCTL_DID1_PKG_SOIC 0x00000000 // SOIC package #define SYSCTL_DID1_PKG_QFP 0x00000008 // LQFP package #define SYSCTL_DID1_PKG_BGA 0x00000010 // BGA package #define SYSCTL_DID1_ROHS 0x00000004 // RoHS-Compliance #define SYSCTL_DID1_QUAL_M 0x00000003 // Qualification Status #define SYSCTL_DID1_QUAL_ES 0x00000000 // Engineering Sample (unqualified) #define SYSCTL_DID1_QUAL_PP 0x00000001 // Pilot Production (unqualified) #define SYSCTL_DID1_QUAL_FQ 0x00000002 // Fully Qualified //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC0 register. // //************************************************************************* #define SYSCTL_DC0_SRAMSZ_M 0xFFFF0000 // SRAM Size #define SYSCTL_DC0_SRAMSZ_2KB 0x00070000 // 2 KB of SRAM #define SYSCTL_DC0_SRAMSZ_4KB 0x000F0000 // 4 KB of SRAM #define SYSCTL_DC0_SRAMSZ_6KB 0x00170000 // 6 KB of SRAM #define SYSCTL_DC0_SRAMSZ_8KB 0x001F0000 // 8 KB of SRAM #define SYSCTL_DC0_SRAMSZ_12KB 0x002F0000 // 12 KB of SRAM #define SYSCTL_DC0_SRAMSZ_16KB 0x003F0000 // 16 KB of SRAM #define SYSCTL_DC0_SRAMSZ_20KB 0x004F0000 // 20 KB of SRAM #define SYSCTL_DC0_SRAMSZ_24KB 0x005F0000 // 24 KB of SRAM #define SYSCTL_DC0_SRAMSZ_32KB 0x007F0000 // 32 KB of SRAM #define SYSCTL_DC0_FLASHSZ_M 0x0000FFFF // Flash Size #define SYSCTL_DC0_FLASHSZ_8KB 0x00000003 // 8 KB of Flash #define SYSCTL_DC0_FLASHSZ_16KB 0x00000007 // 16 KB of Flash #define SYSCTL_DC0_FLASHSZ_32KB 0x0000000F // 32 KB of Flash #define SYSCTL_DC0_FLASHSZ_64KB 0x0000001F // 64 KB of Flash #define SYSCTL_DC0_FLASHSZ_96KB 0x0000002F // 96 KB of Flash #define SYSCTL_DC0_FLASHSZ_128K 0x0000003F // 128 KB of Flash #define SYSCTL_DC0_FLASHSZ_192K 0x0000005F // 192 KB of Flash #define SYSCTL_DC0_FLASHSZ_256K 0x0000007F // 256 KB of Flash #define SYSCTL_DC0_SRAMSZ_S 16 // SRAM size shift #define SYSCTL_DC0_FLASHSZ_S 0 // Flash size shift //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC1 register. // //*************************************************************************** #define SYSCTL_DC1_WDT1 0x10000000 // Watchdog Timer1 Present #define SYSCTL_DC1_CAN1 0x02000000 // CAN Module 1 Present #define SYSCTL_DC1_CAN0 0x01000000 // CAN Module 0 Present #define SYSCTL_DC1_PWM1 0x00200000 // PWM Module 1 Present #define SYSCTL_DC1_PWM0 0x00100000 // PWM Module 0 Present #define SYSCTL_DC1_ADC1 0x00020000 // ADC Module 1 Present #define SYSCTL_DC1_ADC0 0x00010000 // ADC Module 0 Present #define SYSCTL_DC1_MINSYSDIV_M 0x0000F000 // System Clock Divider #define SYSCTL_DC1_MINSYSDIV_100 \ 0x00001000 // Divide VCO (400MHZ) by 5 minimum #define SYSCTL_DC1_MINSYSDIV_66 0x00002000 // Divide VCO (400MHZ) by 22 + 2 = // 6 minimum #define SYSCTL_DC1_MINSYSDIV_50 0x00003000 // Specifies a 50-MHz CPU clock // with a PLL divider of 4 #define SYSCTL_DC1_MINSYSDIV_40 0x00004000 // Specifies a 40-MHz CPU clock // with a PLL divider of 5 #define SYSCTL_DC1_MINSYSDIV_25 0x00007000 // Specifies a 25-MHz clock with a // PLL divider of 8 #define SYSCTL_DC1_MINSYSDIV_20 0x00009000 // Specifies a 20-MHz clock with a // PLL divider of 10 #define SYSCTL_DC1_ADC1SPD_M 0x00000C00 // Max ADC1 Speed #define SYSCTL_DC1_ADC1SPD_125K 0x00000000 // 125K samples/second #define SYSCTL_DC1_ADC1SPD_250K 0x00000400 // 250K samples/second #define SYSCTL_DC1_ADC1SPD_500K 0x00000800 // 500K samples/second #define SYSCTL_DC1_ADC1SPD_1M 0x00000C00 // 1M samples/second #define SYSCTL_DC1_ADC0SPD_M 0x00000300 // Max ADC0 Speed #define SYSCTL_DC1_ADC0SPD_125K 0x00000000 // 125K samples/second #define SYSCTL_DC1_ADC0SPD_250K 0x00000100 // 250K samples/second #define SYSCTL_DC1_ADC0SPD_500K 0x00000200 // 500K samples/second #define SYSCTL_DC1_ADC0SPD_1M 0x00000300 // 1M samples/second #define SYSCTL_DC1_MPU 0x00000080 // MPU Present #define SYSCTL_DC1_HIB 0x00000040 // Hibernation Module Present #define SYSCTL_DC1_TEMP 0x00000020 // Temp Sensor Present #define SYSCTL_DC1_PLL 0x00000010 // PLL Present #define SYSCTL_DC1_WDT0 0x00000008 // Watchdog Timer 0 Present #define SYSCTL_DC1_SWO 0x00000004 // SWO Trace Port Present #define SYSCTL_DC1_SWD 0x00000002 // SWD Present #define SYSCTL_DC1_JTAG 0x00000001 // JTAG Present //************************************************************************** // // The following are defines for the bit fields in the SYSCTL_DC2 register. // //************************************************************************* #define SYSCTL_DC2_EPI0 0x40000000 // EPI Module 0 Present #define SYSCTL_DC2_I2S0 0x10000000 // I2S Module 0 Present #define SYSCTL_DC2_COMP2 0x04000000 // Analog Comparator 2 Present #define SYSCTL_DC2_COMP1 0x02000000 // Analog Comparator 1 Present #define SYSCTL_DC2_COMP0 0x01000000 // Analog Comparator 0 Present #define SYSCTL_DC2_TIMER3 0x00080000 // Timer Module 3 Present #define SYSCTL_DC2_TIMER2 0x00040000 // Timer Module 2 Present #define SYSCTL_DC2_TIMER1 0x00020000 // Timer Module 1 Present #define SYSCTL_DC2_TIMER0 0x00010000 // Timer Module 0 Present #define SYSCTL_DC2_I2C1HS 0x00008000 // I2C Module 1 Speed #define SYSCTL_DC2_I2C1 0x00004000 // I2C Module 1 Present #define SYSCTL_DC2_I2C0HS 0x00002000 // I2C Module 0 Speed #define SYSCTL_DC2_I2C0 0x00001000 // I2C Module 0 Present #define SYSCTL_DC2_QEI1 0x00000200 // QEI Module 1 Present #define SYSCTL_DC2_QEI0 0x00000100 // QEI Module 0 Present #define SYSCTL_DC2_SSI1 0x00000020 // SSI Module 1 Present #define SYSCTL_DC2_SSI0 0x00000010 // SSI Module 0 Present #define SYSCTL_DC2_UART2 0x00000004 // UART Module 2 Present #define SYSCTL_DC2_UART1 0x00000002 // UART Module 1 Present #define SYSCTL_DC2_UART0 0x00000001 // UART Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC3 register. // //************************************************************************* #define SYSCTL_DC3_32KHZ 0x80000000 // 32KHz Input Clock Available #define SYSCTL_DC3_CCP5 0x20000000 // CCP5 Pin Present #define SYSCTL_DC3_CCP4 0x10000000 // CCP4 Pin Present #define SYSCTL_DC3_CCP3 0x08000000 // CCP3 Pin Present #define SYSCTL_DC3_CCP2 0x04000000 // CCP2 Pin Present #define SYSCTL_DC3_CCP1 0x02000000 // CCP1 Pin Present #define SYSCTL_DC3_CCP0 0x01000000 // CCP0 Pin Present #define SYSCTL_DC3_ADC0AIN7 0x00800000 // ADC Module 0 AIN7 Pin Present #define SYSCTL_DC3_ADC0AIN6 0x00400000 // ADC Module 0 AIN6 Pin Present #define SYSCTL_DC3_ADC0AIN5 0x00200000 // ADC Module 0 AIN5 Pin Present #define SYSCTL_DC3_ADC0AIN4 0x00100000 // ADC Module 0 AIN4 Pin Present #define SYSCTL_DC3_ADC0AIN3 0x00080000 // ADC Module 0 AIN3 Pin Present #define SYSCTL_DC3_ADC0AIN2 0x00040000 // ADC Module 0 AIN2 Pin Present #define SYSCTL_DC3_ADC0AIN1 0x00020000 // ADC Module 0 AIN1 Pin Present #define SYSCTL_DC3_ADC0AIN0 0x00010000 // ADC Module 0 AIN0 Pin Present #define SYSCTL_DC3_PWMFAULT 0x00008000 // PWM Fault Pin Present #define SYSCTL_DC3_C2O 0x00004000 // C2o Pin Present #define SYSCTL_DC3_C2PLUS 0x00002000 // C2+ Pin Present #define SYSCTL_DC3_C2MINUS 0x00001000 // C2- Pin Present #define SYSCTL_DC3_C1O 0x00000800 // C1o Pin Present #define SYSCTL_DC3_C1PLUS 0x00000400 // C1+ Pin Present #define SYSCTL_DC3_C1MINUS 0x00000200 // C1- Pin Present #define SYSCTL_DC3_C0O 0x00000100 // C0o Pin Present #define SYSCTL_DC3_C0PLUS 0x00000080 // C0+ Pin Present #define SYSCTL_DC3_C0MINUS 0x00000040 // C0- Pin Present #define SYSCTL_DC3_PWM5 0x00000020 // PWM5 Pin Present #define SYSCTL_DC3_PWM4 0x00000010 // PWM4 Pin Present #define SYSCTL_DC3_PWM3 0x00000008 // PWM3 Pin Present #define SYSCTL_DC3_PWM2 0x00000004 // PWM2 Pin Present #define SYSCTL_DC3_PWM1 0x00000002 // PWM1 Pin Present #define SYSCTL_DC3_PWM0 0x00000001 // PWM0 Pin Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC4 register. // //************************************************************************* #define SYSCTL_DC4_EPHY0 0x40000000 // Ethernet PHY Layer 0 Present #define SYSCTL_DC4_EMAC0 0x10000000 // Ethernet MAC Layer 0 Present #define SYSCTL_DC4_E1588 0x01000000 // 1588 Capable #define SYSCTL_DC4_PICAL 0x00040000 // PIOSC Calibrate #define SYSCTL_DC4_CCP7 0x00008000 // CCP7 Pin Present #define SYSCTL_DC4_CCP6 0x00004000 // CCP6 Pin Present #define SYSCTL_DC4_UDMA 0x00002000 // Micro-DMA Module Present #define SYSCTL_DC4_ROM 0x00001000 // Internal Code ROM Present #define SYSCTL_DC4_GPIOJ 0x00000100 // GPIO Port J Present #define SYSCTL_DC4_GPIOH 0x00000080 // GPIO Port H Present #define SYSCTL_DC4_GPIOG 0x00000040 // GPIO Port G Present #define SYSCTL_DC4_GPIOF 0x00000020 // GPIO Port F Present #define SYSCTL_DC4_GPIOE 0x00000010 // GPIO Port E Present #define SYSCTL_DC4_GPIOD 0x00000008 // GPIO Port D Present #define SYSCTL_DC4_GPIOC 0x00000004 // GPIO Port C Present #define SYSCTL_DC4_GPIOB 0x00000002 // GPIO Port B Present #define SYSCTL_DC4_GPIOA 0x00000001 // GPIO Port A Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC5 register. // //************************************************************************* #define SYSCTL_DC5_PWMFAULT3 0x08000000 // PWM Fault 3 Pin Present #define SYSCTL_DC5_PWMFAULT2 0x04000000 // PWM Fault 2 Pin Present #define SYSCTL_DC5_PWMFAULT1 0x02000000 // PWM Fault 1 Pin Present #define SYSCTL_DC5_PWMFAULT0 0x01000000 // PWM Fault 0 Pin Present #define SYSCTL_DC5_PWMEFLT 0x00200000 // PWM Extended Fault Active #define SYSCTL_DC5_PWMESYNC 0x00100000 // PWM Extended SYNC Active #define SYSCTL_DC5_PWM7 0x00000080 // PWM7 Pin Present #define SYSCTL_DC5_PWM6 0x00000040 // PWM6 Pin Present #define SYSCTL_DC5_PWM5 0x00000020 // PWM5 Pin Present #define SYSCTL_DC5_PWM4 0x00000010 // PWM4 Pin Present #define SYSCTL_DC5_PWM3 0x00000008 // PWM3 Pin Present #define SYSCTL_DC5_PWM2 0x00000004 // PWM2 Pin Present #define SYSCTL_DC5_PWM1 0x00000002 // PWM1 Pin Present #define SYSCTL_DC5_PWM0 0x00000001 // PWM0 Pin Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC6 register. // //************************************************************************* #define SYSCTL_DC6_USB0PHY 0x00000010 // USB Module 0 PHY Present #define SYSCTL_DC6_USB0_M 0x00000003 // USB Module 0 Present #define SYSCTL_DC6_USB0_DEV 0x00000001 // USB0 is Device Only #define SYSCTL_DC6_USB0_HOSTDEV 0x00000002 // USB is Device or Host #define SYSCTL_DC6_USB0_OTG 0x00000003 // USB0 is OTG //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC7 register. // //************************************************************************* #define SYSCTL_DC7_DMACH30 0x40000000 // SW #define SYSCTL_DC7_DMACH29 0x20000000 // I2S0_TX / CAN1_TX #define SYSCTL_DC7_DMACH28 0x10000000 // I2S0_RX / CAN1_RX #define SYSCTL_DC7_DMACH27 0x08000000 // CAN1_TX / ADC1_SS3 #define SYSCTL_DC7_DMACH26 0x04000000 // CAN1_RX / ADC1_SS2 #define SYSCTL_DC7_DMACH25 0x02000000 // SSI1_TX / ADC1_SS1 #define SYSCTL_DC7_DMACH24 0x01000000 // SSI1_RX / ADC1_SS0 #define SYSCTL_DC7_DMACH23 0x00800000 // UART1_TX / CAN2_TX #define SYSCTL_DC7_DMACH22 0x00400000 // UART1_RX / CAN2_RX #define SYSCTL_DC7_DMACH21 0x00200000 // Timer1B / EPI0_WFIFO #define SYSCTL_DC7_DMACH20 0x00100000 // Timer1A / EPI0_NBRFIFO #define SYSCTL_DC7_DMACH19 0x00080000 // Timer0B / Timer1B #define SYSCTL_DC7_DMACH18 0x00040000 // Timer0A / Timer1A #define SYSCTL_DC7_DMACH17 0x00020000 // ADC0_SS3 #define SYSCTL_DC7_DMACH16 0x00010000 // ADC0_SS2 #define SYSCTL_DC7_DMACH15 0x00008000 // ADC0_SS1 / Timer2B #define SYSCTL_DC7_DMACH14 0x00004000 // ADC0_SS0 / Timer2A #define SYSCTL_DC7_DMACH13 0x00002000 // CAN0_TX / UART2_TX #define SYSCTL_DC7_DMACH12 0x00001000 // CAN0_RX / UART2_RX #define SYSCTL_DC7_DMACH11 0x00000800 // SSI0_TX / SSI1_TX #define SYSCTL_DC7_DMACH10 0x00000400 // SSI0_RX / SSI1_RX #define SYSCTL_DC7_DMACH9 0x00000200 // UART0_TX / UART1_TX #define SYSCTL_DC7_DMACH8 0x00000100 // UART0_RX / UART1_RX #define SYSCTL_DC7_DMACH7 0x00000080 // ETH_TX / Timer2B #define SYSCTL_DC7_DMACH6 0x00000040 // ETH_RX / Timer2A #define SYSCTL_DC7_DMACH5 0x00000020 // USB_EP3_TX / Timer2B #define SYSCTL_DC7_DMACH4 0x00000010 // USB_EP3_RX / Timer2A #define SYSCTL_DC7_DMACH3 0x00000008 // USB_EP2_TX / Timer3B #define SYSCTL_DC7_DMACH2 0x00000004 // USB_EP2_RX / Timer3A #define SYSCTL_DC7_DMACH1 0x00000002 // USB_EP1_TX / UART2_TX #define SYSCTL_DC7_DMACH0 0x00000001 // USB_EP1_RX / UART2_RX //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC8 register. // //************************************************************************* #define SYSCTL_DC8_ADC1AIN15 0x80000000 // ADC Module 1 AIN15 Pin Present #define SYSCTL_DC8_ADC1AIN14 0x40000000 // ADC Module 1 AIN14 Pin Present #define SYSCTL_DC8_ADC1AIN13 0x20000000 // ADC Module 1 AIN13 Pin Present #define SYSCTL_DC8_ADC1AIN12 0x10000000 // ADC Module 1 AIN12 Pin Present #define SYSCTL_DC8_ADC1AIN11 0x08000000 // ADC Module 1 AIN11 Pin Present #define SYSCTL_DC8_ADC1AIN10 0x04000000 // ADC Module 1 AIN10 Pin Present #define SYSCTL_DC8_ADC1AIN9 0x02000000 // ADC Module 1 AIN9 Pin Present #define SYSCTL_DC8_ADC1AIN8 0x01000000 // ADC Module 1 AIN8 Pin Present #define SYSCTL_DC8_ADC1AIN7 0x00800000 // ADC Module 1 AIN7 Pin Present #define SYSCTL_DC8_ADC1AIN6 0x00400000 // ADC Module 1 AIN6 Pin Present #define SYSCTL_DC8_ADC1AIN5 0x00200000 // ADC Module 1 AIN5 Pin Present #define SYSCTL_DC8_ADC1AIN4 0x00100000 // ADC Module 1 AIN4 Pin Present #define SYSCTL_DC8_ADC1AIN3 0x00080000 // ADC Module 1 AIN3 Pin Present #define SYSCTL_DC8_ADC1AIN2 0x00040000 // ADC Module 1 AIN2 Pin Present #define SYSCTL_DC8_ADC1AIN1 0x00020000 // ADC Module 1 AIN1 Pin Present #define SYSCTL_DC8_ADC1AIN0 0x00010000 // ADC Module 1 AIN0 Pin Present #define SYSCTL_DC8_ADC0AIN15 0x00008000 // ADC Module 0 AIN15 Pin Present #define SYSCTL_DC8_ADC0AIN14 0x00004000 // ADC Module 0 AIN14 Pin Present #define SYSCTL_DC8_ADC0AIN13 0x00002000 // ADC Module 0 AIN13 Pin Present #define SYSCTL_DC8_ADC0AIN12 0x00001000 // ADC Module 0 AIN12 Pin Present #define SYSCTL_DC8_ADC0AIN11 0x00000800 // ADC Module 0 AIN11 Pin Present #define SYSCTL_DC8_ADC0AIN10 0x00000400 // ADC Module 0 AIN10 Pin Present #define SYSCTL_DC8_ADC0AIN9 0x00000200 // ADC Module 0 AIN9 Pin Present #define SYSCTL_DC8_ADC0AIN8 0x00000100 // ADC Module 0 AIN8 Pin Present #define SYSCTL_DC8_ADC0AIN7 0x00000080 // ADC Module 0 AIN7 Pin Present #define SYSCTL_DC8_ADC0AIN6 0x00000040 // ADC Module 0 AIN6 Pin Present #define SYSCTL_DC8_ADC0AIN5 0x00000020 // ADC Module 0 AIN5 Pin Present #define SYSCTL_DC8_ADC0AIN4 0x00000010 // ADC Module 0 AIN4 Pin Present #define SYSCTL_DC8_ADC0AIN3 0x00000008 // ADC Module 0 AIN3 Pin Present #define SYSCTL_DC8_ADC0AIN2 0x00000004 // ADC Module 0 AIN2 Pin Present #define SYSCTL_DC8_ADC0AIN1 0x00000002 // ADC Module 0 AIN1 Pin Present #define SYSCTL_DC8_ADC0AIN0 0x00000001 // ADC Module 0 AIN0 Pin Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PBORCTL register. // //************************************************************************* #define SYSCTL_PBORCTL_BORIOR 0x00000002 // BOR Interrupt or Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRCR0 register. // //************************************************************************* #define SYSCTL_SRCR0_WDT1 0x10000000 // WDT1 Reset Control #define SYSCTL_SRCR0_CAN1 0x02000000 // CAN1 Reset Control #define SYSCTL_SRCR0_CAN0 0x01000000 // CAN0 Reset Control #define SYSCTL_SRCR0_PWM0 0x00100000 // PWM Reset Control #define SYSCTL_SRCR0_ADC1 0x00020000 // ADC1 Reset Control #define SYSCTL_SRCR0_ADC0 0x00010000 // ADC0 Reset Control #define SYSCTL_SRCR0_HIB 0x00000040 // HIB Reset Control #define SYSCTL_SRCR0_WDT0 0x00000008 // WDT0 Reset Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRCR1 register. // //************************************************************************* #define SYSCTL_SRCR1_COMP2 0x04000000 // Analog Comp 2 Reset Control #define SYSCTL_SRCR1_COMP1 0x02000000 // Analog Comp 1 Reset Control #define SYSCTL_SRCR1_COMP0 0x01000000 // Analog Comp 0 Reset Control #define SYSCTL_SRCR1_TIMER3 0x00080000 // Timer 3 Reset Control #define SYSCTL_SRCR1_TIMER2 0x00040000 // Timer 2 Reset Control #define SYSCTL_SRCR1_TIMER1 0x00020000 // Timer 1 Reset Control #define SYSCTL_SRCR1_TIMER0 0x00010000 // Timer 0 Reset Control #define SYSCTL_SRCR1_I2C1 0x00004000 // I2C1 Reset Control #define SYSCTL_SRCR1_I2C0 0x00001000 // I2C0 Reset Control #define SYSCTL_SRCR1_QEI1 0x00000200 // QEI1 Reset Control #define SYSCTL_SRCR1_QEI0 0x00000100 // QEI0 Reset Control #define SYSCTL_SRCR1_SSI1 0x00000020 // SSI1 Reset Control #define SYSCTL_SRCR1_SSI0 0x00000010 // SSI0 Reset Control #define SYSCTL_SRCR1_UART2 0x00000004 // UART2 Reset Control #define SYSCTL_SRCR1_UART1 0x00000002 // UART1 Reset Control #define SYSCTL_SRCR1_UART0 0x00000001 // UART0 Reset Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRCR2 register. // //************************************************************************* #define SYSCTL_SRCR2_USB0 0x00010000 // USB0 Reset Control #define SYSCTL_SRCR2_UDMA 0x00002000 // Micro-DMA Reset Control #define SYSCTL_SRCR2_GPIOJ 0x00000100 // Port J Reset Control #define SYSCTL_SRCR2_GPIOH 0x00000080 // Port H Reset Control #define SYSCTL_SRCR2_GPIOG 0x00000040 // Port G Reset Control #define SYSCTL_SRCR2_GPIOF 0x00000020 // Port F Reset Control #define SYSCTL_SRCR2_GPIOE 0x00000010 // Port E Reset Control #define SYSCTL_SRCR2_GPIOD 0x00000008 // Port D Reset Control #define SYSCTL_SRCR2_GPIOC 0x00000004 // Port C Reset Control #define SYSCTL_SRCR2_GPIOB 0x00000002 // Port B Reset Control #define SYSCTL_SRCR2_GPIOA 0x00000001 // Port A Reset Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RIS register. // //************************************************************************* #define SYSCTL_RIS_MOSCPUPRIS 0x00000100 // MOSC Power Up Raw Interrupt // Status #define SYSCTL_RIS_USBPLLLRIS 0x00000080 // USB PLL Lock Raw Interrupt // Status #define SYSCTL_RIS_PLLLRIS 0x00000040 // PLL Lock Raw Interrupt Status #define SYSCTL_RIS_MOFRIS 0x00000008 // Main Oscillator Fault Raw // Interrupt Status #define SYSCTL_RIS_BORRIS 0x00000002 // Brown-Out Reset Raw Interrupt // Status //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_IMC register. // //************************************************************************* #define SYSCTL_IMC_MOSCPUPIM 0x00000100 // MOSC Power Up Interrupt Mask #define SYSCTL_IMC_USBPLLLIM 0x00000080 // USB PLL Lock Interrupt Mask #define SYSCTL_IMC_PLLLIM 0x00000040 // PLL Lock Interrupt Mask #define SYSCTL_IMC_MOFIM 0x00000008 // Main Oscillator Fault Interrupt // Mask #define SYSCTL_IMC_BORIM 0x00000002 // Brown-Out Reset Interrupt Mask //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_MISC register. // //************************************************************************* #define SYSCTL_MISC_MOSCPUPMIS 0x00000100 // MOSC Power Up Masked Interrupt // Status #define SYSCTL_MISC_USBPLLLMIS 0x00000080 // USB PLL Lock Masked Interrupt // Status #define SYSCTL_MISC_PLLLMIS 0x00000040 // PLL Lock Masked Interrupt Status #define SYSCTL_MISC_MOFMIS 0x00000008 // Main Oscillator Fault Masked // Interrupt Status #define SYSCTL_MISC_BORMIS 0x00000002 // BOR Masked Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RESC register. // //************************************************************************* #define SYSCTL_RESC_MOSCFAIL 0x00010000 // MOSC Failure Reset #define SYSCTL_RESC_WDT1 0x00000020 // Watchdog Timer 1 Reset #define SYSCTL_RESC_SW 0x00000010 // Software Reset #define SYSCTL_RESC_WDT0 0x00000008 // Watchdog Timer 0 Reset #define SYSCTL_RESC_BOR 0x00000004 // Brown-Out Reset #define SYSCTL_RESC_POR 0x00000002 // Power-On Reset #define SYSCTL_RESC_EXT 0x00000001 // External Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCC register. // //************************************************************************* #define SYSCTL_RCC_ACG 0x08000000 // Auto Clock Gating #define SYSCTL_RCC_SYSDIV_M 0x07800000 // System Clock Divisor #define SYSCTL_RCC_USESYSDIV 0x00400000 // Enable System Clock Divider #define SYSCTL_RCC_PWRDN 0x00002000 // PLL Power Down #define SYSCTL_RCC_BYPASS 0x00000800 // PLL Bypass #define SYSCTL_RCC_XTAL_M 0x000007C0 // Crystal Value #define SYSCTL_RCC_XTAL_4MHZ 0x00000180 // 4 MHz #define SYSCTL_RCC_XTAL_4_09MHZ 0x000001C0 // 4.096 MHz #define SYSCTL_RCC_XTAL_4_91MHZ 0x00000200 // 4.9152 MHz #define SYSCTL_RCC_XTAL_5MHZ 0x00000240 // 5 MHz #define SYSCTL_RCC_XTAL_5_12MHZ 0x00000280 // 5.12 MHz #define SYSCTL_RCC_XTAL_6MHZ 0x000002C0 // 6 MHz #define SYSCTL_RCC_XTAL_6_14MHZ 0x00000300 // 6.144 MHz #define SYSCTL_RCC_XTAL_7_37MHZ 0x00000340 // 7.3728 MHz #define SYSCTL_RCC_XTAL_8MHZ 0x00000380 // 8 MHz #define SYSCTL_RCC_XTAL_8_19MHZ 0x000003C0 // 8.192 MHz #define SYSCTL_RCC_XTAL_10MHZ 0x00000400 // 10 MHz #define SYSCTL_RCC_XTAL_12MHZ 0x00000440 // 12 MHz #define SYSCTL_RCC_XTAL_12_2MHZ 0x00000480 // 12.288 MHz #define SYSCTL_RCC_XTAL_13_5MHZ 0x000004C0 // 13.56 MHz #define SYSCTL_RCC_XTAL_14_3MHZ 0x00000500 // 14.31818 MHz #define SYSCTL_RCC_XTAL_16MHZ 0x00000540 // 16 MHz #define SYSCTL_RCC_XTAL_16_3MHZ 0x00000580 // 16.384 MHz #define SYSCTL_RCC_XTAL_18MHZ 0x000005C0 // 18.0 MHz #define SYSCTL_RCC_XTAL_20MHZ 0x00000600 // 20.0 MHz #define SYSCTL_RCC_XTAL_24MHZ 0x00000640 // 24.0 MHz #define SYSCTL_RCC_XTAL_25MHZ 0x00000680 // 25.0 MHz #define SYSCTL_RCC_OSCSRC_M 0x00000030 // Oscillator Source #define SYSCTL_RCC_OSCSRC_MAIN 0x00000000 // MOSC #define SYSCTL_RCC_OSCSRC_INT 0x00000010 // IOSC #define SYSCTL_RCC_OSCSRC_INT4 0x00000020 // IOSC/4 #define SYSCTL_RCC_OSCSRC_30 0x00000030 // 30 kHz #define SYSCTL_RCC_IOSCDIS 0x00000002 // Internal Oscillator Disable #define SYSCTL_RCC_MOSCDIS 0x00000001 // Main Oscillator Disable #define SYSCTL_RCC_SYSDIV_S 23 //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_GPIOHBCTL // register. // //************************************************************************* #define SYSCTL_GPIOHBCTL_PORTF 0x00000020 // Port F Advanced High-Performance // Bus #define SYSCTL_GPIOHBCTL_PORTE 0x00000010 // Port E Advanced High-Performance // Bus #define SYSCTL_GPIOHBCTL_PORTD 0x00000008 // Port D Advanced High-Performance // Bus #define SYSCTL_GPIOHBCTL_PORTC 0x00000004 // Port C Advanced High-Performance // Bus #define SYSCTL_GPIOHBCTL_PORTB 0x00000002 // Port B Advanced High-Performance // Bus #define SYSCTL_GPIOHBCTL_PORTA 0x00000001 // Port A Advanced High-Performance // Bus //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCC2 register. // //************************************************************************* #define SYSCTL_RCC2_USERCC2 0x80000000 // Use RCC2 #define SYSCTL_RCC2_DIV400 0x40000000 // Divide PLL as 400 MHz vs. 200 // MHz #define SYSCTL_RCC2_SYSDIV2_M 0x1F800000 // System Clock Divisor 2 #define SYSCTL_RCC2_SYSDIV2_2 0x00800000 // System clock /2 #define SYSCTL_RCC2_SYSDIV2_3 0x01000000 // System clock /3 #define SYSCTL_RCC2_SYSDIV2_4 0x01800000 // System clock /4 #define SYSCTL_RCC2_SYSDIV2_5 0x02000000 // System clock /5 #define SYSCTL_RCC2_SYSDIV2_6 0x02800000 // System clock /6 #define SYSCTL_RCC2_SYSDIV2_7 0x03000000 // System clock /7 #define SYSCTL_RCC2_SYSDIV2_8 0x03800000 // System clock /8 #define SYSCTL_RCC2_SYSDIV2_9 0x04000000 // System clock /9 #define SYSCTL_RCC2_SYSDIV2_10 0x04800000 // System clock /10 #define SYSCTL_RCC2_SYSDIV2_11 0x05000000 // System clock /11 #define SYSCTL_RCC2_SYSDIV2_12 0x05800000 // System clock /12 #define SYSCTL_RCC2_SYSDIV2_13 0x06000000 // System clock /13 #define SYSCTL_RCC2_SYSDIV2_14 0x06800000 // System clock /14 #define SYSCTL_RCC2_SYSDIV2_15 0x07000000 // System clock /15 #define SYSCTL_RCC2_SYSDIV2_16 0x07800000 // System clock /16 #define SYSCTL_RCC2_SYSDIV2_17 0x08000000 // System clock /17 #define SYSCTL_RCC2_SYSDIV2_18 0x08800000 // System clock /18 #define SYSCTL_RCC2_SYSDIV2_19 0x09000000 // System clock /19 #define SYSCTL_RCC2_SYSDIV2_20 0x09800000 // System clock /20 #define SYSCTL_RCC2_SYSDIV2_21 0x0A000000 // System clock /21 #define SYSCTL_RCC2_SYSDIV2_22 0x0A800000 // System clock /22 #define SYSCTL_RCC2_SYSDIV2_23 0x0B000000 // System clock /23 #define SYSCTL_RCC2_SYSDIV2_24 0x0B800000 // System clock /24 #define SYSCTL_RCC2_SYSDIV2_25 0x0C000000 // System clock /25 #define SYSCTL_RCC2_SYSDIV2_26 0x0C800000 // System clock /26 #define SYSCTL_RCC2_SYSDIV2_27 0x0D000000 // System clock /27 #define SYSCTL_RCC2_SYSDIV2_28 0x0D800000 // System clock /28 #define SYSCTL_RCC2_SYSDIV2_29 0x0E000000 // System clock /29 #define SYSCTL_RCC2_SYSDIV2_30 0x0E800000 // System clock /30 #define SYSCTL_RCC2_SYSDIV2_31 0x0F000000 // System clock /31 #define SYSCTL_RCC2_SYSDIV2_32 0x0F800000 // System clock /32 #define SYSCTL_RCC2_SYSDIV2_33 0x10000000 // System clock /33 #define SYSCTL_RCC2_SYSDIV2_34 0x10800000 // System clock /34 #define SYSCTL_RCC2_SYSDIV2_35 0x11000000 // System clock /35 #define SYSCTL_RCC2_SYSDIV2_36 0x11800000 // System clock /36 #define SYSCTL_RCC2_SYSDIV2_37 0x12000000 // System clock /37 #define SYSCTL_RCC2_SYSDIV2_38 0x12800000 // System clock /38 #define SYSCTL_RCC2_SYSDIV2_39 0x13000000 // System clock /39 #define SYSCTL_RCC2_SYSDIV2_40 0x13800000 // System clock /40 #define SYSCTL_RCC2_SYSDIV2_41 0x14000000 // System clock /41 #define SYSCTL_RCC2_SYSDIV2_42 0x14800000 // System clock /42 #define SYSCTL_RCC2_SYSDIV2_43 0x15000000 // System clock /43 #define SYSCTL_RCC2_SYSDIV2_44 0x15800000 // System clock /44 #define SYSCTL_RCC2_SYSDIV2_45 0x16000000 // System clock /45 #define SYSCTL_RCC2_SYSDIV2_46 0x16800000 // System clock /46 #define SYSCTL_RCC2_SYSDIV2_47 0x17000000 // System clock /47 #define SYSCTL_RCC2_SYSDIV2_48 0x17800000 // System clock /48 #define SYSCTL_RCC2_SYSDIV2_49 0x18000000 // System clock /49 #define SYSCTL_RCC2_SYSDIV2_50 0x18800000 // System clock /50 #define SYSCTL_RCC2_SYSDIV2_51 0x19000000 // System clock /51 #define SYSCTL_RCC2_SYSDIV2_52 0x19800000 // System clock /52 #define SYSCTL_RCC2_SYSDIV2_53 0x1A000000 // System clock /53 #define SYSCTL_RCC2_SYSDIV2_54 0x1A800000 // System clock /54 #define SYSCTL_RCC2_SYSDIV2_55 0x1B000000 // System clock /55 #define SYSCTL_RCC2_SYSDIV2_56 0x1B800000 // System clock /56 #define SYSCTL_RCC2_SYSDIV2_57 0x1C000000 // System clock /57 #define SYSCTL_RCC2_SYSDIV2_58 0x1C800000 // System clock /58 #define SYSCTL_RCC2_SYSDIV2_59 0x1D000000 // System clock /59 #define SYSCTL_RCC2_SYSDIV2_60 0x1D800000 // System clock /60 #define SYSCTL_RCC2_SYSDIV2_61 0x1E000000 // System clock /61 #define SYSCTL_RCC2_SYSDIV2_62 0x1E800000 // System clock /62 #define SYSCTL_RCC2_SYSDIV2_63 0x1F000000 // System clock /63 #define SYSCTL_RCC2_SYSDIV2_64 0x1F800000 // System clock /64 #define SYSCTL_RCC2_SYSDIV2LSB 0x00400000 // Additional LSB for SYSDIV2 #define SYSCTL_RCC2_USBPWRDN 0x00004000 // Power-Down USB PLL #define SYSCTL_RCC2_PWRDN2 0x00002000 // Power-Down PLL 2 #define SYSCTL_RCC2_BYPASS2 0x00000800 // PLL Bypass 2 #define SYSCTL_RCC2_OSCSRC2_M 0x00000070 // Oscillator Source 2 #define SYSCTL_RCC2_OSCSRC2_MO 0x00000000 // MOSC #define SYSCTL_RCC2_OSCSRC2_IO 0x00000010 // PIOSC #define SYSCTL_RCC2_OSCSRC2_IO4 0x00000020 // PIOSC/4 #define SYSCTL_RCC2_OSCSRC2_30 0x00000030 // 30 kHz #define SYSCTL_RCC2_OSCSRC2_32 0x00000070 // 32.768 kHz #define SYSCTL_RCC2_SYSDIV2_S 23 //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_MOSCCTL register. // //************************************************************************* #define SYSCTL_MOSCCTL_NOXTAL 0x00000004 // No Crystal Connected #define SYSCTL_MOSCCTL_MOSCIM 0x00000002 // MOSC Failure Action #define SYSCTL_MOSCCTL_CVAL 0x00000001 // Clock Validation for MOSC //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGC0 register. // //************************************************************************* #define SYSCTL_RCGC0_WDT1 0x10000000 // WDT1 Clock Gating Control #define SYSCTL_RCGC0_CAN1 0x02000000 // CAN1 Clock Gating Control #define SYSCTL_RCGC0_CAN0 0x01000000 // CAN0 Clock Gating Control #define SYSCTL_RCGC0_PWM0 0x00100000 // PWM Clock Gating Control #define SYSCTL_RCGC0_ADC1 0x00020000 // ADC1 Clock Gating Control #define SYSCTL_RCGC0_ADC0 0x00010000 // ADC0 Clock Gating Control #define SYSCTL_RCGC0_ADC1SPD_M 0x00000C00 // ADC1 Sample Speed #define SYSCTL_RCGC0_ADC1SPD_125K \ 0x00000000 // 125K samples/second #define SYSCTL_RCGC0_ADC1SPD_250K \ 0x00000400 // 250K samples/second #define SYSCTL_RCGC0_ADC1SPD_500K \ 0x00000800 // 500K samples/second #define SYSCTL_RCGC0_ADC1SPD_1M 0x00000C00 // 1M samples/second #define SYSCTL_RCGC0_ADC0SPD_M 0x00000300 // ADC0 Sample Speed #define SYSCTL_RCGC0_ADC0SPD_125K \ 0x00000000 // 125K samples/second #define SYSCTL_RCGC0_ADC0SPD_250K \ 0x00000100 // 250K samples/second #define SYSCTL_RCGC0_ADC0SPD_500K \ 0x00000200 // 500K samples/second #define SYSCTL_RCGC0_ADC0SPD_1M 0x00000300 // 1M samples/second #define SYSCTL_RCGC0_HIB 0x00000040 // HIB Clock Gating Control #define SYSCTL_RCGC0_WDT0 0x00000008 // WDT0 Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGC1 register. // //************************************************************************* #define SYSCTL_RCGC1_COMP2 0x04000000 // Analog Comparator 2 Clock Gating #define SYSCTL_RCGC1_COMP1 0x02000000 // Analog Comparator 1 Clock Gating #define SYSCTL_RCGC1_COMP0 0x01000000 // Analog Comparator 0 Clock Gating #define SYSCTL_RCGC1_TIMER3 0x00080000 // Timer 3 Clock Gating Control #define SYSCTL_RCGC1_TIMER2 0x00040000 // Timer 2 Clock Gating Control #define SYSCTL_RCGC1_TIMER1 0x00020000 // Timer 1 Clock Gating Control #define SYSCTL_RCGC1_TIMER0 0x00010000 // Timer 0 Clock Gating Control #define SYSCTL_RCGC1_I2C1 0x00004000 // I2C1 Clock Gating Control #define SYSCTL_RCGC1_I2C0 0x00001000 // I2C0 Clock Gating Control #define SYSCTL_RCGC1_QEI1 0x00000200 // QEI1 Clock Gating Control #define SYSCTL_RCGC1_QEI0 0x00000100 // QEI0 Clock Gating Control #define SYSCTL_RCGC1_SSI1 0x00000020 // SSI1 Clock Gating Control #define SYSCTL_RCGC1_SSI0 0x00000010 // SSI0 Clock Gating Control #define SYSCTL_RCGC1_UART2 0x00000004 // UART2 Clock Gating Control #define SYSCTL_RCGC1_UART1 0x00000002 // UART1 Clock Gating Control #define SYSCTL_RCGC1_UART0 0x00000001 // UART0 Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGC2 register. // //************************************************************************* #define SYSCTL_RCGC2_USB0 0x00010000 // USB0 Clock Gating Control #define SYSCTL_RCGC2_UDMA 0x00002000 // Micro-DMA Clock Gating Control #define SYSCTL_RCGC2_GPIOJ 0x00000100 // Port J Clock Gating Control #define SYSCTL_RCGC2_GPIOH 0x00000080 // Port H Clock Gating Control #define SYSCTL_RCGC2_GPIOG 0x00000040 // Port G Clock Gating Control #define SYSCTL_RCGC2_GPIOF 0x00000020 // Port F Clock Gating Control #define SYSCTL_RCGC2_GPIOE 0x00000010 // Port E Clock Gating Control #define SYSCTL_RCGC2_GPIOD 0x00000008 // Port D Clock Gating Control #define SYSCTL_RCGC2_GPIOC 0x00000004 // Port C Clock Gating Control #define SYSCTL_RCGC2_GPIOB 0x00000002 // Port B Clock Gating Control #define SYSCTL_RCGC2_GPIOA 0x00000001 // Port A Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGC0 register. // //************************************************************************* #define SYSCTL_SCGC0_WDT1 0x10000000 // WDT1 Clock Gating Control #define SYSCTL_SCGC0_CAN1 0x02000000 // CAN1 Clock Gating Control #define SYSCTL_SCGC0_CAN0 0x01000000 // CAN0 Clock Gating Control #define SYSCTL_SCGC0_PWM0 0x00100000 // PWM Clock Gating Control #define SYSCTL_SCGC0_ADC1 0x00020000 // ADC1 Clock Gating Control #define SYSCTL_SCGC0_ADC0 0x00010000 // ADC0 Clock Gating Control #define SYSCTL_SCGC0_HIB 0x00000040 // HIB Clock Gating Control #define SYSCTL_SCGC0_WDT0 0x00000008 // WDT0 Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGC1 register. // //************************************************************************* #define SYSCTL_SCGC1_COMP2 0x04000000 // Analog Comparator 2 Clock Gating #define SYSCTL_SCGC1_COMP1 0x02000000 // Analog Comparator 1 Clock Gating #define SYSCTL_SCGC1_COMP0 0x01000000 // Analog Comparator 0 Clock Gating #define SYSCTL_SCGC1_TIMER3 0x00080000 // Timer 3 Clock Gating Control #define SYSCTL_SCGC1_TIMER2 0x00040000 // Timer 2 Clock Gating Control #define SYSCTL_SCGC1_TIMER1 0x00020000 // Timer 1 Clock Gating Control #define SYSCTL_SCGC1_TIMER0 0x00010000 // Timer 0 Clock Gating Control #define SYSCTL_SCGC1_I2C1 0x00004000 // I2C1 Clock Gating Control #define SYSCTL_SCGC1_I2C0 0x00001000 // I2C0 Clock Gating Control #define SYSCTL_SCGC1_QEI1 0x00000200 // QEI1 Clock Gating Control #define SYSCTL_SCGC1_QEI0 0x00000100 // QEI0 Clock Gating Control #define SYSCTL_SCGC1_SSI1 0x00000020 // SSI1 Clock Gating Control #define SYSCTL_SCGC1_SSI0 0x00000010 // SSI0 Clock Gating Control #define SYSCTL_SCGC1_UART2 0x00000004 // UART2 Clock Gating Control #define SYSCTL_SCGC1_UART1 0x00000002 // UART1 Clock Gating Control #define SYSCTL_SCGC1_UART0 0x00000001 // UART0 Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGC2 register. // //************************************************************************* #define SYSCTL_SCGC2_USB0 0x00010000 // USB0 Clock Gating Control #define SYSCTL_SCGC2_UDMA 0x00002000 // Micro-DMA Clock Gating Control #define SYSCTL_SCGC2_GPIOJ 0x00000100 // Port J Clock Gating Control #define SYSCTL_SCGC2_GPIOH 0x00000080 // Port H Clock Gating Control #define SYSCTL_SCGC2_GPIOG 0x00000040 // Port G Clock Gating Control #define SYSCTL_SCGC2_GPIOF 0x00000020 // Port F Clock Gating Control #define SYSCTL_SCGC2_GPIOE 0x00000010 // Port E Clock Gating Control #define SYSCTL_SCGC2_GPIOD 0x00000008 // Port D Clock Gating Control #define SYSCTL_SCGC2_GPIOC 0x00000004 // Port C Clock Gating Control #define SYSCTL_SCGC2_GPIOB 0x00000002 // Port B Clock Gating Control #define SYSCTL_SCGC2_GPIOA 0x00000001 // Port A Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGC0 register. // //************************************************************************* #define SYSCTL_DCGC0_WDT1 0x10000000 // WDT1 Clock Gating Control #define SYSCTL_DCGC0_CAN1 0x02000000 // CAN1 Clock Gating Control #define SYSCTL_DCGC0_CAN0 0x01000000 // CAN0 Clock Gating Control #define SYSCTL_DCGC0_PWM0 0x00100000 // PWM Clock Gating Control #define SYSCTL_DCGC0_ADC1 0x00020000 // ADC1 Clock Gating Control #define SYSCTL_DCGC0_ADC0 0x00010000 // ADC0 Clock Gating Control #define SYSCTL_DCGC0_HIB 0x00000040 // HIB Clock Gating Control #define SYSCTL_DCGC0_WDT0 0x00000008 // WDT0 Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGC1 register. // //************************************************************************* #define SYSCTL_DCGC1_COMP2 0x04000000 // Analog Comparator 2 Clock Gating #define SYSCTL_DCGC1_COMP1 0x02000000 // Analog Comparator 1 Clock Gating #define SYSCTL_DCGC1_COMP0 0x01000000 // Analog Comparator 0 Clock Gating #define SYSCTL_DCGC1_TIMER3 0x00080000 // Timer 3 Clock Gating Control #define SYSCTL_DCGC1_TIMER2 0x00040000 // Timer 2 Clock Gating Control #define SYSCTL_DCGC1_TIMER1 0x00020000 // Timer 1 Clock Gating Control #define SYSCTL_DCGC1_TIMER0 0x00010000 // Timer 0 Clock Gating Control #define SYSCTL_DCGC1_I2C1 0x00004000 // I2C1 Clock Gating Control #define SYSCTL_DCGC1_I2C0 0x00001000 // I2C0 Clock Gating Control #define SYSCTL_DCGC1_QEI1 0x00000200 // QEI1 Clock Gating Control #define SYSCTL_DCGC1_QEI0 0x00000100 // QEI0 Clock Gating Control #define SYSCTL_DCGC1_SSI1 0x00000020 // SSI1 Clock Gating Control #define SYSCTL_DCGC1_SSI0 0x00000010 // SSI0 Clock Gating Control #define SYSCTL_DCGC1_UART2 0x00000004 // UART2 Clock Gating Control #define SYSCTL_DCGC1_UART1 0x00000002 // UART1 Clock Gating Control #define SYSCTL_DCGC1_UART0 0x00000001 // UART0 Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGC2 register. // //************************************************************************* #define SYSCTL_DCGC2_USB0 0x00010000 // USB0 Clock Gating Control #define SYSCTL_DCGC2_UDMA 0x00002000 // Micro-DMA Clock Gating Control #define SYSCTL_DCGC2_GPIOJ 0x00000100 // Port J Clock Gating Control #define SYSCTL_DCGC2_GPIOH 0x00000080 // Port H Clock Gating Control #define SYSCTL_DCGC2_GPIOG 0x00000040 // Port G Clock Gating Control #define SYSCTL_DCGC2_GPIOF 0x00000020 // Port F Clock Gating Control #define SYSCTL_DCGC2_GPIOE 0x00000010 // Port E Clock Gating Control #define SYSCTL_DCGC2_GPIOD 0x00000008 // Port D Clock Gating Control #define SYSCTL_DCGC2_GPIOC 0x00000004 // Port C Clock Gating Control #define SYSCTL_DCGC2_GPIOB 0x00000002 // Port B Clock Gating Control #define SYSCTL_DCGC2_GPIOA 0x00000001 // Port A Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DSLPCLKCFG // register. // //************************************************************************* #define SYSCTL_DSLPCLKCFG_D_M 0x1F800000 // Divider Field Override #define SYSCTL_DSLPCLKCFG_D_1 0x00000000 // System clock /1 #define SYSCTL_DSLPCLKCFG_D_2 0x00800000 // System clock /2 #define SYSCTL_DSLPCLKCFG_D_3 0x01000000 // System clock /3 #define SYSCTL_DSLPCLKCFG_D_4 0x01800000 // System clock /4 #define SYSCTL_DSLPCLKCFG_D_64 0x1F800000 // System clock /64 #define SYSCTL_DSLPCLKCFG_O_M 0x00000070 // Clock Source #define SYSCTL_DSLPCLKCFG_O_IGN 0x00000000 // MOSC #define SYSCTL_DSLPCLKCFG_O_IO 0x00000010 // PIOSC #define SYSCTL_DSLPCLKCFG_O_30 0x00000030 // 30 kHz #define SYSCTL_DSLPCLKCFG_O_32 0x00000070 // 32.768 kHz //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SYSPROP register. // //************************************************************************* #define SYSCTL_SYSPROP_FPU 0x00000001 // FPU Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PIOSCCAL // register. // //************************************************************************* #define SYSCTL_PIOSCCAL_UTEN 0x80000000 // Use User Trim Value #define SYSCTL_PIOSCCAL_CAL 0x00000200 // Start Calibration #define SYSCTL_PIOSCCAL_UPDATE 0x00000100 // Update Trim #define SYSCTL_PIOSCCAL_UT_M 0x0000007F // User Trim Value #define SYSCTL_PIOSCCAL_UT_S 0 //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PIOSCSTAT // register. // //************************************************************************* #define SYSCTL_PIOSCSTAT_DT_M 0x007F0000 // Default Trim Value #define SYSCTL_PIOSCSTAT_CR_M 0x00000300 // Calibration Result #define SYSCTL_PIOSCSTAT_CRNONE 0x00000000 // Calibration has not been // attempted #define SYSCTL_PIOSCSTAT_CRPASS 0x00000100 // The last calibration operation // completed to meet 1% accuracy #define SYSCTL_PIOSCSTAT_CRFAIL 0x00000200 // The last calibration operation // failed to meet 1% accuracy #define SYSCTL_PIOSCSTAT_CT_M 0x0000007F // Calibration Trim Value #define SYSCTL_PIOSCSTAT_DT_S 16 #define SYSCTL_PIOSCSTAT_CT_S 0 //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PLLFREQ0 // register. // //************************************************************************* #define SYSCTL_PLLFREQ0_MFRAC_M 0x000FFC00 // PLL M Fractional Value #define SYSCTL_PLLFREQ0_MINT_M 0x000003FF // PLL M Integer Value #define SYSCTL_PLLFREQ0_MFRAC_S 10 #define SYSCTL_PLLFREQ0_MINT_S 0 //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PLLFREQ1 // register. // //************************************************************************* #define SYSCTL_PLLFREQ1_Q_M 0x00001F00 // PLL Q Value #define SYSCTL_PLLFREQ1_N_M 0x0000001F // PLL N Value #define SYSCTL_PLLFREQ1_Q_S 8 #define SYSCTL_PLLFREQ1_N_S 0 //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PLLSTAT register. // //************************************************************************* #define SYSCTL_PLLSTAT_LOCK 0x00000001 // PLL Lock //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DC9 register. // //************************************************************************* #define SYSCTL_DC9_ADC1DC7 0x00800000 // ADC1 DC7 Present #define SYSCTL_DC9_ADC1DC6 0x00400000 // ADC1 DC6 Present #define SYSCTL_DC9_ADC1DC5 0x00200000 // ADC1 DC5 Present #define SYSCTL_DC9_ADC1DC4 0x00100000 // ADC1 DC4 Present #define SYSCTL_DC9_ADC1DC3 0x00080000 // ADC1 DC3 Present #define SYSCTL_DC9_ADC1DC2 0x00040000 // ADC1 DC2 Present #define SYSCTL_DC9_ADC1DC1 0x00020000 // ADC1 DC1 Present #define SYSCTL_DC9_ADC1DC0 0x00010000 // ADC1 DC0 Present #define SYSCTL_DC9_ADC0DC7 0x00000080 // ADC0 DC7 Present #define SYSCTL_DC9_ADC0DC6 0x00000040 // ADC0 DC6 Present #define SYSCTL_DC9_ADC0DC5 0x00000020 // ADC0 DC5 Present #define SYSCTL_DC9_ADC0DC4 0x00000010 // ADC0 DC4 Present #define SYSCTL_DC9_ADC0DC3 0x00000008 // ADC0 DC3 Present #define SYSCTL_DC9_ADC0DC2 0x00000004 // ADC0 DC2 Present #define SYSCTL_DC9_ADC0DC1 0x00000002 // ADC0 DC1 Present #define SYSCTL_DC9_ADC0DC0 0x00000001 // ADC0 DC0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_NVMSTAT register. // //************************************************************************* #define SYSCTL_NVMSTAT_TPSW 0x00000010 // Third Party Software Present #define SYSCTL_NVMSTAT_FWB 0x00000001 // 32 Word Flash Write Buffer // Active //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPWD register. // //************************************************************************* #define SYSCTL_PPWD_P1 0x00000002 // Watchdog Timer 1 Present #define SYSCTL_PPWD_P0 0x00000001 // Watchdog Timer 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPTIMER register. // //************************************************************************* #define SYSCTL_PPTIMER_P5 0x00000020 // Timer 5 Present #define SYSCTL_PPTIMER_P4 0x00000010 // Timer 4 Present #define SYSCTL_PPTIMER_P3 0x00000008 // Timer 3 Present #define SYSCTL_PPTIMER_P2 0x00000004 // Timer 2 Present #define SYSCTL_PPTIMER_P1 0x00000002 // Timer 1 Present #define SYSCTL_PPTIMER_P0 0x00000001 // Timer 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPGPIO register. // //************************************************************************* #define SYSCTL_PPGPIO_P14 0x00004000 // GPIO Port Q Present #define SYSCTL_PPGPIO_P13 0x00002000 // GPIO Port P Present #define SYSCTL_PPGPIO_P12 0x00001000 // GPIO Port N Present #define SYSCTL_PPGPIO_P11 0x00000800 // GPIO Port M Present #define SYSCTL_PPGPIO_P10 0x00000400 // GPIO Port L Present #define SYSCTL_PPGPIO_P9 0x00000200 // GPIO Port K Present #define SYSCTL_PPGPIO_P8 0x00000100 // GPIO Port J Present #define SYSCTL_PPGPIO_P7 0x00000080 // GPIO Port H Present #define SYSCTL_PPGPIO_P6 0x00000040 // GPIO Port G Present #define SYSCTL_PPGPIO_P5 0x00000020 // GPIO Port F Present #define SYSCTL_PPGPIO_P4 0x00000010 // GPIO Port E Present #define SYSCTL_PPGPIO_P3 0x00000008 // GPIO Port D Present #define SYSCTL_PPGPIO_P2 0x00000004 // GPIO Port C Present #define SYSCTL_PPGPIO_P1 0x00000002 // GPIO Port B Present #define SYSCTL_PPGPIO_P0 0x00000001 // GPIO Port A Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPDMA register. // //************************************************************************* #define SYSCTL_PPDMA_P0 0x00000001 // uDMA Module Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPHIB register. // //************************************************************************* #define SYSCTL_PPHIB_P0 0x00000001 // Hibernation Module Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPUART register. // //************************************************************************* #define SYSCTL_PPUART_P7 0x00000080 // UART Module 7 Present #define SYSCTL_PPUART_P6 0x00000040 // UART Module 6 Present #define SYSCTL_PPUART_P5 0x00000020 // UART Module 5 Present #define SYSCTL_PPUART_P4 0x00000010 // UART Module 4 Present #define SYSCTL_PPUART_P3 0x00000008 // UART Module 3 Present #define SYSCTL_PPUART_P2 0x00000004 // UART Module 2 Present #define SYSCTL_PPUART_P1 0x00000002 // UART Module 1 Present #define SYSCTL_PPUART_P0 0x00000001 // UART Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPSSI register. // //************************************************************************* #define SYSCTL_PPSSI_P3 0x00000008 // SSI Module 3 Present #define SYSCTL_PPSSI_P2 0x00000004 // SSI Module 2 Present #define SYSCTL_PPSSI_P1 0x00000002 // SSI Module 1 Present #define SYSCTL_PPSSI_P0 0x00000001 // SSI Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPI2C register. // //************************************************************************* #define SYSCTL_PPI2C_P5 0x00000020 // I2C Module 5 Present #define SYSCTL_PPI2C_P4 0x00000010 // I2C Module 4 Present #define SYSCTL_PPI2C_P3 0x00000008 // I2C Module 3 Present #define SYSCTL_PPI2C_P2 0x00000004 // I2C Module 2 Present #define SYSCTL_PPI2C_P1 0x00000002 // I2C Module 1 Present #define SYSCTL_PPI2C_P0 0x00000001 // I2C Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPUSB register. // //************************************************************************* #define SYSCTL_PPUSB_P0 0x00000001 // USB Module Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPCAN register. // //************************************************************************* #define SYSCTL_PPCAN_P1 0x00000002 // CAN Module 1 Present #define SYSCTL_PPCAN_P0 0x00000001 // CAN Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPADC register. // //************************************************************************* #define SYSCTL_PPADC_P1 0x00000002 // ADC Module 1 Present #define SYSCTL_PPADC_P0 0x00000001 // ADC Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPACMP register. // //************************************************************************* #define SYSCTL_PPACMP_P0 0x00000001 // Analog Comparator Module Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPPWM register. // //************************************************************************* #define SYSCTL_PPPWM_P1 0x00000002 // PWM Module 1 Present #define SYSCTL_PPPWM_P0 0x00000001 // PWM Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPQEI register. // //************************************************************************* #define SYSCTL_PPQEI_P1 0x00000002 // QEI Module 1 Present #define SYSCTL_PPQEI_P0 0x00000001 // QEI Module 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPEEPROM // register. // //************************************************************************* #define SYSCTL_PPEEPROM_P0 0x00000001 // EEPROM Module Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PPWTIMER // register. // //************************************************************************* #define SYSCTL_PPWTIMER_P5 0x00000020 // Wide Timer 5 Present #define SYSCTL_PPWTIMER_P4 0x00000010 // Wide Timer 4 Present #define SYSCTL_PPWTIMER_P3 0x00000008 // Wide Timer 3 Present #define SYSCTL_PPWTIMER_P2 0x00000004 // Wide Timer 2 Present #define SYSCTL_PPWTIMER_P1 0x00000002 // Wide Timer 1 Present #define SYSCTL_PPWTIMER_P0 0x00000001 // Wide Timer 0 Present //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRWD register. // //************************************************************************* #define SYSCTL_SRWD_R1 0x00000002 // Watchdog Timer 1 Software Reset #define SYSCTL_SRWD_R0 0x00000001 // Watchdog Timer 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRTIMER register. // //************************************************************************* #define SYSCTL_SRTIMER_R5 0x00000020 // Timer 5 Software Reset #define SYSCTL_SRTIMER_R4 0x00000010 // Timer 4 Software Reset #define SYSCTL_SRTIMER_R3 0x00000008 // Timer 3 Software Reset #define SYSCTL_SRTIMER_R2 0x00000004 // Timer 2 Software Reset #define SYSCTL_SRTIMER_R1 0x00000002 // Timer 1 Software Reset #define SYSCTL_SRTIMER_R0 0x00000001 // Timer 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRGPIO register. // //************************************************************************* #define SYSCTL_SRGPIO_R14 0x00004000 // GPIO Port Q Software Reset #define SYSCTL_SRGPIO_R13 0x00002000 // GPIO Port P Software Reset #define SYSCTL_SRGPIO_R12 0x00001000 // GPIO Port N Software Reset #define SYSCTL_SRGPIO_R11 0x00000800 // GPIO Port M Software Reset #define SYSCTL_SRGPIO_R10 0x00000400 // GPIO Port L Software Reset #define SYSCTL_SRGPIO_R9 0x00000200 // GPIO Port K Software Reset #define SYSCTL_SRGPIO_R8 0x00000100 // GPIO Port J Software Reset #define SYSCTL_SRGPIO_R7 0x00000080 // GPIO Port H Software Reset #define SYSCTL_SRGPIO_R6 0x00000040 // GPIO Port G Software Reset #define SYSCTL_SRGPIO_R5 0x00000020 // GPIO Port F Software Reset #define SYSCTL_SRGPIO_R4 0x00000010 // GPIO Port E Software Reset #define SYSCTL_SRGPIO_R3 0x00000008 // GPIO Port D Software Reset #define SYSCTL_SRGPIO_R2 0x00000004 // GPIO Port C Software Reset #define SYSCTL_SRGPIO_R1 0x00000002 // GPIO Port B Software Reset #define SYSCTL_SRGPIO_R0 0x00000001 // GPIO Port A Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRDMA register. // //************************************************************************* #define SYSCTL_SRDMA_R0 0x00000001 // uDMA Module Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRHIB register. // //************************************************************************* #define SYSCTL_SRHIB_R0 0x00000001 // Hibernation Module Software // Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRUART register. // //************************************************************************* #define SYSCTL_SRUART_R7 0x00000080 // UART Module 7 Software Reset #define SYSCTL_SRUART_R6 0x00000040 // UART Module 6 Software Reset #define SYSCTL_SRUART_R5 0x00000020 // UART Module 5 Software Reset #define SYSCTL_SRUART_R4 0x00000010 // UART Module 4 Software Reset #define SYSCTL_SRUART_R3 0x00000008 // UART Module 3 Software Reset #define SYSCTL_SRUART_R2 0x00000004 // UART Module 2 Software Reset #define SYSCTL_SRUART_R1 0x00000002 // UART Module 1 Software Reset #define SYSCTL_SRUART_R0 0x00000001 // UART Module 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRSSI register. // //************************************************************************* #define SYSCTL_SRSSI_R3 0x00000008 // SSI Module 3 Software Reset #define SYSCTL_SRSSI_R2 0x00000004 // SSI Module 2 Software Reset #define SYSCTL_SRSSI_R1 0x00000002 // SSI Module 1 Software Reset #define SYSCTL_SRSSI_R0 0x00000001 // SSI Module 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRI2C register. // //************************************************************************* #define SYSCTL_SRI2C_R5 0x00000020 // I2C Module 5 Software Reset #define SYSCTL_SRI2C_R4 0x00000010 // I2C Module 4 Software Reset #define SYSCTL_SRI2C_R3 0x00000008 // I2C Module 3 Software Reset #define SYSCTL_SRI2C_R2 0x00000004 // I2C Module 2 Software Reset #define SYSCTL_SRI2C_R1 0x00000002 // I2C Module 1 Software Reset #define SYSCTL_SRI2C_R0 0x00000001 // I2C Module 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRUSB register. // //************************************************************************* #define SYSCTL_SRUSB_R0 0x00000001 // USB Module Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRCAN register. // //************************************************************************* #define SYSCTL_SRCAN_R1 0x00000002 // CAN Module 1 Software Reset #define SYSCTL_SRCAN_R0 0x00000001 // CAN Module 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRADC register. // //************************************************************************* #define SYSCTL_SRADC_R1 0x00000002 // ADC Module 1 Software Reset #define SYSCTL_SRADC_R0 0x00000001 // ADC Module 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRACMP register. // //************************************************************************* #define SYSCTL_SRACMP_R0 0x00000001 // Analog Comparator Module 0 // Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SREEPROM // register. // //************************************************************************* #define SYSCTL_SREEPROM_R0 0x00000001 // EEPROM Module Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SRWTIMER // register. // //************************************************************************* #define SYSCTL_SRWTIMER_R5 0x00000020 // Wide Timer 5 Software Reset #define SYSCTL_SRWTIMER_R4 0x00000010 // Wide Timer 4 Software Reset #define SYSCTL_SRWTIMER_R3 0x00000008 // Wide Timer 3 Software Reset #define SYSCTL_SRWTIMER_R2 0x00000004 // Wide Timer 2 Software Reset #define SYSCTL_SRWTIMER_R1 0x00000002 // Wide Timer 1 Software Reset #define SYSCTL_SRWTIMER_R0 0x00000001 // Wide Timer 0 Software Reset //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCWD register. // //************************************************************************* #define SYSCTL_RCGCWD_R1 0x00000002 // Watchdog Timer 1 Run Mode Clock // Gating Control #define SYSCTL_RCGCWD_R0 0x00000001 // Watchdog Timer 0 Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCTIMER // register. // //************************************************************************* #define SYSCTL_RCGCTIMER_R5 0x00000020 // Timer 5 Run Mode Clock Gating // Control #define SYSCTL_RCGCTIMER_R4 0x00000010 // Timer 4 Run Mode Clock Gating // Control #define SYSCTL_RCGCTIMER_R3 0x00000008 // Timer 3 Run Mode Clock Gating // Control #define SYSCTL_RCGCTIMER_R2 0x00000004 // Timer 2 Run Mode Clock Gating // Control #define SYSCTL_RCGCTIMER_R1 0x00000002 // Timer 1 Run Mode Clock Gating // Control #define SYSCTL_RCGCTIMER_R0 0x00000001 // Timer 0 Run Mode Clock Gating // Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCGPIO // register. // //************************************************************************* #define SYSCTL_RCGCGPIO_R14 0x00004000 // GPIO Port Q Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R13 0x00002000 // GPIO Port P Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R12 0x00001000 // GPIO Port N Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R11 0x00000800 // GPIO Port M Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R10 0x00000400 // GPIO Port L Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R9 0x00000200 // GPIO Port K Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R8 0x00000100 // GPIO Port J Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R7 0x00000080 // GPIO Port H Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R6 0x00000040 // GPIO Port G Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R5 0x00000020 // GPIO Port F Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R4 0x00000010 // GPIO Port E Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R3 0x00000008 // GPIO Port D Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R2 0x00000004 // GPIO Port C Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R1 0x00000002 // GPIO Port B Run Mode Clock // Gating Control #define SYSCTL_RCGCGPIO_R0 0x00000001 // GPIO Port A Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCDMA register. // //************************************************************************* #define SYSCTL_RCGCDMA_R0 0x00000001 // uDMA Module Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCHIB register. // //************************************************************************* #define SYSCTL_RCGCHIB_R0 0x00000001 // Hibernation Module Run Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCUART // register. // //************************************************************************* #define SYSCTL_RCGCUART_R7 0x00000080 // UART Module 7 Run Mode Clock // Gating Control #define SYSCTL_RCGCUART_R6 0x00000040 // UART Module 6 Run Mode Clock // Gating Control #define SYSCTL_RCGCUART_R5 0x00000020 // UART Module 5 Run Mode Clock // Gating Control #define SYSCTL_RCGCUART_R4 0x00000010 // UART Module 4 Run Mode Clock // Gating Control #define SYSCTL_RCGCUART_R3 0x00000008 // UART Module 3 Run Mode Clock // Gating Control #define SYSCTL_RCGCUART_R2 0x00000004 // UART Module 2 Run Mode Clock // Gating Control #define SYSCTL_RCGCUART_R1 0x00000002 // UART Module 1 Run Mode Clock // Gating Control #define SYSCTL_RCGCUART_R0 0x00000001 // UART Module 0 Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCSSI register. // //************************************************************************* #define SYSCTL_RCGCSSI_R3 0x00000008 // SSI Module 3 Run Mode Clock // Gating Control #define SYSCTL_RCGCSSI_R2 0x00000004 // SSI Module 2 Run Mode Clock // Gating Control #define SYSCTL_RCGCSSI_R1 0x00000002 // SSI Module 1 Run Mode Clock // Gating Control #define SYSCTL_RCGCSSI_R0 0x00000001 // SSI Module 0 Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCI2C register. // //************************************************************************* #define SYSCTL_RCGCI2C_R5 0x00000020 // I2C Module 5 Run Mode Clock // Gating Control #define SYSCTL_RCGCI2C_R4 0x00000010 // I2C Module 4 Run Mode Clock // Gating Control #define SYSCTL_RCGCI2C_R3 0x00000008 // I2C Module 3 Run Mode Clock // Gating Control #define SYSCTL_RCGCI2C_R2 0x00000004 // I2C Module 2 Run Mode Clock // Gating Control #define SYSCTL_RCGCI2C_R1 0x00000002 // I2C Module 1 Run Mode Clock // Gating Control #define SYSCTL_RCGCI2C_R0 0x00000001 // I2C Module 0 Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCUSB register. // //************************************************************************* #define SYSCTL_RCGCUSB_R0 0x00000001 // USB Module Run Mode Clock Gating // Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCCAN register. // //************************************************************************* #define SYSCTL_RCGCCAN_R1 0x00000002 // CAN Module 1 Run Mode Clock // Gating Control #define SYSCTL_RCGCCAN_R0 0x00000001 // CAN Module 0 Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCADC register. // //************************************************************************* #define SYSCTL_RCGCADC_R1 0x00000002 // ADC Module 1 Run Mode Clock // Gating Control #define SYSCTL_RCGCADC_R0 0x00000001 // ADC Module 0 Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCACMP // register. // //************************************************************************* #define SYSCTL_RCGCACMP_R0 0x00000001 // Analog Comparator Module 0 Run // Mode Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCEEPROM // register. // //************************************************************************* #define SYSCTL_RCGCEEPROM_R0 0x00000001 // EEPROM Module Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_RCGCWTIMER // register. // //************************************************************************* #define SYSCTL_RCGCWTIMER_R5 0x00000020 // Wide Timer 5 Run Mode Clock // Gating Control #define SYSCTL_RCGCWTIMER_R4 0x00000010 // Wide Timer 4 Run Mode Clock // Gating Control #define SYSCTL_RCGCWTIMER_R3 0x00000008 // Wide Timer 3 Run Mode Clock // Gating Control #define SYSCTL_RCGCWTIMER_R2 0x00000004 // Wide Timer 2 Run Mode Clock // Gating Control #define SYSCTL_RCGCWTIMER_R1 0x00000002 // Wide Timer 1 Run Mode Clock // Gating Control #define SYSCTL_RCGCWTIMER_R0 0x00000001 // Wide Timer 0 Run Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCWD register. // //************************************************************************* #define SYSCTL_SCGCWD_S1 0x00000002 // Watchdog Timer 1 Sleep Mode // Clock Gating Control #define SYSCTL_SCGCWD_S0 0x00000001 // Watchdog Timer 0 Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCTIMER // register. // //************************************************************************* #define SYSCTL_SCGCTIMER_S5 0x00000020 // Timer 5 Sleep Mode Clock Gating // Control #define SYSCTL_SCGCTIMER_S4 0x00000010 // Timer 4 Sleep Mode Clock Gating // Control #define SYSCTL_SCGCTIMER_S3 0x00000008 // Timer 3 Sleep Mode Clock Gating // Control #define SYSCTL_SCGCTIMER_S2 0x00000004 // Timer 2 Sleep Mode Clock Gating // Control #define SYSCTL_SCGCTIMER_S1 0x00000002 // Timer 1 Sleep Mode Clock Gating // Control #define SYSCTL_SCGCTIMER_S0 0x00000001 // Timer 0 Sleep Mode Clock Gating // Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCGPIO // register. // //************************************************************************* #define SYSCTL_SCGCGPIO_S14 0x00004000 // GPIO Port Q Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S13 0x00002000 // GPIO Port P Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S12 0x00001000 // GPIO Port N Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S11 0x00000800 // GPIO Port M Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S10 0x00000400 // GPIO Port L Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S9 0x00000200 // GPIO Port K Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S8 0x00000100 // GPIO Port J Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S7 0x00000080 // GPIO Port H Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S6 0x00000040 // GPIO Port G Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S5 0x00000020 // GPIO Port F Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S4 0x00000010 // GPIO Port E Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S3 0x00000008 // GPIO Port D Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S2 0x00000004 // GPIO Port C Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S1 0x00000002 // GPIO Port B Sleep Mode Clock // Gating Control #define SYSCTL_SCGCGPIO_S0 0x00000001 // GPIO Port A Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCDMA register. // //************************************************************************* #define SYSCTL_SCGCDMA_S0 0x00000001 // uDMA Module Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCHIB register. // //************************************************************************* #define SYSCTL_SCGCHIB_S0 0x00000001 // Hibernation Module Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCUART // register. // //************************************************************************* #define SYSCTL_SCGCUART_S7 0x00000080 // UART Module 7 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCUART_S6 0x00000040 // UART Module 6 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCUART_S5 0x00000020 // UART Module 5 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCUART_S4 0x00000010 // UART Module 4 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCUART_S3 0x00000008 // UART Module 3 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCUART_S2 0x00000004 // UART Module 2 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCUART_S1 0x00000002 // UART Module 1 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCUART_S0 0x00000001 // UART Module 0 Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCSSI register. // //************************************************************************* #define SYSCTL_SCGCSSI_S3 0x00000008 // SSI Module 3 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCSSI_S2 0x00000004 // SSI Module 2 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCSSI_S1 0x00000002 // SSI Module 1 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCSSI_S0 0x00000001 // SSI Module 0 Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCI2C register. // //************************************************************************* #define SYSCTL_SCGCI2C_S5 0x00000020 // I2C Module 5 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCI2C_S4 0x00000010 // I2C Module 4 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCI2C_S3 0x00000008 // I2C Module 3 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCI2C_S2 0x00000004 // I2C Module 2 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCI2C_S1 0x00000002 // I2C Module 1 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCI2C_S0 0x00000001 // I2C Module 0 Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCUSB register. // //************************************************************************* #define SYSCTL_SCGCUSB_S0 0x00000001 // USB Module Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCCAN register. // //************************************************************************* #define SYSCTL_SCGCCAN_S1 0x00000002 // CAN Module 1 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCCAN_S0 0x00000001 // CAN Module 0 Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCADC register. // //************************************************************************* #define SYSCTL_SCGCADC_S1 0x00000002 // ADC Module 1 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCADC_S0 0x00000001 // ADC Module 0 Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCACMP // register. // //************************************************************************* #define SYSCTL_SCGCACMP_S0 0x00000001 // Analog Comparator Module 0 Sleep // Mode Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCEEPROM // register. // //************************************************************************* #define SYSCTL_SCGCEEPROM_S0 0x00000001 // EEPROM Module Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_SCGCWTIMER // register. // //************************************************************************* #define SYSCTL_SCGCWTIMER_S5 0x00000020 // Wide Timer 5 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCWTIMER_S4 0x00000010 // Wide Timer 4 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCWTIMER_S3 0x00000008 // Wide Timer 3 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCWTIMER_S2 0x00000004 // Wide Timer 2 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCWTIMER_S1 0x00000002 // Wide Timer 1 Sleep Mode Clock // Gating Control #define SYSCTL_SCGCWTIMER_S0 0x00000001 // Wide Timer 0 Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCWD register. // //************************************************************************* #define SYSCTL_DCGCWD_D1 0x00000002 // Watchdog Timer 1 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCWD_D0 0x00000001 // Watchdog Timer 0 Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCTIMER // register. // //************************************************************************* #define SYSCTL_DCGCTIMER_D5 0x00000020 // Timer 5 Deep-Sleep Mode Clock // Gating Control #define SYSCTL_DCGCTIMER_D4 0x00000010 // Timer 4 Deep-Sleep Mode Clock // Gating Control #define SYSCTL_DCGCTIMER_D3 0x00000008 // Timer 3 Deep-Sleep Mode Clock // Gating Control #define SYSCTL_DCGCTIMER_D2 0x00000004 // Timer 2 Deep-Sleep Mode Clock // Gating Control #define SYSCTL_DCGCTIMER_D1 0x00000002 // Timer 1 Deep-Sleep Mode Clock // Gating Control #define SYSCTL_DCGCTIMER_D0 0x00000001 // Timer 0 Deep-Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCGPIO // register. // //************************************************************************* #define SYSCTL_DCGCGPIO_D14 0x00004000 // GPIO Port Q Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D13 0x00002000 // GPIO Port P Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D12 0x00001000 // GPIO Port N Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D11 0x00000800 // GPIO Port M Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D10 0x00000400 // GPIO Port L Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D9 0x00000200 // GPIO Port K Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D8 0x00000100 // GPIO Port J Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D7 0x00000080 // GPIO Port H Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D6 0x00000040 // GPIO Port G Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D5 0x00000020 // GPIO Port F Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D4 0x00000010 // GPIO Port E Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D3 0x00000008 // GPIO Port D Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D2 0x00000004 // GPIO Port C Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D1 0x00000002 // GPIO Port B Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCGPIO_D0 0x00000001 // GPIO Port A Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCDMA register. // //************************************************************************* #define SYSCTL_DCGCDMA_D0 0x00000001 // uDMA Module Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCHIB register. // //************************************************************************* #define SYSCTL_DCGCHIB_D0 0x00000001 // Hibernation Module Deep-Sleep // Mode Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCUART // register. // //************************************************************************* #define SYSCTL_DCGCUART_D7 0x00000080 // UART Module 7 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCUART_D6 0x00000040 // UART Module 6 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCUART_D5 0x00000020 // UART Module 5 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCUART_D4 0x00000010 // UART Module 4 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCUART_D3 0x00000008 // UART Module 3 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCUART_D2 0x00000004 // UART Module 2 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCUART_D1 0x00000002 // UART Module 1 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCUART_D0 0x00000001 // UART Module 0 Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCSSI register. // //************************************************************************* #define SYSCTL_DCGCSSI_D3 0x00000008 // SSI Module 3 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCSSI_D2 0x00000004 // SSI Module 2 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCSSI_D1 0x00000002 // SSI Module 1 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCSSI_D0 0x00000001 // SSI Module 0 Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCI2C register. // //************************************************************************* #define SYSCTL_DCGCI2C_D5 0x00000020 // I2C Module 5 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCI2C_D4 0x00000010 // I2C Module 4 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCI2C_D3 0x00000008 // I2C Module 3 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCI2C_D2 0x00000004 // I2C Module 2 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCI2C_D1 0x00000002 // I2C Module 1 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCI2C_D0 0x00000001 // I2C Module 0 Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCUSB register. // //************************************************************************* #define SYSCTL_DCGCUSB_D0 0x00000001 // USB Module Deep-Sleep Mode Clock // Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCCAN register. // //************************************************************************* #define SYSCTL_DCGCCAN_D1 0x00000002 // CAN Module 1 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCCAN_D0 0x00000001 // CAN Module 0 Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCADC register. // //************************************************************************* #define SYSCTL_DCGCADC_D1 0x00000002 // ADC Module 1 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCADC_D0 0x00000001 // ADC Module 0 Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCACMP // register. // //************************************************************************* #define SYSCTL_DCGCACMP_D0 0x00000001 // Analog Comparator Module 0 // Deep-Sleep Mode Clock Gating // Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCEEPROM // register. // //************************************************************************* #define SYSCTL_DCGCEEPROM_D0 0x00000001 // EEPROM Module Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_DCGCWTIMER // register. // //************************************************************************* #define SYSCTL_DCGCWTIMER_D5 0x00000020 // Wide Timer 5 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCWTIMER_D4 0x00000010 // Wide Timer 4 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCWTIMER_D3 0x00000008 // Wide Timer 3 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCWTIMER_D2 0x00000004 // Wide Timer 2 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCWTIMER_D1 0x00000002 // Wide Timer 1 Deep-Sleep Mode // Clock Gating Control #define SYSCTL_DCGCWTIMER_D0 0x00000001 // Wide Timer 0 Deep-Sleep Mode // Clock Gating Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCWD register. // //************************************************************************* #define SYSCTL_PCWD_P1 0x00000002 // Watchdog Timer 1 Power Control #define SYSCTL_PCWD_P0 0x00000001 // Watchdog Timer 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCTIMER register. // //************************************************************************* #define SYSCTL_PCTIMER_P5 0x00000020 // Timer 5 Power Control #define SYSCTL_PCTIMER_P4 0x00000010 // Timer 4 Power Control #define SYSCTL_PCTIMER_P3 0x00000008 // Timer 3 Power Control #define SYSCTL_PCTIMER_P2 0x00000004 // Timer 2 Power Control #define SYSCTL_PCTIMER_P1 0x00000002 // Timer 1 Power Control #define SYSCTL_PCTIMER_P0 0x00000001 // Timer 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCGPIO register. // //************************************************************************* #define SYSCTL_PCGPIO_P14 0x00004000 // GPIO Port Q Power Control #define SYSCTL_PCGPIO_P13 0x00002000 // GPIO Port P Power Control #define SYSCTL_PCGPIO_P12 0x00001000 // GPIO Port N Power Control #define SYSCTL_PCGPIO_P11 0x00000800 // GPIO Port M Power Control #define SYSCTL_PCGPIO_P10 0x00000400 // GPIO Port L Power Control #define SYSCTL_PCGPIO_P9 0x00000200 // GPIO Port K Power Control #define SYSCTL_PCGPIO_P8 0x00000100 // GPIO Port J Power Control #define SYSCTL_PCGPIO_P7 0x00000080 // GPIO Port H Power Control #define SYSCTL_PCGPIO_P6 0x00000040 // GPIO Port G Power Control #define SYSCTL_PCGPIO_P5 0x00000020 // GPIO Port F Power Control #define SYSCTL_PCGPIO_P4 0x00000010 // GPIO Port E Power Control #define SYSCTL_PCGPIO_P3 0x00000008 // GPIO Port D Power Control #define SYSCTL_PCGPIO_P2 0x00000004 // GPIO Port C Power Control #define SYSCTL_PCGPIO_P1 0x00000002 // GPIO Port B Power Control #define SYSCTL_PCGPIO_P0 0x00000001 // GPIO Port A Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCDMA register. // //************************************************************************* #define SYSCTL_PCDMA_P0 0x00000001 // uDMA Module Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCHIB register. // //************************************************************************* #define SYSCTL_PCHIB_P0 0x00000001 // Hibernation Module Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCUART register. // //************************************************************************* #define SYSCTL_PCUART_P7 0x00000080 // UART Module 7 Power Control #define SYSCTL_PCUART_P6 0x00000040 // UART Module 6 Power Control #define SYSCTL_PCUART_P5 0x00000020 // UART Module 5 Power Control #define SYSCTL_PCUART_P4 0x00000010 // UART Module 4 Power Control #define SYSCTL_PCUART_P3 0x00000008 // UART Module 3 Power Control #define SYSCTL_PCUART_P2 0x00000004 // UART Module 2 Power Control #define SYSCTL_PCUART_P1 0x00000002 // UART Module 1 Power Control #define SYSCTL_PCUART_P0 0x00000001 // UART Module 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCSSI register. // //************************************************************************* #define SYSCTL_PCSSI_P3 0x00000008 // SSI Module 3 Power Control #define SYSCTL_PCSSI_P2 0x00000004 // SSI Module 2 Power Control #define SYSCTL_PCSSI_P1 0x00000002 // SSI Module 1 Power Control #define SYSCTL_PCSSI_P0 0x00000001 // SSI Module 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCI2C register. // //************************************************************************* #define SYSCTL_PCI2C_P5 0x00000020 // I2C Module 5 Power Control #define SYSCTL_PCI2C_P4 0x00000010 // I2C Module 4 Power Control #define SYSCTL_PCI2C_P3 0x00000008 // I2C Module 3 Power Control #define SYSCTL_PCI2C_P2 0x00000004 // I2C Module 2 Power Control #define SYSCTL_PCI2C_P1 0x00000002 // I2C Module 1 Power Control #define SYSCTL_PCI2C_P0 0x00000001 // I2C Module 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCUSB register. // //************************************************************************* #define SYSCTL_PCUSB_P0 0x00000001 // USB Module Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCCAN register. // //************************************************************************* #define SYSCTL_PCCAN_P1 0x00000002 // CAN Module 1 Power Control #define SYSCTL_PCCAN_P0 0x00000001 // CAN Module 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCADC register. // //************************************************************************* #define SYSCTL_PCADC_P1 0x00000002 // ADC Module 1 Power Control #define SYSCTL_PCADC_P0 0x00000001 // ADC Module 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCACMP register. // //************************************************************************* #define SYSCTL_PCACMP_P0 0x00000001 // Analog Comparator Module 0 Power // Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCEEPROM // register. // //************************************************************************* #define SYSCTL_PCEEPROM_P0 0x00000001 // EEPROM Module Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PCWTIMER // register. // //************************************************************************* #define SYSCTL_PCWTIMER_P5 0x00000020 // Wide Timer 5 Power Control #define SYSCTL_PCWTIMER_P4 0x00000010 // Wide Timer 4 Power Control #define SYSCTL_PCWTIMER_P3 0x00000008 // Wide Timer 3 Power Control #define SYSCTL_PCWTIMER_P2 0x00000004 // Wide Timer 2 Power Control #define SYSCTL_PCWTIMER_P1 0x00000002 // Wide Timer 1 Power Control #define SYSCTL_PCWTIMER_P0 0x00000001 // Wide Timer 0 Power Control //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRWD register. // //************************************************************************* #define SYSCTL_PRWD_R1 0x00000002 // Watchdog Timer 1 Peripheral // Ready #define SYSCTL_PRWD_R0 0x00000001 // Watchdog Timer 0 Peripheral // Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRTIMER register. // //************************************************************************* #define SYSCTL_PRTIMER_R5 0x00000020 // Timer 5 Peripheral Ready #define SYSCTL_PRTIMER_R4 0x00000010 // Timer 4 Peripheral Ready #define SYSCTL_PRTIMER_R3 0x00000008 // Timer 3 Peripheral Ready #define SYSCTL_PRTIMER_R2 0x00000004 // Timer 2 Peripheral Ready #define SYSCTL_PRTIMER_R1 0x00000002 // Timer 1 Peripheral Ready #define SYSCTL_PRTIMER_R0 0x00000001 // Timer 0 Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRGPIO register. // //************************************************************************* #define SYSCTL_PRGPIO_R14 0x00004000 // GPIO Port Q Peripheral Ready #define SYSCTL_PRGPIO_R13 0x00002000 // GPIO Port P Peripheral Ready #define SYSCTL_PRGPIO_R12 0x00001000 // GPIO Port N Peripheral Ready #define SYSCTL_PRGPIO_R11 0x00000800 // GPIO Port M Peripheral Ready #define SYSCTL_PRGPIO_R10 0x00000400 // GPIO Port L Peripheral Ready #define SYSCTL_PRGPIO_R9 0x00000200 // GPIO Port K Peripheral Ready #define SYSCTL_PRGPIO_R8 0x00000100 // GPIO Port J Peripheral Ready #define SYSCTL_PRGPIO_R7 0x00000080 // GPIO Port H Peripheral Ready #define SYSCTL_PRGPIO_R6 0x00000040 // GPIO Port G Peripheral Ready #define SYSCTL_PRGPIO_R5 0x00000020 // GPIO Port F Peripheral Ready #define SYSCTL_PRGPIO_R4 0x00000010 // GPIO Port E Peripheral Ready #define SYSCTL_PRGPIO_R3 0x00000008 // GPIO Port D Peripheral Ready #define SYSCTL_PRGPIO_R2 0x00000004 // GPIO Port C Peripheral Ready #define SYSCTL_PRGPIO_R1 0x00000002 // GPIO Port B Peripheral Ready #define SYSCTL_PRGPIO_R0 0x00000001 // GPIO Port A Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRDMA register. // //************************************************************************* #define SYSCTL_PRDMA_R0 0x00000001 // uDMA Module Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRHIB register. // //************************************************************************* #define SYSCTL_PRHIB_R0 0x00000001 // Hibernation Module Peripheral // Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRUART register. // //************************************************************************* #define SYSCTL_PRUART_R7 0x00000080 // UART Module 7 Peripheral Ready #define SYSCTL_PRUART_R6 0x00000040 // UART Module 6 Peripheral Ready #define SYSCTL_PRUART_R5 0x00000020 // UART Module 5 Peripheral Ready #define SYSCTL_PRUART_R4 0x00000010 // UART Module 4 Peripheral Ready #define SYSCTL_PRUART_R3 0x00000008 // UART Module 3 Peripheral Ready #define SYSCTL_PRUART_R2 0x00000004 // UART Module 2 Peripheral Ready #define SYSCTL_PRUART_R1 0x00000002 // UART Module 1 Peripheral Ready #define SYSCTL_PRUART_R0 0x00000001 // UART Module 0 Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRSSI register. // //************************************************************************* #define SYSCTL_PRSSI_R3 0x00000008 // SSI Module 3 Peripheral Ready #define SYSCTL_PRSSI_R2 0x00000004 // SSI Module 2 Peripheral Ready #define SYSCTL_PRSSI_R1 0x00000002 // SSI Module 1 Peripheral Ready #define SYSCTL_PRSSI_R0 0x00000001 // SSI Module 0 Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRI2C register. // //************************************************************************* #define SYSCTL_PRI2C_R5 0x00000020 // I2C Module 5 Peripheral Ready #define SYSCTL_PRI2C_R4 0x00000010 // I2C Module 4 Peripheral Ready #define SYSCTL_PRI2C_R3 0x00000008 // I2C Module 3 Peripheral Ready #define SYSCTL_PRI2C_R2 0x00000004 // I2C Module 2 Peripheral Ready #define SYSCTL_PRI2C_R1 0x00000002 // I2C Module 1 Peripheral Ready #define SYSCTL_PRI2C_R0 0x00000001 // I2C Module 0 Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRUSB register. // //************************************************************************* #define SYSCTL_PRUSB_R0 0x00000001 // USB Module Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRCAN register. // //************************************************************************* #define SYSCTL_PRCAN_R1 0x00000002 // CAN Module 1 Peripheral Ready #define SYSCTL_PRCAN_R0 0x00000001 // CAN Module 0 Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRADC register. // //************************************************************************* #define SYSCTL_PRADC_R1 0x00000002 // ADC Module 1 Peripheral Ready #define SYSCTL_PRADC_R0 0x00000001 // ADC Module 0 Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRACMP register. // //************************************************************************* #define SYSCTL_PRACMP_R0 0x00000001 // Analog Comparator Module 0 // Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PREEPROM // register. // //************************************************************************* #define SYSCTL_PREEPROM_R0 0x00000001 // EEPROM Module Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the SYSCTL_PRWTIMER // register. // //************************************************************************* #define SYSCTL_PRWTIMER_R5 0x00000020 // Wide Timer 5 Peripheral Ready #define SYSCTL_PRWTIMER_R4 0x00000010 // Wide Timer 4 Peripheral Ready #define SYSCTL_PRWTIMER_R3 0x00000008 // Wide Timer 3 Peripheral Ready #define SYSCTL_PRWTIMER_R2 0x00000004 // Wide Timer 2 Peripheral Ready #define SYSCTL_PRWTIMER_R1 0x00000002 // Wide Timer 1 Peripheral Ready #define SYSCTL_PRWTIMER_R0 0x00000001 // Wide Timer 0 Peripheral Ready //************************************************************************* // // The following are defines for the bit fields in the UDMA_STAT register. // //************************************************************************* #define UDMA_STAT_DMACHANS_M 0x001F0000 // Available uDMA Channels Minus 1 #define UDMA_STAT_STATE_M 0x000000F0 // Control State Machine Status #define UDMA_STAT_STATE_IDLE 0x00000000 // Idle #define UDMA_STAT_STATE_RD_CTRL 0x00000010 // Reading channel controller data #define UDMA_STAT_STATE_RD_SRCENDP \ 0x00000020 // Reading source end pointer #define UDMA_STAT_STATE_RD_DSTENDP \ 0x00000030 // Reading destination end pointer #define UDMA_STAT_STATE_RD_SRCDAT \ 0x00000040 // Reading source data #define UDMA_STAT_STATE_WR_DSTDAT \ 0x00000050 // Writing destination data #define UDMA_STAT_STATE_WAIT 0x00000060 // Waiting for uDMA request to // clear #define UDMA_STAT_STATE_WR_CTRL 0x00000070 // Writing channel controller data #define UDMA_STAT_STATE_STALL 0x00000080 // Stalled #define UDMA_STAT_STATE_DONE 0x00000090 // Done #define UDMA_STAT_STATE_UNDEF 0x000000A0 // Undefined #define UDMA_STAT_MASTEN 0x00000001 // Master Enable Status #define UDMA_STAT_DMACHANS_S 16 //************************************************************************* // // The following are defines for the bit fields in the UDMA_CFG register. // //************************************************************************* #define UDMA_CFG_MASTEN 0x00000001 // Controller Master Enable //************************************************************************* // // The following are defines for the bit fields in the UDMA_CTLBASE register. // //************************************************************************* #define UDMA_CTLBASE_ADDR_M 0xFFFFFC00 // Channel Control Base Address #define UDMA_CTLBASE_ADDR_S 10 //************************************************************************* // // The following are defines for the bit fields in the UDMA_ALTBASE register. // //************************************************************************* #define UDMA_ALTBASE_ADDR_M 0xFFFFFFFF // Alternate Channel Address // Pointer #define UDMA_ALTBASE_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the UDMA_WAITSTAT register. // //************************************************************************* #define UDMA_WAITSTAT_WAITREQ_M 0xFFFFFFFF // Channel [n] Wait Status //************************************************************************* // // The following are defines for the bit fields in the UDMA_SWREQ register. // //************************************************************************* #define UDMA_SWREQ_M 0xFFFFFFFF // Channel [n] Software Request //************************************************************************* // // The following are defines for the bit fields in the UDMA_USEBURSTSET // register. // //************************************************************************* #define UDMA_USEBURSTSET_SET_M 0xFFFFFFFF // Channel [n] Useburst Set //************************************************************************* // // The following are defines for the bit fields in the UDMA_USEBURSTCLR // register. // //************************************************************************* #define UDMA_USEBURSTCLR_CLR_M 0xFFFFFFFF // Channel [n] Useburst Clear //************************************************************************* // // The following are defines for the bit fields in the UDMA_REQMASKSET // register. // //************************************************************************* #define UDMA_REQMASKSET_SET_M 0xFFFFFFFF // Channel [n] Request Mask Set //************************************************************************* // // The following are defines for the bit fields in the UDMA_REQMASKCLR // register. // //************************************************************************* #define UDMA_REQMASKCLR_CLR_M 0xFFFFFFFF // Channel [n] Request Mask Clear //************************************************************************* // // The following are defines for the bit fields in the UDMA_ENASET register. // //************************************************************************* #define UDMA_ENASET_SET_M 0xFFFFFFFF // Channel [n] Enable Set //************************************************************************* // // The following are defines for the bit fields in the UDMA_ENACLR register. // //************************************************************************* #define UDMA_ENACLR_CLR_M 0xFFFFFFFF // Clear Channel [n] Enable Clear //************************************************************************* // // The following are defines for the bit fields in the UDMA_ALTSET register. // //************************************************************************* #define UDMA_ALTSET_SET_M 0xFFFFFFFF // Channel [n] Alternate Set //************************************************************************* // // The following are defines for the bit fields in the UDMA_ALTCLR register. // //************************************************************************* #define UDMA_ALTCLR_CLR_M 0xFFFFFFFF // Channel [n] Alternate Clear //************************************************************************* // // The following are defines for the bit fields in the UDMA_PRIOSET register. // //************************************************************************* #define UDMA_PRIOSET_SET_M 0xFFFFFFFF // Channel [n] Priority Set //************************************************************************* // // The following are defines for the bit fields in the UDMA_PRIOCLR register. // //************************************************************************* #define UDMA_PRIOCLR_CLR_M 0xFFFFFFFF // Channel [n] Priority Clear //************************************************************************* // // The following are defines for the bit fields in the UDMA_ERRCLR register. // //************************************************************************* #define UDMA_ERRCLR_ERRCLR 0x00000001 // uDMA Bus Error Status //************************************************************************* // // The following are defines for the bit fields in the UDMA_CHASGN register. // //************************************************************************* #define UDMA_CHASGN_M 0xFFFFFFFF // Channel [n] Assignment Select #define UDMA_CHASGN_PRIMARY 0x00000000 // Use the primary channel // assignment #define UDMA_CHASGN_SECONDARY 0x00000001 // Use the secondary channel // assignment //************************************************************************* // // The following are defines for the bit fields in the UDMA_CHIS register. // //************************************************************************* #define UDMA_CHIS_M 0xFFFFFFFF // Channel [n] Interrupt Status //************************************************************************* // // The following are defines for the bit fields in the UDMA_CHMAP0 register. // //************************************************************************* #define UDMA_CHMAP0_CH7SEL_M 0xF0000000 // uDMA Channel 7 Source Select #define UDMA_CHMAP0_CH6SEL_M 0x0F000000 // uDMA Channel 6 Source Select #define UDMA_CHMAP0_CH5SEL_M 0x00F00000 // uDMA Channel 5 Source Select #define UDMA_CHMAP0_CH4SEL_M 0x000F0000 // uDMA Channel 4 Source Select #define UDMA_CHMAP0_CH3SEL_M 0x0000F000 // uDMA Channel 3 Source Select #define UDMA_CHMAP0_CH2SEL_M 0x00000F00 // uDMA Channel 2 Source Select #define UDMA_CHMAP0_CH1SEL_M 0x000000F0 // uDMA Channel 1 Source Select #define UDMA_CHMAP0_CH0SEL_M 0x0000000F // uDMA Channel 0 Source Select #define UDMA_CHMAP0_CH7SEL_S 28 #define UDMA_CHMAP0_CH6SEL_S 24 #define UDMA_CHMAP0_CH5SEL_S 20 #define UDMA_CHMAP0_CH4SEL_S 16 #define UDMA_CHMAP0_CH3SEL_S 12 #define UDMA_CHMAP0_CH2SEL_S 8 #define UDMA_CHMAP0_CH1SEL_S 4 #define UDMA_CHMAP0_CH0SEL_S 0 //************************************************************************* // // The following are defines for the bit fields in the UDMA_CHMAP1 register. // //************************************************************************* #define UDMA_CHMAP1_CH15SEL_M 0xF0000000 // uDMA Channel 15 Source Select #define UDMA_CHMAP1_CH14SEL_M 0x0F000000 // uDMA Channel 14 Source Select #define UDMA_CHMAP1_CH13SEL_M 0x00F00000 // uDMA Channel 13 Source Select #define UDMA_CHMAP1_CH12SEL_M 0x000F0000 // uDMA Channel 12 Source Select #define UDMA_CHMAP1_CH11SEL_M 0x0000F000 // uDMA Channel 11 Source Select #define UDMA_CHMAP1_CH10SEL_M 0x00000F00 // uDMA Channel 10 Source Select #define UDMA_CHMAP1_CH9SEL_M 0x000000F0 // uDMA Channel 9 Source Select #define UDMA_CHMAP1_CH8SEL_M 0x0000000F // uDMA Channel 8 Source Select #define UDMA_CHMAP1_CH15SEL_S 28 #define UDMA_CHMAP1_CH14SEL_S 24 #define UDMA_CHMAP1_CH13SEL_S 20 #define UDMA_CHMAP1_CH12SEL_S 16 #define UDMA_CHMAP1_CH11SEL_S 12 #define UDMA_CHMAP1_CH10SEL_S 8 #define UDMA_CHMAP1_CH9SEL_S 4 #define UDMA_CHMAP1_CH8SEL_S 0 //************************************************************************* // // The following are defines for the bit fields in the UDMA_CHMAP2 register. // //************************************************************************* #define UDMA_CHMAP2_CH23SEL_M 0xF0000000 // uDMA Channel 23 Source Select #define UDMA_CHMAP2_CH22SEL_M 0x0F000000 // uDMA Channel 22 Source Select #define UDMA_CHMAP2_CH21SEL_M 0x00F00000 // uDMA Channel 21 Source Select #define UDMA_CHMAP2_CH20SEL_M 0x000F0000 // uDMA Channel 20 Source Select #define UDMA_CHMAP2_CH19SEL_M 0x0000F000 // uDMA Channel 19 Source Select #define UDMA_CHMAP2_CH18SEL_M 0x00000F00 // uDMA Channel 18 Source Select #define UDMA_CHMAP2_CH17SEL_M 0x000000F0 // uDMA Channel 17 Source Select #define UDMA_CHMAP2_CH16SEL_M 0x0000000F // uDMA Channel 16 Source Select #define UDMA_CHMAP2_CH23SEL_S 28 #define UDMA_CHMAP2_CH22SEL_S 24 #define UDMA_CHMAP2_CH21SEL_S 20 #define UDMA_CHMAP2_CH20SEL_S 16 #define UDMA_CHMAP2_CH19SEL_S 12 #define UDMA_CHMAP2_CH18SEL_S 8 #define UDMA_CHMAP2_CH17SEL_S 4 #define UDMA_CHMAP2_CH16SEL_S 0 //************************************************************************* // // The following are defines for the bit fields in the UDMA_CHMAP3 register. // //************************************************************************* #define UDMA_CHMAP3_CH31SEL_M 0xF0000000 // uDMA Channel 31 Source Select #define UDMA_CHMAP3_CH30SEL_M 0x0F000000 // uDMA Channel 30 Source Select #define UDMA_CHMAP3_CH29SEL_M 0x00F00000 // uDMA Channel 29 Source Select #define UDMA_CHMAP3_CH28SEL_M 0x000F0000 // uDMA Channel 28 Source Select #define UDMA_CHMAP3_CH27SEL_M 0x0000F000 // uDMA Channel 27 Source Select #define UDMA_CHMAP3_CH26SEL_M 0x00000F00 // uDMA Channel 26 Source Select #define UDMA_CHMAP3_CH25SEL_M 0x000000F0 // uDMA Channel 25 Source Select #define UDMA_CHMAP3_CH24SEL_M 0x0000000F // uDMA Channel 24 Source Select #define UDMA_CHMAP3_CH31SEL_S 28 #define UDMA_CHMAP3_CH30SEL_S 24 #define UDMA_CHMAP3_CH29SEL_S 20 #define UDMA_CHMAP3_CH28SEL_S 16 #define UDMA_CHMAP3_CH27SEL_S 12 #define UDMA_CHMAP3_CH26SEL_S 8 #define UDMA_CHMAP3_CH25SEL_S 4 #define UDMA_CHMAP3_CH24SEL_S 0 //************************************************************************* // // The following are defines for the bit fields in the UDMA_O_SRCENDP register. // //************************************************************************* #define UDMA_SRCENDP_ADDR_M 0xFFFFFFFF // Source Address End Pointer #define UDMA_SRCENDP_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the UDMA_O_DSTENDP register. // //************************************************************************* #define UDMA_DSTENDP_ADDR_M 0xFFFFFFFF // Destination Address End Pointer #define UDMA_DSTENDP_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the UDMA_O_CHCTL register. // //************************************************************************* #define UDMA_CHCTL_DSTINC_M 0xC0000000 // Destination Address Increment #define UDMA_CHCTL_DSTINC_8 0x00000000 // Byte #define UDMA_CHCTL_DSTINC_16 0x40000000 // Half-word #define UDMA_CHCTL_DSTINC_32 0x80000000 // Word #define UDMA_CHCTL_DSTINC_NONE 0xC0000000 // No increment #define UDMA_CHCTL_DSTSIZE_M 0x30000000 // Destination Data Size #define UDMA_CHCTL_DSTSIZE_8 0x00000000 // Byte #define UDMA_CHCTL_DSTSIZE_16 0x10000000 // Half-word #define UDMA_CHCTL_DSTSIZE_32 0x20000000 // Word #define UDMA_CHCTL_SRCINC_M 0x0C000000 // Source Address Increment #define UDMA_CHCTL_SRCINC_8 0x00000000 // Byte #define UDMA_CHCTL_SRCINC_16 0x04000000 // Half-word #define UDMA_CHCTL_SRCINC_32 0x08000000 // Word #define UDMA_CHCTL_SRCINC_NONE 0x0C000000 // No increment #define UDMA_CHCTL_SRCSIZE_M 0x03000000 // Source Data Size #define UDMA_CHCTL_SRCSIZE_8 0x00000000 // Byte #define UDMA_CHCTL_SRCSIZE_16 0x01000000 // Half-word #define UDMA_CHCTL_SRCSIZE_32 0x02000000 // Word #define UDMA_CHCTL_ARBSIZE_M 0x0003C000 // Arbitration Size #define UDMA_CHCTL_ARBSIZE_1 0x00000000 // 1 Transfer #define UDMA_CHCTL_ARBSIZE_2 0x00004000 // 2 Transfers #define UDMA_CHCTL_ARBSIZE_4 0x00008000 // 4 Transfers #define UDMA_CHCTL_ARBSIZE_8 0x0000C000 // 8 Transfers #define UDMA_CHCTL_ARBSIZE_16 0x00010000 // 16 Transfers #define UDMA_CHCTL_ARBSIZE_32 0x00014000 // 32 Transfers #define UDMA_CHCTL_ARBSIZE_64 0x00018000 // 64 Transfers #define UDMA_CHCTL_ARBSIZE_128 0x0001C000 // 128 Transfers #define UDMA_CHCTL_ARBSIZE_256 0x00020000 // 256 Transfers #define UDMA_CHCTL_ARBSIZE_512 0x00024000 // 512 Transfers #define UDMA_CHCTL_ARBSIZE_1024 0x00028000 // 1024 Transfers #define UDMA_CHCTL_XFERSIZE_M 0x00003FF0 // Transfer Size (minus 1) #define UDMA_CHCTL_NXTUSEBURST 0x00000008 // Next Useburst #define UDMA_CHCTL_XFERMODE_M 0x00000007 // uDMA Transfer Mode #define UDMA_CHCTL_XFERMODE_STOP \ 0x00000000 // Stop #define UDMA_CHCTL_XFERMODE_BASIC \ 0x00000001 // Basic #define UDMA_CHCTL_XFERMODE_AUTO \ 0x00000002 // Auto-Request #define UDMA_CHCTL_XFERMODE_PINGPONG \ 0x00000003 // Ping-Pong #define UDMA_CHCTL_XFERMODE_MEM_SG \ 0x00000004 // Memory Scatter-Gather #define UDMA_CHCTL_XFERMODE_MEM_SGA \ 0x00000005 // Alternate Memory Scatter-Gather #define UDMA_CHCTL_XFERMODE_PER_SG \ 0x00000006 // Peripheral Scatter-Gather #define UDMA_CHCTL_XFERMODE_PER_SGA \ 0x00000007 // Alternate Peripheral // Scatter-Gather #define UDMA_CHCTL_XFERSIZE_S 4 //************************************************************************* // // The following are defines for the bit fields in the NVIC_INT_TYPE register. // //************************************************************************* #define NVIC_INT_TYPE_LINES_M 0x0000001F // Number of interrupt lines (x32) #define NVIC_INT_TYPE_LINES_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_ACTLR register. // //************************************************************************* #define NVIC_ACTLR_DISOOFP 0x00000200 // Disable Out-Of-Order Floating // Point #define NVIC_ACTLR_DISFPCA 0x00000100 // Disable CONTROL #define NVIC_ACTLR_DISFOLD 0x00000004 // Disable IT Folding #define NVIC_ACTLR_DISWBUF 0x00000002 // Disable Write Buffer #define NVIC_ACTLR_DISMCYC 0x00000001 // Disable Interrupts of Multiple // Cycle Instructions //************************************************************************* // // The following are defines for the bit fields in the NVIC_ST_CTRL register. // //************************************************************************* #define NVIC_ST_CTRL_COUNT 0x00010000 // Count Flag #define NVIC_ST_CTRL_CLK_SRC 0x00000004 // Clock Source #define NVIC_ST_CTRL_INTEN 0x00000002 // Interrupt Enable #define NVIC_ST_CTRL_ENABLE 0x00000001 // Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_ST_RELOAD register. // //************************************************************************* #define NVIC_ST_RELOAD_M 0x00FFFFFF // Reload Value #define NVIC_ST_RELOAD_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_ST_CURRENT // register. // //************************************************************************* #define NVIC_ST_CURRENT_M 0x00FFFFFF // Current Value #define NVIC_ST_CURRENT_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_ST_CAL register. // //************************************************************************* #define NVIC_ST_CAL_NOREF 0x80000000 // No reference clock #define NVIC_ST_CAL_SKEW 0x40000000 // Clock skew #define NVIC_ST_CAL_ONEMS_M 0x00FFFFFF // 1ms reference value #define NVIC_ST_CAL_ONEMS_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_EN0 register. // //************************************************************************* #define NVIC_EN0_INT_M 0xFFFFFFFF // Interrupt Enable #define NVIC_EN0_INT0 0x00000001 // Interrupt 0 enable #define NVIC_EN0_INT1 0x00000002 // Interrupt 1 enable #define NVIC_EN0_INT2 0x00000004 // Interrupt 2 enable #define NVIC_EN0_INT3 0x00000008 // Interrupt 3 enable #define NVIC_EN0_INT4 0x00000010 // Interrupt 4 enable #define NVIC_EN0_INT5 0x00000020 // Interrupt 5 enable #define NVIC_EN0_INT6 0x00000040 // Interrupt 6 enable #define NVIC_EN0_INT7 0x00000080 // Interrupt 7 enable #define NVIC_EN0_INT8 0x00000100 // Interrupt 8 enable #define NVIC_EN0_INT9 0x00000200 // Interrupt 9 enable #define NVIC_EN0_INT10 0x00000400 // Interrupt 10 enable #define NVIC_EN0_INT11 0x00000800 // Interrupt 11 enable #define NVIC_EN0_INT12 0x00001000 // Interrupt 12 enable #define NVIC_EN0_INT13 0x00002000 // Interrupt 13 enable #define NVIC_EN0_INT14 0x00004000 // Interrupt 14 enable #define NVIC_EN0_INT15 0x00008000 // Interrupt 15 enable #define NVIC_EN0_INT16 0x00010000 // Interrupt 16 enable #define NVIC_EN0_INT17 0x00020000 // Interrupt 17 enable #define NVIC_EN0_INT18 0x00040000 // Interrupt 18 enable #define NVIC_EN0_INT19 0x00080000 // Interrupt 19 enable #define NVIC_EN0_INT20 0x00100000 // Interrupt 20 enable #define NVIC_EN0_INT21 0x00200000 // Interrupt 21 enable #define NVIC_EN0_INT22 0x00400000 // Interrupt 22 enable #define NVIC_EN0_INT23 0x00800000 // Interrupt 23 enable #define NVIC_EN0_INT24 0x01000000 // Interrupt 24 enable #define NVIC_EN0_INT25 0x02000000 // Interrupt 25 enable #define NVIC_EN0_INT26 0x04000000 // Interrupt 26 enable #define NVIC_EN0_INT27 0x08000000 // Interrupt 27 enable #define NVIC_EN0_INT28 0x10000000 // Interrupt 28 enable #define NVIC_EN0_INT29 0x20000000 // Interrupt 29 enable #define NVIC_EN0_INT30 0x40000000 // Interrupt 30 enable #define NVIC_EN0_INT31 0x80000000 // Interrupt 31 enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_EN1 register. // //************************************************************************* #define NVIC_EN1_INT_M 0xFFFFFFFF // Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_EN2 register. // //************************************************************************* #define NVIC_EN2_INT_M 0xFFFFFFFF // Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_EN3 register. // //************************************************************************* #define NVIC_EN3_INT_M 0xFFFFFFFF // Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_EN4 register. // //************************************************************************* #define NVIC_EN4_INT_M 0x000007FF // Interrupt Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_DIS0 register. // //************************************************************************* #define NVIC_DIS0_INT_M 0xFFFFFFFF // Interrupt Disable #define NVIC_DIS0_INT0 0x00000001 // Interrupt 0 disable #define NVIC_DIS0_INT1 0x00000002 // Interrupt 1 disable #define NVIC_DIS0_INT2 0x00000004 // Interrupt 2 disable #define NVIC_DIS0_INT3 0x00000008 // Interrupt 3 disable #define NVIC_DIS0_INT4 0x00000010 // Interrupt 4 disable #define NVIC_DIS0_INT5 0x00000020 // Interrupt 5 disable #define NVIC_DIS0_INT6 0x00000040 // Interrupt 6 disable #define NVIC_DIS0_INT7 0x00000080 // Interrupt 7 disable #define NVIC_DIS0_INT8 0x00000100 // Interrupt 8 disable #define NVIC_DIS0_INT9 0x00000200 // Interrupt 9 disable #define NVIC_DIS0_INT10 0x00000400 // Interrupt 10 disable #define NVIC_DIS0_INT11 0x00000800 // Interrupt 11 disable #define NVIC_DIS0_INT12 0x00001000 // Interrupt 12 disable #define NVIC_DIS0_INT13 0x00002000 // Interrupt 13 disable #define NVIC_DIS0_INT14 0x00004000 // Interrupt 14 disable #define NVIC_DIS0_INT15 0x00008000 // Interrupt 15 disable #define NVIC_DIS0_INT16 0x00010000 // Interrupt 16 disable #define NVIC_DIS0_INT17 0x00020000 // Interrupt 17 disable #define NVIC_DIS0_INT18 0x00040000 // Interrupt 18 disable #define NVIC_DIS0_INT19 0x00080000 // Interrupt 19 disable #define NVIC_DIS0_INT20 0x00100000 // Interrupt 20 disable #define NVIC_DIS0_INT21 0x00200000 // Interrupt 21 disable #define NVIC_DIS0_INT22 0x00400000 // Interrupt 22 disable #define NVIC_DIS0_INT23 0x00800000 // Interrupt 23 disable #define NVIC_DIS0_INT24 0x01000000 // Interrupt 24 disable #define NVIC_DIS0_INT25 0x02000000 // Interrupt 25 disable #define NVIC_DIS0_INT26 0x04000000 // Interrupt 26 disable #define NVIC_DIS0_INT27 0x08000000 // Interrupt 27 disable #define NVIC_DIS0_INT28 0x10000000 // Interrupt 28 disable #define NVIC_DIS0_INT29 0x20000000 // Interrupt 29 disable #define NVIC_DIS0_INT30 0x40000000 // Interrupt 30 disable #define NVIC_DIS0_INT31 0x80000000 // Interrupt 31 disable //************************************************************************* // // The following are defines for the bit fields in the NVIC_DIS1 register. // //************************************************************************* #define NVIC_DIS1_INT_M 0xFFFFFFFF // Interrupt Disable //************************************************************************* // // The following are defines for the bit fields in the NVIC_DIS2 register. // //************************************************************************* #define NVIC_DIS2_INT_M 0xFFFFFFFF // Interrupt Disable //************************************************************************* // // The following are defines for the bit fields in the NVIC_DIS3 register. // //************************************************************************* #define NVIC_DIS3_INT_M 0xFFFFFFFF // Interrupt Disable //************************************************************************* // // The following are defines for the bit fields in the NVIC_DIS4 register. // //************************************************************************* #define NVIC_DIS4_INT_M 0x000007FF // Interrupt Disable //************************************************************************* // // The following are defines for the bit fields in the NVIC_PEND0 register. // //************************************************************************* #define NVIC_PEND0_INT_M 0xFFFFFFFF // Interrupt Set Pending #define NVIC_PEND0_INT0 0x00000001 // Interrupt 0 pend #define NVIC_PEND0_INT1 0x00000002 // Interrupt 1 pend #define NVIC_PEND0_INT2 0x00000004 // Interrupt 2 pend #define NVIC_PEND0_INT3 0x00000008 // Interrupt 3 pend #define NVIC_PEND0_INT4 0x00000010 // Interrupt 4 pend #define NVIC_PEND0_INT5 0x00000020 // Interrupt 5 pend #define NVIC_PEND0_INT6 0x00000040 // Interrupt 6 pend #define NVIC_PEND0_INT7 0x00000080 // Interrupt 7 pend #define NVIC_PEND0_INT8 0x00000100 // Interrupt 8 pend #define NVIC_PEND0_INT9 0x00000200 // Interrupt 9 pend #define NVIC_PEND0_INT10 0x00000400 // Interrupt 10 pend #define NVIC_PEND0_INT11 0x00000800 // Interrupt 11 pend #define NVIC_PEND0_INT12 0x00001000 // Interrupt 12 pend #define NVIC_PEND0_INT13 0x00002000 // Interrupt 13 pend #define NVIC_PEND0_INT14 0x00004000 // Interrupt 14 pend #define NVIC_PEND0_INT15 0x00008000 // Interrupt 15 pend #define NVIC_PEND0_INT16 0x00010000 // Interrupt 16 pend #define NVIC_PEND0_INT17 0x00020000 // Interrupt 17 pend #define NVIC_PEND0_INT18 0x00040000 // Interrupt 18 pend #define NVIC_PEND0_INT19 0x00080000 // Interrupt 19 pend #define NVIC_PEND0_INT20 0x00100000 // Interrupt 20 pend #define NVIC_PEND0_INT21 0x00200000 // Interrupt 21 pend #define NVIC_PEND0_INT22 0x00400000 // Interrupt 22 pend #define NVIC_PEND0_INT23 0x00800000 // Interrupt 23 pend #define NVIC_PEND0_INT24 0x01000000 // Interrupt 24 pend #define NVIC_PEND0_INT25 0x02000000 // Interrupt 25 pend #define NVIC_PEND0_INT26 0x04000000 // Interrupt 26 pend #define NVIC_PEND0_INT27 0x08000000 // Interrupt 27 pend #define NVIC_PEND0_INT28 0x10000000 // Interrupt 28 pend #define NVIC_PEND0_INT29 0x20000000 // Interrupt 29 pend #define NVIC_PEND0_INT30 0x40000000 // Interrupt 30 pend #define NVIC_PEND0_INT31 0x80000000 // Interrupt 31 pend //************************************************************************* // // The following are defines for the bit fields in the NVIC_PEND1 register. // //************************************************************************* #define NVIC_PEND1_INT_M 0xFFFFFFFF // Interrupt Set Pending //************************************************************************* // // The following are defines for the bit fields in the NVIC_PEND2 register. // //************************************************************************* #define NVIC_PEND2_INT_M 0xFFFFFFFF // Interrupt Set Pending //************************************************************************* // // The following are defines for the bit fields in the NVIC_PEND3 register. // //************************************************************************* #define NVIC_PEND3_INT_M 0xFFFFFFFF // Interrupt Set Pending //************************************************************************* // // The following are defines for the bit fields in the NVIC_PEND4 register. // //************************************************************************* #define NVIC_PEND4_INT_M 0x000007FF // Interrupt Set Pending //************************************************************************* // // The following are defines for the bit fields in the NVIC_UNPEND0 register. // //************************************************************************* #define NVIC_UNPEND0_INT_M 0xFFFFFFFF // Interrupt Clear Pending #define NVIC_UNPEND0_INT0 0x00000001 // Interrupt 0 unpend #define NVIC_UNPEND0_INT1 0x00000002 // Interrupt 1 unpend #define NVIC_UNPEND0_INT2 0x00000004 // Interrupt 2 unpend #define NVIC_UNPEND0_INT3 0x00000008 // Interrupt 3 unpend #define NVIC_UNPEND0_INT4 0x00000010 // Interrupt 4 unpend #define NVIC_UNPEND0_INT5 0x00000020 // Interrupt 5 unpend #define NVIC_UNPEND0_INT6 0x00000040 // Interrupt 6 unpend #define NVIC_UNPEND0_INT7 0x00000080 // Interrupt 7 unpend #define NVIC_UNPEND0_INT8 0x00000100 // Interrupt 8 unpend #define NVIC_UNPEND0_INT9 0x00000200 // Interrupt 9 unpend #define NVIC_UNPEND0_INT10 0x00000400 // Interrupt 10 unpend #define NVIC_UNPEND0_INT11 0x00000800 // Interrupt 11 unpend #define NVIC_UNPEND0_INT12 0x00001000 // Interrupt 12 unpend #define NVIC_UNPEND0_INT13 0x00002000 // Interrupt 13 unpend #define NVIC_UNPEND0_INT14 0x00004000 // Interrupt 14 unpend #define NVIC_UNPEND0_INT15 0x00008000 // Interrupt 15 unpend #define NVIC_UNPEND0_INT16 0x00010000 // Interrupt 16 unpend #define NVIC_UNPEND0_INT17 0x00020000 // Interrupt 17 unpend #define NVIC_UNPEND0_INT18 0x00040000 // Interrupt 18 unpend #define NVIC_UNPEND0_INT19 0x00080000 // Interrupt 19 unpend #define NVIC_UNPEND0_INT20 0x00100000 // Interrupt 20 unpend #define NVIC_UNPEND0_INT21 0x00200000 // Interrupt 21 unpend #define NVIC_UNPEND0_INT22 0x00400000 // Interrupt 22 unpend #define NVIC_UNPEND0_INT23 0x00800000 // Interrupt 23 unpend #define NVIC_UNPEND0_INT24 0x01000000 // Interrupt 24 unpend #define NVIC_UNPEND0_INT25 0x02000000 // Interrupt 25 unpend #define NVIC_UNPEND0_INT26 0x04000000 // Interrupt 26 unpend #define NVIC_UNPEND0_INT27 0x08000000 // Interrupt 27 unpend #define NVIC_UNPEND0_INT28 0x10000000 // Interrupt 28 unpend #define NVIC_UNPEND0_INT29 0x20000000 // Interrupt 29 unpend #define NVIC_UNPEND0_INT30 0x40000000 // Interrupt 30 unpend #define NVIC_UNPEND0_INT31 0x80000000 // Interrupt 31 unpend //************************************************************************* // // The following are defines for the bit fields in the NVIC_UNPEND1 register. // //************************************************************************* #define NVIC_UNPEND1_INT_M 0xFFFFFFFF // Interrupt Clear Pending #define NVIC_UNPEND1_INT32 0x00000001 // Interrupt 32 unpend #define NVIC_UNPEND1_INT33 0x00000002 // Interrupt 33 unpend #define NVIC_UNPEND1_INT34 0x00000004 // Interrupt 34 unpend #define NVIC_UNPEND1_INT35 0x00000008 // Interrupt 35 unpend #define NVIC_UNPEND1_INT36 0x00000010 // Interrupt 36 unpend #define NVIC_UNPEND1_INT37 0x00000020 // Interrupt 37 unpend #define NVIC_UNPEND1_INT38 0x00000040 // Interrupt 38 unpend #define NVIC_UNPEND1_INT39 0x00000080 // Interrupt 39 unpend #define NVIC_UNPEND1_INT40 0x00000100 // Interrupt 40 unpend #define NVIC_UNPEND1_INT41 0x00000200 // Interrupt 41 unpend #define NVIC_UNPEND1_INT42 0x00000400 // Interrupt 42 unpend #define NVIC_UNPEND1_INT43 0x00000800 // Interrupt 43 unpend #define NVIC_UNPEND1_INT44 0x00001000 // Interrupt 44 unpend #define NVIC_UNPEND1_INT45 0x00002000 // Interrupt 45 unpend #define NVIC_UNPEND1_INT46 0x00004000 // Interrupt 46 unpend #define NVIC_UNPEND1_INT47 0x00008000 // Interrupt 47 unpend #define NVIC_UNPEND1_INT48 0x00010000 // Interrupt 48 unpend #define NVIC_UNPEND1_INT49 0x00020000 // Interrupt 49 unpend #define NVIC_UNPEND1_INT50 0x00040000 // Interrupt 50 unpend #define NVIC_UNPEND1_INT51 0x00080000 // Interrupt 51 unpend #define NVIC_UNPEND1_INT52 0x00100000 // Interrupt 52 unpend #define NVIC_UNPEND1_INT53 0x00200000 // Interrupt 53 unpend #define NVIC_UNPEND1_INT54 0x00400000 // Interrupt 54 unpend #define NVIC_UNPEND1_INT55 0x00800000 // Interrupt 55 unpend //************************************************************************* // // The following are defines for the bit fields in the NVIC_UNPEND2 register. // //************************************************************************* #define NVIC_UNPEND2_INT_M 0xFFFFFFFF // Interrupt Clear Pending //************************************************************************* // // The following are defines for the bit fields in the NVIC_UNPEND3 register. // //************************************************************************* #define NVIC_UNPEND3_INT_M 0xFFFFFFFF // Interrupt Clear Pending //************************************************************************* // // The following are defines for the bit fields in the NVIC_UNPEND4 register. // //************************************************************************* #define NVIC_UNPEND4_INT_M 0x000007FF // Interrupt Clear Pending //************************************************************************* // // The following are defines for the bit fields in the NVIC_ACTIVE0 register. // //************************************************************************* #define NVIC_ACTIVE0_INT_M 0xFFFFFFFF // Interrupt Active #define NVIC_ACTIVE0_INT0 0x00000001 // Interrupt 0 active #define NVIC_ACTIVE0_INT1 0x00000002 // Interrupt 1 active #define NVIC_ACTIVE0_INT2 0x00000004 // Interrupt 2 active #define NVIC_ACTIVE0_INT3 0x00000008 // Interrupt 3 active #define NVIC_ACTIVE0_INT4 0x00000010 // Interrupt 4 active #define NVIC_ACTIVE0_INT5 0x00000020 // Interrupt 5 active #define NVIC_ACTIVE0_INT6 0x00000040 // Interrupt 6 active #define NVIC_ACTIVE0_INT7 0x00000080 // Interrupt 7 active #define NVIC_ACTIVE0_INT8 0x00000100 // Interrupt 8 active #define NVIC_ACTIVE0_INT9 0x00000200 // Interrupt 9 active #define NVIC_ACTIVE0_INT10 0x00000400 // Interrupt 10 active #define NVIC_ACTIVE0_INT11 0x00000800 // Interrupt 11 active #define NVIC_ACTIVE0_INT12 0x00001000 // Interrupt 12 active #define NVIC_ACTIVE0_INT13 0x00002000 // Interrupt 13 active #define NVIC_ACTIVE0_INT14 0x00004000 // Interrupt 14 active #define NVIC_ACTIVE0_INT15 0x00008000 // Interrupt 15 active #define NVIC_ACTIVE0_INT16 0x00010000 // Interrupt 16 active #define NVIC_ACTIVE0_INT17 0x00020000 // Interrupt 17 active #define NVIC_ACTIVE0_INT18 0x00040000 // Interrupt 18 active #define NVIC_ACTIVE0_INT19 0x00080000 // Interrupt 19 active #define NVIC_ACTIVE0_INT20 0x00100000 // Interrupt 20 active #define NVIC_ACTIVE0_INT21 0x00200000 // Interrupt 21 active #define NVIC_ACTIVE0_INT22 0x00400000 // Interrupt 22 active #define NVIC_ACTIVE0_INT23 0x00800000 // Interrupt 23 active #define NVIC_ACTIVE0_INT24 0x01000000 // Interrupt 24 active #define NVIC_ACTIVE0_INT25 0x02000000 // Interrupt 25 active #define NVIC_ACTIVE0_INT26 0x04000000 // Interrupt 26 active #define NVIC_ACTIVE0_INT27 0x08000000 // Interrupt 27 active #define NVIC_ACTIVE0_INT28 0x10000000 // Interrupt 28 active #define NVIC_ACTIVE0_INT29 0x20000000 // Interrupt 29 active #define NVIC_ACTIVE0_INT30 0x40000000 // Interrupt 30 active #define NVIC_ACTIVE0_INT31 0x80000000 // Interrupt 31 active //************************************************************************* // // The following are defines for the bit fields in the NVIC_ACTIVE1 register. // //************************************************************************* #define NVIC_ACTIVE1_INT_M 0xFFFFFFFF // Interrupt Active //************************************************************************* // // The following are defines for the bit fields in the NVIC_ACTIVE2 register. // //************************************************************************* #define NVIC_ACTIVE2_INT_M 0xFFFFFFFF // Interrupt Active //************************************************************************* // // The following are defines for the bit fields in the NVIC_ACTIVE3 register. // //************************************************************************* #define NVIC_ACTIVE3_INT_M 0xFFFFFFFF // Interrupt Active //************************************************************************* // // The following are defines for the bit fields in the NVIC_ACTIVE4 register. // //************************************************************************* #define NVIC_ACTIVE4_INT_M 0x000007FF // Interrupt Active //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI0 register. // //************************************************************************* #define NVIC_PRI0_INT3_M 0xE0000000 // Interrupt 3 Priority Mask #define NVIC_PRI0_INT2_M 0x00E00000 // Interrupt 2 Priority Mask #define NVIC_PRI0_INT1_M 0x0000E000 // Interrupt 1 Priority Mask #define NVIC_PRI0_INT0_M 0x000000E0 // Interrupt 0 Priority Mask #define NVIC_PRI0_INT3_S 29 #define NVIC_PRI0_INT2_S 21 #define NVIC_PRI0_INT1_S 13 #define NVIC_PRI0_INT0_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI1 register. // //************************************************************************* #define NVIC_PRI1_INT7_M 0xE0000000 // Interrupt 7 Priority Mask #define NVIC_PRI1_INT6_M 0x00E00000 // Interrupt 6 Priority Mask #define NVIC_PRI1_INT5_M 0x0000E000 // Interrupt 5 Priority Mask #define NVIC_PRI1_INT4_M 0x000000E0 // Interrupt 4 Priority Mask #define NVIC_PRI1_INT7_S 29 #define NVIC_PRI1_INT6_S 21 #define NVIC_PRI1_INT5_S 13 #define NVIC_PRI1_INT4_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI2 register. // //************************************************************************* #define NVIC_PRI2_INT11_M 0xE0000000 // Interrupt 11 Priority Mask #define NVIC_PRI2_INT10_M 0x00E00000 // Interrupt 10 Priority Mask #define NVIC_PRI2_INT9_M 0x0000E000 // Interrupt 9 Priority Mask #define NVIC_PRI2_INT8_M 0x000000E0 // Interrupt 8 Priority Mask #define NVIC_PRI2_INT11_S 29 #define NVIC_PRI2_INT10_S 21 #define NVIC_PRI2_INT9_S 13 #define NVIC_PRI2_INT8_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI3 register. // //************************************************************************* #define NVIC_PRI3_INT15_M 0xE0000000 // Interrupt 15 Priority Mask #define NVIC_PRI3_INT14_M 0x00E00000 // Interrupt 14 Priority Mask #define NVIC_PRI3_INT13_M 0x0000E000 // Interrupt 13 Priority Mask #define NVIC_PRI3_INT12_M 0x000000E0 // Interrupt 12 Priority Mask #define NVIC_PRI3_INT15_S 29 #define NVIC_PRI3_INT14_S 21 #define NVIC_PRI3_INT13_S 13 #define NVIC_PRI3_INT12_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI4 register. // //************************************************************************* #define NVIC_PRI4_INT19_M 0xE0000000 // Interrupt 19 Priority Mask #define NVIC_PRI4_INT18_M 0x00E00000 // Interrupt 18 Priority Mask #define NVIC_PRI4_INT17_M 0x0000E000 // Interrupt 17 Priority Mask #define NVIC_PRI4_INT16_M 0x000000E0 // Interrupt 16 Priority Mask #define NVIC_PRI4_INT19_S 29 #define NVIC_PRI4_INT18_S 21 #define NVIC_PRI4_INT17_S 13 #define NVIC_PRI4_INT16_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI5 register. // //************************************************************************* #define NVIC_PRI5_INT23_M 0xE0000000 // Interrupt 23 Priority Mask #define NVIC_PRI5_INT22_M 0x00E00000 // Interrupt 22 Priority Mask #define NVIC_PRI5_INT21_M 0x0000E000 // Interrupt 21 Priority Mask #define NVIC_PRI5_INT20_M 0x000000E0 // Interrupt 20 Priority Mask #define NVIC_PRI5_INT23_S 29 #define NVIC_PRI5_INT22_S 21 #define NVIC_PRI5_INT21_S 13 #define NVIC_PRI5_INT20_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI6 register. // //************************************************************************* #define NVIC_PRI6_INT27_M 0xE0000000 // Interrupt 27 Priority Mask #define NVIC_PRI6_INT26_M 0x00E00000 // Interrupt 26 Priority Mask #define NVIC_PRI6_INT25_M 0x0000E000 // Interrupt 25 Priority Mask #define NVIC_PRI6_INT24_M 0x000000E0 // Interrupt 24 Priority Mask #define NVIC_PRI6_INT27_S 29 #define NVIC_PRI6_INT26_S 21 #define NVIC_PRI6_INT25_S 13 #define NVIC_PRI6_INT24_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI7 register. // //************************************************************************* #define NVIC_PRI7_INT31_M 0xE0000000 // Interrupt 31 Priority Mask #define NVIC_PRI7_INT30_M 0x00E00000 // Interrupt 30 Priority Mask #define NVIC_PRI7_INT29_M 0x0000E000 // Interrupt 29 Priority Mask #define NVIC_PRI7_INT28_M 0x000000E0 // Interrupt 28 Priority Mask #define NVIC_PRI7_INT31_S 29 #define NVIC_PRI7_INT30_S 21 #define NVIC_PRI7_INT29_S 13 #define NVIC_PRI7_INT28_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI8 register. // //************************************************************************* #define NVIC_PRI8_INT35_M 0xE0000000 // Interrupt 35 Priority Mask #define NVIC_PRI8_INT34_M 0x00E00000 // Interrupt 34 Priority Mask #define NVIC_PRI8_INT33_M 0x0000E000 // Interrupt 33 Priority Mask #define NVIC_PRI8_INT32_M 0x000000E0 // Interrupt 32 Priority Mask #define NVIC_PRI8_INT35_S 29 #define NVIC_PRI8_INT34_S 21 #define NVIC_PRI8_INT33_S 13 #define NVIC_PRI8_INT32_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI9 register. // //************************************************************************* #define NVIC_PRI9_INT39_M 0xE0000000 // Interrupt 39 Priority Mask #define NVIC_PRI9_INT38_M 0x00E00000 // Interrupt 38 Priority Mask #define NVIC_PRI9_INT37_M 0x0000E000 // Interrupt 37 Priority Mask #define NVIC_PRI9_INT36_M 0x000000E0 // Interrupt 36 Priority Mask #define NVIC_PRI9_INT39_S 29 #define NVIC_PRI9_INT38_S 21 #define NVIC_PRI9_INT37_S 13 #define NVIC_PRI9_INT36_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI10 register. // //************************************************************************* #define NVIC_PRI10_INT43_M 0xE0000000 // Interrupt 43 Priority Mask #define NVIC_PRI10_INT42_M 0x00E00000 // Interrupt 42 Priority Mask #define NVIC_PRI10_INT41_M 0x0000E000 // Interrupt 41 Priority Mask #define NVIC_PRI10_INT40_M 0x000000E0 // Interrupt 40 Priority Mask #define NVIC_PRI10_INT43_S 29 #define NVIC_PRI10_INT42_S 21 #define NVIC_PRI10_INT41_S 13 #define NVIC_PRI10_INT40_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI11 register. // //************************************************************************* #define NVIC_PRI11_INT47_M 0xE0000000 // Interrupt 47 Priority Mask #define NVIC_PRI11_INT46_M 0x00E00000 // Interrupt 46 Priority Mask #define NVIC_PRI11_INT45_M 0x0000E000 // Interrupt 45 Priority Mask #define NVIC_PRI11_INT44_M 0x000000E0 // Interrupt 44 Priority Mask #define NVIC_PRI11_INT47_S 29 #define NVIC_PRI11_INT46_S 21 #define NVIC_PRI11_INT45_S 13 #define NVIC_PRI11_INT44_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI12 register. // //************************************************************************* #define NVIC_PRI12_INT51_M 0xE0000000 // Interrupt 51 Priority Mask #define NVIC_PRI12_INT50_M 0x00E00000 // Interrupt 50 Priority Mask #define NVIC_PRI12_INT49_M 0x0000E000 // Interrupt 49 Priority Mask #define NVIC_PRI12_INT48_M 0x000000E0 // Interrupt 48 Priority Mask #define NVIC_PRI12_INT51_S 29 #define NVIC_PRI12_INT50_S 21 #define NVIC_PRI12_INT49_S 13 #define NVIC_PRI12_INT48_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI13 register. // //************************************************************************* #define NVIC_PRI13_INT55_M 0xE0000000 // Interrupt 55 Priority Mask #define NVIC_PRI13_INT54_M 0x00E00000 // Interrupt 54 Priority Mask #define NVIC_PRI13_INT53_M 0x0000E000 // Interrupt 53 Priority Mask #define NVIC_PRI13_INT52_M 0x000000E0 // Interrupt 52 Priority Mask #define NVIC_PRI13_INT55_S 29 #define NVIC_PRI13_INT54_S 21 #define NVIC_PRI13_INT53_S 13 #define NVIC_PRI13_INT52_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI14 register. // //************************************************************************* #define NVIC_PRI14_INTD_M 0xE0000000 // Interrupt 59 Priority Mask #define NVIC_PRI14_INTC_M 0x00E00000 // Interrupt 58 Priority Mask #define NVIC_PRI14_INTB_M 0x0000E000 // Interrupt 57 Priority Mask #define NVIC_PRI14_INTA_M 0x000000E0 // Interrupt 56 Priority Mask #define NVIC_PRI14_INTD_S 29 #define NVIC_PRI14_INTC_S 21 #define NVIC_PRI14_INTB_S 13 #define NVIC_PRI14_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI15 register. // //************************************************************************* #define NVIC_PRI15_INTD_M 0xE0000000 // Interrupt 63 Priority Mask #define NVIC_PRI15_INTC_M 0x00E00000 // Interrupt 62 Priority Mask #define NVIC_PRI15_INTB_M 0x0000E000 // Interrupt 61 Priority Mask #define NVIC_PRI15_INTA_M 0x000000E0 // Interrupt 60 Priority Mask #define NVIC_PRI15_INTD_S 29 #define NVIC_PRI15_INTC_S 21 #define NVIC_PRI15_INTB_S 13 #define NVIC_PRI15_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI16 register. // //************************************************************************* #define NVIC_PRI16_INTD_M 0xE0000000 // Interrupt 67 Priority Mask #define NVIC_PRI16_INTC_M 0x00E00000 // Interrupt 66 Priority Mask #define NVIC_PRI16_INTB_M 0x0000E000 // Interrupt 65 Priority Mask #define NVIC_PRI16_INTA_M 0x000000E0 // Interrupt 64 Priority Mask #define NVIC_PRI16_INTD_S 29 #define NVIC_PRI16_INTC_S 21 #define NVIC_PRI16_INTB_S 13 #define NVIC_PRI16_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI17 register. // //************************************************************************* #define NVIC_PRI17_INTD_M 0xE0000000 // Interrupt 71 Priority Mask #define NVIC_PRI17_INTC_M 0x00E00000 // Interrupt 70 Priority Mask #define NVIC_PRI17_INTB_M 0x0000E000 // Interrupt 69 Priority Mask #define NVIC_PRI17_INTA_M 0x000000E0 // Interrupt 68 Priority Mask #define NVIC_PRI17_INTD_S 29 #define NVIC_PRI17_INTC_S 21 #define NVIC_PRI17_INTB_S 13 #define NVIC_PRI17_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI18 register. // //************************************************************************* #define NVIC_PRI18_INTD_M 0xE0000000 // Interrupt 75 Priority Mask #define NVIC_PRI18_INTC_M 0x00E00000 // Interrupt 74 Priority Mask #define NVIC_PRI18_INTB_M 0x0000E000 // Interrupt 73 Priority Mask #define NVIC_PRI18_INTA_M 0x000000E0 // Interrupt 72 Priority Mask #define NVIC_PRI18_INTD_S 29 #define NVIC_PRI18_INTC_S 21 #define NVIC_PRI18_INTB_S 13 #define NVIC_PRI18_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI19 register. // //************************************************************************* #define NVIC_PRI19_INTD_M 0xE0000000 // Interrupt 79 Priority Mask #define NVIC_PRI19_INTC_M 0x00E00000 // Interrupt 78 Priority Mask #define NVIC_PRI19_INTB_M 0x0000E000 // Interrupt 77 Priority Mask #define NVIC_PRI19_INTA_M 0x000000E0 // Interrupt 76 Priority Mask #define NVIC_PRI19_INTD_S 29 #define NVIC_PRI19_INTC_S 21 #define NVIC_PRI19_INTB_S 13 #define NVIC_PRI19_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI20 register. // //************************************************************************* #define NVIC_PRI20_INTD_M 0xE0000000 // Interrupt 83 Priority Mask #define NVIC_PRI20_INTC_M 0x00E00000 // Interrupt 82 Priority Mask #define NVIC_PRI20_INTB_M 0x0000E000 // Interrupt 81 Priority Mask #define NVIC_PRI20_INTA_M 0x000000E0 // Interrupt 80 Priority Mask #define NVIC_PRI20_INTD_S 29 #define NVIC_PRI20_INTC_S 21 #define NVIC_PRI20_INTB_S 13 #define NVIC_PRI20_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI21 register. // //************************************************************************* #define NVIC_PRI21_INTD_M 0xE0000000 // Interrupt 87 Priority Mask #define NVIC_PRI21_INTC_M 0x00E00000 // Interrupt 86 Priority Mask #define NVIC_PRI21_INTB_M 0x0000E000 // Interrupt 85 Priority Mask #define NVIC_PRI21_INTA_M 0x000000E0 // Interrupt 84 Priority Mask #define NVIC_PRI21_INTD_S 29 #define NVIC_PRI21_INTC_S 21 #define NVIC_PRI21_INTB_S 13 #define NVIC_PRI21_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI22 register. // //************************************************************************* #define NVIC_PRI22_INTD_M 0xE0000000 // Interrupt 91 Priority Mask #define NVIC_PRI22_INTC_M 0x00E00000 // Interrupt 90 Priority Mask #define NVIC_PRI22_INTB_M 0x0000E000 // Interrupt 89 Priority Mask #define NVIC_PRI22_INTA_M 0x000000E0 // Interrupt 88 Priority Mask #define NVIC_PRI22_INTD_S 29 #define NVIC_PRI22_INTC_S 21 #define NVIC_PRI22_INTB_S 13 #define NVIC_PRI22_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI23 register. // //************************************************************************* #define NVIC_PRI23_INTD_M 0xE0000000 // Interrupt 95 Priority Mask #define NVIC_PRI23_INTC_M 0x00E00000 // Interrupt 94 Priority Mask #define NVIC_PRI23_INTB_M 0x0000E000 // Interrupt 93 Priority Mask #define NVIC_PRI23_INTA_M 0x000000E0 // Interrupt 92 Priority Mask #define NVIC_PRI23_INTD_S 29 #define NVIC_PRI23_INTC_S 21 #define NVIC_PRI23_INTB_S 13 #define NVIC_PRI23_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI24 register. // //************************************************************************* #define NVIC_PRI24_INTD_M 0xE0000000 // Interrupt 99 Priority Mask #define NVIC_PRI24_INTC_M 0x00E00000 // Interrupt 98 Priority Mask #define NVIC_PRI24_INTB_M 0x0000E000 // Interrupt 97 Priority Mask #define NVIC_PRI24_INTA_M 0x000000E0 // Interrupt 96 Priority Mask #define NVIC_PRI24_INTD_S 29 #define NVIC_PRI24_INTC_S 21 #define NVIC_PRI24_INTB_S 13 #define NVIC_PRI24_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI25 register. // //************************************************************************* #define NVIC_PRI25_INTD_M 0xE0000000 // Interrupt 103 Priority Mask #define NVIC_PRI25_INTC_M 0x00E00000 // Interrupt 102 Priority Mask #define NVIC_PRI25_INTB_M 0x0000E000 // Interrupt 101 Priority Mask #define NVIC_PRI25_INTA_M 0x000000E0 // Interrupt 100 Priority Mask #define NVIC_PRI25_INTD_S 29 #define NVIC_PRI25_INTC_S 21 #define NVIC_PRI25_INTB_S 13 #define NVIC_PRI25_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI26 register. // //************************************************************************* #define NVIC_PRI26_INTD_M 0xE0000000 // Interrupt 107 Priority Mask #define NVIC_PRI26_INTC_M 0x00E00000 // Interrupt 106 Priority Mask #define NVIC_PRI26_INTB_M 0x0000E000 // Interrupt 105 Priority Mask #define NVIC_PRI26_INTA_M 0x000000E0 // Interrupt 104 Priority Mask #define NVIC_PRI26_INTD_S 29 #define NVIC_PRI26_INTC_S 21 #define NVIC_PRI26_INTB_S 13 #define NVIC_PRI26_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI27 register. // //************************************************************************* #define NVIC_PRI27_INTD_M 0xE0000000 // Interrupt 111 Priority Mask #define NVIC_PRI27_INTC_M 0x00E00000 // Interrupt 110 Priority Mask #define NVIC_PRI27_INTB_M 0x0000E000 // Interrupt 109 Priority Mask #define NVIC_PRI27_INTA_M 0x000000E0 // Interrupt 108 Priority Mask #define NVIC_PRI27_INTD_S 29 #define NVIC_PRI27_INTC_S 21 #define NVIC_PRI27_INTB_S 13 #define NVIC_PRI27_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI28 register. // //************************************************************************* #define NVIC_PRI28_INTD_M 0xE0000000 // Interrupt 115 Priority Mask #define NVIC_PRI28_INTC_M 0x00E00000 // Interrupt 114 Priority Mask #define NVIC_PRI28_INTB_M 0x0000E000 // Interrupt 113 Priority Mask #define NVIC_PRI28_INTA_M 0x000000E0 // Interrupt 112 Priority Mask #define NVIC_PRI28_INTD_S 29 #define NVIC_PRI28_INTC_S 21 #define NVIC_PRI28_INTB_S 13 #define NVIC_PRI28_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI29 register. // //************************************************************************* #define NVIC_PRI29_INTD_M 0xE0000000 // Interrupt 119 Priority Mask #define NVIC_PRI29_INTC_M 0x00E00000 // Interrupt 118 Priority Mask #define NVIC_PRI29_INTB_M 0x0000E000 // Interrupt 117 Priority Mask #define NVIC_PRI29_INTA_M 0x000000E0 // Interrupt 116 Priority Mask #define NVIC_PRI29_INTD_S 29 #define NVIC_PRI29_INTC_S 21 #define NVIC_PRI29_INTB_S 13 #define NVIC_PRI29_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI30 register. // //************************************************************************* #define NVIC_PRI30_INTD_M 0xE0000000 // Interrupt 123 Priority Mask #define NVIC_PRI30_INTC_M 0x00E00000 // Interrupt 122 Priority Mask #define NVIC_PRI30_INTB_M 0x0000E000 // Interrupt 121 Priority Mask #define NVIC_PRI30_INTA_M 0x000000E0 // Interrupt 120 Priority Mask #define NVIC_PRI30_INTD_S 29 #define NVIC_PRI30_INTC_S 21 #define NVIC_PRI30_INTB_S 13 #define NVIC_PRI30_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI31 register. // //************************************************************************* #define NVIC_PRI31_INTD_M 0xE0000000 // Interrupt 127 Priority Mask #define NVIC_PRI31_INTC_M 0x00E00000 // Interrupt 126 Priority Mask #define NVIC_PRI31_INTB_M 0x0000E000 // Interrupt 125 Priority Mask #define NVIC_PRI31_INTA_M 0x000000E0 // Interrupt 124 Priority Mask #define NVIC_PRI31_INTD_S 29 #define NVIC_PRI31_INTC_S 21 #define NVIC_PRI31_INTB_S 13 #define NVIC_PRI31_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI32 register. // //************************************************************************* #define NVIC_PRI32_INTD_M 0xE0000000 // Interrupt 131 Priority Mask #define NVIC_PRI32_INTC_M 0x00E00000 // Interrupt 130 Priority Mask #define NVIC_PRI32_INTB_M 0x0000E000 // Interrupt 129 Priority Mask #define NVIC_PRI32_INTA_M 0x000000E0 // Interrupt 128 Priority Mask #define NVIC_PRI32_INTD_S 29 #define NVIC_PRI32_INTC_S 21 #define NVIC_PRI32_INTB_S 13 #define NVIC_PRI32_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI33 register. // //************************************************************************* #define NVIC_PRI33_INTD_M 0xE0000000 // Interrupt Priority for Interrupt // [4n+3] #define NVIC_PRI33_INTC_M 0x00E00000 // Interrupt Priority for Interrupt // [4n+2] #define NVIC_PRI33_INTB_M 0x0000E000 // Interrupt Priority for Interrupt // [4n+1] #define NVIC_PRI33_INTA_M 0x000000E0 // Interrupt Priority for Interrupt // [4n] #define NVIC_PRI33_INTD_S 29 #define NVIC_PRI33_INTC_S 21 #define NVIC_PRI33_INTB_S 13 #define NVIC_PRI33_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_PRI34 register. // //************************************************************************* #define NVIC_PRI34_INTD_M 0xE0000000 // Interrupt Priority for Interrupt // [4n+3] #define NVIC_PRI34_INTC_M 0x00E00000 // Interrupt Priority for Interrupt // [4n+2] #define NVIC_PRI34_INTB_M 0x0000E000 // Interrupt Priority for Interrupt // [4n+1] #define NVIC_PRI34_INTA_M 0x000000E0 // Interrupt Priority for Interrupt // [4n] #define NVIC_PRI34_INTD_S 29 #define NVIC_PRI34_INTC_S 21 #define NVIC_PRI34_INTB_S 13 #define NVIC_PRI34_INTA_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_CPUID register. // //************************************************************************* #define NVIC_CPUID_IMP_M 0xFF000000 // Implementer Code #define NVIC_CPUID_IMP_ARM 0x41000000 // ARM #define NVIC_CPUID_VAR_M 0x00F00000 // Variant Number #define NVIC_CPUID_CON_M 0x000F0000 // Constant #define NVIC_CPUID_PARTNO_M 0x0000FFF0 // Part Number #define NVIC_CPUID_PARTNO_CM4 0x0000C240 // Cortex-M4 processor #define NVIC_CPUID_REV_M 0x0000000F // Revision Number //************************************************************************* // // The following are defines for the bit fields in the NVIC_INT_CTRL register. // //************************************************************************* #define NVIC_INT_CTRL_NMI_SET 0x80000000 // NMI Set Pending #define NVIC_INT_CTRL_PEND_SV 0x10000000 // PendSV Set Pending #define NVIC_INT_CTRL_UNPEND_SV 0x08000000 // PendSV Clear Pending #define NVIC_INT_CTRL_PENDSTSET 0x04000000 // SysTick Set Pending #define NVIC_INT_CTRL_PENDSTCLR 0x02000000 // SysTick Clear Pending #define NVIC_INT_CTRL_ISR_PRE 0x00800000 // Debug Interrupt Handling #define NVIC_INT_CTRL_ISR_PEND 0x00400000 // Interrupt Pending #define NVIC_INT_CTRL_VEC_PEN_M 0x000FF000 // Interrupt Pending Vector Number #define NVIC_INT_CTRL_VEC_PEN_NMI \ 0x00002000 // NMI #define NVIC_INT_CTRL_VEC_PEN_HARD \ 0x00003000 // Hard fault #define NVIC_INT_CTRL_VEC_PEN_MEM \ 0x00004000 // Memory management fault #define NVIC_INT_CTRL_VEC_PEN_BUS \ 0x00005000 // Bus fault #define NVIC_INT_CTRL_VEC_PEN_USG \ 0x00006000 // Usage fault #define NVIC_INT_CTRL_VEC_PEN_SVC \ 0x0000B000 // SVCall #define NVIC_INT_CTRL_VEC_PEN_PNDSV \ 0x0000E000 // PendSV #define NVIC_INT_CTRL_VEC_PEN_TICK \ 0x0000F000 // SysTick #define NVIC_INT_CTRL_RET_BASE 0x00000800 // Return to Base #define NVIC_INT_CTRL_VEC_ACT_M 0x000000FF // Interrupt Pending Vector Number #define NVIC_INT_CTRL_VEC_ACT_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_VTABLE register. // //************************************************************************* #define NVIC_VTABLE_BASE 0x20000000 // Vector Table Base #define NVIC_VTABLE_OFFSET_M 0x1FFFFC00 // Vector Table Offset #define NVIC_VTABLE_OFFSET_S 10 //************************************************************************* // // The following are defines for the bit fields in the NVIC_APINT register. // //************************************************************************* #define NVIC_APINT_VECTKEY_M 0xFFFF0000 // Register Key #define NVIC_APINT_VECTKEY 0x05FA0000 // Vector key #define NVIC_APINT_ENDIANESS 0x00008000 // Data Endianess #define NVIC_APINT_PRIGROUP_M 0x00000700 // Interrupt Priority Grouping #define NVIC_APINT_PRIGROUP_7_1 0x00000000 // Priority group 7.1 split #define NVIC_APINT_PRIGROUP_6_2 0x00000100 // Priority group 6.2 split #define NVIC_APINT_PRIGROUP_5_3 0x00000200 // Priority group 5.3 split #define NVIC_APINT_PRIGROUP_4_4 0x00000300 // Priority group 4.4 split #define NVIC_APINT_PRIGROUP_3_5 0x00000400 // Priority group 3.5 split #define NVIC_APINT_PRIGROUP_2_6 0x00000500 // Priority group 2.6 split #define NVIC_APINT_PRIGROUP_1_7 0x00000600 // Priority group 1.7 split #define NVIC_APINT_PRIGROUP_0_8 0x00000700 // Priority group 0.8 split #define NVIC_APINT_SYSRESETREQ 0x00000004 // System Reset Request #define NVIC_APINT_VECT_CLR_ACT 0x00000002 // Clear Active NMI / Fault #define NVIC_APINT_VECT_RESET 0x00000001 // System Reset //************************************************************************* // // The following are defines for the bit fields in the NVIC_SYS_CTRL register. // //************************************************************************* #define NVIC_SYS_CTRL_SEVONPEND 0x00000010 // Wake Up on Pending #define NVIC_SYS_CTRL_SLEEPDEEP 0x00000004 // Deep Sleep Enable #define NVIC_SYS_CTRL_SLEEPEXIT 0x00000002 // Sleep on ISR Exit //************************************************************************* // // The following are defines for the bit fields in the NVIC_CFG_CTRL register. // //************************************************************************* #define NVIC_CFG_CTRL_STKALIGN 0x00000200 // Stack Alignment on Exception // Entry #define NVIC_CFG_CTRL_BFHFNMIGN 0x00000100 // Ignore Bus Fault in NMI and // Fault #define NVIC_CFG_CTRL_DIV0 0x00000010 // Trap on Divide by 0 #define NVIC_CFG_CTRL_UNALIGNED 0x00000008 // Trap on Unaligned Access #define NVIC_CFG_CTRL_MAIN_PEND 0x00000002 // Allow Main Interrupt Trigger #define NVIC_CFG_CTRL_BASE_THR 0x00000001 // Thread State Control //************************************************************************* // // The following are defines for the bit fields in the NVIC_SYS_PRI1 register. // //************************************************************************* #define NVIC_SYS_PRI1_USAGE_M 0x00E00000 // Usage Fault Priority #define NVIC_SYS_PRI1_BUS_M 0x0000E000 // Bus Fault Priority #define NVIC_SYS_PRI1_MEM_M 0x000000E0 // Memory Management Fault Priority #define NVIC_SYS_PRI1_USAGE_S 21 #define NVIC_SYS_PRI1_BUS_S 13 #define NVIC_SYS_PRI1_MEM_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_SYS_PRI2 register. // //************************************************************************* #define NVIC_SYS_PRI2_SVC_M 0xE0000000 // SVCall Priority #define NVIC_SYS_PRI2_SVC_S 29 //************************************************************************* // // The following are defines for the bit fields in the NVIC_SYS_PRI3 register. // //************************************************************************* #define NVIC_SYS_PRI3_TICK_M 0xE0000000 // SysTick Exception Priority #define NVIC_SYS_PRI3_PENDSV_M 0x00E00000 // PendSV Priority #define NVIC_SYS_PRI3_DEBUG_M 0x000000E0 // Debug Priority #define NVIC_SYS_PRI3_TICK_S 29 #define NVIC_SYS_PRI3_PENDSV_S 21 #define NVIC_SYS_PRI3_DEBUG_S 5 //************************************************************************* // // The following are defines for the bit fields in the NVIC_SYS_HND_CTRL // register. // //************************************************************************* #define NVIC_SYS_HND_CTRL_USAGE 0x00040000 // Usage Fault Enable #define NVIC_SYS_HND_CTRL_BUS 0x00020000 // Bus Fault Enable #define NVIC_SYS_HND_CTRL_MEM 0x00010000 // Memory Management Fault Enable #define NVIC_SYS_HND_CTRL_SVC 0x00008000 // SVC Call Pending #define NVIC_SYS_HND_CTRL_BUSP 0x00004000 // Bus Fault Pending #define NVIC_SYS_HND_CTRL_MEMP 0x00002000 // Memory Management Fault Pending #define NVIC_SYS_HND_CTRL_USAGEP \ 0x00001000 // Usage Fault Pending #define NVIC_SYS_HND_CTRL_TICK 0x00000800 // SysTick Exception Active #define NVIC_SYS_HND_CTRL_PNDSV 0x00000400 // PendSV Exception Active #define NVIC_SYS_HND_CTRL_MON 0x00000100 // Debug Monitor Active #define NVIC_SYS_HND_CTRL_SVCA 0x00000080 // SVC Call Active #define NVIC_SYS_HND_CTRL_USGA 0x00000008 // Usage Fault Active #define NVIC_SYS_HND_CTRL_BUSA 0x00000002 // Bus Fault Active #define NVIC_SYS_HND_CTRL_MEMA 0x00000001 // Memory Management Fault Active //************************************************************************* // // The following are defines for the bit fields in the NVIC_FAULT_STAT // register. // //************************************************************************* #define NVIC_FAULT_STAT_DIV0 0x02000000 // Divide-by-Zero Usage Fault #define NVIC_FAULT_STAT_UNALIGN 0x01000000 // Unaligned Access Usage Fault #define NVIC_FAULT_STAT_NOCP 0x00080000 // No Coprocessor Usage Fault #define NVIC_FAULT_STAT_INVPC 0x00040000 // Invalid PC Load Usage Fault #define NVIC_FAULT_STAT_INVSTAT 0x00020000 // Invalid State Usage Fault #define NVIC_FAULT_STAT_UNDEF 0x00010000 // Undefined Instruction Usage // Fault #define NVIC_FAULT_STAT_BFARV 0x00008000 // Bus Fault Address Register Valid #define NVIC_FAULT_STAT_BLSPERR 0x00002000 // Bus Fault on Floating-Point Lazy // State Preservation #define NVIC_FAULT_STAT_BSTKE 0x00001000 // Stack Bus Fault #define NVIC_FAULT_STAT_BUSTKE 0x00000800 // Unstack Bus Fault #define NVIC_FAULT_STAT_IMPRE 0x00000400 // Imprecise Data Bus Error #define NVIC_FAULT_STAT_PRECISE 0x00000200 // Precise Data Bus Error #define NVIC_FAULT_STAT_IBUS 0x00000100 // Instruction Bus Error #define NVIC_FAULT_STAT_MMARV 0x00000080 // Memory Management Fault Address // Register Valid #define NVIC_FAULT_STAT_MLSPERR 0x00000020 // Memory Management Fault on // Floating-Point Lazy State // Preservation #define NVIC_FAULT_STAT_MSTKE 0x00000010 // Stack Access Violation #define NVIC_FAULT_STAT_MUSTKE 0x00000008 // Unstack Access Violation #define NVIC_FAULT_STAT_DERR 0x00000002 // Data Access Violation #define NVIC_FAULT_STAT_IERR 0x00000001 // Instruction Access Violation //************************************************************************* // // The following are defines for the bit fields in the NVIC_HFAULT_STAT // register. // //************************************************************************* #define NVIC_HFAULT_STAT_DBG 0x80000000 // Debug Event #define NVIC_HFAULT_STAT_FORCED 0x40000000 // Forced Hard Fault #define NVIC_HFAULT_STAT_VECT 0x00000002 // Vector Table Read Fault //************************************************************************* // // The following are defines for the bit fields in the NVIC_DEBUG_STAT // register. // //************************************************************************* #define NVIC_DEBUG_STAT_EXTRNL 0x00000010 // EDBGRQ asserted #define NVIC_DEBUG_STAT_VCATCH 0x00000008 // Vector catch #define NVIC_DEBUG_STAT_DWTTRAP 0x00000004 // DWT match #define NVIC_DEBUG_STAT_BKPT 0x00000002 // Breakpoint instruction #define NVIC_DEBUG_STAT_HALTED 0x00000001 // Halt request //************************************************************************* // // The following are defines for the bit fields in the NVIC_MM_ADDR register. // //************************************************************************* #define NVIC_MM_ADDR_M 0xFFFFFFFF // Fault Address #define NVIC_MM_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_FAULT_ADDR // register. // //************************************************************************* #define NVIC_FAULT_ADDR_M 0xFFFFFFFF // Fault Address #define NVIC_FAULT_ADDR_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_CPAC register. // //************************************************************************* #define NVIC_CPAC_CP11_M 0x00C00000 // CP11 Coprocessor Access // Privilege #define NVIC_CPAC_CP11_DIS 0x00000000 // Access Denied #define NVIC_CPAC_CP11_PRIV 0x00400000 // Privileged Access Only #define NVIC_CPAC_CP11_FULL 0x00C00000 // Full Access #define NVIC_CPAC_CP10_M 0x00300000 // CP10 Coprocessor Access // Privilege #define NVIC_CPAC_CP10_DIS 0x00000000 // Access Denied #define NVIC_CPAC_CP10_PRIV 0x00100000 // Privileged Access Only #define NVIC_CPAC_CP10_FULL 0x00300000 // Full Access //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_TYPE register. // //************************************************************************* #define NVIC_MPU_TYPE_IREGION_M 0x00FF0000 // Number of I Regions #define NVIC_MPU_TYPE_DREGION_M 0x0000FF00 // Number of D Regions #define NVIC_MPU_TYPE_SEPARATE 0x00000001 // Separate or Unified MPU #define NVIC_MPU_TYPE_IREGION_S 16 #define NVIC_MPU_TYPE_DREGION_S 8 //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_CTRL register. // //************************************************************************* #define NVIC_MPU_CTRL_PRIVDEFEN 0x00000004 // MPU Default Region #define NVIC_MPU_CTRL_HFNMIENA 0x00000002 // MPU Enabled During Faults #define NVIC_MPU_CTRL_ENABLE 0x00000001 // MPU Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_NUMBER // register. // //************************************************************************* #define NVIC_MPU_NUMBER_M 0x00000007 // MPU Region to Access #define NVIC_MPU_NUMBER_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_BASE register. // //************************************************************************* #define NVIC_MPU_BASE_ADDR_M 0xFFFFFFE0 // Base Address Mask #define NVIC_MPU_BASE_VALID 0x00000010 // Region Number Valid #define NVIC_MPU_BASE_REGION_M 0x00000007 // Region Number #define NVIC_MPU_BASE_ADDR_S 5 #define NVIC_MPU_BASE_REGION_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_ATTR register. // //************************************************************************* #define NVIC_MPU_ATTR_XN 0x10000000 // Instruction Access Disable #define NVIC_MPU_ATTR_AP_M 0x07000000 // Access Privilege #define NVIC_MPU_ATTR_TEX_M 0x00380000 // Type Extension Mask #define NVIC_MPU_ATTR_SHAREABLE 0x00040000 // Shareable #define NVIC_MPU_ATTR_CACHEABLE 0x00020000 // Cacheable #define NVIC_MPU_ATTR_BUFFRABLE 0x00010000 // Bufferable #define NVIC_MPU_ATTR_SRD_M 0x0000FF00 // Subregion Disable Bits #define NVIC_MPU_ATTR_SIZE_M 0x0000003E // Region Size Mask #define NVIC_MPU_ATTR_ENABLE 0x00000001 // Region Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_BASE1 register. // //************************************************************************* #define NVIC_MPU_BASE1_ADDR_M 0xFFFFFFE0 // Base Address Mask #define NVIC_MPU_BASE1_VALID 0x00000010 // Region Number Valid #define NVIC_MPU_BASE1_REGION_M 0x00000007 // Region Number #define NVIC_MPU_BASE1_ADDR_S 5 #define NVIC_MPU_BASE1_REGION_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_ATTR1 register. // //************************************************************************* #define NVIC_MPU_ATTR1_XN 0x10000000 // Instruction Access Disable #define NVIC_MPU_ATTR1_AP_M 0x07000000 // Access Privilege #define NVIC_MPU_ATTR1_TEX_M 0x00380000 // Type Extension Mask #define NVIC_MPU_ATTR1_SHAREABLE \ 0x00040000 // Shareable #define NVIC_MPU_ATTR1_CACHEABLE \ 0x00020000 // Cacheable #define NVIC_MPU_ATTR1_BUFFRABLE \ 0x00010000 // Bufferable #define NVIC_MPU_ATTR1_SRD_M 0x0000FF00 // Subregion Disable Bits #define NVIC_MPU_ATTR1_SIZE_M 0x0000003E // Region Size Mask #define NVIC_MPU_ATTR1_ENABLE 0x00000001 // Region Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_BASE2 register. // //************************************************************************* #define NVIC_MPU_BASE2_ADDR_M 0xFFFFFFE0 // Base Address Mask #define NVIC_MPU_BASE2_VALID 0x00000010 // Region Number Valid #define NVIC_MPU_BASE2_REGION_M 0x00000007 // Region Number #define NVIC_MPU_BASE2_ADDR_S 5 #define NVIC_MPU_BASE2_REGION_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_ATTR2 register. // //************************************************************************* #define NVIC_MPU_ATTR2_XN 0x10000000 // Instruction Access Disable #define NVIC_MPU_ATTR2_AP_M 0x07000000 // Access Privilege #define NVIC_MPU_ATTR2_TEX_M 0x00380000 // Type Extension Mask #define NVIC_MPU_ATTR2_SHAREABLE \ 0x00040000 // Shareable #define NVIC_MPU_ATTR2_CACHEABLE \ 0x00020000 // Cacheable #define NVIC_MPU_ATTR2_BUFFRABLE \ 0x00010000 // Bufferable #define NVIC_MPU_ATTR2_SRD_M 0x0000FF00 // Subregion Disable Bits #define NVIC_MPU_ATTR2_SIZE_M 0x0000003E // Region Size Mask #define NVIC_MPU_ATTR2_ENABLE 0x00000001 // Region Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_BASE3 register. // //************************************************************************* #define NVIC_MPU_BASE3_ADDR_M 0xFFFFFFE0 // Base Address Mask #define NVIC_MPU_BASE3_VALID 0x00000010 // Region Number Valid #define NVIC_MPU_BASE3_REGION_M 0x00000007 // Region Number #define NVIC_MPU_BASE3_ADDR_S 5 #define NVIC_MPU_BASE3_REGION_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_MPU_ATTR3 register. // //************************************************************************* #define NVIC_MPU_ATTR3_XN 0x10000000 // Instruction Access Disable #define NVIC_MPU_ATTR3_AP_M 0x07000000 // Access Privilege #define NVIC_MPU_ATTR3_TEX_M 0x00380000 // Type Extension Mask #define NVIC_MPU_ATTR3_SHAREABLE \ 0x00040000 // Shareable #define NVIC_MPU_ATTR3_CACHEABLE \ 0x00020000 // Cacheable #define NVIC_MPU_ATTR3_BUFFRABLE \ 0x00010000 // Bufferable #define NVIC_MPU_ATTR3_SRD_M 0x0000FF00 // Subregion Disable Bits #define NVIC_MPU_ATTR3_SIZE_M 0x0000003E // Region Size Mask #define NVIC_MPU_ATTR3_ENABLE 0x00000001 // Region Enable //************************************************************************* // // The following are defines for the bit fields in the NVIC_DBG_CTRL register. // //************************************************************************* #define NVIC_DBG_CTRL_DBGKEY_M 0xFFFF0000 // Debug key mask #define NVIC_DBG_CTRL_DBGKEY 0xA05F0000 // Debug key #define NVIC_DBG_CTRL_S_RESET_ST \ 0x02000000 // Core has reset since last read #define NVIC_DBG_CTRL_S_RETIRE_ST \ 0x01000000 // Core has executed insruction // since last read #define NVIC_DBG_CTRL_S_LOCKUP 0x00080000 // Core is locked up #define NVIC_DBG_CTRL_S_SLEEP 0x00040000 // Core is sleeping #define NVIC_DBG_CTRL_S_HALT 0x00020000 // Core status on halt #define NVIC_DBG_CTRL_S_REGRDY 0x00010000 // Register read/write available #define NVIC_DBG_CTRL_C_SNAPSTALL \ 0x00000020 // Breaks a stalled load/store #define NVIC_DBG_CTRL_C_MASKINT 0x00000008 // Mask interrupts when stepping #define NVIC_DBG_CTRL_C_STEP 0x00000004 // Step the core #define NVIC_DBG_CTRL_C_HALT 0x00000002 // Halt the core #define NVIC_DBG_CTRL_C_DEBUGEN 0x00000001 // Enable debug //************************************************************************* // // The following are defines for the bit fields in the NVIC_DBG_XFER register. // //************************************************************************* #define NVIC_DBG_XFER_REG_WNR 0x00010000 // Write or not read #define NVIC_DBG_XFER_REG_SEL_M 0x0000001F // Register #define NVIC_DBG_XFER_REG_R0 0x00000000 // Register R0 #define NVIC_DBG_XFER_REG_R1 0x00000001 // Register R1 #define NVIC_DBG_XFER_REG_R2 0x00000002 // Register R2 #define NVIC_DBG_XFER_REG_R3 0x00000003 // Register R3 #define NVIC_DBG_XFER_REG_R4 0x00000004 // Register R4 #define NVIC_DBG_XFER_REG_R5 0x00000005 // Register R5 #define NVIC_DBG_XFER_REG_R6 0x00000006 // Register R6 #define NVIC_DBG_XFER_REG_R7 0x00000007 // Register R7 #define NVIC_DBG_XFER_REG_R8 0x00000008 // Register R8 #define NVIC_DBG_XFER_REG_R9 0x00000009 // Register R9 #define NVIC_DBG_XFER_REG_R10 0x0000000A // Register R10 #define NVIC_DBG_XFER_REG_R11 0x0000000B // Register R11 #define NVIC_DBG_XFER_REG_R12 0x0000000C // Register R12 #define NVIC_DBG_XFER_REG_R13 0x0000000D // Register R13 #define NVIC_DBG_XFER_REG_R14 0x0000000E // Register R14 #define NVIC_DBG_XFER_REG_R15 0x0000000F // Register R15 #define NVIC_DBG_XFER_REG_FLAGS 0x00000010 // xPSR/Flags register #define NVIC_DBG_XFER_REG_MSP 0x00000011 // Main SP #define NVIC_DBG_XFER_REG_PSP 0x00000012 // Process SP #define NVIC_DBG_XFER_REG_DSP 0x00000013 // Deep SP #define NVIC_DBG_XFER_REG_CFBP 0x00000014 // Control/Fault/BasePri/PriMask //************************************************************************* // // The following are defines for the bit fields in the NVIC_DBG_DATA register. // //************************************************************************* #define NVIC_DBG_DATA_M 0xFFFFFFFF // Data temporary cache #define NVIC_DBG_DATA_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_DBG_INT register. // //************************************************************************* #define NVIC_DBG_INT_HARDERR 0x00000400 // Debug trap on hard fault #define NVIC_DBG_INT_INTERR 0x00000200 // Debug trap on interrupt errors #define NVIC_DBG_INT_BUSERR 0x00000100 // Debug trap on bus error #define NVIC_DBG_INT_STATERR 0x00000080 // Debug trap on usage fault state #define NVIC_DBG_INT_CHKERR 0x00000040 // Debug trap on usage fault check #define NVIC_DBG_INT_NOCPERR 0x00000020 // Debug trap on coprocessor error #define NVIC_DBG_INT_MMERR 0x00000010 // Debug trap on mem manage fault #define NVIC_DBG_INT_RESET 0x00000008 // Core reset status #define NVIC_DBG_INT_RSTPENDCLR 0x00000004 // Clear pending core reset #define NVIC_DBG_INT_RSTPENDING 0x00000002 // Core reset is pending #define NVIC_DBG_INT_RSTVCATCH 0x00000001 // Reset vector catch //************************************************************************* // // The following are defines for the bit fields in the NVIC_SW_TRIG register. // //************************************************************************* #define NVIC_SW_TRIG_INTID_M 0x000000FF // Interrupt ID #define NVIC_SW_TRIG_INTID_S 0 //************************************************************************* // // The following are defines for the bit fields in the NVIC_FPCC register. // //************************************************************************* #define NVIC_FPCC_ASPEN 0x80000000 // Automatic State Preservation // Enable #define NVIC_FPCC_LSPEN 0x40000000 // Lazy State Preservation Enable #define NVIC_FPCC_MONRDY 0x00000100 // Monitor Ready #define NVIC_FPCC_BFRDY 0x00000040 // Bus Fault Ready #define NVIC_FPCC_MMRDY 0x00000020 // Memory Management Fault Ready #define NVIC_FPCC_HFRDY 0x00000010 // Hard Fault Ready #define NVIC_FPCC_THREAD 0x00000008 // Thread Mode #define NVIC_FPCC_USER 0x00000002 // User Privilege Level #define NVIC_FPCC_LSPACT 0x00000001 // Lazy State Preservation Active //************************************************************************* // // The following are defines for the bit fields in the NVIC_FPCA register. // //************************************************************************* #define NVIC_FPCA_ADDRESS_M 0xFFFFFFF8 // Address #define NVIC_FPCA_ADDRESS_S 3 //************************************************************************* // // The following are defines for the bit fields in the NVIC_FPDSC register. // //************************************************************************* #define NVIC_FPDSC_AHP 0x04000000 // AHP Bit Default #define NVIC_FPDSC_DN 0x02000000 // DN Bit Default #define NVIC_FPDSC_FZ 0x01000000 // FZ Bit Default #define NVIC_FPDSC_RMODE_M 0x00C00000 // RMODE Bit Default #define NVIC_FPDSC_RMODE_RN 0x00000000 // Round to Nearest (RN) mode #define NVIC_FPDSC_RMODE_RP 0x00400000 // Round towards Plus Infinity (RP) // mode #define NVIC_FPDSC_RMODE_RM 0x00800000 // Round towards Minus Infinity // (RM) mode #define NVIC_FPDSC_RMODE_RZ 0x00C00000 // Round towards Zero (RZ) mode //************************************************************************* // // The following definitions are deprecated. // //************************************************************************* #ifndef DEPRECATED //************************************************************************* // // Deprecated defines for the EEPROM register offsets. // //*************************************************************************** #define EEPROM_EEPROMPP_R (((volatile unsigned long )0x400AFFC0)) //*************************************************************************** // // Deprecated defines for the bit fields in the EEPROM_EEPROMPP register. // //*************************************************************************** #define EEPROM_EEPROMPP_SIZE_M 0x0000001F // EEPROM Size #define EEPROM_EEPROMPP_SIZE_S 0 #endif #endif // __LM4F120H5QR_H__	{ "redpajama_set_name": "RedPajamaGithub" }	4,843
Hello there everybody, Silentwisher here with the Silentwisher Photography Podcast, hosted by yours truly. In this episode, we're gonna talk about the upcoming Sigma 105 F1.4 art lens for Sony E-Mount. As you may or may not know, I am a Sony shooter specifically, a Sony A9 shooter, and I really wanna get a hold of this lens. I currently have a 7200 F2.8 G Master and a 55 1.8 Zeiss Sony Lens, and they're both great. I just want to get a hold of this Sigma 105, like it's being called the Bokeh Monster. I mean it's got F1.4 which is gonna be insane in low light and then just something about 100 millimeters/105 millimeters, is just a very satisfying photo to me.Like when I had my 100 to 400 G Master, before I traded it in, I took a picture of one of my friends while we were out, and about at a dock and the picture just turned out great. I loved that focal length, so I might actually just end up skipping the 85-millimeter lenses altogether and just do 105's as far as you know that general range is concerned. I'm looking at the B&H website right now, I'm not sponsored and it says it's aperture range is F1.4 to F16. I personally never go over F8. There's obviously reasons why you might want to, but if you anything about photography, those situations are very limited. So if you haven't seen the pictures of this Sigma 105, its got its own tripod collar. Like it's so big, and I'm assuming heavy. I think there's a weight here, let me look. But it's so big, that it has to have its own tripod collar. You could take it off, but it is huge. Okay, so the lens itself is 3.5, well it's actually 3.62 pounds, so that's pretty heavy for a lens and right now, there's not a whole lot, of images or video or anything covering this. But it's huge, like it's a big lens. The front element is huge. I'm seeing if I could find some information here. It's got a filter thread of 105 millimeters. That's huge and I'm sure any sort of filter that size is gonna be insanely expensive, but yeah, as soon as I get a hold of this, I'm definitely let you guys know and give you my review on it. I actually have a contact with a camera store and he's gonna let me know as soon as he finds out any sort of information on it, me and him are pretty tight and I'll recommend him and his store, well the store he works for in a future episode. But, I'm super excited about this. So if you guys are interested in this lens or just the lens for your native mount, whether that's Canon or Nikon or whatever, let me know and I just can't wait to get a hold of it. That F1.4 is gonna create some really awesome, low light and just in general portraits, so. That'll be it for today's photography podcast. Next episode, well it's gonna have to be a surprise. So if you guys enjoyed it, let me know in the comments and I'll see you guys in the next one, bye, bye. If you'd like to support future episodes, be sure to support me on Patreon. The link will be in the description. Bye guys. In this episode of the Silentwisher Photography podcast, we talk about the Sigma 105 1.4 Bokeh Monster.	{ "redpajama_set_name": "RedPajamaC4" }	2,638
Tag: a.i.c.o. -incarnation- A.I.C.O. -Incarnation- Launches on Netflix by Samantha Ferreira \| Mar 9, 2018 \| News Reporting \| 0 \| This series has finally appeared in its full incarnation! Earlier today, A.I.C.O. -Incarnation-... New Mecha & Weapon Designs Unveiled for A.I.C.O. -Incarnation- Anime Earlier today, the team behind A.I.C.O. -Incarnation- unveiled several new weapon and mecha... Netflix Streams Subtitled A.I.C.O. -Incarnation- Teaser Netflix is giving a look at the latest incarnation of this new series. Earlier today, Netflix... AniWeekly 143: Grand Blue Coin Controversies by Samantha Ferreira \| Jan 14, 2018 \| AniWeekly \| 0 \| Happy Sunday everyone, and welcome to an all-new AniWeekly! It's been a busy week, with... A.I.C.O. -Incarnation- Gets New Staff, Cast, Trailer, Visuals, Premiere Date It looks like Aiko will introduce herself to the world a bit sooner than we expected! Earlier... Netflix Unveils A.I.C.O. -Incarnation- Anime, First PV, Visual, & Crew Announced by Samantha Ferreira \| Aug 2, 2017 \| News Reporting \| 0 \| So, how long until we all groan at an Agent A.I.C.O. joke? Earlier today, Netflix hosted a...	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	7,341
En mecànica clàssica, les equacions de Newton-Euler descriuen la combinació de la dinàmica de transició i de rotació d'un sòlid rígid. Tradicionalment, les equacions de Newton-Euler és l'agrupació conjunta de les dues lleis del moviment d'Euler d'un sòlid rígid en una única equació amb 6 components, que utilitza vectors columna i matrius. Aquestes lleis relacionen el moviment del centre de gravetat d'un sòlid rígid amb la suma de les forces i els parells (també anomenats moments) que actuen en el sòlid rígid. Sistema de referència del centre de massa Respecte un sistema de coordenades que té l'origen en el centre de massa del cos, les equacions de Newton-Euler poden ser expressades en forma matricial com on F = força total que actua sobre el centre de massa m = massa del cos I₃ = la matriu identitat 3×3 acm = acceleració del centre de massa vcm = velocitat del centre de massa τ = parell total que actua al voltant del centre de massa Icm = moment d'inèrcia al voltant del centre de massa ω = velocitat angular del cos α = acceleració angular del cos Sistema de referència qualsevol Respecte un sistema de referència ubicat en un punt P fixe en el cos i que no coincideix amb el centre de massa, les equacions prenen la forma on c és la ubicació del centre de massa expressat en el sistema de referència centrat en el cos, i denota les matrius de producte vectorial antisimètriques. Els termes inercials estan inclosos en la matriu dinèrcia espacial'' mentre que les forces fictícies es tenen en compte en el terme: Quan el centre de massa no coincideix amb el sistema de coordenades (és a dir, quan c és diferent de zero), les acceleracions translacional i angular (a i α''') estan acoblades, de tal manera que totes dues estan associades amb components de força i moment. Aplicacions S'utilitzen les equacions de Newton-Euler com a base de formulacions més complicades en què interactuen diversos cossos (teoria helicoidal) que descriuen la dinàmica de sistemes de sòlids rígids connectats amb juntes i altres restriccions. Es poden solucionar els problemes de diversos cossos utilitzant diversos algorismes numèrics. Referències Vegeu també Lleis del moviment d'Euler d'un sòlid rígid Angles d'Euler Força centrífuga Dinàmica Equacions de la física	{ "redpajama_set_name": "RedPajamaWikipedia" }	4,622
{"url":"https:\/\/forum.sumatrapdfreader.org\/t\/please-help-me-with-rtl-reading\/4529","text":"Forum moved here!\n\neeeE2Eeee\n\nHello, i want to start the first page from the right, so how can I reverse it to get page number 1 on the right and number 2 on the left.\n\nIf you need to permanently reorder the pages or generate a copy of a file with the pages re-ordered then I can recommend pd fLabs pdf-tk server which is a free, as in gratis & GPL command line tool that is available for Windows, Mac and most Linux installations. It can perform a huge range of PDF manipulations say that you had an original file called Fred.pdf and wish to re-order all of the pages as you describe then you can do so with a command like:\n\npdftk A=Fred.pdf shuffle Aeven Aodd output ShuffledFred.pdf\n\n\nYou can then view ShuffledFred.pdf and either replace Fred.pdf with it or discard it.\n\nGitHubRulesOK","date":"2022-05-29 04:42:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.35727187991142273, \"perplexity\": 1371.9817703737863}, \"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-2022-21\/segments\/1652663039492.94\/warc\/CC-MAIN-20220529041832-20220529071832-00568.warc.gz\"}"}	null	null
Q: Force redirect to login when accessing website page odoo 12 I need to force the user to be authenticated when accessing a static page as part of the website module so far what I've done is this: class RestrictAreas(Home): @http.route(['/page1','/page2'],type="http", auth='user', website=True) def test(self): website_page = request.env['ir.http']._serve_page() return website_page it works for page1 and page2 but I need it to be as dynamic, so that controller intercepts any HTTP request for a web page and validates the user. Can I use a regex for a slug or something similar? I am overriding the right method in the controller? thanks in advvance. A: you can overwrite _serve_page() function in ir.http and check if the user is logged in or redirect to login page you don't need to build a custom controller for it, this will handle any static page request and you can build another function for your custom controllers	{ "redpajama_set_name": "RedPajamaStackExchange" }	6,226
You are here: Parliament home page > Parliamentary business > Publications and Records > Committee Publications > All Select Committee Publications > Commons Select Committees > Environment, Food and Rural Affairs > Environment, Food and Rural Affairs Select Committee on Environment, Food and Rural Affairs Minutes of Evidence Examination of Witnesses (Questions 226 - 239) MRS MARIANNE WENNING AND DR TOM BATCHELOR 226. We move on now to evidence from the European Commission and we welcome Dr Tom Batchelor of Ozone Layer Protection and Mrs Marianne Wenning, the Deputy Head of the Climate Change Unit. Can I thank you both very much indeed for coming this afternoon to give evidence to the Committee. I notice that you very patiently sat through the previous evidence session. Having been very closely involved in all the processes involved in the Regulation 2037/2000, is there anything that you have heard from our previous witness that surprised you? Is there anything to which you thought, "My gosh! Is that what we have actually unleashed on the United Kingdom? Is that what the reality is?" Is there anything which really surprised you in what you heard? (Mrs Wenning) Thank you, Mr Chairman, and thank you for having us today to give evidence to this Committee. We have given you also a number of documents— 227. Lots. (Mrs Wenning)—that we will be referring to during the inquiry here. Coming back to your question, I thought it was very interesting to hear the figures of the costs. For example, if you will allow me to comment, normally we are looking at the costs, but we are looking at the costs at the European level, so we are not going into each Member State and what financial implications there are, but we are looking at the European level, so I thought that was interesting. 228. When you formulated the Regulation, did you have discussions with the disposal industry within Europe to talk through the practicalities of achieving the objectives of the Regulation? (Mrs Wenning) Mr Chairman, I am happy to respond to your question, but I am wondering whether it would be useful if we could give you an overview in order to give you our side of the fence, and to put the picture in terms of how we have seen this evolving, the Regulation and any further questions which have been raised. 229. Well, we have, as you will have gathered, got a series of areas that we would like to probe with you and I am very happy if, at the appropriate time, you want to produce any more detailed commentary that might help us, but obviously the line of questioning that the Committee would like to pursue is designed to try and get some kind of overview, hence my question, if you like, arising from the previous evidence which was talking about the practical nature of what this Regulation is designed to achieve and also hence my question in asking you whether you had had any discussions with the waste disposal industry within the European Union about some of these practical points. (Mrs Wenning) It is very difficult for both of us to answer because at the time we both were not around, so I can only speculate. 230. Okay. Let me come to the heart of one of the issues which has concerned us here in the United Kingdom which is the question of what in fact does this Regulation mean. During the course of its negotiations, Article 15 was transmuted into Article 16 and it seems that we come down to one thing, which is this question of practicability actually to do what the Regulation wanted to be done, namely that not just the gases, but the actual CFCs in the foam should also be disposed of and the difference appears to be that the rest of Europe understood that that is what it was about and for quite a long period of time the United Kingdom said that it did not understand what this was about. Could you just help us to understand more clearly if there was any ambiguity in the way that the Regulation was written? Was it entirely clear in your minds what Article 16 meant, that everything which had CFCs inside the fridge, foam and gas, had to be disposed of and perhaps at what point in time did you become entirely clear that that is what it was about? Why is it that the United Kingdom, in your view, had some doubts about this? (Mrs Wenning) Yes, you are right. The main part of the Regulation which is of relevance to these discussions is what is now Article 16 and that Article 16 has three, well it has more, but there are three sub-paragraphs that are important to us. If we look at Article 16.1 of this Regulation, that requires recovery of ozone-depleting substances, such as CFCs, from refrigeration, except domestic refrigerators and freezers, by 1 October 2000. These domestic fridges and freezers are covered by Article 16.2. They have a 15-month grace period after the Regulation came into force to allow Member States to establish facilities for the recovery and the destruction of CFCs. Then we come to Article 16.3 which we consider as the sort of catch-all for ozone-depleting substances that are contained in products that are not mentioned in Article 16.1 and 16.2, under the condition that it is practicable. So foam is the product which is not mentioned under Article 16.1 and not under Article 16.2, so the recovery of ozone-depleting substances, such as CFCs from foam, would be required, if practicable. So for us it was never a question of whether there was the demand to recover CFCs from foam, but always the question: is it practicable? 231. Can I just be clear because in the wording in 16.2 it talks about "controlled substances" and, if I have understood this Regulation correctly, the controlled substance is the CFC. Is that right? (Dr Batchelor) That is correct. 232. So is it right to say that it was an absolute requirement of this Regulation that CFCs in whatever form they were in a refrigerator, whether gas or contained, as our previous witness described, within these bubbles in the foam, they had to be disposed of under controlled circumstances? Is that right? (Dr Batchelor) Yes, that is totally correct. 233. Because the evidence you gave a moment ago seemed to suggest that because there were some CFCs in the foam, they came into 16.3 which says "if [it is] practicable [to do it]". The way I originally read it was that 16.2 gave an absolute requirement that whatever the CFC was within the fridge, it had to be disposed of in a controlled way. (Dr Batchelor) No, we see it slightly differently. There is 16.1 which is the disposal of the refrigerators and freezers and that is from the 1 October 2000, as Marianne has already said. Article 16.2 is purely based on domestic refrigerators and 16.3 is a catch-all for what is a product that is not in either paragraph 2 or paragraph 1, so this Regulation, by covering 16.1, 2 and 3, was very clear to us from the beginning, that no matter which components of a refrigerator you were talking about, no matter how you interpreted it, that required both the recovery of the CFCs from the foam and from the compressor. We spent some time actually with the UK in numerous discussions that we had with them on this issue, both in the Management Committee and in private discussions with the UK. We need to explain a little bit, I think, about the role of the Management Committee in all of this because this is a bit of a mysterious body in its functions. This is one of the main vehicles of communication for Member States and the Commission and when we get together there are typically 50 to 60 people in the room, including the interpreters, so this is a fairly large event, and it occurs twice a year typically. The main point of the Management Committee is to be transparent about some of the finer details of the regulation, and we try to be as sensitive as possible to particular concerns or any kind of struggles with the implementation of the regulation. So at the onset, if you wind your clock back to 1 October 2000, this is a brand new regulation that is hitting the tables and we were very sensitive to try to understand what these concerns were. Just to finish with some comments about the Management Committee, we produce the minutes of this Committee within ten days of the end of the meeting. The minutes can be something like 15 to 16 pages long. In this particular correspondence here, we have supplied to the Committee both the chronology of events and, also, supplied the relevant parts of the Management Committee minutes that appertain to this part of the discussions. I must say that, contrary to Mr Peter Jones' comments, from our perspective the discussions have always been led by DEFRA in the Management Committee meetings and there has always been a very thorough explanation of what the concerns were that were brought to the Management Committee or that were brought to us in other meetings. That kind of explains the Management Committee. 234. When you said that you produce the minutes of the various meetings very quickly, what happens if somebody objects to the minutes and queries them? What I am anxious to try and understand is, was it the case that there was some ambiguity in the way that the discussions at the Management Committee proceeded that could have justifiably caused the United Kingdom to have had doubt about this question of the foam, whereas other Member States seem to be entirely clear? Did any other Member States query whether this applied to the foam or not? (Dr Batchelor) Just on process, Chairman, ten days after the minutes are finished, that is the draft minutes. We then send the minutes out to all the Member States who then have the opportunity to comment on specific parts of the minutes. We use the standard Microsoft Word "tracking procedures" to understand the origin of the comments that come back. We then reformulate the minutes which we then distribute to the members of the committee ten days before the next meeting, so that usually the first item on the agenda is the adoption of the minutes from the previous meeting, the minutes having been corrected by the Member States. In our discussions were there any questions on this? I have to say that the UK was one of the more respondent representations that did get back to us with any comments that they had on the minutes, and I compliment them on getting back to us very quickly on those. We made a number of extensive changes in the minutes to fairly reflect their comments. So the comments that you actually see before you in the chronology of communications and in the pieces that we have taken from the Management meetings reflect not only the views of the UK but, also, the views of the other Member States that are in there. Yes, indeed, there are some points that did create a lot of discussion. Perhaps we could refer, Chairman, to the chronology of communications—if you have a copy there. Under item number 3, for example, we have bullet point 2. This is 23 February 1999. If you could wind your clocks back to 18 months before the regulation came into effect, point 2 here states: "The Commission considers CFC recovery could be difficult. The UK helpfully suggested that Member States might share their experiences so that best practice could be identified. The Netherlands referred to technology already in existence. The Commission concluded that there should be some kind of policy on recovery and destruction and we asked whether the Member States were prepared to send information on how their recovery programmes were working." Mr Chairman, we considered this a first alert, if you like; that this is a very early signal before the regulation came in that said "Look, this is a new facet to the regulation. The indications are that there are Member States that are doing it but this is not easy. So let us have some share of the information here so that those who are not doing this can actually come up to speed." 235. Can I just be clear? Whilst I note what you say about the fact that Member States decided that there could be a use in pooling information on technology, was there any time during the discussion on this regulation where any Member State, including the United Kingdom, said "We do not think it practical to recover CFCs from refrigerators" whether it was there as a gas or incorporated within the foam? (Dr Batchelor) I think there would have been one or two Member States that would have questioned it, but only—if I may say—rather tepidly, nothing with the same heat of the argument that the UK would have brought to the discussions. Chairman: We may return to that in a moment. Paddy Tipping 236. Can I clarify a couple of points? Ms Wenning, you made a comment and Peter Jones made it before, that the regulation applying to the commercial facilities came in 15 months before it applied to domestic refrigerators. So there was a lead-in period and there was some experience to be gained from that commercially. (Mrs Wenning) Yes, absolutely. 237. Secondly, just to reinforce a point that has been made, the minutes of the Management Committee, when they come to the Management Committee for ratification, are agreed? (Mrs Wenning) Yes. 238. There is no doubt. 239. Why is it, in your view, that the United Kingdom had so much difficulty with this? Looking at your schedule of events, on 24 February 1999 there was a fairly extensive discussion about the technology that was around. (Dr Batchelor) We have actually asked ourselves the same question as well. We can only go on the comments that were made to us and, you understand, we can infer certain events might or might not have happened back in the UK. I think one of the first events for us was in October 2000 when, shortly after the regulation came into effect, this is about three weeks afterwards, we had another Management Meeting and DEFRA made it known to us that the UK had not been aware of the considerable export trade of 1-1.5 million domestic refrigerators per year, mainly to developing countries. I think this was brought sharply into focus on 1 October that year when the regulation came into force, that suddenly all these exports were banned. There was a second event, too, Mr Chairman, in that we believe that DEFRA was not aware in January 2001 that in a typical refrigerator two-thirds of the CFCs are contained in the foam and one-third is contained in the compressor cooling circuit. Prepared 25 April 2002	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	4,934
Mike & Molly is een sitcom die op 20 september 2010 van start ging op de Amerikaanse zender CBS, met in de hoofdrollen Billy Gardell en Melissa McCarthy. De laatste won hiervoor in 2011 een Emmy Award voor 'beste vrouwelijke hoofdrol in een komische serie'. In Vlaanderen wordt de serie uitgezonden door ZES, in Nederland door Fox. Uitgangspunt Mike Biggs is een gezette politieagent in Chicago die besluit om wat aan zijn overgewicht te doen. Hij gaat naar een bijeenkomst van de Anonieme Overeters en ontmoet daar Molly Flynn, een basisschoollerares die ook lijdt onder haar overgewicht. Ze nodigt hem uit om over zijn werk te spreken voor haar klas, waarna ze beginnen te daten. Mike krijgt hiervoor adviezen van zijn beste vriend en partner bij de politie Carl McMillan, die zich graag opstelt als ervaren vrouwenversierder, maar zelf inwoont bij zijn oma. Een hindernis die het stel vaak moet nemen om samen te kunnen zijn, wordt gevormd door de capriolen van Molly's dramatisch zus Victoria en haar altijd van een wijnglas voorziene moeder Joyce. Ook mag Mike wat meer tegengas bieden aan zijn eigen moeder, in Molly's ogen. Rolverdeling Billy Gardell - Michael Biggs Melissa McCarthy - Molly Flynn Reno Wilson - Carlton "Carl" McMillan Nyambi Nyambi - Samuel Swoosie Kurtz - Joyce Flynn Katy Mixon - Victoria Flynn Louis Mustillo - Vincent Moranto Cleo King - Rosetta Rondi Reed - Peggy Biggs Externe link Officiële website Programma van CBS Amerikaanse komedieserie	{ "redpajama_set_name": "RedPajamaWikipedia" }	6,689
\section{Introduction} This article is the long-delayed third installment in the authors' study of generalized inductive limits of finite-dimensional C-algebras. The basic study of such generalized inductive limits was begun in \cite{BlackadarKGeneralized}, where the classes of MF algebras, NF algebras, and strong NF algebras were defined, and a number of equivalent characterizations of each class given. In particular, a (necessarily separable) C-algebra is a {\em strong NF algebra} if it can be written as a generalized inductive limit of a sequential inductive system of finite-dimensional C-algebras in which the connecting maps are complete order embeddings (and asymptotically multiplicative in the sense of \cite{BlackadarKGeneralized}). An {\em NF algebra} is a C-algebra which can be written as the generalized inductive limit of such a system, where the connecting maps are only required to be completely positive contractions. An NF algebra is automatically nuclear (and separable). It was shown that a separable C-algebra is an NF algebra if and only if it is nuclear and quasidiagonal. It was not shown in \cite{BlackadarKGeneralized} that the classes of NF algebras and strong NF algebras are distinct. Our second paper \cite{BlackadarKInner} used the notion of inner quasidiagonality to distinguish them. We made the following definition, a slight variation of Voiculescu's characterization of quasidiagonal C-algebras \cite{VoiculescuNote}: \begin{Def} A C-algebra $A$ is {\em inner quasidiagonal} if, for every $x_1,\dots,x_n\in A$ and $\epsilon>0$, there is a representation $\pi$ of $A$ on a Hilbert space ${\mathcal H}$, and a finite-rank projection $P\in\pi(A)''$ such that $\\|P\pi(x_j)-\pi(x_j)P\\|<\epsilon$ $\\|P\pi(x_j)P\\|>\\|x_j\\|-\epsilon$ for $1\leq j\leq n$. \end{Def} Voiculescu's characterization of quasidiagonality is the same with the requirement that $P\in\pi(A)''$ deleted. We then proved that a separable C-algebra is a strong NF algebra if and only if it is nuclear and inner quasidiagonal. The principal shortcoming of this result is that it is often difficult to determine directly from the definition whether a C-algebra is inner quasidiagonal, although we were able to give examples of separable nuclear C-algebras which are quasidiagonal but not inner quasidiagonal, hence of NF algebras which are not strong NF. It is immediate from the definition that a C-algebra with a separating family of quasidiagonal irreducible representations is inner quasidiagonal, and in \cite{BlackadarKInner} we established some very special cases of the converse which were sufficient to yield the examples. The main result of the present article is the full converse in the separable case: \begin{Thm}\label{MThm} A separable C-algebra is inner quasidiagonal if and only if it has a separating family of quasidiagonal irreducible representations. \end{Thm} We thus obtain a characterization of strong NF algebras which is usually much easier to check than the characterization of \cite{BlackadarKInner}: \begin{Cor} A separable C-algebra is a strong NF algebra if and only if it is nuclear and has a separating family of quasidiagonal irreducible representations. \end{Cor} Although our initial interest in inner quasidiagonality was through its connection with generalized inductive limits, it now appears that inner quasidiagonality is of some interest in its own right. In the final section, we list some permanence properties of the class of inner quasidiagonal C-algebras which follow easily from our characterizations: \begin{enumerate} \item[1.] An arbitrary inductive limit (with injective connecting maps) of inner quasidiagonal C-algebras is inner quasidiagonal. \item[2.] A (minimal) tensor product of inner quasidiagonal C-algebras is inner quasidiagonal. \item[3.] The algebra of sections of a continuous field of separable inner quasidiagonal C-algebras is inner quasidiagonal. \item[4.] Inner quasidiagonality is an (SI) property in the sense of \cite[II.8.5]{BlackadarOperator}. \end{enumerate} As a result of (1) and (4), the study of inner quasidiagonal C-algebras can be effectively reduced to studying separable inner quasidiagonal C-algebras, to which Theorem \ref{MThm} applies. \section{Outline of the Proof} To prove Theorem \ref{MThm}, fix a separable inner quasidiagonal C-algebra $A$ and a nonzero $x_0\in A$ (we may assume $\\|x_0\\|=1$ to simplify notation). We will manufacture a quasidiagonal irreducible representation $\pi$ of $A$ with $\pi(x_0)\neq0$. We will construct a sequence $({\mathcal H}_n)$ of finite-dimensional Hilbert spaces, embeddings $I_n:{\mathcal H}_n\to{\mathcal H}_{n+1}$, an increasing sequence $(X_n)$ of finite self-adjoint subsets of the unit ball of $A$ containing $x_0$, with union $X$ dense in the unit ball of $A$, and completely positive contractions $V_n:A\to{\mathcal L}({\mathcal H}_n)$ mapping the closed unit ball of $A$ onto the closed unit ball of ${\mathcal L}({\mathcal H}_n)$, such that, for all $n$: \begin{enumerate} \item[(i)] $\\|I_n^V_{n+1}(x)I_n-V_n(x)\\|<2^{-n-2}$ for all $x\in X_{n+1}$. \item[(ii)] $\\|V_n(xy)-V_n(x)V_n(y)\\|<2^{-n-2}$ for all $x,y\in X_n$. \item[(iii)] $V_n(X_{n+1})$ is $2^{-n-2}$-dense in the unit ball of ${\mathcal L}({\mathcal H}_n)$, i.e.\ for all $z$ in the unit ball of ${\mathcal L}({\mathcal H}_n)$ there is an $x\in X_{n+1}$ with $\\|V_n(x)-z\\|<2^{-n-2}$. \item[(iv)] $\\|V_1(x_0)\\|>3/4$. \end{enumerate} Once this tower constructed, we proceed as follows. Let ${\mathcal H}=\lim_{\rightarrow}({\mathcal H}_n,I_n)$ be the inductive limit Hilbert space, which may be thought of as the ``union'' of the ${\mathcal H}_n$. Let $J_n$ be the natural inclusion of ${\mathcal H}_n$ into ${\mathcal H}$. If $x\in X_m$, then for $n\geq m$ and $\xi,\eta\in J_n{\mathcal H}_n$ we have $$\|\langle(J_{n+1}V_{n+1}(x)J_{n+1}^-J_nV_n(x)J_n^)\xi,\eta\rangle\|$$ $$=\|\langle(J_nI_n^V_{n+1}(x)I_nJ_n^-J_nV_n(x)J_n^)\xi,\eta\rangle\|<2^{-n-2}$$ by (i). So the sequence $(J_nV_n(x)J_n^)$ converges weakly in ${\mathcal L}({\mathcal H})$ to an operator we call $\pi(x)$. For $\xi\in J_m{\mathcal H}_m$, for $n\geq m$ we have $$\\|J_{n+1}V_{n+1}(x)J_{n+1}^\xi\\|\geq\\|J_nJ_n^J_{n+1}V_{n+1}(x)J_{n+1}^\xi\\|$$ $$=\\|J_{n+1}I_n^V_{n+1}(x)I_nJ_n^\xi\\|\geq\\|J_nV_n(x)J_n^\xi\\|-2^{n-2}$$ and thus $\\|\pi(x)\xi\\|\geq\limsup\\|J_nV_n(x)J_n^\xi\\|$. So $J_nV_n(x)J_n^\to\pi(x)$ strongly (cf.\ \cite[I.1.3.3]{BlackadarOperator}). If $x,y\in X$, it follows from (ii) and joint strong continuity of multiplication on bounded sets that $\pi(xy)=\pi(x)\pi(y)$. Since $X$ is dense in the unit ball of $A$ and each $V_n$ is a contraction, $(J_nV_n(x)J_n^)$ converges strongly for each $x\in A$ to an operator on ${\mathcal H}$ we call $\pi(x)$, and $\pi$ is linear, completely positive, contractive, and multiplicative, hence a -representation of $A$ on ${\mathcal H}$. For each $m\in\mathbb{N}$ and $x\in X_m$, we have $\\|\pi(x)J_n-J_nV_n(x)\\|<2^{-n}$ for all $n\geq m$. To show that $\pi$ is irreducible, suppose $\xi,\eta,\zeta\in{\mathcal H}$ are unit vectors and $\epsilon>0$. Choose $m$ with $2^{-m}<\epsilon/4$, and for some $n\geq m$ choose unit vectors $\tilde\xi ,\tilde\eta, \tilde\zeta \in{\mathcal H}_n$ with $\\|\xi-J_n\tilde\xi \\|, \\|\eta-J_n\tilde\eta \\|, \\|\zeta-J_n\tilde\zeta\\|<\epsilon/4$. There is a unitary $u\in{\mathcal L}({\mathcal H}_n)$ with $u\tilde\xi =\tilde\eta$. By (iii), there is an $x\in X_{n+1}$ with $\\|V_n(x)-u\\|<2^{-n-2}$. Since $\\|I_n^V_{n+1}(x)I_n-V_n(x)\\|<2^{-n-2}$, we have $\\|I_n^V_{n+1}(x)I_n-u\\|<2^{-n-1}$. By iteration, $\\|J_n^\pi(x)J_n-u\\|<2^{-n}$ and hence $\\|J_n^\pi(x)J_n\tilde\xi-\tilde\eta\\|<2^{-n}$. Then $$\|\langle\pi(x)\xi-\eta,\zeta\rangle\|\leq\\|\zeta-J_n\tilde\zeta\\|+\|\langle \pi(x)\xi-\eta,J_n\tilde\zeta \rangle\|=\\|\zeta-J_n\tilde\zeta\\|+\|\langle J_n^(\pi(x)\xi-\eta),\tilde\zeta \rangle\|$$ $$\leq \\|\zeta-J_n\tilde\zeta\\|+\\|\xi-J_n\tilde\xi\\|+\\|\eta-J_n\tilde\eta\\|+\|\langle J_n^\pi(x)J_n\tilde\xi-\tilde\eta,\tilde\zeta\rangle\|<\epsilon$$ and so (fixing $\xi$ and $\eta$ and letting $\zeta$ vary) $\eta$ is in the weak closure of $\pi(A)\xi$. To show that $\pi$ is quasidiagonal, let $P_n=J_nJ_n^$ be the projection of ${\mathcal H}$ onto $J_n{\mathcal H}_n$. Then $P_n$ has finite rank, $P_n\to1$ strongly, and, for $x\in X_m$ and $n\geq m$, $$\\|P_n\pi(x)-\pi(x)P_n\\|=\max(\\|(1-P_n)\pi(x)P_n\\|,\\|P_n\pi(x)(1-P_n)\\|)$$ $$=\max(\\|(1-P_n)\pi(x)P_n\\|,\\|(1-P_n)\pi(x^)P_n\\|)$$ $$\\|(1-P_n)\pi(x)P_n\\|\leq\\|\pi(x)P_n-J_nV_n(x)J_n^\\|+\\|P_nJ_nV_n(x)J_n^-P_n\pi(x)P_n\\|$$ $$\leq \\|\pi(x)J_n-J_nV_n(x)\\|+\\|J_nV_n(x)-\pi(x)J_n\\|<2^{-n+1}$$ since $P_nJ_n=J_n$. Similarly, $\\|(1-P_n)\pi(x^)P_n\\|<2^{-n+1}$ since $x^\in X_m$. Finally, note that $\\|J_1V_1(x_0)J_1^\\|=\\|V_1(x_0)\\|>3/4$ and $\\|J_1^\pi(x_0)J_1-V_1(x_0)\\|<1/2$, so $\\|\pi(x_0)\\|>1/4$. \section{Pure Matricial States} In this section $A$ will be a general C-algebra, not the specific C-algebra of Section 2. \begin{definition} \label{def:pure-matr-state} A \emph{pure matricial $n$-state} on a C--algebra $A$ is a completely positive contraction $V\colon A\to {\mathcal{M}}_n=M_n(\mathbb{C} )$ such that there is an irreducible representation $\pi\colon A\to \L (\mathcal{H} )$ and an isometry $I\colon \mathbb{C}^n\to \mathcal{H}$ such that $V(a)=I^\pi(a)I$ for $a\in A$. (Note that $\pi$ and $I$ are uniquely determined up to unitary equivalence (of the $\mathcal{H}$) via the Stinespring dilation.) \end{definition} \begin{rems}\label{rems:pure-matr-state} (i) We have chosen a definition which is easily applicable for our needs. There are other characterizations, which are elementary functional analysis exercises, e.g.\ $V\colon A\to {\mathcal{M}}_n$ maps the \emph{open} unit ball onto the \emph{open} unit ball of ${\mathcal{M}}_n$ and $V$ is an extreme point of the convex set $CPC(A,{\mathcal{M}}_n)$ of completely positive contractions $T\colon A\to {\mathcal{M}}_n$. But, even for $A={\mathcal{M}}_n$ the extreme points of the unital maps in of $CPC(A,{\mathcal{M}}_n)$ are in general not pure $n$-states, i.e.\ are not automorphisms of ${\mathcal{M}}_n$ (for example, if $\phi$ is a pure state of $A$, then the map $x\mapsto\phi(x)1$ from $A$ to ${\mathcal{M}}_n$ is an extreme point of $CPC(A,{\mathcal{M}}_n)$, but not a pure matricial state of $A$). \smallskip \noindent (ii) If one applies the Kadison Transitivity Theorem (see e.g.\ \cite[II.6.1.12]{BlackadarOperator}) to the irreducible representation $\pi$ and the image of $I$, then one can see that the restriction of $V$ to the multiplicative domain $$ A^V:= \{ a\in A\,: \, V(ab)=V(a)V(b) \, \textrm{and} \, V(ba)=V(b)V(a)\,,\, b\in A \} $$ is an epimorphism from $A^V$ onto ${\mathcal{M}}_n$, cf.~ \cite[3.4]{BlackadarKInner}. In particular, $V$ maps the closed unit ball of $A$ onto the closed unit ball of ${\mathcal{M}}_n$, and hence maps the open unit ball of $A$ onto the open unit ball of ${\mathcal{M}}_n$. \smallskip \noindent (iii) Up to unitary equivalence (up to an automorphism of ${\mathcal{M}}_n$) a pure matricial $n$-state is defined by a projection $p$ in the socle of the second conjugate $A^{}$ of $A$ with $pAp\cong {\mathcal{M}}_n$: Consider the support projection $z$ of the normal extension $\overline{\pi}\colon A^{}\to \L(\mathcal{H} )$ of $\pi$ in the center of $A^{}$. The restriction $\varphi$ of $\overline{\pi}$ to $A^{}z$ defines an isomorphism from $A^{}z$ onto $\L (\mathcal{H} )$. Let $p_V:=\varphi^{-1}(II^)$; then $b\mapsto I^\varphi (b)I$ is an isomorphism $\lambda _V$ from $p_VAp_V=p_VA^{}p_V$ onto ${\mathcal{M}}_n$, such that $V(b)=\lambda _V(p_Vbp_V)$ for $b\in A$. Note that $p_V$ is just the support projection of the normal extension $\overline{V}$ of $V$ to $A^{}$. Projections corresponding to disjoint pure matricial states have orthogonal central supports. \smallskip \noindent (iv) A pure matricial $n$-state $V\colon A\to {\mathcal{M}}_n$ always extends to a unital pure matricial $n$-state on the unitization $\tilde{A}$ of $A$ (see (ii)). In particular, $V$ is unital if $A$ is unital. \smallskip \noindent (v) Pure matricial $n$-states are in obvious 1-1-correspondence with those pure states $\eta$ on $A\otimes {\mathcal{M}}_n$ which have the additional property that its (unique) extension $\tilde{\eta}$ to $\tilde{A}\otimes {\mathcal{M}}_n$ satisfies $\tilde{\eta}(1\otimes b)=\tau(b)$, where $\tau$ denotes the tracial state on ${\mathcal{M}}_n$. The correspondence is given by $\eta _V(a\otimes b):=(1/n) \mathrm{Tr}(V(a)b^\top )$ where $b^\top $ denotes the \emph{transposed} matrix of $b$. Not every pure state of $A\otimes{\mathcal{M}}_n$ has this property (cf.\ (i)). \noindent [To see that such an $\eta_V$ extends to a pure state on $\tilde{A}\otimes{\mathcal{M}}_n$, note that $a\otimes b\mapsto (1/n) \mathrm{Tr}(b^\top a)$ defines a pure state $\eta$ on ${\mathcal{M}}_n\otimes {\mathcal{M}}_n$. Conversely, every pure state $\eta$ on $A\otimes {\mathcal{M}}_n$ is given by an irreducible representation $\pi\colon A\to \L (\mathcal{H})$ and a map $I\colon \mathbb{C}^n\to {\mathcal H}$ such that $\eta(a\otimes b)=(1/n)\mathrm{Tr}(V(a)b^\top)$ for $a\in A$ and $b\in {\mathcal{M}}_n$, where $V(a):= I^\pi(a)I$. The condition $\eta(1\otimes b)=(1/n)\mathrm{Tr}(b)$ implies that $I$ is an isometry.] Here is an alternate way of viewing the situation. A pure state on $A\otimes{\mathcal{M}}_n$ is a vector state from an irreducible representation of $A\otimes{\mathcal{M}}_n$. Up to unitary equivalence, every irreducible representation of $A\otimes{\mathcal{M}}_n$ is of the form $\rho\otimes\sigma$, where $\rho$ is an irreducible representation of $A$ on a Hilbert space ${\mathcal H}$, and $\sigma$ is the standard representation of ${\mathcal{M}}_n$ on $\mathbb{C}^n$. Let $\{\zeta_1,\dots,\zeta_n\}$ be the standard basis for $\mathbb{C}^n$. Then every unit vector in ${\mathcal H}\otimes\mathbb{C}^n$ can be written in the form $\sum_{j=1}^n \alpha_j(\xi_j\otimes\zeta_j)$, where the $\xi_j$ are unit vectors in ${\mathcal H}$, $\alpha_j\geq0$, and $\sum \alpha_j^2=1$; the representation is unique if all $\alpha_j>0$. Then the vector state from this vector corresponds to a pure matricial $n$-state on $A$ if and only if the $\xi_j$ are mutually orthogonal and all $\alpha_j$ are equal to $n^{-1/2}$. \smallskip \noindent (vi) Every pure matricial $n$-state $V\colon A\to {\mathcal{M}}_n$ on $A\subset B$ extends to a pure matricial $n$-state $V_e\colon B\to {\mathcal{M}}_n$: Simply extend the pure state $\tilde{\eta}$ on $\tilde{A}\otimes {\mathcal{M}}_n$ to a pure state on $\tilde{B}\otimes {\mathcal{M}}_n$. If $T\colon A\to B$ is completely isometric \emph{and} completely positive then there is a pure matrical state $W\colon B\to {\mathcal{M}}_n$ with $W\circ T=V$. \noindent [Indeed: $T$ extends to a unital completely isometric map $T_1\colon A_1\to B_1$ of the \emph{outer} unitizations $A_1$ and $B_1$. An extremal extension of the extremal state on $T_1(A_1)\otimes {\mathcal{M}}_n \subset B_1\otimes {\mathcal{M}}_n$ -- related to $V$ -- defines the desired extension of $V\circ T^{-1}$ to all of $B_1$ by (iv).] \smallskip \noindent (vii) Since up to unitary equivalence the standard representation of ${\mathcal{M}}_n$ on $\mathbb{C}^n$ is the only irreducible representation of ${\mathcal{M}}_n$, every pure matricial state $V:{\mathcal{M}}_n\to{\mathcal{M}}_m$ (where necessarily $m\leq n$) is the compression of the identity representation to an $m$-dimensional subspace of $\mathbb{C}^n$, i.e.\ there is a unique isometry $I:\mathbb{C}^m\to\mathbb{C}^n$ with $V(x)=I^xI$ for all $x\in{\mathcal{M}}_n$, where ${\mathcal{M}}_n$ and ${\mathcal{M}}_m$ are identified with ${\mathcal L}(\mathbb{C}^n)$ and ${\mathcal L}(\mathbb{C}^m)$ in the standard way. \end{rems} The next result is the crucial technical tool needed for construction of the tower. \begin{Lem}\label{lem:A-to-Bmu} Suppose that $\{ \psi _\mu \colon A\to B_\mu\,; \,\, \mu \in \Gamma \}$ is a separating family of C--algebra homomorphisms. Then for every pure matricial $n$-state $V\colon A\to {\mathcal{M}}_n$, every $\delta >0$ and every finite subset $F\subset A$ there is a $\nu\in \Gamma$ and a pure matricial $n$-state $W\colon B_\nu\to {\mathcal{M}}_n$ such that $$ \\| W\psi _\nu (x)-V(x)\\| <\delta\, \quad \forall \, x\in F\,.$$ \end{Lem} \noindent{\mbox{\textbf{Proof}.\,}} If $n=1$ (the case of pure states) the result is well known (cf.\ \cite[3.4.2(ii)]{DixmierCAlgebras}). For the general case, we may assume $A$ and the $B_\mu$ are unital. Replace $A$ and $B_\mu$ by $A\otimes{\mathcal{M}}_n$ and $B_\mu\otimes{\mathcal{M}}_n$, and $\psi_\mu$ by $\psi_\mu\otimes id$. Let $F\otimes E=\{x\otimes e_{ij}\,:\, x\in F, 1\leq i,j\leq n\}$, where the $e_{ij}$ are the standard matrix units in ${\mathcal{M}}_n$. The pure state $\eta_V$ on $A\otimes{\mathcal{M}}_n$ corresponding to the pure matricial state $V$ on $A$ can be approximated arbitrarily closely (within $\delta/6n^4$ will do) on $F\otimes E$ by a pure state $\theta$ on $B_\nu\otimes{\mathcal{M}}_n$ for some $\nu$. The restriction of $\theta$ to $1\otimes{\mathcal{M}}_n$ is not (obviously) exactly $\tau$, the tracial state on ${\mathcal{M}}_n$, but is at least approximately $\tau$. We must perturb $\theta$ to make the restriction exactly $\tau$. When $\theta$ is represented as a vector state with vector $\sum_{j=1}^n\alpha_j(\xi_j\otimes\zeta_j)$ as in \ref{rems:pure-matr-state}(v), the $\xi_j$ are almost mutually orthogonal and $\alpha_j$ satisfy $\|\alpha_j-n^{-1/2}\|<\delta/6n^4$ for all $j$. Let $\varphi$ be the (pure) state of $B_\nu\otimes{\mathcal{M}}_n$ corresponding to the vector $\sum_{j=1}^n n^{-1/2}(\tilde \xi_j\otimes\zeta_j)$, where the $\tilde \xi_j$ are obtained from the $\xi_j$ by the Gram-Schmidt process. We have $\\|\tilde \xi_j-\xi_j\\|<\delta/3n^3$, so $\\|\varphi-\theta\\|<\delta/2n^2$ and $\\|\varphi(x)-\eta_V(x)\\|<\delta/n^2$ for $x\in F\otimes E$. Then $\varphi=\eta_W$ for some pure matricial $n$-state $W$ on $A$ factoring through $B_\nu$, and $\\|W(x)-V(x)\\|<\delta$ for all $x\in F$. \qed \section{Constructing the Tower} We now construct the tower used in the proof in Section 2, using the next two lemmas: \begin{Lem}\label{fl1} Let $B$ be an inner quasidiagonal C-algebra, $F$ a finite subset of the unit ball of $B$, $b\in F$, and $\epsilon>0$. Then there is a pure matricial state $V:B\to{\mathcal{M}}_n$ for some $n$, such that $\\|V(xy)-V(x)V(y)\\|<\epsilon$ for all $x,y\in F$ and $\\|V(b)\\|>\\|b\\|-\epsilon$. \end{Lem} \begin{proof} In the separable case, this is just (i) $\Rightarrow$ (ii) of \cite[3.7]{BlackadarKInner} (note that there is a misprint in the published statement of \cite[3.16(ii)]{BlackadarKInner}). We give the simple argument, which was omitted in \cite{BlackadarKInner} and which does not require separability. By the definition of inner quasidiagonality, there is a representation $\pi$ of $B$ on a Hilbert space ${\mathcal H}$ and a finite-rank projection $P\in\pi(B)''\subseteq{\mathcal L}({\mathcal H})$ such that $\\|P\pi(x)-\pi(x)P\\|<\epsilon$ for all $x\in F$ and $\\|P\pi(b)P\\|>\\|b\\|-\epsilon$. The central support $Q$ of $P$ in $\pi(B)''$ is Type I and is a sum of finitely many minimal central projections $Q_1,\dots,Q_m$. If $R_1,\dots,R_m$ are minimal projections in $\pi(B)'$ with $Q_j$ the central support of $R_j$, and $P_j=PR_j$, then $\\|P\pi(x)P\\|=\max_j\\|P_j\pi(x)P_j\\|$ for all $x\in B$. Fix $j$ with $\\|P_j\pi(b)P_j\\|>\\|b\\|-\epsilon$. Then $\rho=\pi\|_{R_j{\mathcal H}}$ is irreducible. For $x\in B$, let $V(x)=P_j\rho(x)P_j\in{\mathcal L}(P_j{\mathcal H})\cong{\mathcal{M}}_n$, where $n=dim(P_j{\mathcal H})$; then $V$ is the desired pure matricial state of $B$: $\\|P_j\pi(x)-\pi(x)P_j\\|<\epsilon$ for all $x\in F$, so $$\\|V(xy)-V(x)V(y)\\|=\\|P_j\pi(xy)P_j-P_j\pi(x)P_j\pi(y)P_j\\|$$ $$=\\|P_j(P_j\pi(x)-\pi(x)P_j)\pi(y)P_j\\|<\epsilon$$ for all $x,y\in F$. \end{proof} \begin{Lem}\label{fl2} Let $B$ be a separable inner quasidiagonal C-algebra, $F$ a finite subset of $B$, $V:B\to{\mathcal{M}}_n\cong{\mathcal L}(\mathbb{C}^n)$ a pure matricial state of $B$, and $\epsilon>0$. Then there is a pure matricial state $W:B\to{\mathcal{M}}_m\cong{\mathcal L}(\mathbb{C}^m)$ for some $m$ and an isometry $I:\mathbb{C}^n\to\mathbb{C}^m$ such that \begin{enumerate} \item[(i)] $\\|W(xy)-W(x)W(y)\\|<\epsilon$ for all $x,y\in F$. \item[(ii)] $\\|I^W(x)I-V(x)\\|<\epsilon$ for all $x\in F$. \end{enumerate} \end{Lem} \begin{proof} Let $X=\{x_1,x_2,\dots\}$ be a countable dense subset of the unit ball of $B$, and $X_k=\{x_1,\dots,x_k\}$. For each $k$, apply \ref{fl1} to obtain a pure matricial state $W_k:B\to{\mathcal{M}}_{m_k}$ such that $\\|W_k(xy)-W_k(x)W_k(y)\\|<2^{-k}$ for all $x,y\in X_k$, and $\\|W_k(x_k)\\|>\\|x_k\\|-2^{-k}$. Set $M_k={\mathcal{M}}_{m_k}$, $M=\prod_k M_k$, $J=\oplus_k M_k$. Let $\varphi_k:M\to M_k$ be the $k$'th coordinate map. Then $\{\varphi_k\}$ is a separating family of -homomorphisms on $M$ (since for each $k_0$, $\{x_k\,:\,k\geq k_0\}$ is dense in the unit ball of $B$). The map $\Psi:b\mapsto(W_1(b),W_2(b),\dots)$ from $B$ to $M$ drops to an injective -homo\-morphism $\psi$ from $B$ to $M/J$. By \ref{rems:pure-matr-state}(vi), the pure matricial state $V:B\to{\mathcal{M}}_n$ extends to a pure matricial state, also called $V$, from $M/J$ to ${\mathcal{M}}_n$; $V$ may be regarded as a pure matricial state on $M$ by composing with the quotient map from $M$ to $M/J$. By \ref{lem:A-to-Bmu}, for some $k$ there is a pure matricial state $U$ on $M_k$ with $\\|U(\varphi_k(x))-V(x)\\|<\epsilon$ for all $x\in F$ (where $B$ is identified with $\Psi(B)$). By \ref{rems:pure-matr-state}(vii), there is an isometry $I:\mathbb{C}^n\to\mathbb{C}^{m_k}$ such that $U(y)=I^yI$ for $y\in M_k$. Set $m=m_k$, $W=W_k$ (note that $W_k(x)=\varphi_k(x)$ for $x\in B\subseteq M$). \end{proof} We now construct the tower. Let $A$ and $x_0$ be as in Section 2, and let $X$ be a self-adjoint countable dense subset of the unit ball of $A$ containing $x_0$, closed under multiplication. Enumerate $X$ as $$X=\{x_0,x_0^,x_1,x_1^,\dots,x_n,x_n^,\dots\}$$ Set $X_1=\{x_0,x_0^\}$, and by Lemma \ref{fl1} choose a pure matricial $m_1$-state $V_1:A\to{\mathcal{M}}_{m_1}\cong{\mathcal L}(\mathbb{C}^{m_1})$ of $B=A$ with $F=X_1$, $b=x_0$, and $\epsilon=2^{-3}$, and set ${\mathcal H}_1=\mathbb{C}^{m_1}$. Suppose $X_j,{\mathcal H}_j=\mathbb{C}^{m_j},V_j, I_j$ have been defined for $1\leq j\leq n$. Since $V_n$ maps the closed unit ball of $A$ onto the closed unit ball of ${\mathcal{M}}_{m_n}$, there is a $k_n$ such that $X_{n+1}:=\{x_0,x_0^,\dots,x_{k_n},x_{k_n}^\}$ contains $X_n$ and $X_n^2$ and such that $V_n(X_{n+1})$ is $2^{-n-2}$-dense in the unit ball of ${\mathcal{M}}_{m_n}$. By Lemma \ref{fl2} with $B=A$, $F=X_{n+1}$, $V=V_n$, and $\epsilon=2^{-n-2}$ choose a pure matricial $m_{n+1}$-state $V_{n+1}$ of $A$ and an isometry $I_n:\mathbb{C}^{m_n}\to\mathbb{C}^{m_{n+1}}$. The $X_n,{\mathcal H}_n,V_n,I_n$ satisfy (i), (ii), (iii), and (iv) by construction. The tower thus has all the properties required in Section 2, completing the proof of Theorem \ref{MThm}. \begin{rem} What about the nonseparable case? Many parts of the argument have obvious generalizations to the nonseparable case. The separability hypothesis in Lemma \ref{fl2} should be removable at the cost of some technical complication. However, it is doubtful that the argument in the proof of Theorem \ref{MThm} can be adapted to the nonseparable case. For a slight variation of this argument can be used to give a new proof of the well-known fact that a separable prime C-algebra is primitive; this is known to be false in general for nonseparable C-algebras \cite{WeaverPrime}. \end{rem} \section{Inductive Limits} We do not know whether a general inner quasidiagonal C-algebra has a separating family of quasidiagonal irreducible representations. However, it follows immediately from \cite[3.6]{BlackadarKInner} that every inner quasidiagonal C-algebra is an inductive limit (with injective connecting maps) of separable inner quasidiagonal C-algebras, for which \ref{MThm} holds. Thus the theory of inner quasidiagonal C-algebras can be largely reduced to the separable case. But to complete this reduction, we must know that an arbitrary inductive limit of inner quasidiagonal C-algebras is inner quasidiagonal. In \cite[2.11]{BlackadarKInner}, it was stated that ``it is obvious from the definition'' that an inductive limit of an inductive system of inner quasidiagonal C-algebras (with injective connecting maps) is inner quasidiagonal. In fact, this is not so obvious from the definition, but it is obvious from the equivalence of (i) and (ii) of \cite[3.7]{BlackadarKInner} in the separable case (see \ref{IndLimThm}). So to prove that general inductive limits of inner quasidiagonal C-algebras are inner quasidiagonal, it suffices to remove the separability hypothesis in this equivalence: \begin{Prop}\label{Nonsep37} Let $A$ be a C-algebra. Then $A$ is inner quasidiagonal if and only if the following condition is satisfied: \noindent For every $a_1,\dots,a_n,b\in A$ and $\epsilon>0$, there is a pure matricial state $V$ of $A$ such that $\\|V(a_i)V(a_j)-V(a_ia_j)\\|<\epsilon$ for $1\leq i,j\leq n$ and $\\|V(b)\\|>\\|b\\|-\epsilon$. \end{Prop} The ``only if'' direction follows immediately from Lemma \ref{fl1}. To prove the converse, we will show by induction on $m$ that if the statement in \ref{Nonsep37} holds, the following condition $P(m)$ holds for every $m$. Then, given $a_1,\dots,a_n\in A$ and $\epsilon>0$, applying $P(n)$ with $b_j=a_j$ shows that $A$ is inner quasidiagonal. \bigskip $P(m):$ For every $a_1,\dots,a_n,b_1,\dots,b_m\in A$ and $\epsilon>0$, there are finitely many pure matricial states $V_1,\dots,V_r$ of $A$ such that \begin{enumerate} \item[(i)] $\\|V_k(a_i)V_k(a_j)-V_k(a_ia_j)\\|<\epsilon$ for $1\leq i,j\leq n$, $1\leq k\leq r$. \item[(ii)] $\max_k \\|V_k(b_j)\\|>\\|b_j\\|-\epsilon$ for $1\leq j\leq m$. \item[(iii)] The $V_k$ are pairwise disjoint (the corresponding irreducible representations are pairwise inequivalent). \end{enumerate} Note that $P(1)$ is exactly the condition in the statement of \ref{Nonsep37}. \bigskip Assume $P(m)$ holds (and thus $P(1)$ also holds), and let $a_1,\dots,a_n,b_1,\dots,b_{m+1}\in A$ and $\epsilon>0$. Fix $\delta>0$ such that $\delta<\frac{1}{4}$ and $$2\delta+12\delta(\max_{i,j}\{\\|a_i\\|,\\|b_j\\|\})<\epsilon\,.$$ Let $\pi_k$ ($1\leq k\leq r$) be pairwise inequivalent irreducible representations of $A$ on ${\mathcal H}_k$ with finite-rank projections $p_k\in{\mathcal L}({\mathcal H}_k)$ such that the pure matricial states $U_k(\cdot)=p_k\pi_k(\cdot)p_k$ satisfy \begin{enumerate} \item[(i)] $\\|U_k(x)U_k(y)-U_k(xy)\\|<\delta$ for $x,y\in\{a_1,\dots,a_n,b_1,\dots,b_{m+1}\}$, $1\leq k\leq r$. \item[(ii)] $\max_k \\|V_k(b_j)\\|>\\|b_j\\|-\delta$ for $1\leq j\leq m$. \end{enumerate} If $\pi_k(A)$ contains ${\mathcal K}({\mathcal H}_k)$ (i.e.\ if $\pi_k(A)\cap{\mathcal K}({\mathcal H}_k)\neq\{0\}$), then there is a $c_k\in A_+$, $\\|c_k\\|=1$, with $\pi_k(c_k)=p_k$. If $\pi_k(A)\cap{\mathcal K}({\mathcal H}_k)=\{0\}$, set $c_k=0$. By $P(1)$ there is an irreducible representation $\pi_{r+1}$ of $A$ on ${\mathcal H}_{r+1}$ and a finite-rank projection $p_{r+1}\in{\mathcal L}({\mathcal H}_{r+1})$ such that the pure matricial state $W(\cdot)=p_{r+1}\pi_{r+1}(\cdot)p_{r+1}$ satisfies \begin{enumerate} \item[(i)] $\\|W(x)W(y)-W(xy)\\|<\delta/2$ for $x,y\in\{a_1,\dots,a_n,b_1,\dots,b_{m+1},c_1,\dots,c_r\}$. \item[(ii)] $\\|W(b_{m+1})\\|>\\|b_{m+1}\\|-\delta$. \end{enumerate} If $\pi_{r+1}$ is not equivalent to any $\pi_k$, $k\leq r$, then we can set $V_k=U_k$ for $1\leq k\leq r$ and $V_{r+1}=W$, and we are done (since $\delta<\epsilon$). The difficulty comes when $\pi_{r+1}$ is equivalent to some $\pi_k$, say $\pi_r$ without loss of generality. In this case, there is an isometry $I$ from $p_{r+1}{\mathcal H}_{r+1}$ into ${\mathcal H}_r$ such that $W(\cdot)=I^\pi_r(\cdot)I$. If $\pi_r(A)\cap{\mathcal K}({\mathcal H}_r)=\{0\}$, then (cf.\ \cite{ArvesonNotes}) there is a sequence of isometries $I_t$ from $p_{r+1}{\mathcal H}_{r+1}$ to $(1-p_r){\mathcal H}_r$ such that $$W(x)=\lim_{t\to\infty} I_t^\pi_r(x)I_t\mbox{ for }x\in\{a_1,\dots,a_n,b_1,\dots,b_{m+1}\}\,.$$ For sufficiently large $t$, we may take $V_r(\cdot)=(p_r+I_tI_t^)\pi_r(\cdot)(p_r+I_tI_t^)$ and $V_k=U_k$ for $1\leq k\leq r-1$. The most difficult case is where $\pi_{r+1}$ is equivalent to $\pi_r$ and $\pi_r(A)$ contains ${\mathcal K}({\mathcal H}_r)$. If $q=II^$, then we have $$\\|qp_r-p_r q\\|=\\|q\pi_r(c_r)-\pi(c_r)q\\|<\delta\,.$$ By the following lemma, let $\tilde q$ be a projection in ${\mathcal L}({\mathcal H}_r)$ with $\tilde q p_k=p_k\tilde q$ and $\\|\tilde q -q\\|<3\delta$. Set $\tilde p_r=p_r+\tilde q (1-p_r)$ and $V_r(\cdot)=\tilde p_r\pi(\cdot)\tilde p_r$, and $V_k=U_k$ for $1\leq k\leq r-1$. These $V_k$ have the desired properties, completing the inductive step and thus the proof of \ref{Nonsep37}. \begin{Lem} Let $A$ be a C-algebra, and $p$ and $q$ projections in $A$. If $\\|qp-pq\\|<\epsilon<\frac{1}{4}$, then there is a projection $\tilde q\in A$ with $\\|\tilde q -q\\|<3\epsilon$ and $\tilde q p=p\tilde q$. If $r=p+\tilde q (1-p)$, then for every $x\in A$ we have $$\\|rx-xr\\|\leq 2\\|xp-px\\|+2\\|xq-qx\\|+12\epsilon\\|x\\|\,.$$ \end{Lem} \begin{proof} We may assume $A$ is unital. We have $$\\|q-[pqp+(1-p)q(1-p)]\\|=\\|(1-p)qp\\|=\\|(qp-pq)p\\|\leq\\|qp-pq\\|<\epsilon\,.$$ Also, $$\\|pqp-(pqp)^2\\|=\\|pqp-pqpqp\\|=\\|pq(qp-pq)p\\|<\epsilon$$ and so $\sigma(pqp)\subseteq[1,\gamma]\cup[1-\gamma,1]$, where $\gamma=\frac{1-\sqrt{1-4\epsilon}}{2}<2\epsilon$ since $\epsilon<\frac{1}{4}$. Thus by functional calculus there is a projection $r\in pAp$ with $\\|r-pqp\\|<\gamma<2\epsilon$. Similarly, there is a projection $s\in (1-p)A(1-p)$ with $\\|s-(1-p)q(1-p)\\|<2\epsilon$. If $\tilde q =r+s$, then $$\\|\tilde q -q\\|\leq\\|\tilde q -[pqp+(1-p)q(1-p)]\\|+\\|[pqp+(1-p)q(1-p)]-q\\|<3\epsilon\,.$$ If $x\in A$, then $$\\|rx-xr\\|\leq\\|xp-px\\|+\\|x\tilde q -\tilde q x\\|+\\|xp\tilde q -p\tilde q x\\|$$ $$\leq 2(\\|xp-px\\|+\\|x\tilde q -\tilde q x\\|)\leq 2\\|xp-px\\|+2(2\\|x\\|\\|\tilde q -q\\|+\\|xq-qx\\|)\,.$$ \end{proof} \begin{Cor}\label{IndLimThm} An arbitrary inductive limit (with injective connecting maps) of inner quasidiagonal C-algebras is inner quasidiagonal. \end{Cor} \begin{proof} Let $A=\lim_{\rightarrow}(A_i,\phi_{ij})$, with each $A_i$ inner quasidiagonal. Regard each $A_i$ as a C-subalgebra of $A$. If $a_1,\dots,a_n,b\in A$ and $\epsilon>0$, fix an $A_i$ and elements $\tilde a_1,\dots,\tilde a_n,\tilde b \in A_i$ with $\\|a_j-\tilde a_j\\|<\delta$ for each $j$ and $\\|b-\tilde b\\|<\delta$, where $\delta=\epsilon/3\max(1,\\|a_1\\|,\dots,\\|a_n\\|,\\|b\\|)$. Let $V$ be a pure matricial state of $A_i$ such that $\\|V(\tilde a_j)V(\tilde a_k )-V(\tilde a_j\tilde a_k)\\|<\delta$ for all $j,k$, and $\\|V(\tilde b )\\|>\\|\tilde b \\|-\delta$. Extend $V$ to a pure matricial state $W$ on $A$. Then $\\|W(a_j)W(a_k)-W(a_ja_k)\\|<\epsilon$ for all $j,k$, and $\\|W(b)\\|>\\|b\\|-\epsilon$. Thus $A$ is inner quasidiagonal by \ref{Nonsep37}. \end{proof} \section{Permanence Properties} We finish by recording some other permanence properties of the class of inner quasidiagonal C-algebras. The first one is an easy consequence of the definition of inner quasidiagonality, and could have been noted in \cite{BlackadarKInner}. \begin{Prop}\label{TensProd} The minimal tensor product of inner quasidiagonal C-algebras is inner quasidiagonal. \end{Prop} \begin{proof} If $A$ and $B$ are inner quasidiagonal and $Z=\{z_1,\dots,z_n\}\subseteq A\otimes_{\min}B$, approximate $z_k$ by an element $\sum_{j=1}^{n_k}x_{jk}\otimes y_{jk}$ of the algebraic tensor product $A\odot B$. Let $E=\{x_{jk}\}$ and $F=\{y_{jk}\}$. If $(\pi,P)$ and $(\rho,Q)$ are representations of $A$ and $B$ with projections as in the definition for $E$ and $F$ with sufficiently small $\epsilon$, then $(\pi\otimes\rho,P\otimes Q)$ will be the desired representation for $A\otimes_{\min}B$ and $Z$. \end{proof} It is doubtful that the result holds for maximal tensor products. Note that no separability hypothesis is necessary in \ref{TensProd}. The next property is an immediate consequence of Theorem \ref{MThm}, and essentially generalizes \cite[3.10]{BlackadarKInner}: \begin{Prop} The algebra of sections of a continuous field of separable inner quasidiagonal C-algebras is inner quasidiagonal. \end{Prop} \begin{proof} Let $\langle A(t)\rangle$ be a continuous field of separable continuous trace C-algebras over a locally compact Hausdorff space $X$, and $A$ the C-algebra of continuous sections vanishing at infinity. Each fiber $A(t)$ has a separating family of quasidiagonal irreducible representations by \ref{MThm}, so by composition with the fiber maps from $A$ to the $A(t)$, $A$ also has a separating family of quasidiagonal irreducible representations, hence is inner quasidiagonal. \end{proof} Any C-subalgebra of a quasidiagonal C-algebra is quasidiagonal. This is false for inner quasidiagonality: if $A$ is an NF algebra which is not strong NF (cf.\ \cite[5.6]{BlackadarKInner}), let $\pi$ be a faithful quasidiagonal representation of $A$ on a Hilbert space ${\mathcal H}$; then $\pi(A)+{\mathcal K}({\mathcal H})$ is inner quasidiagonal (by \ref{MThm} or \cite[5.8]{BlackadarKInner}), in fact strong NF, but the C*-subalgebra $\pi(A)$ is not inner quasidiagonal. But we have: \begin{Prop}\label{InnQDSI} Inner quasidiagonality is an (SI) property in the sense of \cite[II.8.5]{BlackadarOperator}. \end{Prop} \begin{proof} This is just a combination of \ref{IndLimThm} and \cite[3.6]{BlackadarKInner}. \end{proof} \bibliographystyle{alpha} \def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$}	{ "redpajama_set_name": "RedPajamaArXiv" }	1,658
Діахронія (від грец. dia — «через» і χρόνος — «час») — історична послідовність розвитку мовних явищ, а також дослідження мови у процесі її історичного розвитку; протилежне синхронії. Фердинанд де Сосюр протиставляв синхронію як вісь одночасності і діахронію як вісь послідовності та вважав, що це протиставлення відповідає протиставленню статики та динаміки, системності та безсистемності. На його думку, є дві абсолютно різні лінгвістики — синхронічна та діахронічна. Джерела Філософія мови Мовознавство	{ "redpajama_set_name": "RedPajamaWikipedia" }	5,427
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\section{Introduction} During the last decade, fractional differential equations(FDEs) \cite{Diethelm,Kilbas,Podlubny} appeared as rich and beautiful field of research due to their applications to the physical and life sciences.The theoretical development of FDEs in the classical Riemann-Liouville's or Caputo sense have been excellently given in \cite{Diethelm,Kilbas,Podlubny,Gorenflo,Miller,Lak3}. For important and interesting works on existence, uniqueness, data dependence and heterogeneous qualitative properties of solution of classical FDEs, we propose the work of Lakshmikantham et al.\cite{Lak3,Lak1,Lak2}, Gejji et al.\cite{Gejji1,Gejji2}, Kai et al.\cite{Diethelm1}, Trujillo et al.\cite{Kucche}, Benchora et al.\cite{Agrawal2}, and the references cited therein. Other interesting works on these aspect can be found in \cite{Agrawal2, Agrawal,Lak4,Wang,Kou,Idczak,Deng,Li,Kilbas2,Gambo,Zhou,Agrawal3,Ahmad,Anber}. The fundamental concept and properties of integrals and derivatives of fractional order of a function with respect to the another function viz. $ \Psi $-Riemann-Liouville integral and derivative have been introduced in \cite[Chapter 2]{Kilbas}. Following the similar approach, Almeida \cite{Almeida} introduced $\Psi$-Caputo fractional derivative and investigated the interesting properties of this operator and extended few preceding study concerned with the Caputo and the Caputo--Hadamard derivative operators. On the other hand, Hilfer \cite{Hilfer} introduced a fractional derivative $\mathcal{D}_{a+}^{\eta ,\nu }(\cdot)$ having two parameters $\eta \in (n-1, n), n \in \mathbb{N}$ and $\nu ~(0\leq \nu \leq 1)$ which in specific gives the Riemann--Liouville and the Caputo derivative operator. Furati et al. \cite{Furati} analyzed nonlinear FDEs with Hilfer derivative operator. Sousa and Oliveira \cite{Vanterler1,Vanterler2} introduced a new definition of fractional derivative viz. $\Psi$--Hilfer fractional derivative and investigated its important properties. Further, it is proved that the $\Psi$--Hilfer derivative is the generalization of many existing fractional derivative operators. In \cite{Vanterler2}, Sousa and Oliveira established generalized Gronwall inequality through the fractional integral with respect to another function and studied the existence, uniqueness and data dependence of solution of Cauchy problem with $\Psi$--Hilfer fractional derivative. For the study relating to existence, uniqueness and stability of different sorts of FDEs involving $\psi$-Hilfer fractional derivative operator, we refer to the work of Sousa et al.\cite{jose1,jose2,jose3} and the references given there in. Motivated by the work of \cite{Diethelm1,Vanterler1,Vanterler2} in this paper, existence along with the interval of existence, uniqueness and continuous dependence of solutions on initial condition have examined for the nonlinear $\Psi$-Hilfer FDEs structured as: \begin{align} ^H \mathcal{D}^{\eta,\,\nu\,;\, \Psi}_{a +}y(t)&= f(t, y(t)),~~ ~t \in \Lambda =[a,a+\xi], ~~\xi>0, ~0<\eta<1, ~0\leq\nu\leq 1,~\label{eq1}\\ \mathcal{I}_{a +}^{1-\zeta\,;\, \Psi}y(a)&=y_a \in \mbox{\Bbb R}, \qquad\zeta =\eta +\nu \left( 1-\eta \right), \label{eq2} \end{align} where $^H \mathcal{D}^{\eta,\nu;\, \Psi}_{a +}(\cdot)$ is the $\Psi$-Hilfer derivative of order $\eta$ and type $\nu$, $\mathcal{I}_{a +}^{1-\zeta;\, \Psi}(\cdot)$ is $\Psi$-Riemann--Liouville integral of order $1-\zeta$ and $f: \Lambda \times \mbox{\Bbb R} \to \mbox{\Bbb R} $ is an appropriate function. Further, by Picard's successive approximation method we derive the existence along with uniqueness of solution for Cauchy type problem \eqref{eq1}-\eqref{eq2}. We have established bound for the error between approximated solution $y_n$ and exact solution $y$ and proved that the difference is approaches to zero when $n$ is very large. The development of Picard's successive approximation technique is utilized to obtain representation formula for the solution of linear Cauchy problem with constant coefficient \begin{align} {}^H\mathcal{D}_{a+}^{\eta,\,\nu\,; \,\Psi}y(t)-\lambda y(t)&=f( t ),~\lambda \in \mbox{\Bbb R}, ~0<\eta<1, ~ 0\leq\nu\leq 1,~t\in \Delta=[a,b].\label{a1} \\ \mathcal{I}_{a+}^{1-\zeta\,; \,\Psi}y(a)&=y_a \in \mbox{\Bbb R}, ~~\zeta =\eta+\nu(1-\eta) \label{a2} \end{align} in form of Mittag--Leffler function, where as the representation formula for the solution of linear Cauchy problem with variable coefficient \begin{align} {}^H\mathcal{D}_{a+}^{\eta,\,\nu\,; \,\Psi}y(t)-\lambda[\Psi(t)-\Psi(a)]^{\mu-1} y(t) &=0,~0<\eta<1, ~0\leq\nu\leq 1,~~\mu>1-\eta ,~ t\in\Delta,\label{a3}\\ \mathcal{I}_{a+}^{1-\zeta\,; \,\Psi}y(a) &=y_a \in \mbox{\Bbb R}, ~~\zeta=\eta+\nu(1-\eta),\label{a4} \end{align} is obtained in form of generalized (Kilbas--Saigo) Mittag--Leffler function. This paper is divided in five sections: In the section 2, we provide some definitions, theorems of $\Psi$-Hilfer fractional derivative and the results which will be utilized throughout this paper. In Section 3, we establish the results pertaining to existence, uniqueness and continuous dependence of solution of \eqref{eq1}-\eqref{eq2}. Section 4, discuss the convergence of Picard's type successive approximations to the solution of \eqref{eq1}-\eqref{eq2}. In Section 5, representation formulas have been obtained for the solution of linear Cauchy problem with constant coefficient and variable coefficient. \section{Preliminaries} \label{preliminaries} We review a few definitions, notations and results of $\Psi$-Hilfer fractional derivative \cite{Vanterler1, Vanterler2}. Let $\Delta=[a,b]$ $(0<a<b<\infty)$ be a finite interval. Consider the space $C_{1-\zeta;\,\Psi}(\Delta,\,\mbox{\Bbb R})$ of weighted functions $h$ defined on $\Delta$ given by \begin{equation} C_{1-\zeta ;\, \Psi }(\Delta,\,\mbox{\Bbb R}) =\left\{ h:\left( a,b\right] \rightarrow \mathbb{R}~\big\|~\left( \Psi \left( t\right) -\Psi \left( a\right) \right) ^{1-\zeta }h\left( t\right) \in C(\Delta,\,\mbox{\Bbb R}) \right\} ,\text{ }0< \zeta \leq 1 \end{equation} endowed with the norm \begin{equation}\label{space1} \left\Vert h\right\Vert _{C_{1-\zeta ;\,\Psi }\left(\Delta,\mbox{\Bbb R}\right) }=\underset{t\in \Delta }{\max }\left\vert \left( \Psi \left( t\right) -\Psi \left( a\right) \right) ^{1-\zeta }h\left( t\right) \right\vert. \end{equation} \begin{definition} Let $\eta>0 ~(\eta \in \mbox{\Bbb R})$, $h \in L_1(\Delta,\,\mbox{\Bbb R})$ and $\Psi\in C^{1}(\Delta,\,\mbox{\Bbb R})$ be an increasing function wth $\Psi'(t)\neq 0$, for all $\, t\in \Delta$. Then, the $\Psi$-Riemann--Liouville fractional integral of a function $h$ with respect to $\Psi$ is defined by \end{definition} \begin{equation}\label{P1} \mathcal{I}_{a+}^{\eta \, ;\,\Psi }h\left( t\right) =\frac{1}{\Gamma \left( \eta \right) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)h(\sigma) \,d\sigma. \end{equation} where,$\mathrm{Q}^{\eta}_{\Psi}(t,s)=\Psi'(s)(\Psi(t)-\Psi(s))^{\eta-1},~ t,s\in \Delta.$ \begin{definition} Let $n-1<\eta <n \in \mbox{\Bbb N} $ and $h ,\Psi\in C^{n}(\Delta,\,\mbox{\Bbb R})$ two functions such that $\Psi$ is increasing with $\Psi'(t)\neq 0,$ for all $\,t\in \Delta$. Then, the $( \mbox{left-sided} )$ $\Psi$-Hilfer fractional derivative $^{H}\mathcal{D}^{\eta,\,\nu\, ;\,\Psi}_{a+}(\cdot)$ of a function $h$ of order $\eta$ and type $0\leq \nu \leq 1$, is defined by \begin{equation}\label{HIL} ^{H}\mathcal{D}_{a+}^{\eta ,\,\nu \, ;\,\Psi }h\left(t\right) =\mathcal{I}_{a+}^{\nu \left( n-\eta \right) \, ;\,\Psi }\left( \frac{1}{\Psi ^{\prime }\left( t\right) }\frac{d}{dt}\right) ^{n}\mathcal{I}_{a+}^{\left( 1-\nu \right) \left( n-\eta \right) \, ;\,\Psi }h\left( t\right). \end{equation} \end{definition} Following results from \cite{Vanterler1, Vanterler2} play key role in proving our main results. \begin{lemma}\label{lema1} If $\eta >0$ and \, $0\leq \mu <1,$ then $\mathcal{I}_{a+}^{\eta \, ;\,\Psi }(\cdot)$ is bounded from $C_{\mu \, ;\,\Psi }\left[ a,b\right] $ to $C_{\mu \, ;\,\Psi }\left[ a,b\right] .$ In addition, if $\mu \leq \eta $, then $\mathcal{I}_{a+}^{\eta \, ;\,\Psi }(\cdot)$ is bounded from $C_{\mu \, ;\,\Psi }\left[ a,b\right] $ to $C\left[ a,b\right] $. \end{lemma} \begin{lemma}\label{lema2} Let $\eta>0$ and $\delta>0$. If $h(t)= \left( \Psi \left( t\right) -\Psi \left( a\right) \right) ^{\delta -1}$, then \begin{equation} \mathcal{I}_{a+}^{\eta \, ;\,\Psi }h(t)=\frac{\Gamma \left( \delta \right) }{\Gamma \left( \eta +\delta \right) }\left( \Psi \left(t\right) -\Psi \left( a\right) \right) ^{\eta +\delta -1}. \end{equation} \end{lemma} \begin{lemma}\label{lema3} Let $\Psi\in C^{1}(\Delta,\mathbb{R})$ be increasing function with $\Psi'(t)\neq 0$, for all $t\in\Delta$. If $\zeta =\eta +\nu \left( 1-\eta \right) $ where, $ 0<\eta <1$ and $0\leq \nu \leq 1,$ then $\Psi$-Riemann-Liouville fractional integral operator $\mathcal{I}_{a+}^{\eta \, ;\,\Psi }\left( \cdot \right): C_{1-\zeta \, ;\,\Psi }\left[ a,b\right] \rightarrow C_{1-\zeta \, ;\,\Psi }\left[ a,b\right]$ is bounded and it is given by : \begin{equation}\label{eq23} \left\Vert \mathcal{I}_{a+}^{\eta \, ;\,\Psi }h\right\Vert _{C_{1-\zeta \, ;\,\Psi }\left[ a,b \right] }\leq M\frac{\Gamma \left( \zeta \right) }{\Gamma \left( \zeta +\eta \right) }\left( \Psi \left( t\right) -\Psi \left( a\right) \right)^{\eta }, \end{equation} where, $M$ is the bound of a bounded function $(\Psi(\cdot)-\Psi(a))^{1-\zeta}h(\cdot)$. \end{lemma} \begin{theorem}\label{lema4} Let $\tilde{u},$ $\tilde{v}\in L_1(\Delta,\,\mathbb{R})$ and $\tilde{g}\in C(\Delta,\,\mathbb{R}) .$ Let $\Psi \in C^{1}(\Delta,\,\mathbb{R}) $ be an increasing function with $\Psi ^{\prime }\left( t\right) \neq 0$, for all $~t\in \Delta$. Assume that \begin{enumerate} \item $\tilde{u}$ and $\tilde{v}$ are nonnegative; \item $\tilde{g}$ is nonnegative and nondecreasing. \end{enumerate} If \begin{equation} \tilde{u}( t) \leq \tilde{v}( t) +\tilde{g}( t) \int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\,\tilde{u}(\sigma)\, \,d\sigma, \end{equation} then \begin{equation}\label{jose} \tilde{u}(t) \leq\tilde{v}( t) +\int_{a}^{t}\overset{\infty }{% \underset{m=1}{\sum }}\frac{\left[\tilde{g}\left( t\right) \Gamma \left( \eta \right) \right] ^{m}}{\Gamma \left( m\eta \right) }\,\mathrm{Q}^{m\eta}_{\Psi}(t,s)\,\,\tilde{v}(\sigma) \,d\sigma,~t\in \Delta. \end{equation} where,\,$\mathrm{Q}^{m\eta}_{\Psi}(t,s)=\Psi'(s)(\Psi(t)-\Psi(s))^{m\eta-1}, ~t, s \in \Delta .$ Further, if $\tilde{v}$ is a nondecreasing function on $\Delta$ then $$\tilde{u}(t)\leq\tilde{v}(t)\,\mathcal{E}_{\eta}\left(\tilde{g}(t) \Gamma(\eta)\left(\Psi(t)-\Psi(a)\right)^{\eta}\right),$$ where, $\mathcal{E}_{\eta}(\cdot)$ is the Mittag-Leffler function of one parameter. \end{theorem} Existence and uniqueness results are proved via following fixed point theorems. \begin{theorem}[\cite{Diethelm1}, Schauder] \label{Schauder} Let $\mathcal{X}$ be a Banach space, let $\mathcal{U}$ be a nonempty convex bounded closed subset of $\mathcal{X}$ and let $\mathcal{A}:\mathcal{U}\rightarrow \mathcal{U}$ be a completely continuous operator. Then $\mathcal{A}$ has at least one fixed point. \end{theorem} \begin{theorem}[\cite{Diethelm1}, Weissinger] \label{Weissinger} Assume $(\mathcal{U},d)$ to be a non empty complete metric space and let $\eta_j\geq 0$ for every $j\in \mbox{\Bbb N}_0$ such that $\overset{\infty}{\underset{j=0}{\sum }}\eta_j$ converges. Furthermore, let the mapping $\mathcal{A}:\mathcal{U}\rightarrow {\mathcal{U}}$ satisfy the inequality $$d(\mathcal{A}^{j}u,\mathcal{A}^{j}v)\leq \eta_{j}\,\,d(u,v)$$ for every $j\in \mbox{\Bbb N}$ and every $u,v \in \mathcal{U}.$ Then, $\mathcal{A}$ has a unique fixed point $u^{}.$ Moreover, for any $u_{0}\in \mathcal{U},$ the sequence $(\mathcal{A}^{j}u_{0})_{j=1}^{\infty}$ converges to this fixed point $u^{}.$ \end{theorem} \begin{definition}[\cite{Gorenflo}] Let $\eta >0,\, \nu >0~ ( ~\eta, \nu \in \mbox{\Bbb R}) $. Then the two parameter Mittag-Leffler function is defined as $$\mathcal{E}_{\eta,\,\nu}(z)=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(k \eta +\nu)}.$$ \end{definition} \begin{definition}[\cite{Gorenflo}]\label{defKS} Let $\eta,\,m \in \mbox{\Bbb R}$ and $l\in\mbox{\Bbb C}$ such that $\eta>0,~m>0$ and $\eta(jm+l)+1\neq-1,-2,-3,\cdots (j=0,1,2,\cdots) $. Then generalized (Kilbas--Saigo) Mittag--Leffler type function of three parameters is defined by $$ \mathcal{E}_{\eta,\,m,\,l}\,(z)= \sum_{k=0}^{\infty} c_k\, z^k $$ where $$ c_{0} =1, ~c_{k}=\prod_{j=0}^{k-1}\frac{\Gamma(\eta[jm+l]+1)}{\Gamma(\eta[jm+l+1]+1)}\,\,(k=1,2,\cdots)$$ and an empty product is assumed to be equal to one. \end{definition} \section{Existence, Uniqueness and Continuous Dependence} The forthcoming theorem establishes the existence of solution along with the interval of existence of the initial value problem (IVP) \eqref{eq1}-\eqref{eq2} using its equivalent fractional Volterra integral equation(VIE) in the weighted space $C_{1-\zeta ; \, \Psi }(\Lambda,\,\mbox{\Bbb R})$. \begin{theorem} [Existence and interval of existence]\label{th3.1} Let $\zeta=\eta+\nu(1-\eta), 0<\eta<1$ and $0\leq\nu\leq 1$. Define $$R_{0}=\left\{(t,y):a\leq t\leq a+\xi,\left\|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_a\right\|\leq k \right\},~\xi>0, ~k>0.$$ where, $\mathcal{H}^{\Psi}_{\zeta}(t,a):=\dfrac{( \Psi (t)-\Psi (a) ) ^{\zeta -1}}{\Gamma (\zeta)}$.~ Let $f:R_0\rightarrow\mbox{\Bbb R}$ is continuous function such that $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,; \,\Psi}(\Lambda,\,\mbox{\Bbb R})$ for every $y\in C_{1-\zeta\,; \,\Psi}(\Lambda,\,\mbox{\Bbb R})$. Let $\Psi\in C^1(\Lambda,\,\mbox{\Bbb R})$ be an increasing bijective function with $\Psi'(t)\neq 0, $ for all $\,t \in \Lambda$ . Then the IVP \eqref{eq1}-\eqref{eq2} possesses at least one solution $y$ in the space $C_{1-\zeta\,; \,\Psi} [a,a+\chi]$ of weighted functions, where $$ \chi=\min\left\{\xi,~\Psi^{-1}\left[\Psi(a)+\left(\frac{k\, \Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\Vert f\right\Vert _{C_{1-\zeta \,; \,\Psi }\left[ a,a+\xi\right]}}\right)^\frac{1}{\eta}\right]-a\right\}. $$ \end{theorem} \begin{proof} The equivalent fractional VIE to the Cauchy problem \eqref{eq1}-\eqref{eq2} in the space $C_{1-\zeta\,; \,\Psi}(\Lambda,\,\mbox{\Bbb R})$ is derived in \cite{Vanterler2} and it is given by \begin{equation}\label{eq3} y (t) =\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f(\sigma,y(\sigma)) \,d\sigma, ~t \in \Lambda. \end{equation} Consider the set defined by, $$ \mathcal{U}=\left\{y\in C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]:\left\Vert y-\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_a \right\Vert_{C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]}\leq k\right\}. $$ Define $\tilde{y}(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}, ~t \in [ a,a+\chi]$. Then $(\Psi(t)-\Psi(a))^{1-\zeta}\, \tilde{y}(t)=\dfrac{y_a}{\Gamma(\zeta)}\in C[a,a+\chi]$ and hence we have, $\tilde{y}(t)\in C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right] $. Further, $$ \left\Vert \tilde{y}-\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_a\right\Vert_{C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]}=0\leq k. $$ Thus, $\tilde{y}\in \mathcal{U}$ and hence $\mathcal{U}$ is non-empty set. Clearly, $\mathcal{U}$ is a convex, bounded and closed subset of space $C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right].$ We define an operator $\mathcal{A}$ on the set $\mathcal{U}$ by $$\mathcal{A}y(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f(\sigma,y(\sigma))\, \,d\sigma.$$ By definition of an operator $\mathcal{A}$, the equation \eqref{eq3} can be written as $$ y=\mathcal{A}y.$$ We analyze the properties of $\mathcal{A}$ so that it admit at least one fixed point. First we show that, $\mathcal{A}\,\mathcal{U} \subseteq \mathcal{U}$. In the view of Lemma \ref{lema1}, $\Psi$-Riemann-Liouville fractional integral operator $\mathcal{I}_{a+}^{\eta \, ;\,\Psi }\left( \cdot \right) $ maps $C_{1-\zeta \, ;\,\Psi }\left[ a,a+\chi\right] $ to $C_{1-\zeta \, ;\,\Psi }\left[ a,a+\chi\right]$, and hence $\mathcal{I}_{a +}^{\eta\,;\, \Psi} f(\cdot,y(\cdot)) \in C_{{1-\zeta};\, \Psi}[a,a+\chi]$ for any $y\in C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]$. Further, $$(\Psi (t)-\Psi (a) )^{1-\zeta }\,\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}=\frac{y_{a}}{\Gamma ( \zeta )}\in C[a,a+\chi],$$ and hence $\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a} \in C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]$. From the above arguments it follows that $ \mathcal{A}y\in C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]$. Next, for any $y\in \mathcal{U}$ and any $x$,~ $ a< x\leq \chi$, we have \begin{align} &\left\Vert \mathcal{A}y -\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_a\right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }}\notag\\ &=\underset{t\in \left[ a,a+x\right] }{\max }\left\|(\Psi (t)-\Psi (a) )^{1-\zeta } \frac{1}{\Gamma ( \eta )}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)(\Psi (\sigma)-\Psi (a) )^{\zeta-1 }\times \right.\notag\\ &\qquad\left.(\Psi (\sigma)-\Psi (a) )^{1-\zeta }f(\sigma,y(\sigma)) \,d\sigma\right\|\notag\\ &\leq(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)(\Psi (\sigma)-\Psi (a) )^{\zeta-1 }\times \notag\\ &\qquad\underset{\varsigma\in \left[ a,\, \sigma\right] }{\max }\left\|(\Psi (\varsigma)-\Psi (a) )^{1-\zeta }f(\varsigma,y(\varsigma))\right\|\,d\sigma\notag\\ &\leq \left\Vert f \right\Vert_{C_{1-\zeta \,; \,\Psi }\left[ a,a+x\right]}(\Psi (a+x)-\Psi (a) )^{1-\zeta }\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(a+x)-\Psi(a))^{\zeta-1}\notag\\ &\leq \left\Vert f \right\Vert_{C_{1-\zeta \,; \,\Psi }\left[ a,a+x\right]}(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi (a+x)-\Psi (a) )^{\eta+\zeta-1 }\notag\\ &=\left\Vert f \right\Vert_{C_{1-\zeta \,; \,\Psi }\left[a,a+x\right]}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi (a+x)-\Psi (a) )^{\eta }. \end{align} Taking $x \rightarrow \chi$ we get, \begin{equation} \label{e4} \left\Vert \mathcal{A}y -\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_a \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+\chi\right] }} \leq \left\Vert f \right\Vert_{C_{1-\zeta \,; \,\Psi }\left[a,a+\chi\right]}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi (a+\chi)-\Psi (a) )^{\eta } \end{equation} Since, $$ \chi=\min\left\{\xi\,\,,\Psi^{-1}\left[\Psi(a)+\left(\frac{k\, \Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\Vert f\right\Vert _{C_{1-\zeta \,; \,\Psi }\left[ a,a+\xi\right]}}\right)^\frac{1}{\eta}\right]-a\right\} $$ we have $$ \chi\leq\Psi^{-1}\left[\Psi(a)+\left(\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\Vert f\right\Vert_{C_{1-\zeta\,; \,\Psi}\left[a,a+\xi\right]}}\right)^\frac{1}{\eta}\right]-a. $$ Further, since $\Psi\in C^1\left( \Lambda,\,\mbox{\Bbb R}\right) $ is a bijective function, $\Psi^{-1}: \mbox{\Bbb R} \to \Lambda$ exists and from above inequality, we have \begin{equation}\label{e5} (\Psi(a+\chi)-\Psi(a))^\eta\leq\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\Vert f\right\Vert _{C_{1-\zeta\, ; \,\Psi }\left[ a,a+\xi\right]}}. \end{equation} Using Eq.\eqref{e5} in Eq.\eqref{e4}, we get $$\left\Vert \mathcal{A}y -\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_a\right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+\chi\right] }}\leq k.$$ We have proved that $\mathcal{A}y\in \mathcal{U} \,\,\text{for any}\,\, y\in \mathcal{U}$. This proves $\mathcal{A} \,\,\text{maps}\,\, \mathcal{U} \,\,\text{to itself}$. Next, we prove that $\mathcal{A}$ is a continuous operator. Let any $\varepsilon>0$. Since $f:R_0\rightarrow\mbox{\Bbb R}$ is continuous and $R_{0}$ is compact set, $f$ is uniformly continuous on $R_{0}$. Thus, there exists $\tilde{\delta}>0$ such that \begin{equation} \left\|f(t,y)-f(t,z)\right\|<\frac{\varepsilon\,\, \Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}},~~ \text{whenever}~ \|y-z\|<\tilde{\delta}. \end{equation} Since, $\Psi$ is continuous on compact set $\Lambda$, we can choose $\delta>0$ such that $( \Psi (t)-\Psi (a) ) ^{1-\zeta }\|y-z\|<\delta$. Therefore, we have \begin{align} \label{eq34} \left\|f(t,y)-f(t,z)\right\|<\frac{\varepsilon\,\, \Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}},~ \text{whenever}~( \Psi (t)-\Psi (a) ) ^{1-\zeta }\|y-z\|<\delta. \end{align} Let any $y,z\in \mathcal{U} $ such that $\left\Vert y-z \right\Vert_{C_{1-\zeta\, ; \,\Psi }\left[ a,a+\chi\right]}<\delta$. Then in the view of \eqref{eq34}, we have $$\left\|f(t,y(t))-f(t,z(t))\right\|<\frac{\varepsilon\,\, \Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}}, ~t\in [ a,a+\chi].$$ Therefore, \begin{align} &\left\Vert \mathcal{A}y-\mathcal{A}z \right\Vert_{C_{1-\zeta\, ; \,\Psi }\left[ a,a+\chi\right]}\\ &\leq\underset{t\in \left[ a,a+\chi\right] }{\max }(\Psi (t)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta )} \int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\left\|f(\sigma,y(\sigma))-f(\sigma,z(\sigma))\right\| \,d\sigma\\ &<\frac{\varepsilon\,\, \Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}}\,\,\,\underset{t\in \left[ a,a+\chi\right] }{\max }(\Psi (t)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta )} \int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\, \,d\sigma\\ &=\frac{\varepsilon\,\, \Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}}\,\,\,\underset{t\in \left[ a,a+\chi\right] }{\max }(\Psi (t)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta+1 )} ( \Psi (t) -\Psi (a)) ^{\eta} \\ &\leq\frac{\varepsilon}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}}\,\,(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}\\ &=\varepsilon. \end{align} Thus, the operator $\mathcal{A}$ is continuous on $\mathcal{U}$. Now, for any $z\in \mathcal{A}(\mathcal{U})=\{\mathcal{A}y: y\in \mathcal{U}\}$ and any $x$ such that $a<x\leq \chi$, we have \begin{align} &\left\Vert z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }} =\left\Vert \mathcal{A}y \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }}\\ &\leq\frac{y_a}{\Gamma(\zeta)}+\underset{t\in \left[ a,a+x\right] }{\max }\left\|\frac{(\Psi (t)-\Psi (a) )^{1-\zeta }}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\times \right.\\ &\qquad(\Psi (\sigma)-\Psi (a) )^{\zeta-1 }(\Psi (\sigma)-\Psi (a) )^{1-\zeta }f(\sigma,y(\sigma)) \,d\sigma\Big\|\\ &\leq\frac{y_a}{\Gamma(\zeta)}+\frac{(\Psi (a+x)-\Psi (a) )^{1-\zeta }}{\Gamma ( \eta ) }\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times \\ &\qquad(\Psi (\sigma)-\Psi (a) )^{\zeta-1 }\underset{\varsigma \in [a,\sigma]}{\max}\left\|(\Psi (\varsigma)-\Psi (a) )^{1-\zeta }f(\varsigma,y(\varsigma)) \right\|\,d\sigma\\ &\leq\frac{y_a}{\Gamma(\zeta)}+{\left\Vert f \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }}(\Psi (a+x)-\Psi (a) )^{1-\zeta }}\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(a+x)-\Psi(a))^{\zeta-1}\\ &=\frac{y_a}{\Gamma(\zeta)}+\left\Vert f \right\Vert _{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }\,\,\left(\Psi (a+x)-\Psi (a)\right )^{1-\zeta }\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi (a+x)-\Psi (a) )^{\eta+\zeta-1 }\\ &=\frac{y_a}{\Gamma(\zeta)}+\left\Vert f \right\Vert _{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }\,\,\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi (a+x)-\Psi (a) )^{\eta }. \end{align} This shows that $\mathcal{A}(\mathcal{U})$ is uniformly bounded subset of $C_{1-\zeta\,; \,\Psi }\left[ a,a+\chi\right] $ and hence it is point wise bounded also. Next, we prove that $\mathcal{A}(\mathcal{U})$ is equicontinuous. For any $t_1,t_2$ such that $a<t_1\leq t_2 \leq a+\chi$ and any $y\in \mathcal{U},$ we have \begin{align} &\|\mathcal{A}y(t_2)-\mathcal{A}y(t_1)\|\\ &\leq\left\|\frac{y_a}{\Gamma(\zeta)}\left\{\left( \Psi (t_2)-\Psi (a) \right) ^{\zeta -1}-( \Psi (t_1)-\Psi (a) ) ^{\zeta -1}\right\} \right\|\\ & \,\,+\left\|\frac{1}{\Gamma(\eta)}\int_{a}^{t_2}\mathrm{Q}^{\eta\,;\,\psi}(t_2,\sigma)\left\|f(\sigma,y(\sigma))\right\|\,d\sigma-\frac{1}{\Gamma(\eta)}\int_{a}^{t_1}\mathrm{Q}^{\eta\,;\,\psi}(t_1,\sigma)\left\|f(\sigma,y(\sigma))\right\|\,d\sigma\right\|\\ &\leq\left\|\frac{y_a}{\Gamma(\zeta)}\left\{\left( \Psi (t_2)-\Psi (a) \right) ^{\zeta -1}-( \Psi (t_1)-\Psi (a) ) ^{\zeta -1}\right\} \right\|\\ &\qquad+\left\|\frac{1}{\Gamma(\eta)}\int_{a}^{t_2}\mathrm{Q}^{\eta\,;\,\psi}(t_2,\sigma)(\Psi (\sigma) -\Psi (a)) ^{\zeta-1}\underset{\varsigma\in [a,\sigma]}{\max}\left\|( \Psi (\varsigma) -\Psi (a)) ^{1-\zeta}f(\varsigma,y(\varsigma))\right\|\,d\sigma\right.\\ &\qquad\left.-\frac{1}{\Gamma(\eta)}\int_{a}^{t_1}\mathrm{Q}^{\eta\,;\,\psi}(t_1,\sigma)(\Psi (\sigma) -\Psi (a)) ^{\zeta-1} \underset{\varsigma\in [a,\sigma]}{\max}\left\|( \Psi (\varsigma) -\Psi (a)) ^{1-\zeta}f(\varsigma,y(\varsigma))\right\|\,d\sigma\right\|\\ &\leq\left\|\frac{y_a}{\Gamma(\zeta)}\left\{\left( \Psi (t_2)-\Psi (a) \right) ^{\zeta -1}-( \Psi (t_1)-\Psi (a) ) ^{\zeta -1}\right\} \right\|\\ &\qquad+\left\Vert f \right\Vert_{C_{1-\zeta \,; \,\Psi }[a,a+\chi]}\left\|\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t_2)-\Psi(a))^{\zeta-1}-\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t_1)-\Psi(a))^{\zeta-1}\right\|\\ &=\frac{y_a}{\Gamma(\zeta)}\left\|\left( \Psi (t_2)-\Psi (a) \right) ^{\zeta -1}-( \Psi (t_1)-\Psi (a) ) ^{\zeta -1} \right\|\\ &\qquad+\left\Vert f \right\Vert_{C_{1-\zeta \,; \,\Psi }[a,a+\chi]}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left\|(\Psi (t_2)-\Psi (a) )^{\eta+\zeta-1 }-(\Psi (t_1)-\Psi (a) )^{\eta+\zeta-1 }\right\|. \end{align} Observe that, the right hand part in the preceding inequality is free from $y$. Thus, using the continuity of $\Psi$, $\|t_2-t_1\|\rightarrow 0 $ implies that $\|\mathcal{A}y(t_2)-\mathcal{A}y(t_1)\|\rightarrow 0$. This proves that $\mathcal{A}(\mathcal{U})$ is equicontinuous. In the view of Arzela-Ascoli Theorem \cite{Diethelm1}, it follows that $\mathcal{A}(\mathcal{U})$ is relatively compact. Therefore, by Schauder's fixed point Theorem \ref{Schauder}, operator $\mathcal{A}$ has at least one fixed point $y \in C_{1-\zeta\,; \,\Psi }\left[ a,a+\chi\right] $ which is the solution of IVP \eqref{eq1}--\eqref{eq2}. \end{proof} In the next theorem, using Lipschitz condition on $f$ and the Weissinger theorem, we establish another existence result for the IVP \eqref{eq1}--\eqref{eq2}, which gives in addition the uniqueness of solution also. \begin{theorem}[Uniqueness] \label{th4.1} Let $\zeta=\eta+\nu(1-\eta),0<\eta<1$, $0\leq\nu\leq 1$ and $$R_{0}=\left\{(t,y):a\leq t\leq a+\xi,\left\|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\right\|\leq k \right\},~\xi>0, ~k>0.$$ where, $\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta-1}}{\Gamma(\zeta)}$. Let $f:R_0\rightarrow\mbox{\Bbb R}$ be a function such that $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,; \,\Psi} (\Lambda,\,\mbox{\Bbb R})$ for every $y\in C_{1-\zeta\,; \,\Psi}(\Lambda,\,\mbox{\Bbb R})$ and satisfies the Lipschitz condition $$ \left \| f(t,u)-f(t,v)\right\|\leq L\|u-v\|,\,\,u,v\in \mbox{\Bbb R}, ~L>0. $$ Let $\Psi\in C^1(\Lambda,\,\mbox{\Bbb R})$ be an increasing bijective function with $\Psi'(t)\neq 0 $ for all $t\in \Lambda$. Then, the IVP \eqref{eq1}-\eqref{eq2} has a unique solution $y$ in the weighted space $C_{1-\zeta\,; \,\Psi} [a,a+\chi]$, where $$ \chi=\min\left\{\xi,~\Psi^{-1}\left[\Psi(a)+\left(\frac{k\, \Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\Vert f\right\Vert _{C_{1-\zeta \,; \,\Psi }\left[ a,a+\xi\right]}}\right)^\frac{1}{\eta}\right]-a\right\}. $$ \end{theorem} \begin{proof} Consider the operator $\mathcal{A}$ used in Theorem \ref{th3.1}. It is as of now demonstrated that $\mathcal{A}:\mathcal{U}\rightarrow \mathcal{U}$ is continuous on closed, convex and bounded subset $\mathcal{U}$ of Banach space $C_{1-\zeta\,; \,\Psi }\left[ a,a+\chi\right]$. We now prove that, for every $n\in \mbox{\Bbb N}$ and every $x$ such that $a<x\leq \chi$, we have \begin{equation}\label{e6} \left\Vert \mathcal{A}^{n}y -\mathcal{A}^{n}z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }}\leq\frac{\Gamma(\zeta)}{\Gamma(n \eta+\zeta)}\left[L\left(\Psi (a+x)-\Psi (a) \right)^{\eta } \right]^n\left\Vert y -z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right]}}. \end{equation} We provide the proof of above inequality by using mathematical induction. By using Lipschitz condition on $f$, for any $x \in (a,\chi]$, we have \begin{align} &\left\Vert \mathcal{A}y -\mathcal{A}z\right\Vert_{{C_{1-\zeta\,; \,\Psi}\left[a,a+x\right]}}\\ &\leq(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\left\|f(\sigma,y(\sigma))-f(\sigma,z(\sigma))\right\| \,d\sigma\\ &\leq L(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times \\ &\qquad(\Psi (\sigma) -\Psi (a)) ^{\zeta-1}(\Psi (\sigma) -\Psi (a)) ^{1-\zeta}\left\|y(\sigma)-z(\sigma)\right\| \,d\sigma\\ &\leq L(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times \\ &\qquad(\Psi (\sigma) -\Psi (a)) ^{\zeta-1}\underset{\varsigma\in \left[ a,\sigma\right] }{\max }\left\|(\Psi (\varsigma) -\Psi (a)) ^{1-\zeta}[y(\varsigma)-z(\varsigma)]\right\| \,d\sigma\\ &\leq L\left\Vert y-z \right\Vert_{C_{1-\zeta \,; \,\Psi }\left[ a,a+x\right]}(\Psi (a+x)-\Psi (a) )^{1-\zeta }\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(a+x)-\Psi(a))^{\zeta-1}\\ &=\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left[L(\Psi (a+x)-\Psi (a) )^{\eta }\right]\left\Vert y-z \right\Vert_{C_{1-\zeta \,; \,\Psi }\left[ a,a+x\right]}. \end{align} Hence, the inequality \eqref{e6} is true for $n=1$. Let us assume that, it is true for $n=k-1$. Now, we prove that inequality \eqref{e6} is also true for $n=k$. Then, we have \begin{align} &\left\Vert \mathcal{A}^{k}y -\mathcal{A}^{k}z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }}\\ &=\underset{t\in \left[ a,a+x\right] }{\max }\left\|(\Psi (t)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma) \left[f\left(\sigma,\mathcal{A}^{k-1}y(\sigma)\right)-f\left(\sigma,\mathcal{A}^{k-1}z(\sigma)\right)\right] \,d\sigma\right\|\\ &\leq L\,\,(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma) \left\|\mathcal{A}^{k-1}y(\sigma)-\mathcal{A}^{k-1}z(\sigma)\right\|\,d\sigma\\ &\leq L\,\,(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times \\ & \qquad(\Psi (\sigma) -\Psi (a)) ^{\zeta-1}\left\Vert \mathcal{A}^{k-1}y-\mathcal{A}^{k-1}z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,\sigma\right] }} \,d\sigma\\ &\leq L\,\,(\Psi (a+x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)(\Psi (\sigma) -\Psi (a)) ^{\zeta-1}\times \\ &\qquad \frac{\Gamma(\zeta)}{\Gamma((k-1) \eta+\zeta)}\left[L\left(\Psi (\sigma)-\Psi (a)\right)^{\eta } \right]^{k-1}\left\Vert y -z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,\sigma\right] }}\,d\sigma\\ &\leq L^k\frac{\Gamma(\zeta)}{\Gamma((k-1) \eta+\zeta)}\,\,(\Psi (a+x)-\Psi (a) )^{1-\zeta }\left\Vert y -z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }} \times\\ &\qquad \mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(a+x)-\Psi(a))^{\eta(k-1)+\zeta-1}\\ &= L^k\frac{\Gamma(\zeta)}{\Gamma((k-1) \eta+\zeta)}\,\,(\Psi (a+x)-\Psi (a) )^{1-\zeta }\left\Vert y -z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }} \times\\ &\qquad\frac{\Gamma((k-1) \eta+\zeta)}{\Gamma(k\eta+\zeta)}(\Psi (a+x)-\Psi (a) )^{k \eta+\zeta-1}\\ &= \frac{\Gamma(\zeta)}{\Gamma(k \eta+\zeta)}\left[L(\Psi (a+x)-\Psi (a) )^\eta\right]^k\left\Vert y -z \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,a+x\right] }}. \end{align} Hence, the inequality \eqref{e6} is true for $n=k$. By mathematical induction the proof of the inequality \eqref{e6} is concluded. Taking $x \to \chi$, we obtain \begin{equation} \left\Vert \mathcal{A}^{n}y -\mathcal{A}^{n}z \right\Vert _{_{C_{1-\zeta; \,\psi }\left[ a,a+\chi\right] }}\leq\frac{\Gamma(\zeta)}{\Gamma(n \eta+\zeta)}\left[L\left(\psi (a+\chi)-\psi (a)\right)^{\eta } \right]^n\left\Vert y -z \right\Vert _{_{C_{1-\zeta; \,\psi }\left[ a,a+\chi\right]}}. \end{equation} Note that, the operator $\mathcal{A}$ satisfy all the conditions of the Weissinger's theorem \ref{Weissinger}, with $$ \eta_n=\frac{\left[L\left(\Psi (a+\chi)-\Psi (a)\right)^{\eta } \right]^n}{\Gamma(n \eta+\zeta)}. $$ Further, by definition of two parameter Mittag--Leffler function, we have $$\sum_{n=0}^{\infty} \,\eta_n= \sum_{n=0}^{\infty} \frac{\left[L(\Psi (a+\chi)-\Psi (a) )^{\eta } \right]^n}{\Gamma(n \eta+\zeta)}=\mathcal{E}_{\eta,\,\zeta}\,\left(L(\Psi (a+\chi)-\Psi (a) )^{\eta }\right),$$ which is convergent series. Thus, by using Weissinger's fixed point theorem \ref{Weissinger}, operator $\mathcal{A}$ has a unique fixed point $y$ in $C_{1-\zeta\,; \,\Psi }\left[ a,a+\chi\right] $, which is a unique solution of the IVP \eqref{eq1}-\eqref{eq2}. \end{proof} \begin{rem} \begin{enumerate} \item The results obtained by Diethelem and Ford {\rm \cite{Diethelm1}} can be regarded as the particular cases of {\rm Theorem \ref{th3.1}} and {\rm Theorem \ref{th4.1} }and can be derived by putting $\nu=1,~a=0 $ and $\Psi(t)=t, ~t\in\Lambda$. \item We have investigated in {\rm Theorem \ref{th3.1}} and {\rm Theorem \ref{th4.1}}, the existence and uniqueness of the Cauchy problem {\rm (\ref{eq1})}-{\rm(\ref{eq2})} involving the $\Psi$-Hilfer fractional derivative. One of the fundamental properties of the $\Psi$-Hilfer fractional derivative is the wide class of fractional derivatives that contain it as particular cases. Thus, in this sense, the respective results obtained by {\rm Theorem \ref{th3.1}} and {Theorem \ref{th4.1}}, are also valid for these fractional derivatives. For examples, some particular cases in the items as follows: \begin{enumerate} \item If we take $\nu\rightarrow 1$ on both sides of {\rm(\ref{eq1})}-{\rm(\ref{eq2})},it reduces to the Cauchy problem with $\Psi$-Caputo fractional derivative and consequently, the {\rm Theorem \ref{th3.1}} and {\rm Theorem \ref{th4.1}}, there are true; \item If we take $\nu\rightarrow 0$ on both sides of the {\rm(\ref{eq1})}-{\rm(\ref{eq2})}, we have the Cauchy problem with $\Psi$-Riemann-Liouville fractional derivative and consequently, the {\rm Theorem \ref{th3.1}} and {\rm Theorem \ref{th4.1}}, there are true; \item Choose $\Psi(t)=t^{\rho}$ and take $\nu\rightarrow 1$ on both sides of {\rm(\ref{eq1})}-{\rm(\ref{eq2})}, we have the Cauchy problem involving the Caputo-Katugampola fractional derivative and consequently, the {\rm Theorem \ref{th3.1}} and {\rm Theorem \ref{th4.1}}, there are true; \item In cases where they involve the fractional derivatives of Hadamard, Caputo-Hadamard, Hilfer-Hadamard or any other that is related to Hadamard's fractional derivative, we have to impose the condition on the parameter $a>0$, since in these fractional derivatives they involve the function $\ln t$ and is not defined when $t=a=0$. \end{enumerate} \end{enumerate} \end{rem} Next, we prove the continuous dependence of solution for the Cauchy type problem \eqref{eq1}-\eqref{eq2} via Weissinger's theorem. \begin{theorem} [Continuous Dependence]\label{th5.1} Let $\zeta =\eta +\nu \left( 1-\eta \right) $ where, $0<\eta <1$ and $0\leq \nu \leq 1.$ Let $f:\left[ a,a +\chi \right] \times \mathbb{R}\rightarrow \mathbb{R} $ be a function such that $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,; \,\Psi} [a,a+\chi]$ for every $y\in C_{1-\zeta\,; \,\Psi} [a,a+\chi]$ and satisfies the Lipschitz condition $$ \left \| f(t,u)-f(t,v)\right\|\leq L\|u-v\|,\,\,u,v\in \mbox{\Bbb R}, ~L>0. $$ Let $y(t)$ and $z(t)$ be the solutions of, the IVPs, \begin{align} {}^H\mathcal{D}_{a+}^{\eta,\,\nu; \,\Psi}y(t)=f(t,y(t) ), \quad \mathcal{I}_{a+}^{1-\zeta\,; \,\Psi}y(a)=y_a \label{e13} \end{align} and \begin{align} {}^H\mathcal{D}_{a+}^{\eta,\,\nu; \,\Psi}z(t)=f(t,z(t) ), \quad \mathcal{I}_{a+}^{1-\zeta\,; \,\Psi}z(a)=z_a \label{e14} \end{align} respectively. Then, \begin{align}\label{cd} \left\Vert y-z\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\leq \left\{1+\Gamma(\zeta)\,\mathcal{E}_{\eta,\,\zeta}\left(L\left(\Psi(a+\chi)-\Psi(a)\right)^\eta \right)\right\} \left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}, \end{align} where, $\tilde{y}_{a}(t)=\dfrac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta -1}}{\Gamma ( \zeta ) }y_{a}$ and $\tilde{z}_{a}(t)=\dfrac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta -1}}{\Gamma ( \zeta ) }z_{a}$. \end{theorem} \begin{proof} Consider the sequences $\{\mathcal{A}^{m}\tilde{y}_{a}\}$ and $\{\mathcal{A}^{m}\tilde{z}_{a}\}$ defined by \begin{align} & \mathcal{A}^{0}\tilde{y}_{a}(t)=\tilde{y}_{a}(t),\\ & \mathcal{A}^{m}\tilde{y}_{a}(t)=\tilde{y}_{a}(t)+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f\left(\sigma,\mathcal{A}^{m-1}\tilde{y}_{a}(\sigma)\right) \,d\sigma, ~(m=1,2,\cdots) \end{align} and \begin{align} & \mathcal{A}^{0}\tilde{z}_{a}(t)=\tilde{z}_{a}(t),\\ & \mathcal{A}^{m}\tilde{z}_{a}(t)=\tilde{z}_{a}(t)+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f\left(\sigma,\mathcal{A}^{m-1}\tilde{z}_{a}(\sigma)\right) \,d\sigma, ~(m=1,2,\cdots)\\respectively. \end{align} Then, for each $x$ with $a<x\leq \chi$, \begin{align} &\left\Vert \mathcal{A}^{m}\tilde{y}_{a}-\mathcal{A}^{m}\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}\\ &=\underset{t\in\left[a,a+x\right]}{\max}\left\|(\Psi(t)-\Psi(a))^{1-\zeta} \Big\{\tilde{y}_{a}(t)-\tilde{z}_{a}(t)\right.\\ &\left.\left.\qquad +\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\left[f\left(\sigma,\mathcal{A}^{m-1}\tilde{y}_{a}(\sigma)\right)-f\left(\sigma,\mathcal{A}^{m-1}\tilde{z}_{a}(\sigma)\right)\right] \,d\sigma \right\}\right\|\\ &\leq \left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}+(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{L}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times\\ &\qquad( \Psi (\sigma) -\Psi (a))^{\zeta-1}\underset{\varsigma\in\left[a,\sigma\right]}{\max}\left\|(\Psi (\varsigma) -\Psi (a))^{1-\zeta}\left\{\mathcal{A}^{m-1}\tilde{y}_{a}(\varsigma)-\mathcal{A}^{m-1}\tilde{z}_{a}(\varsigma) \right\} \right\|\,d\sigma\\ &\leq \left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}+L(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times\\ &\qquad( \Psi (\sigma) -\Psi (a))^{\zeta-1}\left\Vert \mathcal{A}^{m-1}\tilde{y}_{a}-\mathcal{A}^{m-1}\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,\sigma]}\,d\sigma. \end{align} Using the inequality \eqref{e6}, \begin{align} &\left\Vert \mathcal{A}^{m}\tilde{y}_{a}-\mathcal{A}^{m}\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}\\ &\leq \left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}+(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{L}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times\\ &\qquad( \Psi (\sigma) -\Psi (a))^{\zeta-1}\frac{\Gamma(\zeta)}{\Gamma((m-1) \eta+\zeta)}\left[L\left(\Psi (\sigma)-\Psi (a) \right)^{\eta } \right]^{m-1}\left\Vert \tilde{y}_a -\tilde{z}_a \right\Vert _{_{C_{1-\zeta\,; \,\Psi }\left[ a,\sigma\right] }}\,d\sigma\\ &= \left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]} +L^{m}\frac{\Gamma(\zeta)}{\Gamma((m-1) \eta+\zeta)}\left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}\times\\ &\qquad(\Psi(a+x)-\Psi(a))^{1-\zeta}\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}( \Psi (a+x) -\Psi (a)) ^{(m-1)\eta+\zeta-1}\\ &= \left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}\left\{1+L^{m}\frac{\Gamma(\zeta)}{\Gamma((m-1) \eta+\zeta)}(\Psi(a+x)-\Psi(a))^{1-\zeta}\times \right.\\ &\left.\qquad\frac{\Gamma((m-1) \eta+\zeta)}{\Gamma(m \eta+\zeta)} (\Psi(a+x)-\Psi(a))^{m\eta+\zeta-1}\right\}\\ &=\left\{1+{\Gamma(\zeta)}\frac{\left[L(\Psi(a+x)-\Psi(a))^{\eta}\right]^m}{\Gamma(m \eta+\zeta)}\right\}\left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}\\ & \leq \left\{1+{\Gamma(\zeta)}\sum_{k=0}^{m}\frac{\left[L(\Psi(a+x)-\Psi(a))^{\eta}\right]^m}{\Gamma(m \eta+\zeta)}\right\}\left\Vert \tilde{y}_{a}-\tilde{z}_{a}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+x]}. \end{align} Taking limit as $ m\to \infty $ and $ x\to \chi $ and utilizing Theorem \ref{Weissinger}, we get the desired inequality \eqref{cd}. \end{proof} \begin{rem} In the above theorem, in particular if $y_a=z_a$, then we get uniqueness of solution. \end{rem} \section{Picard's Successive Approximations: Nonlinear Case} In this section, we define the Picard's type successive approximations and prove that it converges to a unique solution of nonlinear Cauchy problem \eqref{eq1}-\eqref{eq2}. Further, we obtain an estimation for the error bound. \begin{theorem} \label{th4.2} Let $\zeta=\eta+\nu(1-\eta), 0<\eta<1$ and $0\leq\nu\leq 1$. Define $$R_{0}=\left\{(t,y):a\leq t\leq a+\xi,\left\|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\right\|\leq k \right\},~\xi>0, ~k>0.$$ where, $\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta-1}}{\Gamma(\zeta)}$. Let $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,; \,\Psi} [a,a+\chi]$ for every \,$y\in C_{1-\zeta\,; \,\Psi} [a,a+\chi]$. Let $\Psi\in C^1\left([a,a+\chi],~\mbox{\Bbb R} \right)$ be an increasing bijective function with $\Psi'(t)\neq 0$, for all $\, t \in [a,a+\chi]$, where $$ \chi=\min\left\{\xi,~\Psi^{-1}\left[\Psi(a)+\left(\frac{k\, \Gamma(\eta+\zeta)}{\Gamma(\zeta)M}\right)^\frac{1}{\eta}\right]-a\right\} $$ and $M$ is constant such that $$ \|\left( \Psi (t)-\Psi (a) \right) ^{1-\zeta}f(t)\|\leq M,~ \forall \,t \in [a,a+\chi]. $$ Then the successive approximations defined by \begin{align}\label{e7} y_0(t)&=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\notag \\ y_n(t)&=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a} +\mathrm{I}_{a+}^{\eta\,; \,\Psi}f\left(t,y_{n-1}(t)\right), n=1,2,\cdots. \end{align} satisfies the following conditions: \begin{enumerate} \item [{\rm(a)}]$y_{n}\in C_{1-\zeta;\, \Psi}[a,a+\chi], ~n=0,1,2,\cdots$ ; \item [{\rm(b)}] $(\Psi(t)-\Psi(a))^{1-\zeta}\|y_{n}(t)-y_0(t)\|\leq \dfrac{M\, \Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta}, ~t \in [a, a+\chi], ~n=0,1,2,\cdots.$ \end{enumerate} \end{theorem} \begin{proof} (a) Since $y_0(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}$, we have $( \Psi (t)-\Psi (a) ) ^{1-\zeta}y_0(t)=\dfrac{y_a}{\Gamma(\zeta)} \in C[a,a+\chi]$ and hence $y_0\in C_{1-\zeta\,; \,\Psi}[a,a+\chi]$. Further, by Lemma \ref{lema1}, $\Psi$--Riemann fractional integral operator $\mathcal{I}_{a+}^{\eta \,; \,\Psi }\left( \cdot \right) $ maps $C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right] $ to $C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]$ and hence $\mathcal{I}_{a +}^{\eta;\, \Psi} f(\cdot,y_{n-1}(\cdot)) \in C_{{1-\zeta};\, \Psi}[a,a+\chi]$ for any $y_{n-1}\in C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi\right]$ for $n=1,2,\cdots$. This proves, $y_n\in C_{1-\zeta\,; \,\Psi}[a,a+\chi]~ \forall \, n\in \mbox{\Bbb N}$.\\ (b) For any $t \in [a, a+\chi] $ and $n \in \mbox{\Bbb N} $, we have \begin{align} \|y_{n}(t)-y_{0}(t)\|&=\left\|\frac{1}{\Gamma(\eta)}\int_{a}^{t} \mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma, y_{n-1}(\sigma))\,d\sigma\right\|\\ &\leq \frac{1}{\Gamma(\eta)}\int_{a}^{t} \mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\times\\ &\qquad\left\|(\Psi(\sigma)-\Psi(a))^{1-\zeta}f(\sigma, y_{n-1}(\sigma))\right\|\,d\sigma\\ &\leq M\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t)-\Psi(a))^{\zeta-1}\\ &= M \frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta+\zeta-1}. \end{align} Therefore, \begin{equation} (\Psi(t)-\Psi(a))^{1-\zeta}\|y_{n}(t)-y_0(t)\|\leq \frac{M\, \Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta}, ~t \in [a, a+\chi], ~n=0,1,2,\cdots. \end{equation} \end{proof} \begin{theorem}\label{theorem4.2} Let $\zeta=\eta+\nu(1-\eta), 0<\eta<1$ and $0\leq\nu\leq 1$. Define $$R_{0}=\left\{(t,y):a\leq t\leq a+\xi,\left\|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\right\|\leq k \right\},~\xi>0, ~k>0.$$ where, $\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta-1}}{\Gamma(\zeta)}$. Let $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,; \,\Psi} [a,a+\chi]$ for every $y\in C_{1-\zeta\,; \,\Psi} [a,a+\chi]$ and satisfies the Lipschitz condition $$ \left \| f(t,u)-f(t,v)\right\|\leq L\|u-v\|,\,\,u,v\in \mbox{\Bbb R}, ~L>0. $$ Let $\Psi\in C^1\left([a,a+\chi],~\mbox{\Bbb R} \right)$ be an increasing bijective function with $\Psi'(t)\neq 0$,for all $\, t \in [a,a+\chi]$, where $$ \chi=\min\left\{\xi,~\Psi^{-1}\left[\Psi(a)+\left(\frac{k\, \Gamma(\eta+\zeta)}{\Gamma(\zeta)M}\right)^\frac{1}{\eta}\right]-a\right\} $$ and $M$ is constant such that $$ \|\left( \Psi (t)-\Psi (a) \right) ^{1-\zeta}f(t)\|\leq M,~ \forall \,t \in [a,a+\chi]. $$ Then the successive approximations defined by \eqref{e7} converges to the unique solution $y$ of the Cauchy type problem \eqref{eq1}-\eqref{eq2} in the weighted space $C_{1-\zeta\,; \,\Psi} [a,a+\chi]$. \end{theorem} \begin{proof} We give the proof of the theorem in the following steps. \noindent \\ {{Step 1}:} In the above theorem we have already proved that $y_n\in C_{1-\zeta\,; \,\Psi}[a,a+\chi]$,~$\forall \, n\in \mbox{\Bbb N}$. \noindent \\ {{Step 2}:} We show that the sequence $\{y_n\}$ converges to a function $y\in C_{1-\zeta\,; \,\Psi}[a,a+\chi]$ with respect to the norm $\\|\cdot\\|_{C_{1-\zeta\,; \,\Psi}[a,a+\chi]}$. Observe that $y_n$ can be written as \begin{equation}\label{e8} y_n=y_0+\sum_{k=1}^{n}(y_k - y_{k-1}) \end{equation} which is partial sum of series \begin{equation}\label{e9} y_0+\sum_{k=1}^{\infty}(y_k - y_{k-1}). \end{equation} Therefore, to show that sequence $\{y_n\}$ is convergent, we prove that the series \eqref{e9} is convergent. For any $x \in [a,a+\chi]$, consider the space $C_{1-\zeta\,; \,\Psi}[a,x]$ having norm \begin{equation}\label{space1} \left\Vert h\right\Vert _{C_{1-\zeta\,; \,\Psi }\left[ a,x\right] }=\underset{t\in \left[ a,x\right] }{\max }\left\vert \left( \Psi \left( t\right) -\Psi \left( a\right) \right) ^{1-\zeta }h\left( t\right) \right\vert. \end{equation} By mathematical induction, we now prove , for each $x\in [a,a+\chi]$ and $y_j\in C_{1-\zeta\,; \,\Psi}[a,x]$, \begin{equation}\label{e10} \left\Vert y_{n+1}-y_{n} \right\Vert_{_{C_{1-\zeta\,; \,\Psi }\left[ a,x\right] }}\leq \frac{ M \Gamma(\zeta)} {L}\frac{\left[L(\Psi(x)-\Psi(a))^\eta\right]^{n+1}}{\Gamma((n+1)\eta+\zeta)}, ~n\in \mbox{\Bbb N}. \end{equation} Using Lemma \ref{lema3}, we obtain \begin{align} &\left\Vert y_{1}-y_{0}\right\Vert_{_{C_{1-\zeta\,; \,\Psi }\left[ a,x\right] }}\\ &=\underset{t \in [a,x]}{\max}\left\|(\Psi (t)-\Psi (a) )^{1-\zeta }\left\{ y_{0}(t)+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma,y_{0}(\sigma)) \,d\sigma-y_{0}(t)\right\}\right\|\\ &=\underset{t \in [a,x]}{\max}\left\|(\Psi (t)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma,y_{0}(\sigma)) \,d\sigma\right\|\\ &=\left\Vert \mathcal{I}_{a+}^{\eta\,; \,\Psi}f\left(\cdot,y_{0}(\cdot)\right)\right\Vert_{_{C_{1-\zeta\,; \,\Psi }\left[ a,x\right] }}\\ &\leq M\,\,\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left(\Psi(t)-\Psi(a)\right)^{\eta}\\ &\leq M\,\,\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left(\Psi(x)-\Psi(a)\right)^{\eta}, \end{align} which is the inequality \eqref{e10} for $n=0$. Now, assume that the inequality \eqref{e10} is hold for $n=k$. We prove it is also hold for $n=k+1$. In fact, \begin{align} &\left\Vert y_{k+2}-y_{k+1}\right\Vert_{C_{1-\zeta\,; \,\Psi }\left[ a,x\right] }\\ &\leq L\underset{t \in [a,x]}{\max}(\Psi (t)-\Psi (a) )^{1-\zeta } \frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\times \\ &\qquad( \Psi (\sigma) -\Psi (a)) ^{\zeta-1}( \Psi (\sigma) -\Psi (a)) ^{1-\zeta}\left\| y_{k+1}(\sigma)-y_{k}(\sigma)\right\| \,d\sigma\\ &\leq L(\Psi (x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{x}\mathrm{Q}^{\eta:\Psi}(x,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\zeta-1}\left\Vert y_{k+1}-y_{k}\right\Vert_{{C_{1-\zeta\,; \,\Psi }\left[ a,\sigma\right] }}\,d\sigma\\ &\leq L(\Psi (x)-\Psi (a) )^{1-\zeta }\frac{1}{\Gamma ( \eta ) }\int_{a}^{x}\mathrm{Q}^{\eta:\Psi}(x,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\zeta-1}\times \\&\qquad\frac{ M \Gamma(\zeta)} {L}\frac{\left[L(\Psi(\sigma)-\Psi(a))^\eta\right]^{k+1}}{\Gamma((k+1)\eta+\zeta)}\,d\sigma\\ &=M\,\,L^{k+1}\,\,\frac{\Gamma(\zeta)}{\Gamma((k+1)\eta+\zeta)}(\Psi(x)-\Psi(a))^{1-\zeta}\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}( \Psi (x) -\Psi (a)) ^{(k+1)\eta+\zeta-1}\\ &=M\,\,L^{k+1}\,\,\frac{\Gamma(\zeta)}{\Gamma((k+1)\eta+\zeta)}(\Psi(x)-\Psi(a))^{1-\zeta}\times\frac{\Gamma((k+1)\eta+\zeta)}{\Gamma((k+2)\eta+\zeta)}(\Psi(x)-\Psi(a))^{(k+2)\eta+\zeta-1}\\ &=\frac{M\,\,\Gamma(\zeta)}{L}\frac{\left( L\left(\Psi(x)-\Psi(a)\right)^{\eta}\right) ^{k+2}}{\Gamma((k+2)\eta+\zeta)}. \end{align} Thus, inequality \eqref{e10} is true for $n=k+1$. Hence, by using principle of mathematical induction the inequality \eqref{e10} holds for each $n\in \mbox{\Bbb N}$ and $x \in [a,a+\chi]$. Taking $x \to a+\chi$ in \eqref{e10}, we get, $$ \left\Vert y_{k}-y_{k-1}\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\leq\frac{ M \Gamma(\zeta)} {L}\frac{\left[L(\Psi(a+\chi)-\Psi(a))^\eta\right]^{k}}{\Gamma(k\eta+\zeta)}. $$ Therefore, \begin{align} \sum_{k=1}^{\infty}\left\Vert y_{k}-y_{k-1}\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]} &\leq\frac{ M \Gamma(\zeta)} {L}\sum_{k=1}^{\infty}\frac{\left[L(\Psi(a+\chi)-\Psi(a))^\eta\right]^{k}}{\Gamma(k\eta+\zeta)}\\ &\leq\frac{ M \Gamma(\zeta)} {L}\left[\mathcal{E}_{\eta,\,\zeta}\left(L\left(\Psi(a+\chi)-\Psi(a)\right)^\eta\right)-\frac{1}{\Gamma(\zeta)}\right]. \end{align} This proves the series $y_0+\sum_{k=1}^{\infty}\left(y_k - y_{k-1}\right)$ is convergent in the space $C_{1-\zeta\,; \,\Psi}[a,a+\chi]$. Let us suppose that $$\tilde{y}=y_0+\sum_{k=1}^{\infty}(y_k - y_{k-1}).$$ Therefore, \begin{equation}\label{e11} \left\Vert y_{n}-\tilde{y}\right\Vert_{C_{1-\zeta\,; \,\Psi}[a,a+\chi]}\rightarrow 0~ \text{as}\,\,n\rightarrow \infty. \end{equation} \noindent {{Step 3}:} $\tilde{y}$ is solution of fractional integral equation \eqref{eq3}. In fact, we have \begin{align} &\left\Vert\mathcal{I}_{a+}^{\eta\,; \,\Psi}f\left(\cdot,y_{n}(\cdot)\right)-\mathcal{I}_{a+}^{\eta\,; \,\Psi}f\left(\cdot,\tilde{y}(\cdot)\right)\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\\ &\leq L\left\Vert y_{n}-\tilde{y}\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\underset{t \in [a,a+\chi]}{\max}(\Psi (t)-\Psi (a) )^{1-\zeta }\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}( \Psi (t) -\Psi (a)) ^{\zeta-1}\\ &\leq L\left\Vert y_{n}-\tilde{y}\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\underset{t \in [a,a+\chi]}{\max}(\Psi (t)-\Psi (a) )^{1-\zeta}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi (t)-\Psi (a) )^{\eta+\zeta-1}\\ &\leq L\,\,\ \frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi (a+\chi)-\Psi (a) )^{\eta} \left\Vert y_{n}-\tilde{y}\right\Vert_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}. \end{align} Using \eqref{e11}, we obtain \begin{equation}\label{e12} \left\Vert\mathcal{I}_{a+}^{\eta\,; \,\Psi}f\left(\cdot,y_{n}(\cdot)\right)-\mathcal{I}_{a+}^{\eta\,; \,\Psi}f\left(\cdot,\tilde{y}(\cdot)\right)\right\Vert_{C_{1-\zeta\,;\,\Psi}[a, a+\chi]}\rightarrow 0\,\,\, \text{as}\,\, n\rightarrow \infty. \end{equation} Therefore, taking limit as $n \rightarrow \infty$ in \eqref{e7}, we get \begin{align} \tilde{y}(t)&=y_0(t)+\mathcal{I}_{a+}^{\eta\,; \,\Psi}f\left(t,\tilde{y}(t)\right)\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_a+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma,\tilde{y}(\sigma)) \,d\sigma . \end{align} This proves, $\tilde{y}\in C_{1-\zeta\,; \,\Psi}[a, a+\chi]$ is solution of \eqref{eq3}. \noindent {{Step 4}:} Uniqueness of the solution.\\ Let $y(t)$\,\,and\,\,\,${y}^\ast(t)$ be any two solutions of the IVP \eqref{eq1}--\eqref{eq2} and consider the function defined by $z(t)=\left\|y(t)-{y}^\ast(t)\right\| $. Then, we get \begin{align} z(t)&=\left\|y(t)-{y}^\ast(t)\right\|\\ &=\left\|\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\left[f(\sigma,y(\sigma))-f(\sigma,{y}^\ast(\sigma))\right] \,d\sigma\right\|\\ &\leq\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\left\| f(\sigma,y(\sigma))-f(\sigma,{y}^\ast(\sigma)) \right\|\,d\sigma\\ &\leq \frac{L}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\left\| y(\sigma)-{y}^\ast(\sigma) \right\|\,d\sigma\\ &\leq \frac{L}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)z(\sigma)\,d\sigma. \end{align} Using Gronwall inequality given in the Theorem \ref{lema4}, we get $$ z(t)\leq 0\times \mathcal{E}_{\eta}[L \left(\Psi(t)-\Psi(a)\right)^{\eta}].$$ Therefore $z(t)=0$ and we have $ y(t)={y}^\ast(t)$. \end{proof} \begin{rem} For $\eta=1,\, \nu=1$ and $\Psi(t)=t$ the above theorems includes the results of Coddington {\rm[\cite{Coddington},Chapter 5]} for the ordinary differential equations. \end{rem} \begin{theorem} Let $\{y_n\}$ be the sequence of Picard's successive approximation defined by \eqref{e7} and $y$ is the solution of the IVP \eqref{eq1}-\eqref{eq2}. Then the error $ y -y_{n}$ satisfies the condition \begin{equation}\label{eqn} \\| y -y_{n}\\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\leq\frac{M\Gamma(\zeta)}{L}\left( \mathcal{E}_{\eta ,\, \zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\sum_{k=0}^{n}\frac{(L(\Psi(a+\chi)-\Psi(a))^{\eta})^{k}}{\Gamma(k\eta+\zeta)} \right). \end{equation} \end{theorem} \begin{proof} From the proof of the Theorem \ref {theorem4.2}, we have $$y=y_0+\sum_{k=1}^{\infty}(y_{k} - y_{k-1})$$ and $$y_{n}=y_0+\sum_{k=1}^{n}(y_{k} - y_{k-1}).$$ Hence $$ y -y_{n}=\sum_{k=n+1}^{\infty}(y_{k} - y_{k-1}).$$ From the above relations, in view of \eqref{e10}, we have \begin{align} &\\| y -y_{n}\\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\\ &\leq \sum_{k=n+1}^{\infty}\\|y_{k}-y_{k-1}\\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\\ &\leq \sum_{k=n+1}^{\infty} \frac{M \Gamma(\zeta)}{L}\frac{\left( L (\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k }}{\Gamma(k\eta+\zeta)}\\ &\leq\frac{M \Gamma(\zeta)}{L}\left(\sum_{k=0}^{\infty} \frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}-\sum_{k=0}^{n} \frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k }}{\Gamma(k\eta+\zeta)}\right)\\ &\leq \frac{M\Gamma(\zeta)}{L}\left( \mathcal{E}_{\eta ,\, \zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\sum_{k=0}^{n}\frac{(L(\Psi(a+\chi)-\Psi(a))^{\eta})^{k}}{\Gamma(k\eta+\zeta)} \right). \end{align} \end{proof} \begin{rem} From the inequality \eqref{eqn}, it follows that \begin{small} \begin{align} &\lim\limits_{n\rightarrow \infty} \\| y -y_{n}\\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\\ &\leq \frac{M\Gamma(\zeta)}{L}\lim\limits_{n\rightarrow \infty}\left(\sum_{k=0}^{\infty} \frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}-\sum_{k=0}^{n} \frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k }}{\Gamma(k\eta+\zeta)}\right) \\ &= \frac{M\Gamma(\zeta)}{L}\left(\mathcal{E}_{\eta ,\, \zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\sum_{k=0}^{\infty}\frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k }}{\Gamma(k\eta+\zeta)}\right)\\ &= \frac{M\Gamma(\zeta)}{L}\left(\mathcal{E}_{\eta ,\, \zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\mathcal{E}_{\eta,\,\zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)\\ &=0. \end{align} This implies the sequence $\{y_n \}$ of successive approximation converges to the solution $y $ of the problem \eqref{eq1}-\eqref{eq2} as $ n \rightarrow \infty.$ \end{small} \end{rem} \section{Picard's Successive Approximations: Linear Case} Here we derive the representation formula for the solution of linear Cauchy problem with constant coefficient and variable coefficients which extend the results of \cite[Chapter 7]{Gorenflo}. \subsection{Linear Cauchy Type problem with Constant Coefficient} \begin{theorem} \label{th6.1} Let $f\in C_{1-\zeta\,; \,\Psi}\left( \Delta,\,\mbox{\Bbb R}\right) ]$ and $\lambda\in \mbox{\Bbb R}$. Then, the solution of the Cauchy problem for FDE with constant coefficient involving $ \Psi $-Hilfer fractional derivative \eqref{a1}-\eqref{a2} is given by \begin{align} y\left( t\right)&=y_a(\Psi(t)-\Psi(a))^{\zeta-1} \,\mathcal{E}_{\eta,\,\zeta}\left( \lambda(\Psi(t)-\Psi(a))^{\eta}\right) \\ &\qquad+\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\,\mathcal{E}_{\eta,\,\eta}\left( \lambda(\Psi(t)-\Psi(\sigma))^{\eta}\right) \,f(\sigma)\,\,d\sigma\notag \end{align} where,~$\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)=\Psi'(\sigma)(\Psi(t)-\Psi(\sigma))^{\eta-1} $ \end{theorem} \begin{proof} The linear Cauchy problem \eqref{a1}-\eqref{a2} is equivalent to \begin{align}\label{e63} y (t) &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{\lambda}{ \Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y(\sigma)\,d\sigma+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma. \end{align} Employing the method of successive approximation we derive the solution of \eqref{e63}. Consider the sequences $\{y_{m}\}$ defined by \begin{align} y_0(t)&=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\\ y_m(t)&=y_0(t)+\frac{\lambda}{ \Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y_{m-1}(\sigma) \,d\sigma\\ &\qquad+\frac{1}{ \Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma, ~(m=1,2,\cdots). \end{align} By method of induction, we prove that \begin{small} \begin{align}\label{e18} y_m(t)=y_{a}\sum_{k=0}^{m}\frac{\lambda^k( \Psi (t) -\Psi (a)) ^{k\eta+\zeta-1}}{\Gamma ( k\eta+\zeta )}+\int_{a}^{t}\Psi ^{\prime }(\sigma)\sum_{k=0}^{m-1}\frac{\lambda^{k}( \Psi (t) -\Psi (\sigma)) ^{(k+1)\eta-1}}{\Gamma ((k+1)\eta )} f(\sigma)\,d\sigma. \end{align} \end{small} For the case $m=1$, we have \begin{align} y_1(t)&=y_0(t)+\frac{\lambda}{ \Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y_{0}(\sigma)\,d\sigma+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma\\ &=y_0(t)+\frac{\lambda}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\frac{( \Psi (\sigma)-\Psi (a) ) ^{\zeta -1}}{\Gamma ( \zeta ) }y_{a}\, \,d\sigma\\ &\qquad+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma\\ &=y_0(t)+y_{a}\frac{\lambda}{\Gamma ( \zeta ) }\, \mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t)-\Psi(a))^{\zeta-1}+ \mathcal{I}_{a+}^{\eta\,; \,\Psi}f(t)\\ &=y_{a}\,\frac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta -1}}{\Gamma ( \zeta ) }+y_{a} \, \frac{\lambda}{ \Gamma ( \eta +\zeta) } ( \Psi (t) -\Psi (a)) ^{\eta+\zeta-1}\\ &\qquad+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma. \end{align} This is the equation \eqref{e18} for $m=1$. Now, we assume that the equation \eqref{e18} is hold for $m=j$ and prove that it is also true for $m=j+1$. In fact, \begin{small} \begin{align} y_{j+1}(t) &=y_0(t)+\frac{\lambda}{ \Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y_{j}(\sigma) \,d\sigma+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma\\ &=y_0(t)+\frac{\lambda}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma) \left\{ y_{a}\sum_{k=0}^{j}\frac{\lambda^k ( \Psi (\sigma)-\Psi (a) ) ^{k\eta+\zeta -1}}{\Gamma (k\eta+ \zeta ) }\right.\\ &\left.\qquad+\int_{a}^{\sigma}\Psi ^{\prime }(\tau)\sum_{k=0}^{j-1} \frac{\lambda^{k}}{\Gamma((k+1) \eta)} ( \Psi (\sigma)-\Psi(\tau))^{(k+1)\eta-1} f(\tau)d\tau) \right \}\,d\sigma+\mathcal{I}_{a+}^{\eta\,; \,\Psi}f(t)\\ &=y_0(t)+y_{a}\sum_{k=0}^{j}\frac{ \lambda^{k+1} }{\Gamma (k\eta+ \zeta ) }\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}\\ &\qquad +\sum_{k=0}^{j-1}\frac{ \lambda^{k+1} }{\Gamma ((k+1)\eta )} \frac{1}{\Gamma ( \eta )}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma) \times\\ &\qquad \qquad \left\{\int_{a}^{\sigma}\Psi ^{\prime }(\tau)( \Psi (\sigma)-\Psi(\tau))^{(k+1)\eta-1} f(\tau)d\tau \right \}\,d\sigma+\mathcal{I}_{a+}^{\eta\,; \,\Psi}f(t). \end{align} \end{small} Changing the order of integration in the second last term, \begin{small} \begin{align} & y_{j+1}(t) \\ &=y_{a}\frac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta -1}}{\Gamma ( \zeta ) }+y_{a}\sum_{k=0}^{j}\frac{ \lambda^{k+1} }{\Gamma ((k+1)\eta+\zeta)}(\Psi(t)-\Psi(a))^{(k+1)\eta+\zeta-1}\\ &\qquad +\sum_{k=0}^{j-1}\frac{ \lambda^{k+1} }{\Gamma ((k+1)\eta )}\frac{1}{\Gamma ( \eta )}\int_{a}^{t}\Psi ^{\prime }(\tau) \left\{\int_{\tau}^{t}\Psi ^{\prime }(\sigma)( \Psi (\sigma)-\Psi(\tau))^{(k+1)\eta-1}(\Psi (t) -\Psi (\sigma)) ^{\eta-1} f(\sigma)\,d\sigma \right \} f(\tau)d\tau\\ &\qquad+\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma\\ &=y_{a}\sum_{k=0}^{j+1}\frac{ \lambda^{k} }{\Gamma (k\eta+ \zeta)}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}\\ &\qquad +\sum_{k=0}^{j-1}\frac{ \lambda^{k+1} }{\Gamma ((k+1)\eta )}\frac{1}{\Gamma ( \eta )}\int_{a}^{t}\frac{\Gamma(\eta)\Gamma((k+1)\eta)}{\Gamma((k+2)\eta)}( \Psi (t)-\Psi(\tau))^{(k+2)\eta-1} \Psi ^{\prime }(\tau) f(\tau)d\tau\\ &\qquad +\frac{1}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma) \,d\sigma\\ &=y_{a}\sum_{k=0}^{j+1}\frac{ \lambda^{k} }{\Gamma (k\eta+ \zeta)}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}\\ &\qquad+\int_{a}^{t}\left\{\sum_{k=0}^{j-1}\frac{ \lambda^{k+1} }{\Gamma((k+2)\eta)}(\Psi(t)-\Psi(\sigma))^{(k+2)\eta-1} +\frac{( \Psi (t) -\Psi (\sigma)) ^{\eta-1}}{\Gamma(\eta)}\right\} \Psi ^{\prime }(\sigma) f(\sigma) \,d\sigma\\ &=y_{a}\sum_{k=0}^{j+1}\frac{ \lambda^{k} }{\Gamma (k\eta+ \zeta)}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}+\int_{a}^{t}\Psi ^{\prime }(\sigma)\sum_{k=0}^{j}\frac{ \lambda^{k} }{\Gamma((k+1)\eta)}(\Psi(t)-\Psi(\sigma))^{(k+1)\eta-1} f(\sigma) \,d\sigma. \end{align} \end{small} This proves equation \eqref{e18} is true for $m=j+1$. By mathematical induction the equation \eqref{e18} is true for all $m\in\mbox{\Bbb N}$. Taking limit as $m\rightarrow \infty$, on both sides of \eqref{e18}, we get the solution of the Cauchy problem \eqref{a1}-\eqref{a2} as \begin{align} y(t)&=\lim\limits_{ m\rightarrow \infty}y_m(t)\\ &=y_a\sum_{k=0}^{\infty}\frac{\lambda^k( \Psi (t) -\Psi (a)) ^{k\eta+\zeta-1}}{\Gamma ( k\eta+\zeta )}+\int_{a}^{t}\Psi ^{\prime }(\sigma)\sum_{k=0}^{\infty}\frac{\lambda^{k}( \Psi (t) -\Psi (\sigma)) ^{(k+1)\eta-1}}{\Gamma ((k+1)\eta )} f(\sigma)\,d\sigma\\ &=y_a( \Psi (t) -\Psi (a)) ^{\zeta-1}\mathcal{E}_{\eta,\,\zeta}\left( \lambda(\Psi(t)-\Psi(a))^{\eta}\right) \\ &\qquad+\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma) \mathcal{E}_{\eta,\,\eta}\left( \lambda(\Psi(t)-\Psi(\sigma))^{\eta}\right) f(\sigma)\,d\sigma. \end{align} \end{proof} \subsection{Linear Cauchy Type problem with Variable Coefficient} \begin{theorem} \label{th7.1} Let $f\in C_{1-\zeta\,; \,\Psi}\left( \Delta,\mbox{\Bbb R}\right) ,~\lambda \in \mbox{\Bbb R}~\text{and}~\mu>1-\eta $. Then, the solution of Cauchy problem for homogeneous FDE with variable coefficient involving $ \Psi$-Hilfer fractional derivative \eqref{a3}-\eqref{a4} is given by \begin{align} y\left( t\right) =\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\, \mathcal{E}_{\eta,\,1+\frac{\mu-1}{\eta},\,\frac{\mu+\zeta-2}{2}} \left( \lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right). \end{align} where, $\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left( \Psi (t)-\Psi (a) \right) ^{\zeta-1}}{\Gamma(\zeta)}$ \end{theorem} \begin{proof} The equivalent fractional integral of \eqref{a3}-\eqref{a4} is \begin{equation}\label{e21} y(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{\lambda}{ \Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\mu-1}y(\sigma) \,d\sigma. \end{equation} We consider the sequences $\{y_{m}\}$ of successive approximation defined by \begin{align} y_0(t)&=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\\ y_m(t)&=y_0(t)+\frac{\lambda}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\mu-1}y_{m-1}(\sigma) \,d\sigma, ~(m=1,2,\cdots), \end{align} and derive the solution of \eqref{e21}. By mathematical induction, we prove that \begin{align}\label{e22} y_{m}(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\{1+\sum_{k=1}^{m}c_k\,\left( \lambda( \Psi (t) -\Psi (a)) ^{\eta+\mu-1}\right) ^k \right \}, \end{align} where \begin{align} c_k&=\prod_{j=0}^{k-1}\frac{\Gamma(j(\eta+\mu-1)+\mu+\zeta-1)}{\Gamma(j(\eta+\mu-1)+\eta+\mu+\zeta-1)}\\ &=\prod_{j=0}^{k-1}\frac{\Gamma(\eta[j(1+\frac{\mu-1}{\eta})+\frac{\mu+\zeta-2}{\eta}]+1)}{\Gamma(\eta[j(1+\frac{\mu-1}{\eta})+\frac{\mu+\zeta-2}{\eta}+1]+1)}. \end{align} For the case $m=1$, we have \begin{align} y_1(t) &=y_0(t)+\frac{\lambda}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\mu-1}y_0(\sigma) \,d\sigma\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{\lambda}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\mu-1}y_{a}\frac{( \Psi (\sigma)-\Psi (a) ) ^{\zeta -1}}{\Gamma ( \zeta ) } \,d\sigma\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+y_{a}\frac{\lambda}{\Gamma ( \zeta ) }\,\, \mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t)-\Psi(a))^{\mu+\zeta-2}\\ &=y_{a}\frac{( \Psi (t)-\Psi (a) ) ^{\zeta -1}}{\Gamma ( \zeta ) }+y_{a}\,\,\frac{\lambda}{\Gamma(\zeta)}\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}(\Psi(t)-\Psi(a))^{\eta+\mu+\zeta-2}\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\{1+\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}\left( \lambda( \Psi (t) -\Psi (a)) ^{\eta+\mu-1}\right) \right \}. \end{align} This is the equation \eqref{e22} for $m=1$. Now, we assume that the equation \eqref{e22} is hold for $m=j$. we prove it is also hold for $m=j+1$. In fact, \begin{align} & y_{j+1}(t)\\ &= y_0(t)+\frac{\lambda}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\mu-1}y_{j}(\sigma) \,d\sigma\\ &= y_0(t)+\frac{\lambda}{\Gamma ( \eta ) }\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\mu-1} \times\\ & \qquad \frac{( \Psi (\sigma)-\Psi (a) ) ^{\zeta -1}}{\Gamma (\zeta)}y_{a}\,\left\{1+\sum_{k=1}^{j}c_{k}\left( \lambda( \Psi (\sigma) -\Psi (a)) ^{\eta+\mu-1}\right) ^{k} \right \} \,d\sigma\\ &= y_0(t)+y_{a}\frac{\lambda}{\Gamma (\zeta)}\frac{1}{\Gamma ( \eta )} \int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{\mu+\zeta-2} \,d\sigma\\ &\qquad+y_a\sum_{k=1}^{j}\frac{c_{k}\,\,\lambda^{k+1}}{\Gamma (\zeta)}\frac{1}{\Gamma ( \eta )}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)( \Psi (\sigma) -\Psi (a)) ^{(k+1)\mu+k\eta+\zeta-k-2} \,d\sigma\\ &=y_0(t)+y_{a}\frac{\lambda}{{\Gamma(\zeta)}}\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t)-\Psi(a))^{\mu+\zeta-2}\\ &\qquad+y_a\sum_{k=1}^{j}\frac{c_{k}\,\,\lambda^{k+1}}{\Gamma (\zeta)}\,\,\mathcal{I}_{a+}^{\eta\,; \,\Psi}(\Psi(t)-\Psi(a))^{(k+1)\mu+k\eta+\zeta-k-2}\\ &=y_{a}\frac{(\Psi(t)-\Psi(a))^{\zeta-1}}{\Gamma(\zeta)}+y_a\frac{\lambda \,\Gamma(\mu+\zeta-1)}{\Gamma(\zeta)\Gamma(\eta+\mu+\zeta-1)}(\Psi(t)-\Psi(a))^{\eta+\mu+\zeta-2}\\ &\qquad+y_a\sum_{k=1}^{j}\frac{c_{k}\,\,\lambda^{k+1}}{\Gamma (\zeta)}\frac{\Gamma((k+1)\mu+k\eta+\zeta-(k+1))}{\Gamma((k+1)(\mu+\eta)+\zeta-(k+1))}(\Psi(t)-\Psi(a))^{(k+1)(\eta+\mu)+\zeta-k-2}\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\{1+\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right.\\ &\qquad~\left.+\sum_{k=1}^{j}c_{k}\,\,\frac{\Gamma((k+1)\mu+k\eta+\zeta-(k+1)}{\Gamma((k+1)(\mu+\eta)+\zeta-(k+1)}\lambda^{k+1}(\Psi(t)-\Psi(a))^{(k+1)(\eta+\mu-1)}\right\}\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\{1+\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1} \right.\\ &\qquad~+\left.\sum_{k=1}^{j}c_{k}\frac{\Gamma(k(\eta+\mu-1)+\mu+\zeta-1)}{\Gamma(k(\eta+\mu-1)+\eta+\mu+\zeta-1)}\lambda^{k+1}(\Psi(t)-\Psi(a))^{(k+1)(\eta+\mu-1)}\right\}\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a} \left\{1+\sum_{k=1}^{j+1}c_{k}{\left( \lambda(\Psi(t)-\Psi(a))^{(\eta+\mu-1)}\right) }^{k}\right\} \end{align} where, $$c_k=\prod_{j=0}^{k-1}\frac{\Gamma(j(\eta+\mu-1)+\mu+\zeta-1)}{\Gamma(j(\eta+\mu-1)+\eta+\mu+\zeta-1)} .$$ This proves equation \eqref{e22} is true for $m=j+1$. By mathematical induction the equation \eqref{e22} is true for all $m\in\mbox{\Bbb N}$. Taking limit as $m\rightarrow \infty$, on both sides of \eqref{e22}, we get the solution of the Cauchy problem \eqref{a3}-\eqref{a4}, given by \begin{align} y(t)&=\lim\limits_{ m\rightarrow \infty}y_m(t)\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\{1+\sum_{k=1}^{\infty}c_k[\lambda( \Psi (t) -\Psi (a)) ^{\eta+\mu-1}]^k \right \}\\ &=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a} \,\mathcal{E}_{\eta,\,1+\frac{\mu-1}{\eta},\,\frac{\mu+\zeta-2}{2}}\left( \lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right). \end{align} where $\mathcal{E}_{\eta,\,1+\frac{\mu-1}{\eta},\,\frac{\mu+\zeta-2}{2}}(\cdot)$ is generalized (Kilbas--Saigo) Mittag--Leffler type function of three parameters. \end{proof} \section{Concluding remarks} In the present paper, we are able to provide a brief study of the theory of FDEs by means of the $\Psi$-Hilfer fractional derivative. We examined the existence along with the interval of existence, uniqueness, dependence of solutions and Picard's successive approximations in cases: nonlinear and linear. The existence results pertaining to Cauchy problem (\ref{eq1})-(\ref{eq2}), was obtained through the theorems of Schauder and Arzela-Ascoli. On the other hand, we can prove the uniqueness and continuous dependence of the problem (\ref{eq1})-(\ref{eq2}) by making use of mathematical induction and Weissinger's fixed point theorem. In addition, we investigate Picard's successive approximation in the nonlinear case for the Cauchy problem (\ref{eq1})-(\ref{eq2}) and in the linear cases: (\ref{a1})-(\ref{a2}) with constant coefficients and (\ref{a3})-(\ref{a4}) with variable coefficients, making use of the Gronwall's inequality and mathematical induction. It should be noted that the results obtained in the space of the weighted functions $C_{1-\zeta;\,\Psi}\left( \Delta,\mbox{\Bbb R}\right) $ are contributions to the fractional calculus field, in particular, the fractional analysis. The question that arises is: will it be possible to perform the same study in space $L_{p}([0,1],\mathbb{R})$ with the norm $\|\|(\cdot)\|\|_{p,\,\eta}$, involving $\Psi$-Hilfer fractional derivative and Banach's fixed point theorem \cite{vanterler3}? If the answer is yes. What are the restrictions for such results, if any? These issues and others, are studies that are in progress and will be published in the future.	{ "redpajama_set_name": "RedPajamaArXiv" }	3,015
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THE _Girl's_ GUIDE TO _(Man)_ HUNTING THE _Girl's_ GUIDE TO _(Man)_ HUNTING JESSICA CLARE HEAT New York THE BERKLEY PUBLISHING GROUP Published by the Penguin Group Penguin Group (USA) Inc. 375 Hudson Street, New York, New York 10014, USA Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.) • Penguin Books Ltd., 80 Strand, London WC2R 0RL, England • Penguin Group Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd.) • Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty. Ltd.) • Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi—110 017, India • Penguin Group (NZ), 67 Apollo Drive, Rosedale, Auckland 0632, New Zealand (a division of Pearson New Zealand Ltd.) • Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England This book is an original publication of The Berkley Publishing Group. This is a work of fiction. Names, characters, places, and incidents either are 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. The publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content. Copyright © 2012 by Jessica Clare. Cover photograph by Kostudio. Cover design by Lesley Worrell. All rights reserved. No part of this book may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author's rights. Purchase only authorized editions. HEAT and the HEAT design are trademarks of Penguin Group (USA) Inc. PUBLISHING HISTORY Heat trade paperback edition / May 2012 Library of Congress Cataloging-in-Publication Data Clare, Jessica. The girl's guide to (man)hunting / Jessica Clare.—Heat trade paperback ed. p. cm. ISBN: 978-1-101-56910-8 I. Title. PS3603.L353G57 2012 813'.6—dc23 2011037775 PRINTED IN THE UNITED STATES OF AMERICA 10 9 8 7 6 5 4 3 2 1 ALWAYS LEARNING PEARSON _For Cindy and Jane. This book is entirely your fault, and I mean that in a good way._ # Table of Contents ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE TEN ELEVEN TWELVE THIRTEEN FOURTEEN FIFTEEN SIXTEEN SEVENTEEN EIGHTEEN NINETEEN # ONE Like everything else bad that happened in Miranda Hill's life, rear-ending old Mrs. Doolittle was purely the fault of Dane Croft. She could have sworn that she'd recognized the broad shoulders, tight ass, and familiar swagger of her nemesis walking into the local coffeehouse. Her most hated enemy. The man who had ruined her life. In fact, she'd been so busy craning her neck to see if it really was Dane Croft that she hadn't paid attention to the stoplight...and had plowed right into the car in front of her. Yet another thing she could add to the list of reasons why she hated him. Miranda put her pickup in park and slid out of the cab to look at the damage she'd caused to the other car. Mrs. Doolittle drove a Buick that was older than Miranda herself, and the thing was built like a tank—a big, powder blue tank. The bumper wasn't even dinged, not that Mrs. D cared. The old woman crawled from the belly of the tank and scowled at her. "You hit my car, Miranda." If Mrs. Doolittle had a cane, she probably would have shaken it in Miranda's face. "What on earth were you thinking, girl?" Miranda gave Mrs. D an apologetic look and self-consciously tugged at the high collar of her pink sweater set. "I'm so sorry, Mrs. Doolittle. I was just...distracted." She was still distracted, actually. Her gaze strayed to the Kurt's Koffee on the far side of the street, but the windows were tinted and impossible to see into. The elderly woman peered at her. "Young lady, were you using the Twitters while you were driving? You know—" "No Internet," Miranda blurted, tugging on her collar again. "I just wasn't paying attention. I thought I saw...something." _Someone._ A car pulled up behind them. No surprise, given that most of the streets in downtown Bluebonnet were single lanes, with just enough room in the city square to park in front of one of the two restaurants. She waved for the driver to go around them, and then continued apologizing to Mrs. D, even as they exchanged insurance information. Anything to get out of the street and appease her curiosity. She kept glancing at the coffeehouse as she scribbled down her contact numbers. Finally, Mrs. D was on her way, satisfied. Miranda pulled her truck into a parking space across the street and sprinted toward the coffee shop, but didn't go inside. Instead, she pressed her hands to the glass and peered in. A few people were seated, but she didn't see the man she was looking for. No Dane Croft. Was she crazy? Had she imagined that she saw him? Chewing on her lip, Miranda straightened the front of her sweater set in the reflection and then went inside. "Well, well, well, if it isn't the Boobs of Bluebonnet," said Jimmy Langan from behind the counter. Jimmy was the town rebel, with purple, red, and black Rasta braids, a face that had never seen a tan, and enormous ear gauges that he'd probably regret when he was seventy. He grinned at her, giving Miranda the up-and-down look that she'd become far too accustomed to in the past nine years. "What can I do for you?" "Shut up, Jimmy," she said. Three weeks. She could deal with the jokes and the sneaking glances at her breasts for three more weeks. Moving past the counter, she peered down the hallway at the restrooms. No Dane Croft. She resisted the urge to open the door, and instead wandered back to the counter. "Is anyone in there?" "You want me to go and check under the stalls for feet?" Jimmy said dryly. "Well, no," she stammered, her hand going to the collar of her sweater. "Maybe." She hesitated, reluctant to say the name of the man she was looking for. If she even so much as uttered Dane's name, the rumors would start flying all over town again. _You know that nice Miranda Hill? She never quite got over Dane Croft. She was asking about him in Kurt's Koffee. Poor thing._ _Remember that man in the photos with Miranda Hill? She's still sweet on him. I heard she's still got the hots for him and that's why she hasn't married._ _The town librarian? She's a slut. Want to see the pictures? She spent seven minutes in heaven with Casanova Croft back when they were both in high school. They even took photos of it. Just search for "Boobs of Bluebonnet" on the Internet and you'll see them._ Miranda clutched the collar of her demure sweater even harder. "So what kind of customers have you had today?" Jimmy shrugged lazily, adjusting the thick black-frame glasses on his pasty, scruffy face. He'd been a stoner back when they'd graduated from high school together, and he was a stoner still. Asking him to remember the customers he'd had that morning might be beyond his pot-riddled memory. "Couple soy lattes, couple double espressos, a venti mocha frap with double Splenda..." Great, just what she needed: a rundown of coffee orders. She feigned interest, her eyes skimming the restaurant as Jimmy rattled off a litany of special requests. "And a certain someone you might recall," Jimmy added slowly, his gaze dropping to her breasts. "We went to high school with him." She crossed her arms over her chest, doing her best to hide what the underwire minimizer wouldn't. Her heart was thudding hard in her chest, but she forced herself to be nonchalant about the information. "Oh? Someone from high school? Who's that?" To her surprise, he reached behind the counter and pulled out a brown and green pamphlet. "You remember Dane Croft? Casanova Croft? Star of the Las Vegas Flush?" The guy she'd been making out with in the closet? The one with his hand on her boobs and the other down her pants for all eternity thanks to a few ill-timed photos and the magic of the Internet? Who'd left the next day to be drafted into the NHL and become a star while she'd been stuck in town as her mother had a nervous breakdown? The Casanova Croft who'd been booted out of the NHL six years later for sleeping with the coach's wife? Life-ruiner and all-around jerk? Yeah, she knew who he was. "I'm familiar with the guy." "He's moved back to town," Jimmy said, offering her the pamphlet. "Him and two other guys we went to high school with are starting a business here. Something about survival training classes. They bought the Daughtry Ranch on the outskirts of town." "The Daughtry Ranch?" Miranda echoed, taking the pamphlet from him and forcing her shaking fingers to open it. The Daughtry Ranch was ten thousand acres of private property, and when old Mr. Daughtry had died without an heir, the ranch had gone up for auction. No one in town knew who'd ended up buying it. Sure enough, there in the picture on the pamphlet were three men she recognized: Grant Markham, Colt Waggoner, and her nemesis—Dane Croft. The three of them were dressed in black T-shirts and camouflage pants, and the top of the brochure proudly proclaimed, "Wilderness Survival Expeditions: Bushcraft Training for Corporate and Military Groups." Survival training? The Dane Croft she remembered was a hard-partying playboy who refused to do anything that didn't involve beer or girls—or both. She remembered Grant and Colt—one was a jock and one had been the richest guy in her class. Both had moved away when they'd graduated, just like Dane. And now they were back...just like Dane. Could today possibly get any worse? She tucked the brochure into her pocket, feeling faint. "Thanks, Jimmy. Can I get a green tea latte, please?" "Sure," Jimmy said lazily, his gaze sliding to her breasts again. "Venti, grande, or tall? Iced or hot? Two percent, whole, skim, or soy?" Miranda had her phone out, dialing, and ignored Jimmy. Her other hand fluttered back to her pocket repeatedly, touching the brochure again and again. "Right. I'll just make something up," he drawled, then turned away to make her drink. Beth Ann picked up the office phone on the second ring. "California Dreamin'," she answered in a chirpy voice. "We do waxes, haircuts, highlights, and perms. Can I make you an appointment?" "It's me," Miranda hissed into the receiver, covering the phone and turning away in case Jimmy planned on listening in. "You're never going to believe who's back in town." "Who?" "Dane Croft," Miranda gritted. There was a long pause on the other end of the line. " _The_ Dane Croft? The Vegas Flush player? The one we went to high school with?" "That's him—" "The one who put his hand down your pants—" "Beth Ann!" "I'm clearing my lunch appointments," Beth Ann declared. "Be here in twenty minutes and we'll talk." For Beth Ann, a "talk" usually involved waxing Miranda's eyebrows, a trim for Miranda's split ends, and a manicure. They'd been friends ever since the fifth grade, and if there was one thing that Miranda knew about Beth Ann, it was that she liked her hands busy while she chatted. Her small salon was nearly empty at noon on a Wednesday, and Miranda waited patiently as Beth Ann unlocked the back room that housed a tanning bed and let in a teenage blonde. "I keep telling Candy that she's going to look like a handbag by the time she's thirty, but she won't listen to me," Beth Ann said with a shrug, returning to the barber chair Miranda sat in. "And the tanning bed brings in almost as much money as manicures do." She spun the chair around, turning Miranda toward the mirror, and flung the pink satin styling cape over her clothes. "Now, honey, tell me your problems." "My problem is Dane Croft," Miranda said, digging under the cape and pulling the brochure out. She held it toward Beth Ann. "He's moved back to town—permanently. And he's started a survival business with Grant Markham and Colt Waggoner." "Survival business?" Beth Ann tucked a lock of perfectly highlighted blond hair behind her ear and gave Miranda an odd look in the mirror. "That doesn't sound like the Dane Croft we went to high school with." "It's him—look at the picture." Miranda slumped in the salon chair, wishing this day would start over again. Beth Ann's eyebrows rose as she stared at the pamphlet. "Professional survival services? That's kind of strange." "I know," said Miranda flatly. "Mmm. Just look at them. They've all filled out rather...nicely, don't you think?" Miranda scowled and snatched the pamphlet back, glancing at the photo again. All three men were tall and fit, she supposed. Dane's arms were especially toned with muscle. He had a dark tan and his black hair was cut incredibly short. The white smile on his face was as familiar as her own. He actually looked like a hunky, Hollywood version of a survival instructor. That made her feel worse. "This is just awful." "Why is it awful?" She began to comb out Miranda's long, dark brown hair and trim the ends. "This is the perfect time for him to come back. You're leaving for that big job in the city in three weeks, remember? You only have to avoid him until then." And she sighed. Miranda ignored Beth Ann's sigh. She'd heard enough of them to feel permanently guilty about the fact that she wanted to leave Bluebonnet behind for a job in Houston. A job with real benefits and a chance to move up the corporate ladder. A job that could lead anywhere, maybe even chief information officer. Or higher. Miranda Hill, the Boobs of Bluebonnet, would have a fancy title and an even fancier job. She could actually do something with her master's in Library Science instead of just re-cataloging books and taking complaints from old ladies who wanted the "dirty vampire books" removed from the shelves. "This is my chance to do something, Bethy. To get out of town. To be something other than the Boobs of Bluebonnet." "It's what you've always wanted," agreed Beth Ann. "It doesn't mean that it won't make me sad to see you go." Miranda regarded her friend through the mirror, watching as Beth Ann clipped her ends with careful, precise fingers. "I know. I'll come back and visit you all the time." In the mirror, Beth Ann gave her a wry smile. "Sure you will." Miranda glared down at the pamphlet and the three tanned, attractive men on the cover. "You know, I was hoping for three quiet weeks to relax and get things settled. My last day at the library was yesterday. My apartment in Houston is leased. The house is almost packed. I've got nothing to do for the next three weeks except stare at this picture and stew. Except every time I look at this, I see them." "Three sexy beasts?" "Not them. The _pictures_." The images were ingrained into her memory. If she lived to be eighty, she'd never forget one single detail of those grainy, horrible photos—her torso facing the camera, an expression of complete and utter abandon on her face. Her T-shirt pushed up around her neck, her breasts facing the camera. Dane's mouth on her neck and his hand down the front of her panties. Then the picture of her kneeling in front of him, as if she was about to give him a blow job. She'd never known that there'd been a camera in the closet. Or that he'd pack up and leave town practically the next day to join the NHL, without a single word to her. Miranda had been forced to pretend that she wasn't hurt by his abandonment, but abandonment had soon given way to horror as soon as the pictures surfaced. And with a town as small as Bluebonnet...everyone talked. She hadn't slept with Dane, but that didn't matter. She'd tried going to the police when the pictures first went up, but her mother had been so upset and sheriff had looked at her like she was trash, and she'd dropped the entire thing rather than acknowledge that the pictures were of her and Dane. At the time, she'd hoped it would all just go away. No such luck. Everyone in town assumed she had slept with Dane, blown him in the closet at a party, and they looked at her like she was dirt. In their eyes, she _was_ dirt. The town slut. It had taken patience, a stiff upper lip, and years of a quiet existence as the town's librarian before she'd managed to grasp a semblance of her reputation again. Beth Ann put down the scissors and leaned over the back of the chair, smiling into the mirror at Miranda's frown. "Well, you've got three weeks to burn, and your infamous ex is back in town. You can pretty much do what you want and you won't be here to suffer the repercussions. So what do you want to do? TP his house? Key his car? I'm sure we can think up something totally juvenile and completely satisfying." Miranda stared down at the pamphlet, at Dane's confident smile. But what she saw? Pictures of herself on a webpage, in e-mails forwarded to thousands of people, tossed up on the Internet and forever linked to her name. Pictures of his hand down her pants, her breasts angled at the camera like twin beacons. And she stared at Dane's casual, confident brochure smile again. _Professional survival training_ , the pamphlet read. _Casanova Croft, kicked off of the Las Vegas Flush for sleeping with the owner's wife._ Professional survival training. _Professional_. "I think I want revenge," Miranda blurted, then turned to stare up at her friend. "I know it's not rational, and I don't even care. Is that crazy?" "Not at all," soothed Beth Ann. "What did you have in mind?" Miranda held up the brochure, an idea forming. "I want to ruin his career like he did mine." "I'm listening, honey." Miranda flipped open the pamphlet. "They're just starting a business, right? What if pictures of Dane Croft surfaced on the Internet? Naked pictures of him? Naked, compromising pictures of him?" The idea began to grow in her mind, and she jumped out of the chair, almost trembling with excitement. "Naked, compromising pictures of him in a _survival situation_?" Her best friend's blond brows furrowed together. "And where would you get such pictures?" "I'd take them myself." Beth Ann raised an eyebrow. "And just how are you going to do that?" Miranda held up the brochure triumphantly. "I'm going to sign up for a survival course and use the legendary Boobs of Bluebonnet against him. Casanova Croft won't stand a chance." "Are you sure that's wise?" "I've never been wise around Dane Croft," Miranda said, thinking of the last time she'd seen him. _"Seven minutes in heaven," Chad announced, shoving Miranda and Dane toward his bedroom closet. Giggling teenagers surrounded them, and Miranda felt her cheeks heat with embarrassment, but she didn't let go of Dane's hand._ _Dane nudged Chad and grinned. "Do me a favor, bro, and skip the timer."_ _Chad smirked._ _She could have protested, said she wasn't that kind of girl, but she said nothing, not even when the door shut behind them. She wanted to be that kind of girl with Dane._ _Chad's closet smelled like sweaty football gear and dirty clothes. It was crammed full of boxes and clothing on hangers, the single flickering lightbulb overhead not offering much in the way of light. She wrinkled her nose at the musty smell of the closet and waited, her breath catching. Would Dane make a move on her tonight? They'd flirted for weeks, held hands for the last one, and kissed under the bleachers. Given time, she knew she wanted him to be the one to take her overdue virginity._ _But time was the one thing they didn't have. They'd graduated earlier that evening and after the cap and gown ceremony, they'd headed to Chad's for the last senior fling._ _It was now or never._ _She gestured at the light overhead as it flickered again. "Should we turn that off?"_ _"Leave it on. I like looking at you." Dane's hand gave hers a squeeze and he smiled at her. "You okay?"_ Yes, _she wanted to say._ I'm fine. Did you have a nice time at graduation? _But it came out as a whimper, the words lodged in her throat._ _Dane chuckled at that. "I guess I should be telling you 'Happy eighteenth birthday,'" he said. "You're as old as me now."_ _Eighteen, and they'd be going off to college soon. The thought ran through her mind, urgent curls of heat rushing through her. Instead of responding, she pulled him close and began to kiss him instead, her mouth seeking his._ _"Whoa," Dane whispered, but his hands went to her ass and hepulled her against him, grinding his hips against her own. His tongue slid into her mouth, delving deep and tasting her in the sweetest kiss she'd ever had. His mouth pulled away from hers after a long moment and he breathed hard in her ear. "Damn, Miranda."_ _Her own breath thrilled at that, and she slid her leg between his...and stumbled, landing on him._ _He cursed, trying to shift his weight, pinned between a row of jackets and a stack of boxes._ _"Sorry," she whispered meekly, shaking her high-heeled boot. "I think my shoe got caught in his helmet."_ _They fumbled in the cramped quarters, and Miranda grabbed hold of a shelf and pulled herself up, then turned to remove the football helmet from her boot._ _Dane shifted behind her, his hands sliding around her waist. "That's better," he whispered against her neck. Something tickled at her waist, where her shirt rode up—his fingers._ _Her hand covered his, and she moved it farther up under her shirt, quivering with pleasure. "Touch me, Dane. Please."_ _"Love to," he whispered in her ear, and pressed a kiss against her neck, making her squirm. "You are the hottest damn thing in this town, Miranda Hill."_ _"You know it, Dane Croft," she whispered, craning her neck so his tongue could glide along her throat. Heat pulsed through her body. She didn't protest when his hands slid to her shirt and pulled it over her head in the near darkness. She even unhooked her own bra, since his fingers fumbled at her back for a long moment. But then his hands were cupping her breasts, his fingers warm against her skin. Fingers teased her nipples and she gasped, lifting her arms and twining them around his bent head._ _From behind her, he pressed a kiss against her bare shoulder and she could feel his erection against her jeans. His fingers tweaked her nipples again, and her breath caught in response. "Dane," she whispered. "God, do that again."_ _"I'll do even better," he said against her neck. One hand grasping her full breast, his other slid down her belly and undid the button on her jeans. Her entire body tensed, tingles of excitement running through her. Was he going to touch her...there?_ _His fingertips slid into her panties, brushed the curls of her sex, and she let out a whimper of delight. Two seconds later, his fingertips slipped into her panties. One finger swept past the lips of her sex, grazed her clit. Oh yes. His hand squeezed her breast at the same time that he stroked her there, and her entire body stiffened, the anticipation of being in the closet with him rushing her toward an orgasm—_ Click. _Miranda froze in place. Dane continued to finger her, biting at her shoulder, and she pulled away from him, sliding his hand out of her panties. "Did you hear that?"_ _His hands reached for her, brushed against her breasts again. "Didn't hear anything."_ _"I thought I heard a noise," she said softly, staring at the closet door. It was still shut, and the doorknob didn't move. Overhead, the light flickered again. Nothing. Maybe she was imagining things. Paranoid at being caught. If she listened hard, she could hear her classmates giggling in the other room, waiting for them to emerge._ _She started to protest, but he bit her shoulder and pleasure crashed over her, and she didn't protest when his hand slid back into her panties once more._ *** * * Looking back, she had been so very, very dumb. She should have guessed that Dane would have hidden a camera in that damn closet. Should have guessed that he'd want all his buddies to see that he'd gotten into curvy Miranda Hill's panties and made her writhe against his hand in a closet. She hadn't blown him, either, but no one would believe that from looking at the photos. And she should have guessed that he'd disappear as soon as the NHL came calling. Who was she to him? No one, it seemed, but a quickie in the closet. # TWO After leaving Beth Ann's salon, she headed over to her mother's store, Hill Country Antiques. The store looked as ramshackle as ever, the wooden sign listing a bit too much on one side, windows dusty and full of clutter. Antiques stores came in different flavors—from austere and highbrow to cluttered and junky. Her mother's store was definitely on the junky side. More thrift and yard sale than actual antiques, it was a cornucopia of bizarre odds and ends that nevertheless managed to bring in a decent income for her mother. "Hi, hon," her mother called when Miranda entered, the cowbell on the door clanging against the glass. "You're just in time." "Oh? In time for what?" "They had a storage unit sale over in Livingston, and Marilou picked up someone's old unit for fifty bucks!" Her mother said, moving to the front of the store and sashaying past Miranda. She flipped the store sign to CLOSED. "I get to split everything in there with her, but we've got to clean it out before the end of the day. I could use an extra set of hands, too." "I can't," Miranda said with a grimace, gesturing at her car. "I need to stop by the library and pick up my last check. Sorry." It was a bit of a white lie, but she really didn't want to go and spend the day picking through someone else's junk. The last time her mother had bought a storage unit, they'd found nothing but endless rows of comic-book boxes, their contents eaten by mice. "I'm about to head out of town for a week or so." "Out of town?" Her mother looked surprised. "Where are you going?" "Oh, just checking some stuff out in Houston," Miranda lied. "But I wanted to let you know that I'm not going to be answering my phone for a few days. I'll swing by when I'm back, okay?" "But—" She froze, waiting. In the past, any small thing that interrupted her mother's daily routine would be met with crying, anxiety, and comments about Miranda's reputation about town. She'd had a nervous breakdown when the pictures had hit the Internet nine years ago, and it had taken a lot of time and patience and support to get her mother steady again. Now that things were going well, Miranda was getting out of Bluebonnet once and for all. She knew Tanya was having a hard time adjusting to the fact that her daughter was finally leaving the nest, and things had been fragile for the past few weeks. "—who's going to help me clean out the storage unit?" Thank goodness. Miranda leaned in and kissed her mother on the cheek. "I'll see if Beth Ann can send her little sister Lucy over. I'm sure she'll help for a few bucks. Now, I've got to go, Mom. I'll talk to you next week." "'Bye, hon," her mother said absently as they went out to their separate cars. Miranda got into her truck, waved at her mother, and backed out, heading toward the library. Well. That had gone better than she'd expected. She turned down Main Street and waited at the town's only stoplight. Absently playing with her collar, she thought about her plans for this week. She'd need some camping clothing, should toss out the stuff in the fridge, maybe see if— A car honked next to her. Miranda glanced over, and immediately wished she hadn't. Two men sat in the car, both a few years younger than her. She knew their families. Had seen them around town. Both were grinning at her in that way that told her they'd seen her half naked. Seen the photos. "Hey, Boobs," one called with a leer. "I've got an overdue library book. Wanna come to my house and get it?" Next to him, the passenger began to pump his fist in front of his open mouth, mimicking a blow job. Cheeks flaring with heat, she turned away, just in time to hear both men erupt in laughter. The light turned green and she floored the pedal, surging forward and down the street. She couldn't _wait_ to be done with this town. "Remember, Dane. Hands off the clientele." Colt said the words with a grin and gave the game controller in his hands a twist, staring at the TV screen. "This is our make-or-break moment, and I need you to have your head in the game." "Thanks, coach," Dane said sarcastically to his friend, stuffing a pair of spare socks into his bag. "Glad to have you riding my tail." Colt glanced over at Dane, looking away from the TV screen for a brief moment. "I'd better be the _only_ thing riding your tail this next week." Ah, friends. If he didn't like the guy so much, he'd be tempted to deck him. Dane ignored Colt's gibes and double-checked his survival pack one more time as they waited in the Daughtry Ranch's rec room before meeting the clients that would be gathering shortly. They were taking a few moments to unwind before being "on" for the rest of the week. And while Colt chose to play a video game to get in the right mind-set for the trip, Dane felt better looking over their gear one more time. The sounds of a cheering audience erupted from the television, and Dane's head snapped up. Sure enough, Colt was playing a hockey video game. It set his nerves on edge, watching the pixelated players skate around the ice. It reminded him of his old life, which he didn't appreciate as he was trying to start the new one. "Do you have to do that shit right now?" Colt didn't look up from the screen. "Yes." Dane snorted and moved to check his bag again, turning away from the screen. He didn't need distractions right now; he needed to be ready. This inaugural training needed to go perfectly. First he double-checked the survival supplies he'd be bringing for the group: matches, flint, needles and thread, fish hooks and line, a compass, snare wire, a flexible saw, a medical kit, flares, and a utility knife. At Grant's request, he'd also packed six military MREs and a satellite phone in case the corporate guys couldn't hack it out in the wild. As the "wild" went, the Daughtry Ranch was pretty tame in comparison to where Dane and Colt had spent their survival missions, or the times that they'd roughed it off the grid, but it was perfect for the business. Hechecked his pack one more time. Dane felt comfortable viewing the small amount of survival gear, the familiar anticipation edging through his body and drowning out any lingering irritation from Colt's joking. He lived for this. He loved it—pitting himself against the wilderness and using his skills to survive. It centered him. When he was out there in the wild, Dane could find peace in himself, no matter what was bothering him. No one but him, nature, the land...and six neophytes looking to him for direction, he added wryly. Still, he doubted they'd be able to take the enjoyment of the experience away from him. This was part of who he was now. And it was why he'd lived off the grid ever since he'd left hockey behind. He was a new man, with a new life, and he liked himself now. The challenge of living off the land appealed to him. The simplicity of a survival situation couldn't be beat. Just you and nature. You didn't need electricity or television or telephones to survive. All you needed was skill and perseverance. He liked that much better than modern society. He slung the light pack over his shoulder and gave Colt a friendly clap on the shoulder. "I assure you, man. The last thing I want to do is touch a woman right now." Not when their business was just about to take off. "Some things are more important." "I'm just making sure," Colt drawled. "Everyone already thinks that your dick rules your business decisions. We need to prove them wrong if this has a hope of succeeding." It irritated him that Colt was right. That everyone thought that his cock was in charge of his brain. Dane rubbed his jaw, grimacing at the thought. Back in his hockey days, he'd been a different person. Headstrong and reckless beyond belief, he'd played so hard and carelessly that he'd managed to score two serious concussions in a row, and when another man would have paid attention to the doctors and been more cautious, he'd gone back on the ice as soon as he'd had the okay...and walked right into concussion number three in a play-off game. Tensions had already been high at that point, and that particular concussion was a career-ender. He was just injured far too often, and he was a good player, but not a great one. The coaches didn't want to take a risk on him. And then Samantha Kingston—the wife of the team's owner—had approached him. She liked younger players. He'd turned her down, but she'd turned to the tabloids to salvage her wounded pride, and "Casanova Croft" was born. She'd used him and made him look like a jackass, and it didn't matter how good a player he'd been. He'd turned into "that creep who nailed the boss's wife." His contract wasn't renewed, and a free agent with too many injuries was too big a risk for most teams to take on. Combine that with his tabloid notoriety, and no one would touch him. It hadn't helped that his past was full of a string of C-list actresses who were interested in dating a professional athlete—the latest trendy fashion accessory. The tabloid notoriety—on top of his world crashing down on him—became too much. When he'd started getting offers for sex tapes, he realized just how fucked-up his life had become. He'd fled, with nowhere to turn. Colt had contacted him, invited him to take a monthlong survival course with him to clear his head. He'd gone reluctantly, expecting nothing but a month of no phone calls from anyone. Going on the survival trip had been the best thing to ever happen to him. Forced to use his wits and skills to survive...it had been life changing. Nothing had been easy—no shelter, no supplies, no showers. At first he'd hated it—and Colt—for dragging him out into the Alaskan wilderness. But then things had changed. He'd learned to like making things with his hands, trapping his own food. It gave him a feeling of intense satisfaction. Dane had discovered a new passion, one that surpassed the adrenaline of even the most exciting play-off game. When they'd finished the trip, Colt had suggested that Dane join him at his lodge in Alaska, completely off the grid. They'd lived there for a year—no electricity, no running water, no food storage—nothing except what they could catch and take care of on their own. It had been rough and incredibly difficult. It had been bliss. He would have kept living off the grid indefinitely—not exactly hiding inasmuch as keeping a low profile—if Grant hadn't visited him and Colt in their cabin in Alaska to get away for a few weeks. Colt had invited him—the marine wasn't much for chit-chat, but he knew Grant was struggling, even years after the death of his wife. Once in Alaska, the three friends had quickly fallen into an old, easy camaraderie. Though Grant didn't share quite the same enthusiasm as Dane and Colt for wood smoke fires and catching game for dinner, the time spent roughing it at the cabin had given Grant an interesting idea. A survival business—run by the three of them. Colt and Dane could handle the trainings, and Grant would handle the business. They'd work for themselves and answer to no one. Neither Dane nor Grant needed the money, but the challenge of the business intrigued all three of them. Now Dane found himself back in his hometown, and avoiding everyone there. If there was anyone who hadn't forgotten Casanova Croft, it was the people of Bluebonnet. He'd been leery to return to town, expecting the worst. As far as he knew, he was the only person from town who had ever had fifteen minutes of fame, and he expected harassment. So far, though, so good. He kept a low profile, and for the most part, the usually nosy citizens of Bluebonnet had left him alone. Just as he liked it. Well, he wasn't being totally honest about not wanting to look up anyone. One particular person did spring to mind, but he was pretty sure Miranda Hill wanted nothing to do with him. The last time he'd spoken to her had been in person, and when he'd tried calling her from NHL training camp, she'd ignored his calls—or worse, made her crazy mother answer them. After getting chewed out by Mrs. Hill three times in a row, he got the picture. He'd stopped calling, and stopped caring. There were always more girls willing to throw themselves at a hockey player, especially a hotshot up-and-comer. He'd eventually forgotten about Miranda Hill, the one that had gotten away. Well, sort of. And if he'd had a thing for long brown hair and girls with a soft Southern drawl, that was just how it went. Miranda Hill had probably moved away long ago. Maybe she'd gotten married and shot out five kids in five years like his cousin Tara had, and now spent her time chain-smoking and watching daytime TV. Either way, it was best if Miranda Hill remained a memory. So, no, he wasn't going to look her up. The business's only other employee—their coordination assistant, Brenna James—showed up a moment later with her clipboard and a beaming smile on her face. "Guess whose clients are here? Are you two ready?" Colt kicked up out of his chair, putting down his game controller. "Soon." Always so chatty, that Colt. "I'm ready," Dane said, grabbing the two packs. Colt had packed his hours ago in preparation for the trip, but Dane had delayed, waiting until the last moment. Almost as if he was delaying the inevitable. Colt gave him a serious look. "Hope so." Irritation surged in Dane and he ignored his friend's well-meaning look. The guys either trusted him or they didn't. He could keep his dick in his pants. It wasn't like he was some oversexed nutjob waiting to jump out of the bushes at the first pretty girl that passed by. Not anymore. Grant appeared in the room, grinning. He carried a champagne bottle and three glasses. "This is it, boys. Our big inaugural class. You ready?" Dane was starting to wish that everyone would quit asking him if he was ready. "Gonna be a good one," he said, and rubbed his hands together. "You ready to sit on your ass and soak up the profits from Colt's and my hard work?" Grant rolled his eyes and removed the foil from the top of the champagne bottle. "More like, you two get to have a weeklong vacation in the woods and I have to hold down the fort and do all the busywork. There are a million things to be done between now and when you guys get back, and Brenna's not going to be much help." "You like busywork," Colt said. "You make more for yourself just so you have shit to do." Grant popped the cork on the champagne. "Time to celebrate." Colt looked at the champagne with distaste. "You shoulda brought beer." "Beer isn't for celebrating," Grant said, ignoring Colt's sour mood. He poured a glass for Dane and handed it over. Dane took it, but he only half paid attention until the other two men raised their drinks. "To success," Grant said. "Success," Colt echoed. _To a week of proving to his buddies that his dick didn't run his life_. "To success," Dane said, and then downed the champagne. The two men emerged from the lodge and Dane squinted up at the sun. It was perfect weather for their first excursion—sixty-five degrees with no rain. The rain could come later in the week, but today? Today was perfect. This was a good omen, Dane decided, his mood light. Piece of cake. Brenna steered him toward one of two lines of waiting clients—men in camo clothes who had probably never been more "outdoors" than a corporate gym. He put on his camera-ready smile and began the meet and greet—okay, some things weren't all that different from hockey. The first guy was a small business owner, the next a lawyer looking to send his team of attorneys through the training if they liked the class. Dane hadn't believed Grant when he said that corporations would pay big bucks for this sort of thing, but sure enough, every single man he shook hands with was testing out the class for a corporate club or toastmasters or a professionals group. No pressure. Dane shook the hands of five men before coming to the end of the line and his sixth and final "student" for the week. To his surprise, the person that came out from behind the parked jeep was none other than Miranda Hill, the girl he'd left behind nine years ago. The one that had gotten away. The one he'd fantasized about for years. He stared at her in surprise. "Miranda?" She tilted her head, shiny brown hair sliding over her shoulder. "Don't I get a handshake, too?" "What are you doing here?" She looked exactly the same as she had nine years ago—same great hair, same dark doe eyes, same amazing figure with an even better rack. This had to be a test from Colt and Grant. "I'm, uh." He glanced back at Colt, openly skeptical that his fantasy girl had somehow shown up on the first day of the new business, but Colt was busy greeting his own students and wasn't looking in his direction. "I'm a little busy right now." She pulled a baseball cap over her hair and smiled at him. Well...damn. Miranda wasn't anything like his cousin Tara. If anything, she looked better than she had when he'd left nine years ago. Her slim figure was lush with curves, and she had a healthy tan. She wore a high-necked maroon top with a pair of scruffy jean shorts and beat-up sneakers. A bag was tossed over her shoulder and she was looking at him expectantly. He didn't know what to say. "Dane," Brenna said between gritted teeth. She poked him in the arm with her oversized pen. "Miranda has signed up for the survival training. You'll be her instructor this week." Damn. He looked at the welcoming smile gracing her mouth, the casual hand on her hip, and a rush of memories flooded through him. Her soft mouth on his, the feel of her skin underneath his hands. The eager teenager had turned into an amazingly sexy woman. He looked into her smiling eyes and the curve of her mouth and felt his cock stir. Hell. The week had just gotten a lot longer. # THREE This was starting to feel like a mistake. Miranda kept her nervousness hidden, though she shifted on her feet repeatedly as the survival class gathered and the two instructors talked in low voices in the distance. This had seemed like a great idea a few days ago. It hadn't been easy to get into the class at the last minute, but she'd made up some sort of excuse that her new job wanted some team-building work on her resume, forked over the ridiculous amount of money for the weeklong training, and passed the preliminary physical with flying colors. Easy enough. Her goal was simple. Find Dane Croft, flirt up a storm, and use her feminine wiles to hook him. If it were any other man, she'd have concerns about playing the seductress, but Casanova Croft was legendary for his exploits. The man was a poon hound, and she intended to use that against him. She'd get him dancing to her tune, get him a little compromised, and then let the camera in her backpack do the damage. This week, she was going to let her evil side rule things. Good Miranda was definitely going to be shoved to a back burner. Center stage? Evil Miranda. Now, looking at the people surrounding her, this didn't seem like the brightest idea she'd ever had. The class was small—six people to an instructor. Five men lined up next to her, and all seemed ready to go and eager to spend the next week in the wild. Four of the five were dressed in camouflage, and one had even painted his face with black stripes under each eye, as if he were expecting to run a few downs of football after hiking. They'd also gone overboard on the gear. Since they'd been instructed to pack extremely light, she'd decided to wear comfortable clothing over her sexiest underwear. After all, she didn't want to seem too obvious. Her hiking boots were just beat-up jogging shoes, for example. But the others seemed like they had cleaned out the local sporting goods store, and their shoes were clean and crisp and had probably been taken out of the box minutes before they arrived here. All of the men in her group—in both groups, really—were relatively fit and lean and likely in their late thirties or forties. In addition to being the youngest one here, Miranda was also the only female other than the assistant, who took everyone's waivers and wrote up their information on a clipboard. It was an acute, disturbing feeling. A week alone in the wilderness with six guys and just her? It had all the makings of a bad porno. Out of the corner of her eye, Miranda saw the second instructor approach the group of waiting clients. He was wearing a black T-shirt that had the survival school's logo on the back. When he turned at someone's question, Miranda recognized him, and not just from the photo. Colt Waggoner hadn't changed much since high school. He'd gotten taller, but he was still lean and muscled. Instead of wearing the sloppy, oversized T-shirts she remembered him in, he was dressed sharply, his T-shirt tucked into camouflage pants and shiny boots on his feet. As the client addressed him, Colt stood with his hands clasped behind his back. "No, sir," Colt replied in a crisp voice to the man's low question. "No outside electronics." The man looked at everyone else nervously. "Oh, well. I just thought I'd ask." Colt gave him a crisp nod and then began to walk past the group. He stopped and paused in front of Miranda, recognition dawning on his face. "Miranda. What are you doing here?" She pulled out a crumpled brochure and waved it in front of her, feeling like an idiot under Colt's piercing stare. "Thought I'd take the class. How are you doing, Colt? It's good to see you." "Fine," he said stiffly, then tilted his head. "You're still in Bluebonnet?" "Haven't left," she said awkwardly. Ever. Oh God. _Please don't ask me why I stayed. Please don't ask me why I stayed._ "I'm sorry," he said in a short, clipped voice. His hands clasped behind his back, and his "relaxed" pose was still stiffer than most. Military, maybe? "Sorry?" "Sorry that you haven't left. This town is a joke." A surprised laugh erupted from her throat. "So it is. What brings you back, then?" "Business," he said. "Grant's here, too. We—" "Colt," Dane yelled from behind him. "Hey, Colt. C'mere." Colt tilted his head again, a little, and he didn't turn to look at Dane. "If you'll excuse me, Miranda. Nice to see you again." "You, too," she said faintly. "Welcome back." Colt turned and trotted off to Dane's side up the hill. Dane leaned in close, saying something in a rapid-fire, angry tone. Miranda couldn't make it out, though. Dane said something, and both men turned and looked back at her. Then they spoke again. To her surprise, Colt patted the front of Dane's pants and said something. Dane swung a punch, but Colt ducked out of the way, smirking. Dane didn't look amused—he looked pissed. When Dane gestured sharply in her direction, a bit of a smile curved Miranda's mouth. Well, at least that was something. Anger was better than nothing. When she'd first extended her hand for Dane to shake, there had been a blank look on his face, as if he didn't quite know what to make of seeing her here. She had to admit, she didn't know what to make of him either. A smirk she'd have expected. A lecherous grin she'd have expected. The baffled look he'd given her? Not so much. Brenna paused in front of Miranda, peering down at her clipboard. "Do you have your registration packet with you?" She handed the paperwork to Brenna and was given a red bandana in return. "You're going to be on the red team," Brenna announced. "The red instructor will be your leader for the next week. Wear your bandana at all times, as we're going to have a few team-versus-team challenges later in the week." "Got it," Miranda said in a meek voice. "And my instructor is Dane?" The assistant glanced up and gave her a searching look, a hint of a frown on her face. "I'm a hockey fan," Miranda hastily explained, lying through her teeth. "Plus, he and I go way back. High school and all that." She didn't point out that she'd gone to high school with Colt Waggoner, too. "You're not here for hockey or class reunions. You're here for survival training," Brenna said. "If that's going to be a problem, I can switch instructors—" "No!" Miranda squeaked, hiding the red bandana behind her back. "Not a problem at all. I just happened to notice it." "Well, unnotice it if you can. Mr. Croft doesn't care to discuss hockey," Brenna said, and peeked around, then leaned in to whisper. "You cause any trouble, and I'll move you to the other team." She gave Miranda a wide smile. "Okeydoke?" Jeez. Miranda nodded. "No hockey. Got it." "I'm glad we had this little talk." Brenna beamed at her and then moved down the line to the first person on the blue team. "Got your paperwork?" "She's a bit much, don't you think?" The man next to Miranda chuckled. "Glad she's not going to be our instructor, or this would really be a long week." Miranda gave a sheepish smile to the man. He was a tall, bordering-on-skinny guy who wore black-framed square glasses that hid his pale face. He seemed nice enough—strong jaw, thick sandy blond hair, and a friendly smile. Kind of cute, if you were into nerds. Shame she'd always had a thing for jocks. "I'm Pete. And you are..." He switched his bandana to his left hand and extended his right for a handshake. "Miranda," she said, shaking his hand and trying to do her best not to peer around him. He was blocking her view of Dane. "So what do you do, Miranda?" Pete asked with an easy smile. "Public relations? Pharmaceutical sales?" She gave him an odd look. "I'm a librarian." He laughed at that, as if anything she said was hysterically funny. "Really? Pretty young woman like you moldering in a library? I wouldn't have figured it." Okay, that was definitely flirting. Miranda stared at Pete for a moment, unsure how to answer. Flirt back? Ignore the flirting? She settled for polite small talk. "I take it back—I used to be a librarian. I'm taking a new job down in Houston in a few weeks as Chief Information Officer at a start-up electronics company. What is it you do, Pete?" "I own Hazardous Waste Games in Austin," he said, his smile widening with pride. "Oh, wow," she said, her attention drawn back to him. "You own a company?" "A billion-dollar company," he agreed proudly. "We make the biggest first-person shooter MMO in PC gaming." Like she knew what that was. Miranda gave him a hesitant smile. "Wow. Biggest first, uh, shooter. That's great." He nodded, glancing around the clearing in front of the lodge. "Our next project involves survival skills. I thought I'd check out the scene and see what it's like. Get a little first-person experience of my own." "Good idea," she enthused, but her interest was rapidly waning despite his friendliness. Dane was marching back toward the group, a resigned look on his face and a red bandana wrapped around his hand. Her breath expelled from her chest in a whoosh of relief—she hadn't realized how tense she'd been. If she'd been moved to the blue team...she'd have been screwed. And not in a pleasant way. Now that Dane was approaching, she could look her fill at him again. Her memories of him from high school had been vague and steamy—she'd recalled a tall, lanky boy with dark, messy hair, shoulders that seemed too wide for his lean body, and an easy smile. The man who paused in front of the team seemed to be the same, but different. The Dane she'd had her hands on nine years ago had been lean—this man was nothing but solid muscle. His biceps bulged from underneath the sleeves of his black T-shirt and were tanned a delicious shade of bronze. Back in high school, Dane had almost been too pretty—with a beautiful mouth, perfect nose, lean face, and piercing green eyes. The Dane before her still had the beautiful mouth and piercing green eyes, but his face had filled out, and his nose had been broken several times and had a large ridge in the middle proclaiming that he'd been in fights while on the ice. He had a scar just above it, and another one on his chin. She'd have thought the scars would make him less attractive, but for some reason they broke up the delicacy of his face and made him dangerous. There was even a third whisper-thin white scar running through his left eyebrow that gave him a rakish look. The rumpled black hair she remembered had been cut short and clung to his head in a thick cap that made her fingers itch to stroke it. And as he stood in front of them, she admired his shoulders. Still broad and blatant with muscles, but the rest of his body seemed to have caught up, and the entire picture was mouthwatering indeed. She felt a bit dismayed at the sight of him. Why couldn't he have been more torn-up looking? Why couldn't his face be covered with hockey injuries, his nose broken beyond all hope, and his cheekbones crushed like a boxer's? Why did he have to have those scars that made him look so damn...delicious? _All the better to seduce him,_ Evil Miranda whispered in her ear. Hm. Evil Miranda definitely had a point. This was all about seduction, and it'd be a lot easier to seduce a man when he was easy on the eyes. Heck, if he was easy on the eyes, it'd make it a pleasure for her to seduce him, rather than a chore. She could be down with that. Dane turned to say something to Brenna, and Miranda's gaze slid to his tight ass, outlined in his camo shorts. Definitely easy on the eyes, all right. She felt a little hot and breathless just looking at the way his hips narrowed. He turned and his gaze flicked to Miranda, catching her staring at his ass. A blush crossed her face and then she winked at him. To her vast delight, that seemed to fluster him even more, as if he hadn't expected that kind of reaction from her. Evil Miranda was delighted with that response. Right then, she decided to let Evil Miranda take the reins this week. "Welcome, everyone, to your week of survival training," Dane said in a low voice that made her thighs quiver. Heat flared, settling low in her hips. "For the next week, you're going to learn how to live out in the wild on your own. It's not going to be easy. You'll be sleeping on the ground, catching your own food, and learning the best ways to move about in the bush. We're going to have a team challenge against the blue team, and at the end of the week, you're going to have to survive on your own for a day using the training I give you. Understand?" Survive on her own for a day? Did that mean she wouldn't be spending the full week with Dane? She hid her frown. Okay, then, _six_ days to seduce the man and get naked photos of him. She could handle six days. Evil Miranda would just have to work a little faster. "The land we're going to be surviving on is the private property of the Daughtry Ranch. You're surrounded by ten thousand acres of nothing but trees and wildlife. We own this ranch, so anything you can bring down to serve as food, do so. No sport killing—this is to teach you how to survive, understand?" He cast a stern glare over the group, arms crossed over his chest. No one moved. "Now," he declared, "we're all going to empty our packs and I'm going to make sure you're not smuggling in anything to make this easier on you. Our instructions said to bring a utility knife"—he ticked the words off on his fingers—"a change of clothes, extra socks, and three Ziploc bags. Nothing else." Miranda stiffened, her hands tightening on the straps of her backpack. She was going to have to show the contents of her bag? Oh crap. This could be awkward. Or embarrassing. Or both. As she hesitated, Dane smiled at the first person in their small line, took his backpack, and upended it on the ground. She groaned inwardly as the man's gear came spilling out and Dane began to pick through it. "Not allowed," he said, pushing aside the first item. "Not allowed, not allowed." Oh yes. This was definitely going to be bad. She watched as the man—Will, she thought his name was—stiffened and looked as if he were about to mutiny before the class even started. Assistant Brenna was right at Dane's side, taking the beef jerky, cell phone, and travel thermal blanket pouch that were handed to her. "You'll get this stuff back when the class is done," she said in a take-no-arguments kind of voice. "Not a moment sooner." "I need that phone. My company is securing a deal this week—" "I'm sorry, did you want to remove yourself from the class?" Brenna said with a cheerful, innocent smile, waving the phone in front of his face. "Because if so, I'd be happy to refund you the tuition—minus your deposit, of course." "No, ma'am," Will said in a resigned voice. He gave the phone a last longing look and then sighed, stepping back into line. Dane smiled and clapped the man on the back, leaning in and murmuring a few words of encouragement that Miranda couldn't make out. Whatever it was, it had the desired effect—Will perked up again and gave Dane a rueful smile. He'd always been good at charming people. The prick. Even so, she squirmed a little, imagining him leaning in and telling her that she'd been a naughty girl. Even just standing in line, she was getting turned on by Dane's presence. Evil Miranda was going to have a field day this week. Miranda's nerves grew taut, her grip on her backpack tight as Dane went down the line. A few people grimaced and gave up their contraband—emergency flashlights, granola bars, another cell phone—and Pete had seemed extremely put out to relinquish bug spray. Still, he did so, falling prey to Dane's easy charm even as Brenna snatched the contraband from him. And then Dane was standing in front of her. She swallowed hard as he put his hand out. "Well, uh," she said, stalling. "How about I promise you that I don't have any contraband and we'll call it even?" He raised an eyebrow at her, his friendly smile curving into a frown. "Is there a particular reason why you don't want me to check your bag, Ms...." Miranda gave him a smile. "You know who I am." "Nine years ago I did," he admitted. "A lot happens in nine years." "It's still Hill. No marriage, no divorce." She wondered if he would think that was a good thing or a bad thing. And before he could respond, she offered, "And I'll show you my bag, but it has to be in private. You know. Because of girl stuff." That scarred eyebrow went sky-high. He looked down at her for a long moment, and Miranda kept a casual smile on her face, though her skin was prickling from nervousness. If he dumped her bag out on the grass in front of everyone...well, it was going to be a _really_ long week. "All right," he agreed after a long, tense moment. "Let's go behind that tree and you can show me." She nodded and headed in that direction, feeling a wild sense of relief. No public humiliation. That was good. They moved behind a large juniper tree, the bushy spread of branches allowing for privacy. Dane gave her a speculative look as they paused, and he crossed his arms over his chest. "Now, what's this about?" "What makes you think this is about something?" She opened her eyes wide, feigning innocence. He arched an eyebrow. "Because this all seems a little coincidental, don't you think? My ex-girlfriend from high school shows up for my first survival class, and she's just as beautiful as I remembered?" Her nipples hardened at the husky, almost teasing tone of his voice, and she placed a hand on the tree behind her, steadying herself. _Down, girl._ She felt conflicted, staring up into his rugged face. Was it possible to still be insanely attracted to the man even if she hated him? She'd only been around him for five minutes and already her body was perking up in response. So he thought she was beautiful? "This isn't a setup. I assure you." At least, it wasn't in the way he thought it was. He gave her an "I'm waiting" look and didn't budge from his spot in front of her. "It doesn't explain why you won't let me see what's inside your bag." Ah yes. That answered it. She _could_ be physically attracted to a man and still want to kick him in the balls. Mentally steeling herself to put her plan into action, she shrugged the bag off her shoulder and then held it out to him. "Here you go." Like the others, Dane took her pack, unzipped the top, and then dumped the contents onto the ground. Lingerie fluttered to the grass: red silk bras, black lace thongs, and a pair of pink silk panties—her favorite—with a ruffle across the derriere. On top of it all lay her camera. He stared down at the mix for a long moment, and then looked back at her, astonished. "What's all this?" "Survival gear," Miranda said, her voice slightly husky with nervousness. She forced herself to step forward, her body trembling with nervousness, and laid a hand on his chest. Oh my. A really, really defined pectoral. That was new, and very welcome. Her mouth went dry and she glanced up at him, where he stood stock-still, looming over her. Her voice lowered to a whisper. "All those panties are terribly essential to my survival this week." His body tensed under her hand, and she waited. Waited for him to push her away and scowl in her direction, all business...or for him to respond to her touch. Dane looked down at her hand, small on his chest. Then he looked back at her, his expression inscrutable. His voice was low. "What's going on, Miranda?" This was her moment. She knotted her fingers in his shirt and tugged him forward, and was gratified to see that he let himself be pulled forward. Her breasts bounced against his rock-hard chest and she felt a thrill of excitement, a spurt of adrenaline, and a heady rush of desire. She tilted her head back and looked up at him, her gaze sliding to his mouth. "We never slept together nine years ago, Dane Croft. Do you remember? You were supposed to take my virginity and you were a no-show." "I remember lots of things about you, Miranda Hill," he said in a low, husky voice. "Miranda Hill," he said, repeating her name. "No marriage, no divorce. Just Miranda Hill." Her toes curled and she pulled him a little closer. His mouth was now so close to hers that his breath fanned against her cheek, warm and sweet. "I heard you were back in town," she whispered, and she could smell the wonderful, crisp scent of him—all man and musk and just a hint of sweat and fresh outdoors. "And I wanted to see if maybe we could pick up where we left off, and see how that works out." Then she deliberately licked her lips, her mouth so close to his that her tongue grazed his lips. He groaned and moved forward—just an inch or two, but enough that his mouth brushed hers, and she felt his tongue brush against her own. Her lips parted and she welcomed the caress. When his tongue dipped into her mouth and thrust, she felt it all the way to her core, and her sex began to pulse with desire. Her lips parted wider, and her tongue met his next thrust, tangling with his. He still kissed as well as she remembered. Better, she thought as he stroked against her tongue in a move that made her nerves thrill and her pussy clench. Someone in the distance coughed, and Dane froze against her. She stroked her tongue along his lips, determined to win the contest of wills, and was rewarded when he pulled away from her grasp, dazed. "What's the matter, Dane?" Her voice was soft, playful. "Or is it time for the strip search?" He pulled her hand off his chest and glanced over back through the trees, where the others still waited for them. Dane swore lightly under his breath and raked a hand over his crisp skullcap of hair. Then he knelt and pocketed her camera and began to stuff panties back into her bag, scowling. "This week isn't about hooking up, Miranda." _Says you_ , she thought, but said nothing. That merest brush of a kiss had inspired her. Nine years ago, she'd never gotten to hook up with Dane, and she thought she'd use this week to torment him, drive him mad with lust, and then get the pictures she wanted. But now that he'd pocketed her camera, she had a new idea...one that involved a side benefit for her. She'd always wanted to sleep with Dane. At least, when she was younger, she'd wanted it. He'd been the only man who had ever been able to bring her to an orgasm—and they hadn't even slept together. After he hadn't shown up to take her virginity and the photos went online, she'd grown a bit of a hang-up about sex. She'd lost her virginity to the first nonlocal boy she'd dated, just because it wouldn't get around town. That had been an awkward, embarrassing situation for them both. Her relationships after that hadn't gotten much better. She'd bed-hopped between several men during her college years, all equally unable to get her to the point at which her brain would turn off and she could relax and orgasm. She _couldn't_ relax, couldn't come, couldn't enjoy the moment, and every relationship of hers usually ended a night or two after sex was introduced. After a while, she'd given up on dating, sure that her issues with Dane and her mother's issues with men had permanently ruined any chance of a normal relationship. She'd stopped dating...and had bought a vibrator instead. But just being in Dane's presence already had her turned on more than any man she'd been near in years. And she remembered that the time she and Dane had been petting in the closet, she'd come hard and come fast. She wondered if she could do it again—have an orgasm with a man. It shouldn't be hard, but it seemed that every woman on earth could manage one but her. Maybe she just needed the right man to experiment on, even if it was Dane Croft, her most hated enemy. This week was the perfect time to find out, she thought as she gave him another predatory look. He finished stuffing the last of the clothing back into her bag and patted his pocket with her camera. All right, her plans would have to change a little. Seduce and fuck the man this week. Have fun. Use him for sex—and, hopefully, orgasms. Then, when the week was over, she'd invite him back to her place and take the pictures then. That still worked. She smiled wickedly. Evil Miranda approved of using men. Dane scowled and shoved the bag back into her hands. "The memo said for clothing you can live in for the next few days. That means hike, eat, sleep, and possibly swim in your clothing." Evil Miranda came out to play again. She leaned forward and trailed a finger down the front of Dane's chest. "I can hike, sleep, and swim in my lacy panties," she said, her voice a low purr. "But eating in my lacy panties? That's going to have to be left to you." Then she licked her lips and smiled seductively at him. Dane swore under his breath again, and then bolted back for the others, adjusting himself as he did. Evil Miranda, one; Dane, zero. # FOUR Well, damn. Dane checked the equipment in his backpack one more time, determined not to stare at Miranda as she sauntered back to the small group, her rounded hips swaying in a completely seductive motion that had his cock erect. Hell. Five minutes into their inaugural class and he was already in trouble. Of all the women in the entire world, Miranda had to be here in his class. Approaching him. Kissing him, her intentions clear. _I wanted to see if maybe we could pick up where we left off._ The woman had packed herself an entire bag full of lingerie for a weeklong survival trip in the woods, an obvious signal of her intent. And...damn. He pictured Miranda's curvy figure in a particular pair of ruffled panties, sitting in his lap at the campfire... Then he promptly put it out of his mind, thinking of other things. Unsexy things, cold showers, anything to get his mind back in the game. He couldn't let a pretty flame from his past distract him from his job. His eyes narrowed. Unless...that was why she was here. Maybe Miranda had volunteered to tempt him, bait for a very sexy trap. She didn't seem like the type to sign up for a weeklong outdoor excursion, he thought, looking over at her smiling, too-cheerful face. Maybe Colt and Grant had set her up to be in his group. Colt had been awfully chummy with her back during the introductions. He remembered seeing the two of them in a conversation, and Miranda had laughed. The sound had scraped on his already ragged nerves, and he'd barked Colt's name to drag him away from her. When Colt had come back, he'd teased Dane. _Now it's gonna be extra-hard for you to keep this soldier at ease_ , he'd said, and patted the front of Dane's pants. He'd tried to deck the guy in response. Still regretted missing. Maybe she was in on this with Colt. Grant, too. Maybe they'd pulled out the sexiest girl in Bluebonnet and had hired her to come to the training class to test his willpower. Maybe they didn't trust him to keep his dick in his pants like they said they did. That certainly made more sense than anything else he could think up. And it pissed him off. Dane watched Miranda smile at the other men, her hand playing with the high neckline of her T-shirt. He frowned. She wasn't exactly dressed like a seductress. For one, her shirt was modest to the point of puritanical. For two, she didn't wear a bit of makeup, and her hair was simple—long, straight, and smooth. Not that she needed makeup to look good—her fresh skin and pretty brown eyes had smiled at him in a way that reminded him he hadn't had sex with a woman in a very long time. Well then, if Miranda was a plant to test his willpower, it was one test that Dane intended to ace. They'd kissed, but that had been a mistake. He wouldn't let it happen again. Time to get down to business. Miranda toyed with the collar of her shirt, a nervous habit from years past. At her side, Pete smiled at her in a way that seemed a little too friendly. He kept looking over at her and grinning, and she gave him a faint smile back because what else was she supposed to do? Ignore him? She wondered, briefly, if he had seen the waterfall of lingerie that had cascaded out of her pack when Dane turned it over, and she cringed. It'd be even worse if he'd seen her come on to their instructor and kissed him. That would make this week really, really awkward. "So have you ever been camping before?" Pete asked her as Dane moved back toward the group. "I have to admit, I didn't expect to see a pretty girl here on the team, but it's a welcome surprise." Keeping the polite smile on her face, Miranda turned back to Dane, trying not to show her irritation. "I don't see why it's so unbelievable that a woman wants to go on a survival trip. I was in Girl Scouts. They do camping, though not quite like this." He chuckled at her in a condescending way. "Well, get ready for the ride of your life." She glanced over at Pete. "Why? Have you done one of these trips before?" Pete flushed a little. "Oh. Uh, no. I've just heard that it's going to be a crazy week, that's all." He seemed so very uncomfortable at being put on the spot that she gave him another smile to make him feel better. So far Pete seemed to be a lot worse with people than she was. In the distance, the blue team began to hike into the woods, angling away from the enormous wooden cabin that served as the lodge and headquarters. Dane turned back to his team, clapped his hands together once, and rocked on his heels. "Are we ready to begin?" She noticed he didn't look over at her. Instead, he made eye contact with every man on their small team and then swung away before his gaze could reach hers. Miranda bristled at that, twisting the red bandana around her wrist. Plenty of time to win him over. Lots and lots of time to seduce him. She just needed to be patient. "I have a few rules before we move into the woods," Dane began. His hands moved to his hips and he stood before them, legs apart and arms akimbo, larger than life. "For starters, this is not going to be an easy trip. We're going to be doing a lot of hiking. Some swimming. We're going to learn how to make traps and tie knots. We're going to fish for our dinner and forage what we can't find in the streams. We're going to learn to build shelters and how to start and maintain a fire. It's going to be very cold at night and very warm in the daytime. You're going to sweat. You're going to get dirty. And if you're a little too fancy-pants for this sort of thing, it's best that you leave now." Miranda watched Dane curiously. Did he...did he just blush when he said the word _fancy-pants_? His gaze slid over to her, and then back away again, and she felt a surge of triumph. He was thinking about _her_ _panties_. She let a smile curve her mouth. Well, now. "Most of all, we're going to learn what it's like to work as a team." He turned to them and started to pass out PowerBars. Three were handed to Miranda, and she stared down at them, then at the canteen that was handed to her. It didn't look like much water. "Those are all of the supplies you're going to have this week," Dane announced. "The first day's meal and drinks are on me. The rest is up to you." One person groaned. It might have been Pete. To Dane's credit, he ignored it and continued speaking. "We're going to assign partners for this week, and sometimes you and your buddy will pair up for challenges. Are you with me?" At their nods, he gestured. "Now, each of you pick a buddy and let's get going." Pete turned and gave Miranda a hopeful look. Well, why not. Miranda gave him a thumbs-up and an awkward smile in return. He looked the least woods-capable, and she could always hope he would bail early, leaving her with no choice but to buddy up with Dane. "All right," Dane said when they were standing next to their partners. He pointed at the two older men that had teamed up. "George and Jamie will walk at the front of the line. We're going to do this single file, since it makes things easiest. Next will be Steve and Will." He gestured at the other male-male team, and then finally at her and Pete. "Pete and Miranda can take up the rear. Miranda, you in front of Pete, since that's safest. Call out if you two start to get left behind." "I know how to walk," she said in a bristling tone, offended. God, the man really was a jerk. "Just because I'm a girl doesn't mean that I'm incompetent." "No, of course not," Dane said. "You're right. You take up the rear. Pete, you walk in front of Miranda." Well, that wasn't exactly better. She suspected that she was getting stuck with end-of-the-line duty so Dane wouldn't have to look at her. Figured. "All right, let's go." Dane waved them forward and then turned, beginning to march through the trees. He pulled out a satellite phone, murmured something into it, and then shut it off. One by one, they fell into place behind him. Miranda picked up the rear, her light pack bouncing against her shoulders as they walked. The ground was uneven and a bit rocky in places, and pebbles scattered under their feet. "Now, one thing I should warn you guys about this ranch," Dane said, glancing back at them as they walked. "Five years ago, this ranch was the biggest emu farm in the South before the owner died and the assets were sold off. The emu were moved to different breeders, but there're one or two still trotting around on the property. Just warning you in case you run across one this week." One of the businessmen—Miranda thought his name was Steve—stopped. "What the heck is an emu?" "Big, giant bird," Dane said casually. "About five feet tall. Kind of like an ostrich, but with a nasty temper. Just be on the lookout." "You mean to tell me that you've got overgrown birds out here?" Steve sputtered. "Are you even sure that's safe?" "Not sure if it's safe," Dane drawled. "But there's not a lot that's 'safe' in the wild, if you catch my drift. You're here to learn how to take care of yourself in the wilderness, and maybe that includes a lesson on emu-avoiding, maybe not." His big shoulders braced and then he began to jog up an incline. "Come on, let's pick up the pace. We've got a long way to go tonight before we hit the area I've designated for our camp." They had no choice but to follow their leader. One by one, they followed Dane up the incline, Miranda's sneakers sliding on the gravel a little. "You all right back there?" Dane called. "I'm fine," Miranda called back, her tone a little strained. This was already getting tiresome—she could walk just as well as any of the men. "Don't you worry about me." "Oh, I won't," Dane called back. "Let's go, men. And woman." And with that, he began to jog into the woods. That day, Miranda began to have a reluctant appreciation for Dane's athleticism. They jogged for a bit—mostly to get away from the headquarters cabin and into the wild. Once the lodge had been out of sight for about fifteen minutes, Dane slowed the group to a brisk hike. He led the way, climbing over rocks and brush like he was born to the wild. The rest of the team followed behind him, much slower and far clumsier. Two of the older businessmen were panting and frowning, but all kept up with their instructor as he skirted them up a dry creek bed, grasping tree roots for handholds. The others followed close behind, and when it came to Miranda's turn, Pete offered a hand down to her. She ignored it and clambered up the side of the embankment on her own. "I'm fine, thanks." Her voice was crisper than it needed to be, but she was getting irritated at him and it was only day one. And what a long day one it was. She was covered in sweat—they all were—and tendrils of her hair were sticking to her face, but she was keeping up with the men and that was just fine. Pete seemed to be struggling, his steps slowing as they hiked. As they walked, Dane began to chat with the students in his group, his voice casual. It had always been easy for Dane to make friends, Miranda remembered, and he seemed right in his element. The men told him about their jobs, their families, and previous hunting trips they'd been on. When it came to Pete's turn, he immediately began to brag about his company. "I run Hazardous Waste Games. It's a billion-dollar company—" Miranda rolled her eyes and tuned him out as he rattled on about shooters and gaming. "—but that's me," Pete eventually concluded with a grin. "Married to the job unless I can find the right woman to make a new man out of me." He grinned back at Miranda, obviously thinking that his business assets made up for his lack of other assets. She made a face and hoped he couldn't see it. His chalk-pale face was flushed a florid red, and his hair was stuck to his forehead. Big circular sweat stains had bloomed under his arms, and he was beginning to smell. He needed a woman to make a new man out of him? It sure as shit wasn't going to be her. As if sensing her thoughts, Dane spoke. "What about you, Miss Hill?" She jerked, startled. "Oh, I'm not interested in finding a husband, thank you. I'm here to learn survival skills." The men snickered. "I meant," Dane said in a patient voice. "What about you? Tell us about your family." "You know my family, Dane Croft," Miranda said. "My mother, Tanya, runs the local antiques store, and my daddy was a no-good trucker who only showed up every few years. I'm surprised you forgot." Before he could reply to her jab, she hurried on. "But other than that, it's just me. No husband, no kids." Two and a half weeks and she'd be working at a massive corporation, doing what she was always meant to do. Everything between now and then wasn't important. It was just finishing off the story of Old Miranda, the Boobs of Bluebonnet. Soon she'd just be a local legend. Her eyes narrowed at Dane's back. A local legend with a really good ending, she decided. When the sun was high in the sky, the group made it to the area Dane had designated as the first campsite. The men grumbled and joked about how tired they were. Dane had half expected Miranda to complain, but she was the only one on his small team that took the rigorous exercise in stride. Her face was flushed and sweaty, but she remained strong and calm, and he was reassessing his initial thoughts about her. With those bright red bras and silky panties, Dane had thought she'd be a girly girl. Maybe not. He was sweating, too, his body aching in a good way at the exercise. The weather was perfect. Just cool enough to make the day pleasant, and warm enough to take the edge off of the night. He inhaled the fresh air and grinned to himself. He was enjoying the time in the woods, even if his team was not. But then again, it was his job to make them believers, and he had a week to do so. He pulled a piece of paper out of his pocket as they rested, mentally going over his notes. This week was more than just survival skills—it was about building teamwork. Dane was not exactly an expert on teamwork. He'd sucked at it in a jersey, and just because he had survival skills didn't mean he could make people work together. Colt and Grant had insisted, though—corporate sponsors wanted practical applications for survival skills, and couching fire-making and shelter-building as team exercises was the way to do it. He'd have to go by their rules. On his crib sheet, he read Brenna's flowery, bubbly handwriting, the _I_ 's dotted with hearts. _Blindfolds for team-building. Make them do tasks together. Rope challenges. Team challenge on day 3._ Right. He turned to address his team. "We're going to make camp here tonight. And as we set up the camp, I'm going to show you how to do the basics. I'll build you a basic shelter and show you how to make fire and we'll boil drinking water. Then I'm going to set snares and show you how to fish, and we'll forage for what we can eat tonight." All eyes were on him, their faces expectant. He continued. "And you'll need to pay attention to these lessons, because tomorrow? It's going to be on you guys to do it all yourselves. Understand?" They nodded. "This doesn't seem like much of a team exercise," the nerd with glasses complained. "It sounds like you're just showing us stuff. How are we supposed to apply that to our lives in the corporate world?" His first nonbeliever. Grant had told him to prepare for this sort of thing. Didn't make him have to like the guy, though. Dane put on his most charming smile and tried to recall the man's name. He stared at him for a long moment. "We just got here, Pete," Miranda murmured, shooting Dane a meaningful look. "I'm sure this all has a purpose." Pete. He'd remember that now. "It does have a purpose," Dane agreed. "Which I was just about to get to, if you'd give me a moment." Though his words were harsh, he kept the smile on his face. "Sorry," Pete said, but he wasn't looking at Dane when he said it. His gaze was on sweaty, too-cute-for-this-trip Miranda, with her high-necked shirt that was sticking to her rather admirable cleavage and outlining it in a way that was far more blatant than a low neckline would have been. _Concentrate, Dane. You've seen plenty of breasts in your life, and that particular pair is here specifically to distract you._ Dane cleared his throat. "Like I was saying...this afternoon I will show you how to do six different skills. Each of you should pay attention, of course, but each of these skills will be the sole responsibility of one of you on the team. It will be your job to handle that particular chore for the group all week, and it will also be your job to show the others how to do your job. As you learn each other's skills, you'll see that all hands are needed to have a fully integrated campsite, just like a team. Understand?" "Yes," they chorused back to him. "But first...we're going to do a different kind of exercise," Dane said. "Can we get something to drink first?" Pete piped up, an edge of a whine in his voice. "I'm freaking parched and I drank everything in my canteen already." This was going to be a very long week with Pete around. "There's a stream nearby," he explained. "However, that stream is filled with bacteria. Any water you drink is going to have to be boiled first, unless you'd like to be on the receiving end of a nasty strain of giardia. And to boil water, what do we need first?" "A cooking pot?" Miranda said helpfully. It was the first statement she'd directed to him since they'd left the lodge area. "This week, you'll be stringing your canteen on a tripod and boiling the water over the fire," he said. Miranda gave him a slow smile, as if they were sharing a secret. "Fire, then. We need a fire first." That husky, playful timbre in her voice jogged his memory, and he caught himself grinning at her despite himself. When she smiled, he couldn't resist her. But he guessed that Grant and Colt knew that, too, and that was why they'd sent her on this trip. His friends needed to butt the hell out. His mood taking a sour turn once more, Dane gestured at the woods. "That's right. First we need a fire, and to make a fire, we need wood. Since we are going to have a fire going all night, we're going to need a lot of wood. And here's how we're going to get it." He clasped his hands together again, and then gestured at each of the mini-teams. "The three pairs will split up and go in different directions. One of the partners will be blindfolded, using the team bandanas that we've provided for you. The other partner will be instructing you on where the wood is and guiding you without touching you. When your arms are full, you'll return to camp, drop off your load, and then the blindfolded partners will switch. You'll have equal time under the blindfold, and equal time to be the guide. Understand?" "How do we know we won't get lost?" George said. "We don't know our way around these woods." Dane had been prepared for this, and he pulled out a package of wristbands from his backpack. "These are GPS trackers that emit an electronic signal. If you get lost, I can find you. No worries." He passed them out to the teams and one by one, they strapped the GPS trackers on their wrists. As he handed Miranda hers, he noticed there was a tiny crease in her smooth brow, as if she were unhappy with this turn of events. "There a problem?" he said in a low voice to her. She looked up, startled. "Oh. No, no problem." She quickly strapped her tracker to her wrist and turned to Pete. "You can be blindfolded first, though, okay?" Pete shrugged. "Fine with me." The teams blindfolded their partners and began to set off in the woods. One of the businessmen—Steve—barked orders to his partner in a booming voice that echoed in the woods. Dane made a mental note to have a chat with Steve later and discuss why it was bad to talk at supersonic levels in the forest, especially when some of the tasks coming up would require stealth and avoiding the other team. His ears strained, and he could pick up Miranda's soft, husky voice. "Left, Pete, left," she was saying as she guided him past one tree and then another. "Two steps forward—I, no, two, Pete. Pete. Pete! Look out—" He watched her cringe as her partner ran into a low-hanging branch. They seemed to be having a bit of trouble, so Dane planted the team flag in the center of camp, and then jogged over to trail behind Miranda and Pete. "You've got to give me better directions than that," Pete was complaining to Miranda, stretching his hands out. "My directions are perfectly fine," she argued with him. "Or they would be if you'd actually listen to them. When I tell you two steps, I mean two steps, okay?" "Two steps," Pete agreed. He leaned forward, his hands searching as he moved. "Is there any wood nearby at least? Miranda thought for a moment as Pete's hands flailed. "There's a fallen limb about three paces to your left," she began. Pete immediately did a complete about face and plunged toward her. His hands landed squarely on Miranda's breasts. Shit. Dane strode forward, ready to break the two of them apart. It was day one and already it was looking like he'd have a sexual harassment suit on his hands. This was bad. This was very, very bad. And judging from the startled—yet pleased—look on Pete's face, it wasn't entirely unintentional. That did not surprise Dane. The nerd had been gunning for sexy Miranda since she'd showed up. He was, however, completely surprised when Miranda shoved Pete away and decked him square across the jaw. # FIVE This was not shaping up as she'd planned. Miranda trembled with anger as Dane stepped between her and Pete. He stretched an arm out to keep Miranda off of the fallen man, not that he needed to. She'd flattened him with one well-placed punch. Her hand throbbed like mad but that was all right—it could swell like a balloon and she wouldn't take back the hit. Pete had totally deserved it. He pulled off his blindfold and stared up at the two of them, his glasses askew over his eyes and his expression shocked. His hand moved up to rub his jaw. "You hit me!" She resisted the urge to clutch her collar closed, to make sure that there was not an ounce of cleavage exposed. Instead, she forced her hands down to her sides, and they clenched there like fists. "You grabbed me," she hissed back at him. "Don't you _ever_ do that again." "It was a mistake," Dane said, turning to face her. He put a hand on her shoulder and shifted, blocking her sight of Pete sprawled on the ground. "Let's calm down about this, all right?" There was a thread of concern in his voice, and his brow was furrowed as he looked down at her, as if he hadn't quite anticipated having a woman on his team and didn't know what to do with her. "You okay?" She nodded, biting off any angry words. Instead, she crossed her arms over her chest protectively and stalked back to the camp flag, where they'd left their packs. "Pete," Dane said in that easy voice. "Why don't you go and gather firewood on your own? I'll talk with Miranda and make sure there's no problem." "Why should there be any problems?" Pete said defensively, his long fingers swiping at the grass stuck to his shirt. "She's the one that hit me." "Go on," Dane said equally pleasantly, though Miranda doubted his tone was sincere. It sounded a little forced. She glanced over out of the corner of her eye and watched Dane help the other man up, brushing off his clothes. They glanced over at her before Pete shrugged and headed off into the woods, leaning over to pick up a fallen branch. At that, Dane turned and began to walk back toward her. Miranda worried at the tense set of his shoulders and the frown on his face. Shit. This wasn't working the way she planned. Her temper had gotten away from her when that creep had touched her boobs. This wouldn't work. She needed to clear her head and concentrate. She wouldn't be able to seduce Dane if she got sent home for fighting with her partner. When Dane returned, she tilted her head and offered him a half smile, her hand fluttering back to her collar protectively. "Sorry. Knee-jerk reaction." "You sure you're okay?" He said, not moving from in front of her. He scratched his head, rubbing at the closely cropped hair in a gesture she remembered from high school. "Do you want to go back? I can take you to the lodge if it's going to bother you being out here with six men. I don't want any problems this week." "I'll be fine. He just took me by surprise and I reacted." Dane looked skeptical. "Look," she said and took a step forward. The smile curved her mouth, and she forced her voice to be teasing again. "If anyone gives me any trouble, I'll let them apologize before I hit them. I've only got one good hand left." She tried to put her hand on his chest, bridge the space between them. He stopped her, catching her hand in his, and examined her knuckles. His fingertips brushed over them and he pulled her hand close to his face. "Did you hurt yourself?" She watched him curiously, her gaze focused on her hand in his. She could feel the rough callus of his hands against hers, and she had to admit that it sent a tingle through her. "I'm all right. I took a self-defense class in college. I know how to punch a creep without hurting myself." Dane glanced up at her and gave her a wry smile, rubbing her knuckles with his thumb. "Can you refrain unless it's absolutely necessary? That creep paid the same two grand you did for the survival trip. And if he needs to be hit again, I might have to be the one to do it." She laughed, and then was horrified at the giggle that escaped her throat. Evil Miranda _never_ giggled. She should have given a sultry chuckle instead. But it made Dane smile, and so she moved forward to lay her hand on his chest again, giving him a very interested look. "Want to kiss my boo-boos and make them better?" He released her hand as if he'd been burned, turning to the campsite. "We need to start building the fire. Bring over some of that wood, will you?" Miranda resisted the urge to put her hands on her hips, frustrated at his skittishness. "Fine," she said, careful to hide the annoyance she felt. She had Dane alone for a few precious minutes, and she needed to capitalize on it. She glanced over at him as he crouched near the spot he'd selected for the fire, tossing a few rocks out of the area. Her gaze slid to his ass, tight in his shorts. And she got an idea. Moving to the firewood, Miranda bent at the knees and crouched in a way that would make her shorts ride low and expose her black lace thong and the pale expanse of skin it drew across. She glanced over her shoulder, but Dane was facing the other direction. Dammit. She turned and looked at the biggest log—no more than the size of her arm around. She could pick that up no problem, but doing so wouldn't serve her purposes. So she feigned a deep sigh. "I think this one is too heavy. What do you think?" She glanced over her shoulder again to see if he was looking at her. This time, he was. He turned and she saw his expression change from exasperation at her weakness to...something else. His gaze slid down to her skimpy black thong, evident over the low waist of her shorts, and she resisted the urge to stand up and cover her backside and then clutch at her neckline. Evil Miranda would _not_ approve of covering up. He seemed to swallow hard. "Are you wearing a thong?" "What, this old thing?" she drawled, and gave her hips a little wiggle that caused her to wobble slightly on her feet. "Of course." As if she never left her house in anything but Victoria's Secret's tawdriest. He moved closer to her, his gaze still on the expanse of flesh she was exposing. "You do realize that's not exactly camping appropriate?" She shrugged and tossed her long hair over her shoulder, watching as his gaze flicked to it, then back down to her thong. "It's a little tight against my skin," she admitted in a coy, secretive voice. "I don't suppose you could...help me with it?" Even that was a little forward. Very Evil Miranda. Good Miranda would have been totally mortified. But Dane moved down next to her, his voice low and husky in her ear as he spoke. "Sure thing." She arched her back and closed her eyes, anticipating his touch. His hand grasped the back of her panties roughly, and her eyes flew open at the uncomfortable wedgie. Before she could protest, he pulled the thong away from her skin. She heard the click of his knife, and then the chopping hiss as he sliced through the fabric. Dane let go of her and she fell forward, the fabric loose between her legs. Her hand flew to her bottom and sure enough, he'd cut the thong square up the backside. The crotch of it dangled against her leg. She gasped. "What did you just do?" "We're on a survival trip, Miranda." The look in his cool green eyes was no-nonsense. "First rule of survival is to be as comfortable as you can, and you should remember that." He stood and turned away, then extended a hand toward her. "Or wasn't that what you had in mind when you asked me to help you with them?" His eyes were knowing as they looked at her, glimmering with amusement. Dane knew her game and he wasn't going to make it easy for her. She glared at him. He gestured at the wood behind her. "Hand me one of the lighter logs, would you?" "I can't believe you just cut my panties!" "Probably a good thing you packed a whole bag of them, then, eh?" Forget revenge and ruining his life. She was going to kill him before the week was out. Forcing a tight smile to her face, she slapped a piece of wood into his hand. "Good thing," she echoed sourly. "I'd hate to have to go without." His laugh strangled off and he gave her an intense look, and then stomped away. Evil Miranda two; Dane Croft zero. Miranda was going to be the death of him this week. Dane couldn't get the sight of her thong out of his mind. He tried not to think about her panties when the others returned with enough wood for the evening. He tried not to think about it when he cut a few longer branches and began to show Steve (who had been designated as the shelter expert) how to build a shelter. He tried not to think about the pretty thong or the sweet curves of skin it laced over as he showed Will how to strain water using a waterproof hat. He tried not to think about it when he showed Miranda how to build a fire using the bow method and her long, glossy hair swung over her shoulder as her arms worked. And he really, really tried not to think about the fact that he hadn't had sex in more than three years. But he thought about it anyhow. It was like the image was burned into his memory and no matter what he tried to focus on, the vision of it kept rising in his head. Of the soft globes of her ass, pale against the thong, making him think of how she'd look riding his cock, that delicious ass bouncing as he pumped into her. He'd been hot for the teenage Miranda, but the Miranda nine years later was burning a hole in his mind and making his dick primed for sex 24/7. _Correction_ , he thought, his mind going back to that damned thong. Nothing but sex with _her_. No one else. _You've been waiting nine years for this. And she wants it—wants you._ They banked the fire and prepared for sleep. It was early in the evening but the team was drooping and the sun was down. They looked exhausted and tired, and even water and their PowerBars couldn't perk them up. "We'll call it an early night," he announced, throwing a heavy log over the fire's coals, steering his gaze away from Miranda, where she looked slightly sweaty and disheveled and completely and utterly delicious. "Be ready to get going first thing in the morning, as we're going to change campsites and the real survival will begin." One of the men groaned, but the others got up and stretched, heading to the larger of the two shelters. It was a lean-to with a thick amount of deadfall tossed on the wind-facing side and crosshatched with leaves and branches. The ground had been smoothed and the packs lay underneath. "How's this going to work?" Steve had said earlier. "We have one shelter, six guys, and a woman." They had ended up building Miranda a separate small lean-to a few feet away so she didn't have to huddle with the men. She went and sat under it, adjusting her pack before lying down on the hard ground and using it as a pillow. As Dane watched, she crossed her arms over her chest and seemed to huddle in, bracing against the cold. She looked small and alone as the other men piled into the larger shelter, and Dane couldn't help himself—he went over to check on her. "You going to be okay over here?" "No problem." She gave him a tired thumbs-up, then hugged her arms close again. He hesitated. "If you get too cold, just crawl in with us. We'll be fine thanks to body heat." "I'd rather not, but thanks for the offer," she said, closing her eyes to end the conversation. "I'm good here." Dane nodded and went back to join the others. He glanced over at Miranda one last time, but she had her back to him. Dane's offer was the perfect opportunity, Miranda thought as she shivered in her shelter. Sleep wasn't happening—not with the chill bite on the ground and the fact that she really didn't have anything much warmer to wear. This camping trip was an exercise in survival, but the next time she opted to "survive" she'd pack some thermal underwear first. As it was, her light jacket wasn't helping much. The men didn't seem to have the same problem—she could hear them snoring quietly. Occasionally one would move around, shifting to try and get comfortable. The fire popped, but other than that, the camp was silent. It gave her a lot of time to think. And the main thing she thought about? Her battle plan to seduce Dane. The fact that he'd sliced her thong off had thrown her for a loop. He'd seemed interested—really interested—but he'd given her the worst kind of brush-off with that insulting move. Did he not find her attractive? Was that it? In high school, he'd been a flirt and had teased all the girls—including her. When he played hockey for the Vegas Flush, she'd heard all kinds of rumors about whom he was dating—London Harris, the sexy socialite who loved to be in the tabloids. Molly Sun, the starlet with the yellow curls and huge breasts. Susie Lynn Jacobs, the ingénue country singer. The list read like an issue of _People_ 's Most Beautiful. And he was totally ignoring her. Miranda's hand went to her collar and she grasped it, making sure that it didn't gape open. Men flirted with her occasionally—at the library, at the county fair, when she'd gone for her job interview. Even Pete had shown interest. Why was Dane not interested? She was practically throwing herself at him. Well, she amended, "throwing herself at him" was relative. As she was not the best flirter in the world, she acknowledged that maybe she wasn't doing enough to show her interest. Maybe he thought she was just being a tease? It was time to be totally straight with Dane and show him exactly what she wanted. Sucking in a deep breath, she got up from her lean-to and approached the men's. They were bundled together like puppies—all neat and lined up, their feet hanging out of the shelter. She studied them for a minute, looking for a familiar form. Dane was on the far edge of the shelter, lying on his side. She approached and knelt beside him. All was silent in the camp, and Miranda inched closer to Dane, studying him as he slept. His shoulders seemed broader than ever from this angle, his hips narrow and tapered. She moved in and gave him a slight shake. Nothing. She frowned and slid her hand onto Dane's pant leg, just above the knee. He didn't stir. She grew bolder. Her hand moved to his groin. She cupped his cock in her hand and sighed at the warmth—and weight—there. Very nice. Dane stirred, and she felt him jerk awake. Felt him wake up below, too, just as she slid her hand away. _Count on a little late-night fondling to wake a man up,_ she thought wryly. "Miranda?" he whispered in a strangled voice. "Dane," she said, kneeling close and leaning low toward him. "Can I talk to you?" "Everything okay?" She waved for him to keep his voice down. "Yes. I just...want to talk. Away from camp." He squinted up at her, then at the campsite. "We can talk in the morning," he said, rubbing his eyes. "Go back to sleep." God, the man was obtuse. She was tempted to reach over and grab his cock again, because that seemed to be the only thing he was listening to. "Dane, I need..." she paused for a minute, thinking. Then she lied, "I need your help. Something bit me." His eyes flew open at that, and he stared up at her, then got to his feet. "Something bit you?" he whispered again. "Where?" _Might as well go all in,_ she thought. "In a personal spot." Dane swore under his breath and then raked a hand over his closely trimmed hair. He reached into his backpack and grabbed one of the emergency flashlights and the first-aid kit, then gestured for her to follow him out of camp. When they got into the edge of the woods, out of the clearing and away from the camp, he clicked the flashlight on. "Now, show me where—" She rushed forward and put her hands over the flashlight, hiding the beam. Miranda glanced over where the others slept. No one had stirred—good. "Can you turn that off? I don't want the others finding us." He gave her an exasperated look. "How am I supposed to tell where you've been bitten if I can't see it?" "I'll show you," she said. "You can feel it on my skin. Just, please. Don't wake up the others." If he did, her window of opportunity tonight was going to end up a big fat failure. A long moment ticked past, and then Dane sighed, clicked off the flashlight, and turned to her. "All right." She could see the silhouette of him in the moonlight, highlighting the wide sweep of his shoulders, and she felt another excited thrill pass through her. When was the last time she'd been this hyped up to touch a guy? Answer: nine years. "Show me, then," he said in a low voice. She took his hand and instead of placing it on her skin, she began to walk farther into the woods, to put as much distance between camp and the two of them as possible. She expected Dane to protest, but he only followed her lead, his large, warm hand loose in hers. When they'd walked a good distance away and made it deeper into the woods, she stopped under a tree. "Here's good." "You said something bit you?" Dane's voice was skeptical, as if he wasn't sure what she was up to anymore. Was that amusement in his voice? Oh, she hoped so—if he was furious, she'd never get to put her hands all over him. _And I'll never get my revenge,_ she added at the last moment, a bit ashamed that it hadn't been the first thought to spring to mind. "It did," she lied, pulling his hand against her stomach under her shirt. She hid the shiver of desire she felt at his fingers against her bare skin. "Let me show you." His chuckle rumbled low. "I'm guessing it's not a snake, then, if we're trekking all over the woods." "Not a snake," she quickly agreed, looking up at him. God, his cheekbones looked amazing in the moonlight. She'd forgotten how incredibly sexy Dane was, how much looking at him made her wet with excitement. "Does the skin burn?" Oh, _absolutely_. Her hand pressed over his where it rested on her flat stomach, just above her belly button. His fingers brushed against her stomach, sending a pulse of heat through her body. His whisper grew intimate, as if the fact that he was touching her and they were very, very alone had suddenly occurred to him. "I don't feel anything, Miranda." "It's lower," she lied, her eyes watching his moonlit face, waiting to see if he'd take the bait. They stood so close together that she could feel his warm breath on her neck. Dane paused for a long, long moment, and then gave her a knowing look. "Lower?" She nodded, not trusting her voice. _Touch me_ , she wanted to whisper. _Please. Everything hinges on you touching me._ His fingers brushed at her waistband. "Lower?" "Yes," she said. Her hand clutched at his shirt as she waited. He looked up at her and his hand moved boldly down her belly. His hand slid into her shorts. "Where are your panties?" "Some guy cut them off of me earlier," she said, her voice husky. His hand skimmed past the curls of her sex and slipped between the folds of her pussy in a sudden move that had her gasping. "Down here?" he asked in a husky voice. "Is this where you're burning? Because you're certainly wet." His fingers brushed against her clit and her body stiffened in a hot rush of desire. He rubbed the slick bud with his fingertips. "I'm starting to think you haven't been bitten at all." She clung to his shoulders, her fingernails digging into his skin as his hand flexed in her shorts, small gasps erupting from her throat. "You—you have me all figured out, it seems," she managed to whisper, the whisper turning into a whimper as his clever fingers gave her clit another stroke. Her head tilted back and she leaned against the tree, her hips bucking against his hand. Oh my God. That was so good. If he could just keep touching her there— His hand started to pull away and she gave a small cry, her hand moving down to rest over his. "What are you playing at, Miranda?" he said low in her ear, his face pressing against her neck. He didn't move his hand from her hot, wet sex, but his fingers had stopped their rubbing of that most delicious of spots. She could have wept in disappointment. He stared down at her, his gaze intense. "Is this some sort of trick?" "Trick? No," she said, tilting her face toward his. "I'm just...I..." What could she tell him that he would possibly believe? The words froze in her mouth. His lips were inches away from hers and she longed to move her face closer to his, kiss him, feel that tongue stroke into her mouth in each conquering sweep. But his lips were firm and hard with anger. He wouldn't kiss her back. "Then why don't you tell me what's going on? You've been after me ever since we laid eyes on each other, Miranda. And while I'm flattered, I have to wonder what your game is." Shit. Well, okay, maybe she was being obvious—too obvious. She stared up at him, acutely aware of his hand still down her shorts. One nice squirm and she bet she could get his fingers to brush against her clit again—but how humiliating would that be? To try and get off against a man who wasn't responding? So she took a deep breath and pulled her hand off of his, placing it on his T-shirt. That wasn't much better—she could feel the finely corded ropes of muscle in his arms, and that made her think of his hand down her shorts all over again. She was getting wetter just thinking about it. "I..." His fingers twitched against her clit, a little prompting motion, and he leaned in toward her, pinning her between his hard body and the tree. "Well?" "I...um..." She stalled, thinking hard. Then she bit her lip and confessed the truth—or at least part of it. "I can't have an orgasm." That was clearly not the answer he'd been expecting. He frowned down at her, and then his fingers gave a little swirling motion against her clit, eliciting another shuddering gasp from her. "Really? Because you seem to be responding to my touch pretty well." His voice has dipped husky again, and she could have celebrated. He was listening to what she had to say. She fought a surge of excitement. Her fingers dug into his shoulders again and she gave a little trembling gasp when he slid a finger farther down, away from her clit. One thick digit brushed against the opening of her sex. Her knees threatened to collapse. "I can't have an orgasm. With a man. In bed." She was finding it hard to concentrate, his finger making small little circles against the opening of her sex, where she was wettest, tickling her in the most erotic fashion. He leaned even closer to her, her breasts pressing against his chest, and she lifted her face to his, startled to see his face looming so close that she could practically see beard stubble. His lips were close to her own. "Women, then?" "What? No." Her hips rocked against his hand and she whimpered. It was so hard to concentrate. "What do you mean?" "I mean...I can't shut my brain off during sex. And when I heard you were back in town, I remembered..." "That night in the closet?" he said huskily. "Back at graduation?" She flinched, thinking of the camera. "I remember that," he said in a low rumble, and his mouth dipped against her neck, pressing a light kiss there. "How my hand had been on you, just like this, and you came all over my fingers." She shuddered at that, pleasure washing over her. "I remember that," she murmured. "You didn't have a problem coming apart in my arms then," he said, and his finger slipped deep inside her, giving a gentle thrust. "I know," she said, her breathing coming hard and fast. She wanted to lift her leg around his hips, to grind her hips against his hand, to do...something. But she was pinned between him and the tree. "But that was a long time ago. I've had—trouble—since then." Trouble was putting it mildly. More like counting tiles on the ceiling while her boyfriend of the moment tried unsuccessfully to elicit a reaction from her. His mouth brushed against her throat again, and she could feel the hint of beard stubble scrape her skin. "So you want to try again? With me?" She nodded. "See if it's me, or if it's them." Dane's face lifted away from her neck and hovered an inch from her lips. "Miranda Hill," he said in a low, husky voice. "You should definitely know that it's them." "Is that a yes?" "It's a definite yes." And he kissed her again, his lips descending on her own parted ones, his tongue sliding into her mouth with a powerful thrust that his finger deep inside her mimicked. Her hips rocked with that, and she made a sound of need low in her throat. His tongue thrust again, in time with his finger, and she began to feel that slow, wonderful build that only seemed to come with her vibrator. Her mouth parted wider under his, and when his tongue stroked into her mouth again, her tongue caressed it, submitting to him, telling him how much she wanted him. He gave one last thrust of his finger into her sex and then broke off the kiss, staring down at her. His finger slid from her pussy and brushed her clit again, sending electric currents of desire racing through her. "I suppose heavy petting isn't going to cut it?" "It might," she admitted, clinging to him. She didn't want his hands to move away from where they were, setting all her nerve endings aflame. "We can keep trying. I don't mind." Dane pressed another long, searing kiss to her mouth, and then his hand slid from her skin. She could have wept with disappointment at that. He chuckled at her reaction. "I'm not going anywhere just yet. Take your top off. I want to see those pretty breasts. I've been thinking about them nonstop all day." The words sent a shiver of delight racing through her, and a dash of fear. _There won't be a camera here_ , she told herself, fighting the urge to check her surroundings. Dane wouldn't have known that she was trying to seduce him, wouldn't know to wire this area. It was all in her imagination. But her moves were mechanical and jerky as she gripped the hem of her shirt and pulled it over her head, then tossed it on the ground. Her black bra cups stood out in stark contrast against her pale skin in the moonlight, and he groaned at the sight. His fingers brushed against the curve of her breast and she shuddered. "Are you sure you want to do this, Miranda? We barely know each other anymore." She gave a small, nervous laugh and slid her hands under his shirt, feeling the warmth of his rippled, taut muscles under her fingertips. "You should have thought of that before you had your hand in my shorts." There was a flash of white in the darkness—his grin, she realized—and he pulled his shirt off, tossing it next to hers on the ground. Her mouth went dry at the sight of him. She remembered Dane from high school—tall and rangy with a wide triangle of too-lean shoulders, and a boyish chest. Nine years had changed that; he rippled with muscle now, his body thick with it. His chest was still almost hairless, and she saw a tattoo of a twisting playing card on one shoulder—the logo for the Las Vegas Flush, his old team. Her fingers brushed against it, then down his chest, exploring his body. Not an inch of fat anywhere. She supposed if she had to use a man for revenge, this was the best sort of specimen. "Do you approve?" he said, one palm moving to cup one of her breasts. "Been nine years since we saw each other with our tops off." "It's...acceptable," she teased, her fingertips scraping over one taut nipple, and then she gasped when he used the same motion on her. "And what about me? Do I pass muster?" His fingers slid to her bra strap, easing it down her shoulder, and he kissed the bared flesh of her shoulder. "You are the sexiest thing I have seen in a long time." That made the liquid heat surge back through her body again and she trembled a little, leaning in and sliding her hands to his waist when his kiss on her shoulder turned into his tongue dancing along her collarbone. Heat built and began to throb between her legs again, her sex getting wet with need once more. His hand reached behind her back and Dane unhooked her bra. Before she could squeak her surprise, his mouth captured her own again, and he kissed away any sound she might have made, the delicious stroke of his tongue hypnotizing her once more. This time, when she moved to lift a leg against his, he encouraged it, letting her slide over his braced thigh and straddle it. Her dark hair fell forward over her shoulders and she watched him, breathless, waiting to see what he thought of her breasts. He curled his fingers around one heavy globe and brushed his thumb against the nipple. "When we were in high school, I thought you had the most amazing breasts I had ever seen." He stared down, seemingly fascinated as a nipple puckered against his touch, and then palmed the other one. "Hasn't changed. Damn, Miranda. You are beautiful." That sent a quiver of pleasure through her body and she rocked her hips against his thigh, lifting her arms to wrap around his neck and pull him in for a kiss again. She wanted to press her breasts against his chest, to feel her nipples against that hot wall of muscle. He didn't release her breasts from his grasp, just continued to brush the peaks with his thumbs, turning them into twin points of heat as he kissed her, his tongue sweeping and stroking against her mouth. She had thought she'd be the aggressor in the encounter, but as soon as he'd realized what she wanted sexually, he'd become the dominant one...and she found it incredibly arousing. He leaned back a little, letting his mouth trail down her collarbone and working his way downward, pressing kisses against her skin, sliding down until he reached her breasts and his mouth flicked against one nipple. That brought a gasp of pleasure to her lips, and she arched against his thigh again, rubbing her sex against the hard angle of it. "Damn," Dane swore against her skin, then pressed a kiss there. "You have the most amazing breasts. I could spend hours here." But before she could tell him that was fine with her, he moved up and captured her mouth in another hot, short kiss. "Take your shorts off, Miranda. Now." He took a step backward and she complied, undoing the laces that held her cargo shorts up and shimmying them down her thighs. She dropped them to the ground and stepped out of them, her hands going back to his clothing. Miranda slid her fingers to his fly. Dane groaned at that, letting her fingers dance along the zipper of his shorts, undoing it and helping him slide them down his legs. They were quickly followed by his boxers, and his cock was exposed. It was a nice specimen, she had to admit—no wonder Dane had been so popular with women. His cock was long and thick and utterly beautiful, with a large crown and just a hint of a curve to it where it jutted from his body. Definitely a lot larger than her last boyfriend's, she realized with pleasure. She glanced down at his naked legs and noticed he still wore his hiking boots, just like she still wore her sneakers. "Should I take off my shoes?" "No," he said, and when she reached for his cock, he wrapped her fingers in his and turned her around, until her back was resting against his chest. "Not tonight. The ground is wet." "But—" His arms wrapped around her from behind and he nuzzled her neck again. "You're thinking too much." His hand trapped one breast, his other sliding to her sex as he pressed more kisses on her neck and shoulder. She forgot all about shoes. Then his fingers touched her clit again. She clung to his hands, riding his motions as he thrust against her bare backside, his fingers mimicking the rocking of his hips. His fingers slipped into her sex and thrust again. "You ready for this, Miranda? Ready for me?" "More than ready," she breathed. "I can tell you are," he said against her neck, his fingers thrusting into her pussy one last time. "You're so wet you're soaking." She shivered at that, and didn't protest when he moved away from her and bent to retrieve the first-aid kit. He pulled a box of condoms out and stared at it in surprise. "Fucking Brenna packed an entire box of condoms," he rasped. "Don't know if that's just fucked-up or brilliant." He ripped the box of condoms open and pulled one from the rest. He glanced back to her. "Put your hands on the tree." Her hands rested on the bark of the tree and she glanced over her shoulder, watching him. He rolled on the condom and put his hands on her hips, pulling her backward a little. He nudged her knees apart until she was spread wide, her outstretched arms bracing against the tree for balance, her legs cocked open, her body bent over slightly. Miranda's entire body tensed, waiting. Dane's hips brushed against hers and she felt the head of his cock nudge against her sex, seeking entrance. She sucked in a breath, anticipating the hard slide of him into her body. Nothing happened for a long moment, and she was about to protest when she felt him sink in. She whimpered in response. She hadn't had sex in a while and he felt...big. "So fucking tight," he said in a low voice; his teeth gritted. "Feels so good." Her fingers dug into the bark of the tree and she bit her lip, willing her body to relax. She flexed her hips, encouraging him to move, but Dane wouldn't be rushed. He anchored her hips against his and gave a slight pump before withdrawing. She gave a whimper of distress at this, not liking the fact that he drew away. Her body was on fire with need, her blood throbbing in her veins, and she was so tempted to just reach down and start to play with her own clit so she could go over the edge that he'd been teasing away from her— With a swift motion, Dane thrust again, and she forgot what she was thinking. A deep push, and he rotated his hips slightly as he pressed deeper into her, this time not stopping until she'd taken all of him. She gave a low moan at the sensation of being filled—desire began to move through her faster, and she gave a little squirm, flexing her hips and encouraging him to fuck her. "Tell me what you want, Miranda," he said in a low voice, leaning over her until he covered her, his stomach pressing against her back. "Tell me what you want and I'll give it to you." Her breath rasped in her throat, and she swallowed hard, her thighs quivering when he gave another small thrust, more of a tease than what she wanted. "I want you to fuck me," she breathed. She was rewarded with a hard thrust, but then he stopped again. "Please, Dane," she said, flexing her hips so she could slide along his cock. "I want you to fuck me harder. Faster." Dane thrust again, and again, two sharp bolts that made her toes curl with pleasure. Then he slowed, pumping in and out, slow and methodical and so very good. The constant, steady motion of his body rocking into her own drove her utterly insane with need. He'd thrust, then pull back, then thrust again, all slow and unhurried, and she raised her hips roughly against him, a wordless request. He seemed to realize that she needed something more. "Is this how you like it, Miranda? Or do you want it rougher?" "Rougher," she breathed. "I need you to drive me crazy. Make me think of nothing but sex." The next thrust was so hard she nearly bounced off her feet, and she gave a small cry and clung to the tree. "Oh God," she said. "Like that. _Yes_." He gave her another wild thrust, and then began to pound into her as hard as he could, ignoring all their playful games, and she could have cried out with delight. This was what she wanted and expected from her first encounter with Dane. For him to make love to her, rough and wild, forcing her to concentrate on the slam of his body against her own, the slap of his balls against her pussy as he thrust over and over again. She felt the familiar elusive heat building, and began to cry out softly with each successive thrust, until her cries came together so quickly that she was doing little more than keening her need into the forest, despite Dane's halfhearted attempts to shush her. And still it wasn't enough. She felt the build but couldn't quite seem to find the edge. Couldn't quite get there, even as he pumped harder and harder. Frustration began to make her building orgasm ebb, and she fought even harder to bring it back, but it was no good. "Dane," she said, her voice sobbing the word. "I need...I can't..." "I've got you," he said against her ear, and she felt his hand slide between her legs, felt him stroke her clit. "Come for me." He'd wait for her to come. The thought exploded in her mind at the same time as the orgasm did. She immediately shattered, an involuntary cry escaping her throat, her muscles locking in the onset of the most intense orgasm she'd ever had. He said something but she didn't hear it, the blood roaring through her ears as he drove into her a few more times and came himself, his body stiffening behind hers. Miranda panted, still clutching the tree. She should turn around, give him a flirty look. Toss him an airy smile and thank him for proving a point for her. But she was so stunned in the aftermath of the orgasm that she was...well, she was just a little lost for words. She remembered their make-out session in the closet all those years ago. She remembered her previous boyfriends, unable to bring her to orgasm, who'd eventually given up on satisfying her in bed when she remained cold and unable to respond. She remembered buying a vibrator just because she'd been so frustrated, and it had helped, but it was nothing like...this. Holy shit. One thing was clear—she _totally_ had to do this again. # SIX Well, fuck. Dane hitched his pants around his hips again, trying not to look over at Miranda. It hadn't even been twenty-four hours and he'd broken his word. The first piece of tail that had come swinging at him, and he'd gone after her with grabby hands. Colt and Grant were going to kill him. Murder him dead and bury him in the woods. And he'd deserve it. He scrubbed a hand over his face, thinking hard. It just wasn't fair. Of all the women they'd decided to rope in for their little scheme, did it have to be Miranda? And did she have to look as good as—no, better than—she had in high school? He glanced over at her, where she remained propped up against the tree. She hadn't moved. Waiting to gloat at him? He pulled his shirt down over his pants and studied her face. Miranda wore the most blissed-out expression he'd ever seen. Her mouth hung open slightly, her lips curved into a smile, and she stared dreamily out into the forest. Her clothes were still on the ground, her naked breasts bare in the moonlight. She'd propped her arms up over her head and simply leaned against the tree, as if she needed time to contemplate everything. That was unexpected. The sight of her languid smile made his body surge with lust again, and he felt the urge to reach over and pull her away from that tree and against him, and see if he could make that contented look change to one of desire again. But he couldn't. She was off-limits. Clenching one fist to rein himself in, Dane waited for Miranda to say something. And when she didn't move, he took the initiative. "Well?" Those blissful, sleepy eyes turned to him, and she gave him a satisfied look. "Well, what?" "What now?" He scowled at her. Was she going to run to Colt and Grant and tell them that he hadn't been able to keep it in his pants after all? She blinked, then looked at him, really looked at him for the first time. "Right. We're done here, aren't we?" She tugged her long hair over her breasts and bent over to pick her shorts back up. "We need to talk—" he began. "No, we don't," she said cheerfully. "What about your bite?" he said. Her gaze moved back to his mouth and she looked confused—and turned on—for a moment. Hell. His body hardened in response. "You said you were bitten...?" "Oh!" Her expression changed, and he could have sworn she was blushing in the dark. "That's right." She looked up at him, sheepish. "I lied." He'd guessed that. "So it was all a plot to drag me out here into the woods and have sex with me?" She tied the laces on her shorts and gave a throaty giggle as she put her shirt back on. "Sort of. Yeah. I guess it was, wasn't it?" "And that's why we need to talk about this. I need to know what you plan on saying to the others—" She yawned, patting her mouth delicately. "I'm not going to tell them anything. That would ruin any fun we plan on having this week." Fun? This week? Did she intend for them to do this again? He stared at her in surprise. So this wasn't some sort of trap after all? Or was it, and he just couldn't seem to figure it out? Miranda moved away from him and gestured to the camp. "I'm going to head back now. You hang back a few minutes, and then follow me in so it doesn't look like we were together." He said nothing. "Oh, for heaven's sake," she said, then moved back to him, grabbed the front of his shirt, and pulled his mouth down against hers for a quick, hot kiss. Her tongue flicked against his, and then before he could react, she pulled away again. "Like I said, we're not telling anyone about this. All right?" And then she walked away, leaving Dane behind her in the woods, dumbfounded and staring. What the hell had just happened? Had she just used _him_? For sex? That...was unexpected. * * * Always a light sleeper, Miranda stirred awake before the others. The sun was just beginning to purple the edges of the sky. After she'd returned to her solitary shelter, Dane had paced around camp. She saw that he'd bunked down with the rest of the men in the shelter, his body slightly apart from theirs. The others still snored. She stretched in her bunk, feeling incredible physically...and incredibly torn emotionally. She was a bit sore between her legs and her palms were scratched up from how hard she'd gripped the tree last night, but...her body felt alive and the blood sang in her veins. And it was all due to that bastard Dane, which was why she was a little torn at the moment. She'd slept with her worst enemy. Told him her biggest secret—that she couldn't have an orgasm with a man. Just blabbed it and left herself completely vulnerable to him. It was an uneasy feeling. Part of her hadn't expected to get so caught up in the moment. She'd suspected it would end up like every other sexual encounter she'd ever had: She'd come on to the man, they'd kiss, and when it progressed past kissing, her body would turn off like a switch and she'd spend the next ten minutes waiting for him to be done and faking an orgasm. But last night? She hadn't had to fake anything. He'd caught her off guard and had knocked down all her defenses. And she'd come. So incredibly hard that it had nearly made her see stars. Miranda wasn't sure if she should be happy that she'd had such intense sex, or devastated that she'd responded like a wanton to a man she professed to hate. She shouldn't be upset about that, though, right? She was on this trip to be strong and aggressive and take charge of her life...and to ruin his. So she'd used him a little last night. It was payback, in a sense, for the way he'd destroyed her reputation. And so she got an utterly amazing orgasm out of it—the first one that hadn't involved her vibrator in nine years. So what? She didn't regret it. In fact...she wanted to do it again. Using a man for her needs had been so very gratifying. There'd been no emotion involved, just animal attraction. Even this morning, she knew that any rational woman would be riddled with guilt, but, she admitted to herself, her guilt was not nearly as great as that pleased part of her that had enjoyed herself, enjoyed using Dane, and wanted to do it all over again. After all, she had an entire week left in this class. Why _not_ enjoy herself? Explore what Dane's body had to offer, and then discard him like yesterday's trash once she'd gotten her revenge? It sounded good to her. A little mercenary, but she didn't care. After all, it was her life. She was going to take something for herself, damn it. So she didn't have her camera with her. She'd have a week of naughty, illicit fucking, and top it off when she got back home and exacted her revenge. Sounded good to her. Feeling a bit wicked this morning, she moved to Dane's side and lay down next to him, a mere few inches away from his face. Miranda stared at his sleeping features. His mouth was parted slightly and the lines of his face were eased, making him look younger than his twenty-seven years. Her gaze skimmed the scars on his face—one on his nose just above the break, the small one through his eyebrow, and a longer one slashing across his chin and up to his lip. She was so close she could see the stubble on his face. It wouldn't take much for her to lean over and kiss him awake. So she did, because she was all about taking what she wanted this week. His mouth was relaxed against hers, and she felt the stubble of his cheek graze against her smooth skin. She kept the kiss gentle at first, pressing her mouth against his lightly, sucking on his lower lip and tasting its softness. His lower lip was full and it gave him a slightly pouty look that she'd gone wild over as a teenager. She'd loved tugging on that sultry lower lip with her teeth, and she remembered that he'd liked it, too. She did it now, heat curling through her body when she felt his tongue brush against her mouth as he awoke and responded to her kiss. He was kissing her back. Encouraged, she let her tongue stroke into his mouth, at first coaxing and then bolder. His mouth relaxed against hers, and her tongue swept inside, tangling against his. When his tongue touched hers in return, the stroke hard and sure, her nipples tightened with pleasure at the sensation. His hand tangled in her hair, and instead of kissing her back, he pulled her away. His expression was dismayed. "Miranda?" "Hush," she said, tracing a finger along his jaw and leaning in for another kiss. Kissing Dane had been a delicious experience. She hadn't wanted to enjoy it, but...she had. Quite a bit. It was further proof that no matter how awful a person Dane was, she was attracted to him. The moment she'd placed her lips on his, her body had immediately flared with need. It had been a long time since she'd been in a relationship with a man—since college. And she was suddenly keenly aware of the lack and seemed to want to make up for lost time. If revenge had to involve kissing Dane a lot, then she'd gladly turn herself over to the task. She slid a little closer, trying to take his mouth with her own again. His hand remained firmly anchored in her hair, trying to hold her in place. She was trapped an inch away from his face. "Miranda, what are you doing?" "Trying to kiss you," she whispered, her gaze focused on his mouth. It had thinned out into a hard line that she was very tempted to kiss into softness. His confusion just turned her on more. "Let go of my hair and I can get back to it." Instead of doing as she asked, Dane sat up and rolled away. She gave him a frown as he stood up, but it smoothed away when he reached for her hand. Miranda placed it in his and allowed him to help pull her off the ground. Immediately he began to tug her away from the camp, into the woods. An excited flutter started low in her belly, her pulse throbbing in all the right places. Were they going to go hide in the woods and have sex again? Could she handle a morning orgasm? She suddenly liked the thought of that very much, and licked her lips in anticipation. Oh boy, could she ever. Who knew that her revenge would be so much _fun_? There was a large tree with low-hanging branches at the edge of camp, and Dane pulled her toward it, his hand tight on her own. When they were behind the branches, she reached for him again, tilting her face up to his, a slight smile on her mouth. "Good morning, sunshine," she said and slid a hand along the front of his chest. Oh yum. She could feel washboard muscles and well-defined pectorals through the fabric of his shirt. Miranda resisted the urge to rub her hand all over his chest and explore his physique. He pulled her hand off his chest and frowned down at her. "I think we need to talk." She stiffened. That had not been the response she was expecting. Didn't most men like a sexually aggressive woman? "What's there to talk about?" "This!" He gestured at her and then at his mouth. "The kissing and the...you know. Last night in the forest." He dropped his voice and looked over at the campsite to make sure the others weren't awake. "You and me. What the hell is going on here, Miranda?" She rolled her eyes. "Do we really have to analyze it? I wanted to have sex with you to see what it'd be like, and so I did. Satisfied?" Dane shook his head, frowning. "Whatever it is Colt and Grant set you up to do, you won, all right? No sense rubbing salt in the wound." That made her flinch. She released his hand as if he were diseased and took a quick step away. " _Excuse_ me?" He crossed his arms over his chest, his mouth firming in disapproval. "I know Colt and Grant are behind this. I mean, it's pretty obvious. Our first week on the job and who happens to show up? My high school crush, looking just as amazing as the last time I saw her, and wanting nothing more than to jump my bones. Convenient, isn't it?" Dane glared down at her, as if he was cursing the fact that he was attracted to her. His hot gaze raked over her body. "They're obviously paying you to test my willpower. And it's obvious that I failed." Her jaw dropped. Her hand went to her collar and she nervously tugged it up closer to her neck. "Well," Miranda said nervously. "That's a first." "What is?" "I've never been called a whore in one sentence, and flattered in the next. I'm not quite sure whether I should be insulted or amused, though I have to admit that I'm leaning toward insulted." He rubbed a hand over his mouth, and it curved in a slow smile that irritated the shit out of her. "If it makes you feel any better, if I weren't on the job, I'd take you back into the woods again and fuck a few more orgasms out of you." His eyes smoldered down at her. "Since that's clearly what you're here for." The arrogant piece of shit! Did he think she was all over him because someone had paid her money? Her hand clenched even harder. So much for enjoying kissing Dane—all she wanted right now was to kick him in the groin. Hard. "That's sweet," she said in a dry voice. Her arms crossed over her chest and she gave him an angry, mulish look. "But if you call me a whore again, Pete's not going to be the only one nursing a bruised jaw. Understand me?" "Are you mad at me? I was trying to flatter you. I haven't seen you in nine years and I have to admit that you look amazing." Dane's smile tilted. "Sorry if that offends you." "That's not the part that offends me," she bit out. "It's the part where you keep saying someone paid me to sleep with you." His easy grin disappeared entirely. "You mean Colt and Grant didn't approach you to...come to me?" A sour taste built in her mouth. "I said all of three words to Colt yesterday. I haven't talked to Grant since high school. Why would they pay me to come out on a survival trip?" Dane remained silent for a long moment. "That...wasn't what I meant. Did they ask you to hit on me?" "No!" Her voice raised an outraged octave. "Why would I sleep with you for money?" A hand covered her mouth. Dane pressed his hand over her lips, his palm warm. He glanced back at the camp to see if the others had awakened from the sound of their argument, but they slept on. Dane reached over and touched her arm, pulling her closer so they could talk quietly, his hand falling from her mouth. "Miranda..." he began, then stopped, studying her face for a long moment. She felt the insane urge to tug at her collar. "If you're not doing this because Colt or Grant put you up to it, then why did you sleep with me?" "Jeez, I didn't realize I was that bad of a lay." His gaze dropped to her mouth. "Not in the slightest," he said, his voice dipping to a husky tone. His thumb brushed against her lower lip. "But you can't think I wouldn't be suspicious of your motives. You packed an entire bag of lingerie for a survival trip." "I told you why," she said defensively, pulling away. "I wanted to see if I could have a...you know"—she waved a hand—"with you, or if I was totally broken. And now that I know that I can...come"—she still blushed at saying the word—"with a man and not just a battery-powered toy, I'm satisfied. I don't need you any longer." She was lying through her teeth, of course, to see if he would take the bait. "Were you?" he said in a low, husky voice. His gaze remained on her mouth and he brushed his thumb over her lip again, tugging her mouth open a little. "Satisfied with that? Because it was just a quick meeting in the woods, Miranda. If you think that's the best I've got to offer, you're sorely mistaken." Her knees trembled a little at the thought. "Oh?" A slow smile curved his mouth, and he pulled her closer to him. His hand caressed her cheek and stroked down her neck, playing with the neckline of her shirt. Touching her as if he owned her—possessed her. The thought made her weak. "That was just one tiny little orgasm," he said. "I bet if you give me the chance, I could make you come two, maybe three times in a row." "Three times?" Her breath caught a little at the thought. She'd read romance novels where the heroine screamed the entire time they had sex, and movies made it look like an hours-long marathon, of course, but her experiences had been sadly underwhelming in that respect. Even her rendezvous with Dane—while mind-blowing in itself—had been a brief excursion. "Is that normal?" "It is when you're in bed with me," he said softly, and his hand slid down her back, fingers tickling her spine before resting on the small of her back. "Don't tell me you've never had a man lick that sweet pussy of yours for hours, making you come so often that your legs won't support you any longer?" Her legs were having difficulty supporting her body right now. She felt weak, boneless like he described, and the urge to lean against him and sink into his warmth was nearly overwhelming. She realized he was watching her, waiting for her answer, and she shook her head. "It's none of your business, Dane, but I've had plenty of oral sex." That was a lie. She'd had some, but when it became obvious that she was just uncomfortable instead of enjoying herself, it was quickly taken off the table. After a while, she'd given up on oral altogether. "Lots. So don't you worry about me." "And no orgasms? I'm heartbroken at the thought. Those must have been some shitty boyfriends." A laugh escaped her throat, and she quickly muffled it again, then gave him a skeptical look. "What makes you think I'm interested in another round, Dane? Maybe I sampled the goods and found them lacking." "You didn't," he said confidently. His hand slid back up her spine and he cupped the base of her neck, pulling her close and tilting her head back. "Did you? I bet if I dropped to the ground and kissed that pussy of yours, you'd let me do whatever I wanted to you." Dear God, it was true. Her fingers wrapped in his shirt and she clung to him. Dane was taller than her by a few inches, but he'd bent near and their faces were so close together that she could smell the musk of his skin and see the shadow of a beard on his chin. He licked his lips as if he were still thinking about putting his mouth on her pussy, and her entire body tingled in response, a throb starting low in her sex. His other hand slid to her ass, pulling her body against him. Her nipples grazed his chest and she gasped. His mouth brushed against her own, giving her the most fleeting of kisses. "How about," he said softly, "I drop to my knees right now and give you a taste?" "Do it," she breathed, her heart pounding in her breast. He grinned at her and slid his hands to her waist. As she watched him, he dropped to his knees, his face going to the cradle of her thighs and— "Hey," called a voice back at the camp. "Anyone seen Dane?" Dane jerked upright, nearly knocking her over in his haste to get to his feet. The look in his eyes had gone from sexy to wildly paranoid in a flash. With a sigh, she realized that she wasn't going to get what he'd promised after all. "Guess we'll never find out if you're full of hot air, will we? Shame." The look he gave her was hot, and he pulled her in for a quick, fierce kiss. "Tonight. You don't say a thing, I don't say a thing, and we meet again tonight." Then he disappeared into the bushes, trudging his way back to camp. Tonight? She curled her toes in anticipation at the thought, then sighed. The blood still throbbed in her veins and tonight was a very, very long day away. She waited a few minutes, then sauntered back to camp herself, tying the strings on her shorts as if to make it look like she'd been in the woods for a different reason. Dane looked up as she reentered the camp and gave her a light wave, as if just now seeing her. What a faker. "Morning, Miranda. Sleep well?" "Like a baby." She said the lie with a grin, and moved to the far side of the camp to get her pack. As she walked, she let her hips roll, and swung her long, tangled hair over her shoulder, knowing that he was watching. Dane wanted to see her tonight. That thrilled her just a little too much to fit in with her plans, and she frowned to herself. If this revenge idea was going to work, she needed to keep control of the situation. And Dane had just taken control a few minutes ago. Worse, she'd been ready and willing to give it to him. She was going to need to be tougher if she wanted to get revenge on Casanova Croft, rather than just be one of his conquests. # SEVEN Once the entire team was awake, they ate a small breakfast of the last of their carefully saved PowerBars, boiled water to refill their canteens, and then broke camp. The shelter was disassembled, the fire put out, and their bags repacked. Then they shouldered their packs to move on to the next location. Dane showed them how to make a fire bundle that would keep an ember smoldering while they walked, and he handed it to Miranda to carry. "It's very important that you keep this going at all times," he said to her, his face utterly serious. "If you keep it smoldering, you can rekindle a fire with just a touch." She stared at him, then at the fire bundle, her thoughts on things other than fire. Was his meaning what she thought it was...? She decided to test that theory. "I think I can manage to keep a spark burning," she said in a low, husky voice, and licked her lips. "Even if it requires working it a little to fan the flame. It'll be worth it, don't you think?" The look in his eyes grew hot, and she knew they were thinking the same thing. "See that you do. I'd hate to have to start all over, though...I'd be happy to show you how to tend it again, if need be." Her pulse fluttered and she tried not to blush. Show her how to tend her fires, indeed. As the team hiked, Dane pointed out flora and fauna to them. He showed them a poisonous spider, demonstrated which nuts were edible, and gave them a nature lesson as they walked. Occasionally he'd pick up a bit of tinder or a particular leaf, his eyes constantly scanning the environment. It was actually really interesting information, and Miranda would have appreciated it if she hadn't been quite so distracted all the time. The vision of Dane's tight ass muddied her thoughts of revenge, as did the incredible sex they'd had last night. And the conversation this morning. She couldn't stop thinking about three orgasms. Three. It seemed like a sinful concept to her, when she struggled so hard to have just one in bed. Here he was offering her three. Her pussy clenched at the thought of him between her legs, languidly licking her to orgasm, and she felt the slick wetness of arousal. Her spark was definitely going to keep going all day. By the time they were able to sneak away tonight, she'd probably come as soon as he said hello to her. And then he'd give her a smug smile, showing her that he was the shit and she was just another dumb girl who had fallen for his prowess in the sack... But even as she said it to herself, the puzzle pieces didn't fit. Last night, when she'd come on to him, he hadn't acted like it was his due, or that he'd expected it. He'd seem shocked—and then flattered. And then, he was just as turned on as she was. That didn't match the womanizing flirt she remembered and the tabloids had loved to harass. The man in her mind was far more self-centered and smug than this one seemed. Had Dane learned humility at some point? Had Casanova Croft had all the womanizing arrogance beaten out of him? She doubted it. They hiked over a ridge and Dane suddenly staked the flag in the ground. "This will be our next campsite." Miranda glanced around, but there didn't seem to be anything particularly special about this spot. They'd passed the stream again about five minutes ago, so maybe that was why—nearby water in easy access. The team dumped their packs on the ground and began to stretch, Pete wiping his brow. The gamer CEO was in worse shape than the rest of them; he began sweating as soon as they started hiking and didn't stop until he went to sleep. "What do we do now?" George said, looking expectantly at Dane. Dane grinned and clapped the man on the back. "Now you're going to get a chance to show me the survival skills that you've picked up. I showed you some things last night and this morning, and I want to see what you've learned." With that simple command, the team set to work. George left to go into the woods and begin setting up traps, and Pete headed off with the fishing line to do his task. He looked uncomfortable at the thought, but no one offered to help him with it. After the disastrous incident involving her boobs, Pete had been reassigned from water and firewood to fishing (to get him away from her, she suspected). She couldn't say she was sorry for it. So now it was Will who was assigned to help her with the firewood, and while she waited for him to bring back the first bundle, she examined their new campsite. There was a small, scorched area that told her someone had built a fire here in the past, and she ran her shoe over it, looking for small rocks that would pop when heated. Nothing. Holding her fire bundle in her hands, she glanced around. Dane stood nearby, arms crossed, leaning against a tree. He'd been muttering something into his satellite phone. Even though he was concentrating on something else, his gaze was on her, watching her as she worked around the campsite. For some reason that made her blush. _Dane Croft,_ she thought. So very, very arrogant and self-assured. So certain that she'd take him up on the three orgasms he'd offered her that evening, now that he was assured she wouldn't run and tell his partners that he was sleeping with the clientele. She wanted to put him in his place...or kiss him. Right now she couldn't decide which, and loathed herself for it. She was supposed to hate the man, not think about his mouth on her body all day. Not blush when he looked in her direction. Where was her righteous indignation? Or was she starting to lose her edge? Scowling at the thought, she turned back to the fire. She needed to focus on work, not on sex. Starting a fire would distract her. There was a scatter of larger rocks nearby and she spent some time gathering them and forming a circle to ring the fire and keep it from spreading. Once that was done, she dug out the small area inside the rocks and then began to place the wood on it, stacking it the way she had been shown as Will came back with armful after armful. Poor Will had a thankless task—yesterday they'd realized just how much wood was needed to keep a fire going all night, and she suspected he'd gotten the raw end of the deal...and she'd gotten the easy job that kept her at camp and close at hand. Miranda's mouth soured at that. Protecting the girl? Or keeping her close for other reasons? Frowning to herself, she stacked the wood and then sat back to examine her fire pit. Not bad. She'd gathered some fallen leaves and dried grasses to use as tinder as they'd hiked. Her pockets had bulged with the material and now she pulled it out and began to set it at the base of the fire. Her fire bundle had been carefully tended all day, and every time she'd coaxed a bit of smoke out of the coals, she'd grown aroused all over again, thinking of Dane and his promises to her. _I've had plenty of oral sex._ _And no orgasms? I'm heartbroken at the thought. Those must have been some shitty boyfriends._ She pictured Dane between her legs, her hands rubbing on his too-short hair as he kissed the lips of her sex. Her nipples went erect just thinking about it and she squirmed, clenching the fire bundle tightly. "Miranda, before you begin," Dane said, interrupting her thoughts. She looked up and jerked backward when she realized that Dane was standing over her, his crotch at eye level. Damn. Was he thinking what she was thinking? She looked up at him and licked her lips, confused. A bolt of desire crossed his face and he glanced around to make sure the others weren't watching, and then he crouched next to her. "Stop that, Miranda," he whispered. "Stop what?" "Stop looking at me like you want me to throw you on the ground and fuck that dazed expression off of your face." His voice was husky, as if he'd been thinking about the same thing. "Hours to go before sunset and it's not a good idea to let the others know what we're planning." "Oh," she said, and frowned at him, resisting the urge to give him a shove. "If you don't want me thinking about later, then don't shove your junk in my face, all right?" Sure, she'd gotten a glimpse of the package last night, but when he practically pushed it into her face? She couldn't help but think of other things. And to make it worse, he was definitely...well equipped. _Of course he was_ , she thought sourly to herself. Dane Croft had been built like an Adonis and she was being swayed by his good looks and godlike smiles. She hated herself for being so very shallow. He chuckled and patted her on the shoulder, standing up again. "That's better. Now, can I see your fire bundle?" She slapped the bundle into his hand, then winced at his surprised expression. It wasn't like it could hurt him anyhow. The fire bundle was nothing more than a long piece of rubber pulled from the inside of a shoe that had been wrapped tightly around an ember buried in packed tinder. The bundle had been tied tightly with a shoelace. He'd shown them how to make it—to carry fire from campsite to campsite without having to make it all over again. It was extremely important to survival, Dane had said, and Miranda had treated it so. She'd kept a careful eye on it, blowing on it from time to time to stoke the embers again. It had smoked and smoldered all day long. Just like the desire that still burned through her body. And it irritated Miranda that Dane had decided to just waltz over and put his hands all over her bundle. "It's still lit," she pointed out. "Give it back." "Can't do that," he said with a smile. As she watched, he carefully unwrapped the fire bundle and exposed the ember, then poured water over it. She sputtered in shock and tried to snatch it back from Dane. "What are you doing? That's our fire!" "It is," he agreed with a grin. "Or I should say, it was." "I worked hard on that," she blurted. "I kept it going all day long." Was this some sort of message he was trying to send to her? If so, she was not amused. "Or does this mean you're no longer interested in nurturing my spark, Dane Croft?" Her voice had risen to a rather loud level, and he winced and gestured for her to lower her voice. When he'd glanced around and had determined that no one was listening to them, Dane looked back at her. "Doesn't mean that at all, Miranda, and you know it." Actually, she _didn't_ know it. "This course is about survival," he said a little louder, and handed her the wet fire bundle. "And I need to know that you can make a fire on your own. So no bundle today." "You are a horrible man." Dane only laughed and smiled down at her with a satisfied expression. "You won't be saying that tonight, I promise." Flustered, she knelt next to the fire pit. Her hands searched through the wood, trying to recall what he'd taught her. Focus on work, she told herself. Not Dane. Think about fire, not about his mouth on her body. So she sat back and concentrated, gathering her thoughts. She needed to make a bow. After a few moments of searching, she found a long piece for a bow and a second piece of soft pine that would be suitable for a baseboard. She examined the wood for a moment more, and then glanced over at Dane. He crouched near the fire pit, looking like he had nothing better to do than to sit and harass her. "You can quit hovering," she pointed out. "Don't you have someone else on this team to bug?" He grinned, seemingly unbothered by her prickly attitude. "Fire's important. Once I've established that you can get a spark going all on your own, I'll check on the others." She wasn't going to touch that double entendre with a ten-foot pole. "You're going to be waiting a while if you think I'm going to spark anything with you sitting there staring at me." He didn't move. Miranda rolled her eyes in exasperation. "If you're going to stay here, hand me your knife, then." He did. "I should make you get your own knife." She rolled her eyes again and used his knife to make a notch in the baseboard, like he'd showed her. Once that was done, Miranda handed his knife back and began to pull the laces off of her shoe to use to string the bow. He was watching her, and it made her nervous. Made her think about sex again, and that wouldn't do. She needed a distraction. "So, Dane," she began as she tied one end of the laces to her chosen stick. "What made you decide to run a survival school? I have to admit it's not what I pictured for you." His easy grin began to fade a little, and he hesitated for a moment, as if choosing his words carefully. When he answered, it was simple and direct. "I enjoy it. I spent the last year living off the grid." "Living off the grid?" she asked, finishing her bow and testing the cord. It was tight, with just enough slack to wrap around a stick. Hopefully that would do. "What does that mean?" "No electricity, no running water, no power," Dane explained, his gaze on her hands as she began to set up the fire-making implements. "Just you and the wild. Colt and I had a cabin in Alaska that we built. It was..." He paused, thinking. "It was nice." "Not a lot of girls up in the wilds of Alaska," she teased. "Were you pinch-hitting for the other team or just doing a lot of masturbating?" He laughed at that. "You have a filthy mind." "What? Admit it—that's the first thing you thought about, too." Dane grinned. "I was there to camp. As for masturbating, nah. It wasn't on my mind at all. By the time I got to Alaska, I was pretty much done with dating. It was nice to have a vacation from everything in my life." "You, done with women?" She laughed. She looped a stick through the bow and aimed it over the notch she'd carved in the baseboard. "That doesn't sound like Casanova Croft at all." His look became shuttered immediately. "Yeah, well, sometimes what you get isn't always what you want." Before she could comment on that, he reached over and corrected her hands. "Hold it like this. And don't forget to put your tinder under the notch so your ember has something to fall on." She looked at him in wary surprise. His voice had been cold, efficient. Gone was the warm, teasing note. What had she said that was so wrong? Miranda put a bit of tinder under the baseboard and swallowed down the defensive feeling. She was here to fuck—and fuck with—Dane Croft this week, and if she pissed him off, she could kiss her revenge good-bye. Irritated at herself, she began to saw the bow, turning the spindle and creating friction against the baseboard. It was harder than Dane had made it look, and she gave it another rough tug, causing the spindle to twist again. An uncomfortable silence fell, the only sound the sawing of her spindle against the wood. After a few minutes of watching her work, Dane glanced over at her again. "So, what about you?" She glanced up, still sawing at the bow and turning the spindle. It was hard to concentrate on the conversation, especially when she was trying so hard to get enough friction to create a spark in the small notch she'd carved in the baseboard. Crap—why did she get the fire-making task? This was hard. Concentrating on her task, she didn't look up. "What about me?" "You wanted to be an editor or something, right? How come you never left town? Bluebonnet's not exactly a hotbed of activity." His voice was wry. "I couldn't wait to get away from here." She didn't like where this was heading. So she remained silent, hoping he'd continue talking until he moved long past what she had or hadn't done with her life. But he paused, waiting for her to respond. "Journalist," she finally offered, her arms beginning to ache from sawing at the fire-making bow. How long did she have to keep doing this before she got a spark? She didn't even have smoke yet. Frustrated, she sawed it harder. "And you weren't the only one who wanted to leave." "So why didn't you?" She was going to start throwing a temper tantrum if she didn't get a wisp of smoke, she really was. So she just sawed harder, her teeth gritted. "Couldn't." "How come?" She didn't answer. He wouldn't let it go. "Did you have to help your mother with her store? She still runs that antiques shop, right?" That was a little too close to the ugly truth. What sort of game was he playing? Did he want her to come out and admit that the pictures he'd taken had ruined her life? Was this some sort of nasty revenge for somehow offending him? Reminding her who she was? Putting the slut of Bluebonnet back in her place? She threw the fire-making implements down and stood up. "I need to take a walk." "Miranda, what—" She whirled around to face him, glaring. "Leave me alone. Understand? I need to take a walk, and not with you." With that, she turned and stomped out of the camp. Christ, but that woman was prickly. Dane stared after Miranda, wondering at her explosion and subsequent exit from camp. What exactly was she hiding that made her so upset? He was tempted to ask one of the other men, but they wouldn't know anything about her either, being out of towners. Anyone in Bluebonnet could have told him the truth, he suspected. Everyone in town knew everyone else's business. And Miranda's was apparently unpleasant business, at least in her mind. He stared down at the tools she'd dropped on the ground. Then he moved to go after her. "Dane! Look! I got dinner!" Pete held a fish aloft, trotting back through the woods. "I caught something!" Dane glanced at Pete, then back at the woods, then sighed and turned back to him. The man's forehead was beaded with sweat and his pants were splotchy with water. He held aloft a fish, about a foot in length. "Good job," Dane said absently, glancing at where Miranda had disappeared one last time before turning back to Pete. "Get a flat rock and I'll show you how to scale it." Pete gave him a funny look. "I have to scale it?" He chuckled at the other man's expression. "Only if you plan on eating it. You're going to have to gut it, too." The gamer looked a bit green at the thought, and Dane wondered how he'd managed to catch the fish if the thought of touching it was so revolting. He nodded at the fish. "Here, give it to me and I'll show you how to do it this one time, but after this, it's on you. Understand?" Pete seemed reluctant to hand the fish over, but did so after a moment, and Dane immediately saw the problem. "This fish is dead," he pointed out, angling his face away from the smell. "Very, very dead. Several days dead." Pete crossed his arms over his chest. "Is that a problem?" Dane held it toward Pete's face, watching as the other man flinched away. "Do you want to eat it?" "Well, no." He held it back out to Pete. "Take this out there and bury it somewhere. You're supposed to be catching live fish, not scavenging dead ones. Leave that for the coyotes." The other man suddenly looked panicked. "There are coyotes?" "Don't worry about the coyotes," Dane told him. "They're terrified of people. You're more likely to see a unicorn than a coyote out here. Now head back out and actually fish. _In_ the water. _With_ line and bait. I'm going to go find Miranda." "Speaking of Miranda," Pete said, his voice low and thoughtful. "You guys know each other?" The hair on the back of Dane's neck prickled at Pete's question. "We went to high school together. Why?" "She single?" Hot jealousy speared through him. He resisted the urge to bite off that no one was going to be touching Miranda but him. They were supposed to be keeping things a secret. Clenching his hands, he reached for a piece of kindling and began to snap it into smaller pieces. "I didn't ask her. Why?" Pete gave him a smug look. "She was checking me out the other day. I thought I might see if she's interested in going out when we get out of this little hellhole called nature." For some reason, that really irked Dane. Nature wasn't hell. And to think that Miranda had been checking the skinny creep out...he didn't buy it. "Unless you're planning on tapping that ass?" Pete said, interrupting his thoughts. "I've noticed the way you've been looking at her." His jaw tightened. The urge to suddenly pound Pete's face in washed over him, and he clenched his fists. "No, I'm not," he lied. In that moment, he missed hockey and the ability to punch the hell out of your opponent. "She's just an old friend." He couldn't say yes— _Sure, I slept with Miranda last night and she was wild. It was hot as hell, and I plan on doing it again. I want to see the expression on her face when I show her how to come again. I want to see the expression on her face when I put my mouth on her sweet pussy, and her expression when I feed my cock into her body._ He couldn't say any of that. And even if he thought about Miranda's sassy little thong or her curving smile or the way she'd made those soft, surprised little cries of pleasure when he'd pounded into her, as if she hadn't been expecting to enjoy it so much. He couldn't say a damn thing. This was business, and Miranda was business, and no matter how much he might like for it to be otherwise, it couldn't be. Pete adjusted his glasses and smiled. "Excellent. Then you don't mind if I go after her?" If his jaw gritted any harder, his teeth were going to snap. "Not during survival week." "Oh, after, of course." Pete stared off into the woods where Miranda had disappeared. "I wouldn't want to see her before she could take a nice, long shower." Fucking asshole. As if Miranda smelled bad. Just the opposite, in fact. She'd smelled like the woods—wood smoke and the wind and just a hint of sweat—and he'd found it incredibly appealing. This little creep wouldn't know what was appealing if it decked him in the face. Hiding his anger, Dane pointed at the dead fish. "You need to get rid of that and catch a real fish. Got me?" The other man gave him a reluctant nod and then headed back away from camp, muttering under his breath. He swiped at the branches as he walked, the actions of a petulant child and not a grown man. Dane gave it two days before Pete bailed out on the class entirely. _Good_. The man was acting like a brat and the class would only get harder. That was one of the things he appreciated about Miranda, he thought as he turned in the opposite direction and began walking. She didn't complain about the class, about being unable to shower or sweating in the dirt and sleeping on the ground. When he'd seen that bag full of lingerie, he'd been worried that she would be a huge pain in the ass this week. But...she wasn't. She actually seemed to be enjoying herself in the outdoors, and he was enjoying her presence as well. Then again, he hadn't expected to have sex with her. It made him a little uncomfortable to think that he'd automatically assumed that she'd been a plant from Colt and Grant—she had been so offended at the thought that he knew she was sincere. He shouldn't have slept with her. Shouldn't have, and yet...he couldn't resist. When her gaze went soft, he wanted to bury himself deep inside her and make love until morning. Still, he wasn't entirely sure Miranda's motives were innocent. Why would a woman who liked lingerie and sexy things want to spend a week in the wilderness? Things didn't add up, he decided. Either Miranda had a really killer dual personality—girly-girl of the backwoods—or he was missing some vital element. For the life of him, he couldn't figure out what. As he contemplated the Miranda situation, he walked through the woods, idly noticing the play of footprints in the dirt. Though he wasn't the best tracker, it wasn't hard to see that someone had come this way. He touched a broken twig and knelt by the ground. The hard dirt hadn't seen rain for a few days and showed the wavy lines of a boot sole perfectly. Judging by the size of the shoe, it wasn't a man unless the guy was packing some seriously dainty feet. He followed the footprints, thinking about his small class. With one glaring exception, they'd been interested and willing to learn. The others were corporate hounds—it was easy to spot the type, as they were aggressive and driven. The desire to succeed was clear. Tracking wasn't on the week's menu, but he thought of Miranda's face and the way it lit up when things clicked and she learned something new. Maybe he'd show her a few things when he found her. After she'd cooled down, that is. She'd almost had her fire—a few more minutes of sawing and she'd have had a spark for sure. Her quick mind had picked up on the implements and had followed his instructions almost to the letter. She'd been so close...until they'd started talking about town. About Bluebonnet. Then she'd bailed on him. He ducked under a tree branch and scanned the woods as he thought about their conversation. Dane had immediately steered it once they'd gotten to his personal life. He liked Miranda but he didn't feel like sharing why he'd left the NHL—no matter what she'd heard. No one believed him anyhow. They liked the tabloid version of things far too much. That he was a pussy hound who couldn't turn down a woman. That he was a user. He had been, once upon a time...until someone had used him. Then he'd changed. He wasn't the old Dane anymore, and he got tired of trying to prove it to everyone he met. Of course, Miranda had been equally prickly. She'd bristled as soon as he started asking her why she hadn't left town. Was there a boyfriend still in Bluebonnet? Someone she'd stuck around for? A surge of jealousy tore at his thoughts. Was that the reason she'd packed all the sexy panties? Had her hands on his cock as soon as they were alone together? To make someone jealous? Dane frowned as he spotted another set of footprints near the stream. He approached on the far bank, his movements quiet and stealthy with years of practice. And there she was. Miranda stood in the creek, hip deep, her back to him. She wasn't totally naked. Under the thick, spill of dark brown hair that cascaded down her back, a thin black bra strap stretched over her shoulder. From his viewpoint on the bank, he could see the creamy small of her back, perfect in its symmetry and the way it dipped inward just above her bottom. He groaned at the sight of her rounded ass as she leaned forward, exposing the heart-shaped flesh. Definitely a thong. He barely caught it peeking between the cleft of her full, firm buttocks. Damn. Dane stared. Miranda had the most singularly perfect ass he'd ever seen. He closed his eyes and leaned against the tree he was gripping for support. Hell. She was bathing. And here he was, standing on the bank and peeking at her like some sort of creepy pervert. His hand slid over the hard rise of his cock in his shorts, and he swore. He was a creepy, _turned on_ pervert watching her bathe. She might enjoy flirting with him and their midnight trysts, but he was pretty sure she'd hate the thought of him spying on her like some hormonal teenager. Yet he couldn't stop staring at that perfect ass, and he thought of how she'd felt last night with her sweet pussy wrapped around his cock, clenching him deeper every time he thrust. Of the cries she'd made—so turned on and so very surprised that she'd been so lost in the moment. Of the look on her face when she'd finally come, as if he'd just handed her a million dollars. Fuck. If he got any harder, he was going to charge into the water after her and forget all about the "tonight" part of their next meeting. He'd take her on the creek bank, in broad daylight, and wouldn't care who saw them. The heel of his hand rubbed down the front of his cock again, need surging through him. Before last night it had been two long years since he'd been with a woman. He hadn't missed one in all that time. There was always something to distract him—chopping wood, hunting, a ten-mile hike through the snow back when they'd lived in the cabin...and when all else failed, there was his hand. Last night should have gotten her out of his system. Quenched the urge so he could stop thinking with his dick and get back to his job. But as he watched Miranda raise her pale arms to her back and toy with the clasp of her bra, he knew that it was going to be close to impossible to think about anything but taking Miranda again. And he palmed his cock once more. # EIGHT She couldn't do it. Miranda's fingers trembled on her bra clasp, ruining the sensual movement she was going for. She tried again, squeezing her eyes closed and focusing on undoing that one stupid strap, but every time she came close, her fingers locked. The pictures on the Internet flashed through her mind over and over again. Her kneeling before Dane. Her breasts thrust against the camera in another shot. That triangle-shaped mole under her left breast had identified her even if her face hadn't been in the picture...and it had been. Her expression in the photos had been contorted in rapture, and she remembered Dane's big hand toying with her nipples. She'd loved his touch. Nine years later, she still loved his touch, though she hated herself a little for it now. She'd recovered from her tantrum walking through the woods, realizing that she wouldn't be able to keep Dane trotting after her if he was mad at her. So she'd calmed down and sat on the banks of the stream, staring at it as she tried to think up her next battle plan. After a moment, the solution had become glaringly obvious, of course. Strip naked in the stream until Dane ran across her, then seduce him again. Keep the control firmly on her side. Never let him get the upper hand in their relationship. Keep him guessing, above all else. Except...her hands weren't cooperating. Her subconscious had a strict sense of modesty even if Evil Miranda was trying to shed it. Last night she'd been able to provoke and flirt with Dane, and hadn't thought twice about shedding her clothes. In the daylight? It was different—more exposed. And knowing that Dane was on the bank watching her? Made her even more nervous. Last night it had been dark and he'd only been able to get quick glimpses of her body in the moonlight. They'd kept some of their clothes on even as they'd had sex. Standing here in the stream in nothing but a thong and lacy bra in broad daylight felt...naked. Was he looking at how much her body had changed in the past nine years? Comparing her breasts to that old photo? Her fingers twitched and she opened her eyes, blowing her long bangs off of her forehead in frustration. Why was this hard? She'd hit on him last night and he'd given her the screwing she'd asked for. He'd made her come so hard that her eyes had nearly crossed. This was just a stupid bra and striptease. Why was it such an issue for her? _You're being a baby_ , she told herself, even as her fingers clamped down on the front of her bra, clutching it to her chest protectively. Anyone could be watching her, not just Dane. What if someone else had stumbled upon her bathing and it wasn't Dane at all? What if he had a camera, too? Oh God. What if— A noise to her left got her attention—had Dane moved to the other side of the bank? Or was there truly another watcher here at the stream? Miranda's hand slid forward over her the cups of her bra, protecting her breasts from prying eyes, and she turned. A gigantic bird stood on the bank, about two feet away from her. It looked like a giant ostrich with enormous, round black eyes and a nasty beak. It ruffled its feathers in alarm at the sight of her, the long neck rearing back as if it were about to peck her eyes out. It squawked at her, the sound angry and strident. Miranda yelped. She stumbled backward. She lost her footing and skidded into the water up to her neck. Gasping, she struggled to regain her balance and then continued to slide away as the bird squawked again and ruffled its massive black wings. The thing paced on the bank, storklike legs twitching nervously. Shit! Could birds swim? Did emus attack people? "Miranda," Dane said in a low voice behind her. "Careful." Screw careful. She turned and vaulted for the opposite bank, toward Dane. Forgetting about her state of undress or the fact that she was in danger of losing her wet bra, she plowed toward him. When he extended a hand to help her out of the water, she grasped it and hauled herself onto the bank. The thing across from them trumpeted in alarm, and Miranda yelped again. She didn't stop at climbing out of the water, and began to climb up Dane himself. She leapt onto him, her legs locking around his waist and her arms around his shoulders. Her only thought was to get away from that damn bird, and Dane was safe. Dane wouldn't let it eat her. Once she'd climbed on top of him, she squeezed her eyes shut, breathing hard and waiting for the stupid thing to attack. A long moment passed in which she could only hear her own rapid panting, and then she heard a disgruntled cluck from the bird, a ruffle of feathers, and then nothing else. Daring greatly, she peeked across the bank, and sure enough, the bird was leaving. She exhaled loudly in relief. "Miranda?" Dane said in a weird voice. "Are you all right?" She looked down at him. Dane's face was pressed between her breasts. Her bra straps had fallen low on her shoulders, the entire garment slipping down several inches. Her nipples were barely covered by the damp cups and were clearly outlined. Worse than that, she was clutching his head to her breasts in an effort to anchor her body on the high point that she'd climbed in her distress...aka, him. As she took stock of her body parts, she realized that her legs were still wrapped around him. She was pretty sure those were his hands all over her ass, too. "Hi there," she said brightly, trying not to blush. "Guess what? I found an emu." "Looks like it," he said in a husky voice, his hot gaze on her face. Her bra strap slipped farther down her shoulder, and she shrugged a little, trying to get it to move back up. His gaze focused on the strap. As she watched, Dane moved forward and his mouth brushed against her arm. She shivered as he caught the bra strap in his teeth and began to slide it slowly up one pale shoulder. The breath escaped her lungs. She watched him gently use his teeth to slide her bra strap into place, scarcely daring to breathe. His hands still clutched at her ass, his fingers digging into her flesh in a way that turned her on. "Dane," she said softly, the word a breathy plea. While she'd hoped to tease him a little—okay, a lot—with her bathing, the reality was almost overwhelming to her senses. And it made her hungry for more. After her bra strap was back in place, his mouth remained on her shoulder, and he very softly pressed a kiss to her collarbone. Miranda shivered, her hands clasping his neck harder. She wanted to be the one in charge—to tell him where and how to kiss her. But at the feathery kisses he was pressing to her skin, she shuddered and let him lead. It felt so good. With a small sigh of pleasure, she tilted her head to the opposite side, moving her wet hair and exposing her neck to him. He took the suggestion. His mouth moved to the base of her neck, and he pressed a light kiss against the hollow next to her collarbone, then teased the spot with his tongue. That gentle lick caused her pussy to flare with need, and she gasped, pulling him closer. "Dane," she murmured again, her voice softer than before. He unhooked her legs from around his waist and slid her to the ground, and the lovely skin contact was broken. Miranda gave a disappointed sigh as her first foot touched down. Her sigh was swallowed up when he continued to hold her other leg behind the knee. It forced her to hang on to his neck, pressing her sex against his body in a blatant fashion. Pressing her right up against his cock, which, even through his pants, was clearly hard and ready. She gasped at the sensation, her gaze flicking up to his. Dane's eyes were hooded and sleepy with desire, mere slits of green in his tanned face. There was a hint of a smile tugging at the line of his mouth, and she watched that mouth as he turned and leaned forward. Her ass rested on a low-hanging tree branch, and she was trapped between the tree and Dane's massive form. Not that she wanted to escape. She wanted him to lean closer, to put those lips on her skin again. She got her wish. Dane's mouth swooped onto her own, capturing her gasp before she could release it. His tongue dove into her mouth, giving her a hard, possessive flick. It danced along her own tongue, twining briefly before darting to graze against her parted lips. He pulled away just long enough for her to catch her breath, and then he was sucking on her lower lip, as if he could devour her whole. His fingers gripped behind her knee and pulled her closer, and she felt his hips rotate ever so slightly. The subtle gesture pressed his erection against the core of her sex, sending liquid heat flooding through her body. She felt her pussy grow slick with desire, the thin scrap of her panties not leaving her much protection against the abrasive fabric of his jeans—something that turned her on as well. Gasping her pleasure, Miranda dug her fingernails into his shoulders, leaning in to take his mouth in her own again. As their tongues locked, his hips pressed against her own, and she felt the hard length of him pushing, pushing against her core. Her mouth devouring his, her hand slid between them and rubbed along the hard ridge of his cock in the jeans, and she heard him groan with need. The sound, raw and full of hunger, made her toes curl with utter pleasure. His mouth pulled away from hers again, as if it were a fight between the two of them to take control and a kiss was a surrender. Instead, his mouth slid to her chin, tasting her skin there and then sinking lower, to her throat, her collarbone, and her breastbone, teasing the sensitive flesh there where the curve of her breast began. Miranda suddenly wished her bra had fallen off entirely. She wanted his mouth on her nipples; tugging at them with his teeth was driving her crazy. His mouth slid over her bra and his teeth nibbled at her sensitive skin, then he stopped with an odd expression on his face. "Tastes like stream water," he said. That was all the warning she got before he tugged at the cup and exposed her breast, and then his mouth was on her nipple. His tongue rasped against the aching tip, eliciting a gasp and a writhing arch from her. "Much better," he said huskily, the words blowing across her nipple before he dove at it again. A strangled sound of delight escaped her throat and Miranda bucked her hips against his jeans-clad cock, throwing her head back at the sensations. God, he knew just how to touch her. Her arm locked around his neck, and she thrust her breast against his mouth, her breath coming in harsh little gasps. Taking his cue from her, his lips brushed against her nipple, grazing it. She watched him as he nuzzled her breast, as if he simply enjoyed the feeling of her skin, then watched as his mouth closed over the tip, and she felt the delicious scrape of his teeth against her nipple as he took it between them and bit down gently. A cry broke from her throat. "You like that? Miss sweet, innocent Miranda likes it a bit rougher? You have no idea how incredibly sexy that is." His eyes were slits of desire, and he plumped her breast with one hand, making her nipple poke out, a beacon for his mouth. But instead of biting down like she wanted, he glanced back at her. "Or do I remember wrong? What do you want, Miranda?" "Oh," she breathed, pushing her nipple back into his mouth when he pulled away. "Please, Dane." "Please what?" His lips brushed against her nipple as he spoke, and his mouth grazed the creamy white flesh around it. "Please...lick you gently?" As if to prove a point, he licked the tip of her breast and then blew on it, his hot gaze moving back to her face. She whimpered in distress. That had felt good, but it wasn't what she wanted. "Dane, please." "Please...bite you?" He leaned in and his teeth flicked against her nipple, more suggestion than action. "Yes," she breathed, a shiver of excitement lancing down her body. "Bite. Please." In response, she strained against him, trying to offer him her breast in suggestion. God, she needed him inside her. Right now. Right fucking now. Her legs locked around him tighter, pulling her wet pussy against his cock even harder. "Oh God, bite me everywhere. Make me come like you promised." He gave a low growl in his throat. "I promised I'd make you come tonight, Miranda. Or were you just too impatient to wait?" He pushed her harder against the tree, and she felt one of his hands slide away from her ass and brush against her sex, pinned between them. The flutters of excitement turned to pulses of need, and her fingernails dug into his back. Oh lord, that felt amazing. Her eyes closed as he moved back toward her breast, anticipation surging. "Touch me. Don't make me wait." His fingertip grazed past her panties, one thick digit sliding against the silky heat of her pussy, and he groaned against her breast. "You are fucking soaked." She could feel it—feel how totally slick her pussy was, how his rough finger slid along the heated flesh. Her hips twisted, and she desperately needed him to touch her right— Her eyes flew open just as he sank one thick finger into her pussy, the gasping cry threatening to escape her throat suddenly swallowed by Dane's mouth on hers. "Hello?" A voice called in the distance. "Miranda? Dane?" Miranda's orgasm built, so very, very close to the edge, and she bucked her hips—to find that she was suddenly the only one participating. Dane had frozen against her, and though he hadn't moved, his eyes had gone wide, and his mouth was on hers only to keep her cries from escaping. When she pushed her hips against his hand, he didn't move. Her orgasm fizzled at the look on his face: chagrined and a bit ashamed. Ashamed of the thought of being caught with her by one of his clients. For some reason, that made her feel...dirty. Her memory filled with mental images of the photos in the closet, and she winced, shame slamming into her like a brick to the head. Her hand pushed at his, trying to break free. "That's Pete," he whispered into her ear, disentangling his hands from her body and lowering her so her feet rested on the ground. His gaze focused on the woods, not her face. "We can't be seen together. I'll meet him and distract him away while you get dressed." She said nothing. He leaned over and gave her a rough, brisk kiss. "I'll see you after dark." With that, he adjusted the crotch of his jeans with his hand, shook himself as if to clear his mind, and headed off into the woods, leaving her here alone. So he _did_ want to have sex with her. He just didn't want to have sex with her and have others know about it. Fair enough. If they found out about the two of them before the week was out, her chances at getting her revenge would be slim. After all, what was the point of ruining someone's reputation with dirty photos if she'd already been caught making out with him? He was right—they had to remain a secret from the others on the survival trip. So why did she feel so freaking dirty all of a sudden? To Miranda's surprise, Dane had turned out to be an excellent teacher. Once she returned to camp, she was able to get a roaring fire going. They ate fish that night and a few berries that Jamie had foraged in the woods for them. And the shelter was a bit lackluster but the evening was warm, and they had spent their time in front of the fire, sharing stories and listening to Dane's camping tales. He had stories of when he and Colt had run into a black bear in Alaska, of being stuck in the woods and completely lost and following train tracks for a week before finding civilization again, of fishing and foraging odd things. Of making stew out of squirrel or whatever else they could find when times were lean. Of hiking across the wild with nothing but a multi-tool. Dane's stories were told with a zealous enthusiasm that she found easy to like. He seemed to honestly enjoy the survival thing, she realized. The sparkle in his eyes wasn't only due to the firelight. He seemed...happy. It was weird. The Dane she remembered had been a smug teenage boy who'd always been rushing to his next hockey practice. She'd loved that sulkiness as a teenager, found it irresistibly sexy. The adult Miranda was drawn to the enthusiastic Dane, though. The man who went after what he wanted with both hands and approached the wilderness with an obvious pleasure that turned her on just to see. The new Dane was incredibly sexy. The reasonable, confident man who tended the fire and showed Steve where the shelter had a weakness. The man who took everything in stride, complimented his team when they did a good job, and encouraged them when they did not. He was like a big, grown-up Boy Scout with a wicked, naughty side, she thought, remembering the way he'd slid a finger deep inside her mere hours ago. Just thinking about it made her want to stick her hand down her panties and play with her clit. She was really, really turned on and she'd gotten no payoff. And every time Dane looked over at her, she couldn't help but think of their conversation earlier. Three orgasms tonight. He'd promised her three. She watched him across the fire, saying nothing. Her gaze went to his hands, watching them move as he retied a hunting knot. Her mind dwelled on how those hands had stroked her wet flesh, and how they had felt against her body. How they had gripped her hips so tightly as he'd pounded into her. God, this day was taking _forever_ to be over. This day was going to last forever. Dane glanced at the sun—still too high in the sky for his liking—as he showed George a deadfall snare for the seventh time that day. The older man meant well, but he wasn't quick to pick up on the basics, and Dane regretted not assigning the trap-making to Steve instead, who had shown a lot of competence in all the tasks he'd been given. Not that he was thinking about much of anything. His mind kept going to Miranda. How she'd looked with her sexy body wet and glistening from the water. How she'd crawled all over him, and how hot and tight she'd been when he'd slid a finger deep inside her. He'd gone into this thinking that he could be strong against the lure of an old girlfriend, but at the first opportunity to touch her, he'd caved. Like a weak, starving man offered food, he'd abandoned his obligations and had sex with her, if only to get her out of his mind once and for all. But he'd underestimated how scorchingly hot she'd be after all this time, and how naughty and wicked she'd been in his arms. And he couldn't quite get it out of his head, which was why he kept messing up the trap as he tried to show George. She was distracting him. Though she sat completely across from him in the camp, the blazing fire separating the two of them, he could feel her gaze on him. Every time he looked up, she was watching him with those big brown eyes. Her eyes would watch his hands, and he could almost guess what she was thinking. And damn if it wasn't going to make him hard, right in the middle of camp. Dane cleared his throat and shifted, making excuses about going to get more firewood. He returned a few minutes later, his desire under control once more, and dropped the armful of logs in camp. While he was gone, Pete had sat down next to Miranda, and the young CEO was talking her ear off. She seemed uninterested, though she was smiling politely, her legs drawn up to her chest as she sat. It was clear that Pete was totally smitten with her. Dane picked up a piece of wood and began to break off the smaller branches, rounding off the wood so it would burn evenly. He moved toward the fire, listening to their conversation. "So like I was saying," Pete said, his body turned toward Miranda. "There's a huge awards show next month. E3 convention. It's a big thing in gaming. I haven't designated a date yet, though there are tons of women who would kill to go." Miranda's smile was polite but distracted. "Sounds like a nice time," she said absently. "I was wondering if you'd want to check it out," he said in a lower voice. "I could get you in as long as you were with me." Dane's hands tightened on the log and it threatened to snap. _As long as she slept with him,_ Pete meant. He was just pussyfooting around the concept. Either Miranda didn't hate the idea or it was too subtle for her. She gave Pete an absent smile. "We'll see." And her gaze stole back to Dane's hands, and she clutched her knees a little tighter to her chest, the look in her eyes going soft again. Fuck. She was thinking about their interlude by the stream earlier. He recognized that soft, melting look in her eyes. She'd had the same expression on her face right before they'd been interrupted. Fuck, fuck. The memory of that seared through his mind again. "We need more wood," he barked at the group and walked away again before they could catch his returning erection. Eventually, the camp turned in and went to bed. Dane didn't sleep, his body tense and aware of the others. In the middle of the night, when the other men were snoring, Dane got up from the shelter and began to poke at the fire. He was intensely aware of his surroundings, of the quiet hum of the forest, the snores of the men piled into the makeshift shelter, and of Miranda's small lean-to on the far side of the fire. He was especially attuned to her actions, and when her eyes opened and she sat up, giving him an expectant look, he put a finger to his lips. A glance backward showed that the others were still asleep, which was perfect. She yawned and gestured that she was going to go out to the woods first, and for him to follow. He nodded, and watched her pick her way across the campsite and toward a large landmark tree. It gave him a small bit of pleasure to notice that she'd been utterly silent as she'd crossed the campsite—Miranda was smart and clever, and she knew that the snap of a twig could potentially blow their cover. He liked that about her—that she used her brain. That she knew this required stealth and silence. They were on the same page. Dane adjusted the fire for a bit longer, then moved to the woodpile, pretending to be unsatisfied with the size of it. No one was awake, but he felt better acting out the ruse just in case. He placed another log on the fire, and then left camp in the opposite direction from Miranda. Anyone who woke up might think he was heading out in search of more wood. Miranda would know better. Once the campfire was out of sight, he tracked back and headed in Miranda's direction. He saw her waiting ahead in a clearing, her arms hugging warmth close to her body. She had her back to him, and he silently brushed a hand along her spine, alerting her to his presence. To his gratification, she didn't yelp in alarm. She simply turned and gave him a shadowy smile in the darkness, and slipped her hand in his. "Lead on," she whispered. He did. He knew these woods well; even in the darkness, the paths were familiar to him. Her hand gripped his tightly and she let him lead the way. When they were in deep enough that he knew the others wouldn't be able to find them, he paused and cupped her chin, tilting her face up to his for a brief kiss. His lips grazed her mouth, and he could feel his cock harden just thinking about what they were about to do. "Give me a moment to set up, all right?" he said. Her expression grew puzzled. "Set up?" He pulled out a small plastic packet and ripped it open. "Grant insisted that we pack some basic survival supplies just in case someone on the team couldn't hack it. I brought an insulated blanket so you won't have to lie on the ground." "Oh," she said softly. "Thank you." Dane unwrapped it, wincing at the wrinkling noise in the quiet woods. He spread out the blanket—made of a silver, crinkly sort of cellophane, it was the loudest sort of blanket, but it was the only one he had available. It'd have to do. He got down on his knees to spread it on the ground, and then when it was flat, he turned to Miranda and offered her his hand. She placed her hand in his, trembling a little. He could feel the shivers working through her body. The evening was cool, but he suspected nervousness on her part. "You okay?" he asked softly. "Of course," she said quickly, pulling her hand back out of his and sitting on the blanket promptly. "But..." She hesitated, and he watched her bite her lip in the moonlight. "I'm warning you, this might be more difficult than you think with the whole three-times thing." So that was it? She was worried that she wouldn't be able to perform? That was a bit of role reversal. He swallowed the laugh that threatened to rise in his throat, guessing that she wouldn't appreciate that. "Miranda, you're thinking too much about this. If it takes all night to get you to relax, then it takes all night. All right?" "Sure," she said lamely, and he knew she didn't believe him. He nodded at her shoes. "Why don't you take those off? Get comfortable?" She did so, kicking them off and wiggling her feet in her socks. He moved onto the blanket with her and she stiffened, so he pulled back again. This was...different. When Miranda had come on to him, she'd been confident, sexy, and wild. But he'd told her his plans and she'd stewed all day on what he wanted to do to her. Now he could see it was a mistake that he'd let her anticipate their meeting tonight. Gone was the confident woman from earlier—in her place was a skittish girl who seemed to be terrified that she was going to somehow prove one of them wrong and disappoint them both. He knew her mind would be working hard on the fact that she needed to have an orgasm—more than one—to please him. In other words, his sexy promise had completely stressed her out. Not exactly the desired reaction. He sat next to her and took her hand in his. "Hey." She looked at him, wary. "What is it?" "I promise to come all over your face if you take too long on your end. Deal?" He said it purely to ease the tension from her shoulders, the strange worry that he'd somehow put there by promising her pleasure. Miranda giggled, and the sound washed over him like an electric current, lighting up his nerve endings. That soft, sultry little giggle made his cock instantly hard. But he ignored that, studying her face. "Gosh, thanks," she said sarcastically, but her face was lit up in a smile. He liked seeing that smile. Dane leaned forward and put a hand at the base of her head, pulling her forward. His mouth captured hers in a kiss—hot, wet, and open. She gasped and stiffened against him, her mind clearly starting up. He didn't want that. Apparently when she thought too much, she worried about her own responses and whether they were right or not. He just wanted her to _respond_ , not to think about responding. So he stroked his tongue into her mouth, a wicked, hard thrust designed to take her off guard and remind her what they were here for. She stiffened for a moment, and then he felt her melt against him. Her hands moved to his chest and shoulders, fingers curling into his shirt as she held on. His tongue thrust into her mouth again, then he rubbed it against hers, teasing her, coaxing with every slow, deep plunge of his tongue. Each thrust was a reminder of what he was going to do to her very shortly, and he wanted her to know it. Make her feel every deep lick into her mouth straight to her pussy. With each breath, she gasped, making soft little noises in the back of her throat. He liked that—he liked that a lot, but he continued to kiss her, not rushing a thing. Instead of the deep, searing kisses, he changed tactics on her. With one last lick into her mouth, he started nibbling at her lips, tasting the plump softness, appreciating the way that she responded to his kisses. And he slowly, slowly pulled her forward with each kiss, until he hauled her into his lap and forced her to straddle him. He expected her to stiffen again, but she didn't—she eagerly straddled him and her hips rocked against his, her hands going to his neck as she began to kiss him back with eager intensity. And when he pressed another light kiss to her mouth, she made a noise of frustration and took the initiative, moving away from the soft kisses and back to a deep, searing one, her tongue seeking out his. Fuck yes. He pressed down on her hips, letting her feel the hard length of him against her, and was rewarded with a low moan in her throat. "Feel that? I've been walking around hard all day, thinking about you and your sweet little mouth, that long hair, your tight pussy. I can't wait to taste you." She stiffened against him, and he felt the uncertainty wash over her. Damn. Back to square one. He gave her another deep kiss, but he could tell she was pulling back again. The only way he was going to get her over this ridiculous fear was to show her she had nothing to be afraid of. "Miranda," he whispered, leaning in to kiss her mouth lightly again. "I'm going to show you that you can come as often as you want to. Just because you dated a bunch of fools in the past with sausages for fingers doesn't mean that you're the problem. Understand?" She shook her head, and that sexy fall of dark hair brushed over her shoulders. "What if it was a fluke?" she whispered. "What if we try tonight and we can't make it happen?" She sounded so brokenhearted at the thought that his chest ached. In that instant, he decided that he was going to make her come if it killed him, just to prove to her that she wasn't the problem. And if it took until dawn, then he was going to enjoy every last minute of it. "You're focusing on the wrong things," he told her, using every ounce of willpower he had not to thrust up against her hips. Tonight was going to be about her, not him. "It's not about how fast you can come. It's about how much you enjoy yourself until you do. Understand me?" She gave a small snort in the darkness. "I'm not stupid, Dane. I know what sex is about." "Of course you do. And that's why you're going to let me show you how to enjoy yourself, aren't you?" He grinned at her and rocked forward, tipping them both onto the crinkling blanket until Miranda was pinned beneath him and he lay on top of her. Her eyes widened as his weight settled over her and she stared up at him. "You don't have to worry about me. I'm going to enjoy myself." Dane grinned down at her. "Good. Now, can I take your top off?" She hesitated a moment, then nodded. When he reached for the hem of her shirt, she wiggled and helped him pull it off. Her bra was another lacy confection, this one so sheer that he could see her dusky nipples in the moonlight through the fabric. He leaned down and brushed his mouth over one, feeling the fabric abrade against the sensitive peak. Miranda's breath sucked in and her hands went to his shoulders, her body tensing slightly under him. She'd liked that. Just that small reaction made him feel like the fucking king of the world. "Tonight you're going to let me take control, Miranda. I'm going to decide what you like, and I'm going to take that and use it to pleasure you, understand?" She rolled her eyes at him and attempted to get up, but he took her arm and pinned it above her head, his grip gentle but firm. "Understand?" he said softly. "This is about letting me have control over you. About me giving you pleasure." Her body trembled under his, and her hips flexed in an automatic gesture of need. She widened her eyes, staring up at him, and slowly nodded. "All yours," she said. "For tonight." "Very good," he murmured, and slid a hand over her shoulders, feeling the warmth of her creamy skin, enjoying the way the moonlight played over her body. He could feel her heart pounding under his hand, and he glanced up at her face. She had the same melting, soft expression in her eyes that he'd recognized from before—the one where she was lost in desire. Good, that was exactly what he wanted to see from her. His fingers hooked to the front of the pretty bra. There was a tiny bow at the front, and he tugged at the fabric beneath it. "You've got another one of these, right?" A frown creased her brow. "Why?" He'd take that as a yes. Dane snagged his finger under the fabric and ripped it straight down the center. The flimsy fabric split in half, exposing her breasts to the moonlight. "Because it's in my way." "You Neanderthal," she said, but a laugh bubbled up in her throat and she halfheartedly tried to slap his hand away. He kept her other arm pinned and just grinned at her, his hand moving quickly to cup one of the breasts. It was perfectly rounded, and he felt the hard little nipple scrape against his palm as he touched her. Her laugh died in her throat, her eyes widening just a little at the caress, her expression going soft. Dane cupped her breast, feeling the weight of it against his palm, and put his other hand on her opposite breast, watching her reaction. She didn't move, her arm still above her head as if he yet pinned it. Her cheeks were flushed with desire and her breathing had sped up slightly. That was good. That was very good. He brushed a thumb across one nipple, finding it hard and peaked. Her breath sucked in at the small touch, so he repeated it, rubbing the tip back and forth with his thumb. She arched underneath him, her lips parting, her eyes closing in ecstasy. He continued to rub at the nipple with one hand, enjoying the small whimpers that poured forth from her, and bent over the other peak. He brushed his lips against it, then nibbled at the tip. It hardened underneath his lips and he lapped at it, then he swirled his tongue while she moaned in response. Her back arched under him, pressing her breasts harder against his hands, and he felt her hips give a tiny instinctive buck. How had Miranda thought she was unresponsive in bed? How had her boyfriends of the past not been able to wring orgasms from her? Were they idiots who hadn't cared if she came or not? Or had they always let her have control of the situation, never realizing that what she truly craved was to be the one out of control? The one off guard? He'd discovered that when she was thinking, she worried. His goal had been to stop that thinking, and all it had taken was kisses on the soft skin and attention to her breasts. And her responses were delicious—already she writhed under him, needing more. And damn if it wasn't making him as hard as a rock. His cock was thick and heavy in his shorts, straining against his clothing, but he ignored it. He wanted to bring her to where she needed to be before he even thought about himself. And right now, she needed him touching her. He pressed another kiss on the tip of her nipple, enjoying the way her skin prickled in a thousand tiny goose bumps in reaction. So responsive. He bit lightly at the pale flesh, then kissed the nip away, replacing his mouth with his hand and cupping her breast once more, his thumbs grazing the nipples to keep her fevered. And as he touched her, he lowered his mouth to the cleft between her breasts, kissing the soft skin there. Her response was a small sigh of pleasure, her fingers tightening in his shirt. She liked to be kissed. He decided in that moment that Miranda Hill needed a lot more kissing. He pressed a kiss lower, on her ribs and belly, then another, and another, enjoying the silky feeling of her skin. Her body was perfect in the moonlight, softly rounded but sleek, all curves and delicate skin. He nibbled at her stomach, enjoying the flutter of her laughter. "That tickles," she whispered. "Good." He kissed her sweet belly because he could, and because he liked tickling her. His thumbs brushed over her nipples again, and he felt her chuckle turn into a rasping inhalation of pleasure. He could sit here and play with her gorgeous breasts for hours, he mused to himself, moving back to her breasts and replacing one hand with his mouth again. The nipple looked sadly neglected and he leaned in to bite at it. She jumped and shivered again. "Such pretty skin," he told her, plumping both breasts together so he could switch back and forth between the nipples. Even as he ran his tongue over one tip, she faltered underneath him, and he noticed her hands sliding off of his shirt. When he bit her nipple again, she gasped, but it wasn't the sexy, excited little gasp that she'd given before. Miranda was thinking again. What on earth was she thinking about? He lifted his head and moved forward to press a kiss to her mouth. "Miranda?" She gave him a hesitant smile, but said nothing. His hands moved over her breasts and he pinched both nipples at once, causing her body to jolt. The blurry, dazed look returned to her eyes, and then quickly faded again. "What are you thinking, Mir?" She bit her lip—fuck, that did amazing things to his cock when she did that—and then glanced away. He pinched both her nipples again, harder—and she yelped. "Miranda," he said in a husky voice. "Don't make me bite you into a confession." She shivered at that, but he couldn't tell if it was a good shiver or a bad one. "I was just...I'm enjoying myself," she said after a moment. "Really." But what? He watched her face, then watched her expression as his thumbs gently brushed over the tips. That flutter of excitement passed over her face again, then quickly disappeared, followed by the faintest line between her eyebrows, as if she were concentrating very hard and somehow failing. Did she think she needed to come already? Because hell, they were just getting started. He slid back down over her, pressing kisses to her neck and collarbone before returning to her breasts. Damn, he liked her breasts. He plumped one breast in his hand and then licked the nipple like he would an ice cream. "So," he said casually. "I knew a girl once who could get off on nipple play." Her expression became bewildered, then flustered. Her hands pushed at him. "Why are you telling me this right now, you creep?" Dane pinned her arm over her head again, his face leaning in close to hers. Her arm over her head left her breast high and arching, and he reached out to casually play with the nipple, toying with the tip, enjoying the goose bumps his touch left behind on her skin. "I said that I knew one girl, Miranda. _One._ She was one of the locker room bunnies that would show up and hope to bang a player. My buddies passed her around for a while. She'd show up at every home game, looking to score, and she'd come over the smallest touch. And she came a _lot_. She was noisy as hell, too. She really liked having her breasts touched, and she'd come as soon as someone touched her there." Miranda's glare could have melted steel. She'd gone completely stiff in his arms. "One time I went in to the locker room and noticed that she was in there, waiting for one of the guys to pay attention to her, and in the meantime, she was rubbing herself up against a doorknob, and getting off." Her glare turned into a smothered laugh. His own smile returned, and he leaned in to give her a gentle kiss. "She was the only girl I'd ever known who could get off on some guy touching her nipples. Most girls require a lot more work and a lot more foreplay, and I don't want you thinking there's something wrong with you because you can't get off on doorknobs." She laughed, that sultry little giggle that she always tried to smother but leaked out anyhow. His cock went even harder at the sound. He was going to lose control if he didn't start thinking about something unsexy. Like hockey. That always made his dick want to shrivel. "Understand what I'm saying?" he said huskily, teasing that perky nipple. Fuck, he loved touching her breasts. He could caress them all night, and he told her so. Her eyes widened at that, and the soft, melting look returned to her eyes. "When I touch you, Miranda," he said softly, "I don't expect you to shatter instantly. I expect you to enjoy it and tell me what you like. Understand?" She bit her lip again, and nodded. "I wasn't a virgin the other night, you know. I realize that I seem like a big nervous idiot, but I just want you to know—" "You don't have to explain to me," he said huskily. "Just let me touch you. I enjoy touching you." Miranda hesitated, then relaxed underneath him. Clearly Miranda wasn't used to men moving slowly in bed. That was a shame. Maybe she rushed them, trying to speed along the inevitable. Not him. He could spend all night playing with her breasts, teasing her body just to watch her reactions. But if he was losing her, that wasn't good. He sat up, and moved his hands to her belt. Her body tensed underneath him, her eyes wide as she watched him undo the button on her shorts and lower the zipper. Her stomach moved up and down with her excited breathing, but when he looked up at her face, she showed anticipation...and a hint of nervousness. If she'd had oral sex, she hadn't had it often, he decided. And he determined right then and there that she was going to enjoy it with him. He lifted her hips off the ground and pulled the shorts down off her legs, tossing them aside. All that was left was a tiny pair of panties low on her hips, a charming pink ruffle on the waistband. As if sensing his thoughts, her hands flew to the panties protectively. "You can't rip these. They're my favorites." Dane grinned down at her. "Better take them off fast, then." She lifted her hips and slid them down with a quick shimmy, then pushed them aside. He moved between her legs, forcing them to part, and pulled his shirt off, throwing it on the blanket next to her clothing. He wanted to take his shorts off, but right now that was the only thing keeping him from sliding between her legs and fucking her right away, so he left them on. He groaned when her fingertips slid over his stomach, tracing his muscles. "You have an amazing chest," she said in a soft, low voice filled with wonder. "It looks so perfectly formed—like a sculpture." Her fingertips trailed to his belly button and brushed the line of dark hair on his stomach, moving to the waistband of his pants. "Not yet," he said, his voice husky. He slid down on the crinkling blanket, kissing her belly again. The quiver returned to her body, her skin reacting with shivering goose bumps every time he kissed it. He kissed her belly button, then moved lower, and she began to tense underneath him. He reached up and took her hand in his, twining her fingers with his own even as he kissed the soft skin lower. "You just tell me if you want me to stop, all right?" She laughed, the sound a little nervous. "Why would I ever tell you to stop?" Ah, bravado. He recognized it well. He said nothing, simply continued to kiss the soft crease between thigh and pelvis, running his tongue along it in a way that made her body tremble. Then he placed one hand over the warmth of her mound, and she jumped slightly. Her fingers clenched in his. "Sorry," she whispered. "I'm jumpy tonight." "No worries," he said, then brushed his fingers over the trim brown curls hiding her sex. He ran a finger over the edge of the lips, from the front all the way down to where it led into the deep well of her body, then ran his finger back up again. She was soaked, her sex slick and wet, and his fingertip was damp just from her wetness. The body shivers hadn't stopped, and her fingers clenched his tightly—but when he looked up, she still had the soft look in her eyes. Good. "You have a lovely pussy, Miranda," he said. "So soft and shy. Been waiting all day to kiss it." Her back arched slightly, as if anticipating the kiss, but he simply continued to run his finger over the seam of her sex, before dipping in a little farther and sliding through the slick wetness there. Her breath hissed out of her lungs, and when, again and again, he'd raise that skimming finger up to her clit, her hips would lift in need. He didn't give her what she wanted, though—he'd flick over her skin and then sweep down to the warmth of her core, brushing his fingertip there as well. She moaned. Her fingers twisted in his, clenching hard. "Please," she breathed. He slid a finger closer, rubbing in the slick wetness of the hood of her sex, circling close to her clit—close enough to torment but too far for her to get satisfaction from it. "Please what, Miranda?" "Please touch me," she breathed. "There." "Where?" he teased, his voice husky. "Tell me where you want me to put my fingers." Miranda arched, her mouth working silently. He watched her, his cock jumping in response every time she gasped. He continued to run his finger along the wetness, teasing and coaxing the words out of her. "On my clit, please." He slid a wet fingertip around her clit, circling it once, enjoying the gasping cry that it wrung out of Miranda. "Like that?" "Oh God," she moaned, her fingers clutching his hard. "Again." He decided to do one better. Taking one of her legs, he let go of her hand and pulled her leg over his shoulder. Then he did the same for the other leg, until he knelt with his face inches from that hot, wet pussy, and her legs were over his shoulders. "Want me to touch you again?" he murmured against that hot, damp flesh. His mouth watered, waiting to taste her. She moaned a response, her fingers digging into the thin plastic blanket. He'd take that as a yes. Leaning down, he nuzzled against her sex and then swept his tongue over her clit. Her entire body tensed and she shuddered hard, her thighs clenching against his face. He could hear the blanket crinkling madly from her hands fisting in it. He lifted his head to watch her. "Did you like that, Miranda?" She gave a jerky nod, her hips quivering so close to his mouth. "Tell me if you want more." Again, a jerky nod. "Well?" "Just fucking lick me, Dane," she snarled at him. "Absolutely," he said in a low, pleased voice, and gave her another slow, sensual taste from the heated well of her sex all the way back up to her clit. Her legs trembled again, and her breath was coming in short, sharp pants. Excellent. His own cock was rock hard, his shorts painful against the hard length of him. Soon, he told himself. She was close. Not there yet, but closer. It was time to think about hockey again. He started to mentally run down penalties in his head. Icing. Boarding. High sticking. She flexed underneath him, impatient. He gave her another sweeping lick and then settled against her clit, slowly circling it with the tip of his tongue, coaxing it with teasing flicks before circling it again. Miranda's entire body was tense with desire, and every time he licked, her entire body shivered and a small, breathy little cry erupted from her throat that drove him wild. He wanted to stop and ask her if she was enjoying herself, but making her think was off-limits. Dane swirled his tongue around her clit and sucked lightly, and he was rewarded with her sharp inhale. He increased his efforts, swirling his tongue around her clit faster and faster, brushing against the small button repeatedly as her cries increased. Her hands fisted against his short hair and her hips bucked against his tongue. "Oh...like that," she cried. Hell yeah. He increased the attention, licking and sucking at her clit rapidly. She continued to whimper, her fingers desperately digging against his hair, scratching at his scalp as she tried to find purchase. After a few minutes, she began to push his head down and raise her hips against his tongue, her cries turning into short, sexy little pants that had no words, just whimpers. He increased the pressure, his next lick stroking hard against her clit, and then sucked hard, and then sucked again. A startled cry erupted from her throat and she came, wetness on his lips and tongue as her entire body trembled and undulated underneath him, her legs clamped tightly around his shoulders, her breath whistling out of her lungs in a low, slow, deep cry that seemed to go on forever. Holy fuck, that was sexy. She'd come so hard. How had she ever thought herself unable to come with a man? He gave her clit one last slow lick, and was rewarded with a long, shivering aftershock that quivered through her body. She was still up the mountaintop, still lost in pleasure. God. He started to think about penalties again. High sticking. Slashing. Spearing. Fuck, no, spearing made him think of how badly he needed his cock deep inside her— "I..." She panted, her hands falling back to her breasts and she struggled to think of something to say. Her eyes were wide. "I...oh wow." He kissed the inside of her thigh, his own body tense with need. "Like that, did you?" "That was incredible." She stared up at him in surprise. "I mean, last night was really good but that one...wow." Her knee was close to his mouth and so pretty he reached out and kissed it, too. "That," he told her, "was one. I promised you three." She licked her lips, the glazed expression returning to her face. "I...but..." Dane slid a finger through her drenched sex again, enjoying her latest shudder of response. He slid the digit into the hot well of her pussy, noticing how she spasmed against his finger when he thrust. The low moan began to build back up in her throat again, a moan that he felt all the way down to his cock. Damn. He was so close to coming in his pants. He pressed his forehead against her belly, trying hard to get his body back under control again. High sticking, he reminded himself. Icing. Hooking. Holding. Hand pass. Fuck it, why did all the penalties sound totally fucking dirty when he needed them to keep his mind off of sex? He forced himself to concentrate on teams, instead. Montreal Habs. Los Angeles Kings. Vancouver Canucks... When he was safe from going over the edge again, he grasped her breast and teased the nipple, thrusting his other finger deeper into her wet pussy. She was hot and slick with need, and her moans turned quickly to gasps. He moved his mouth back down to her pussy, flicking his tongue against her clit in time with the thrust of his finger, and when she began to quiver hard, he added a second finger, twisting them and repeatedly thrusting them into her wet sex, mimicking the fucking he was dying to give her. It didn't take long for her to go over again. She started to tense under him, her muscles clenching as if she were preparing for the next orgasm, when suddenly her entire body stiffened and she sucked in a long, deep breath. Her pussy contracted—hard—against his fingers, and he smiled against her clit. He gave it one last, long lick. "That's two." She looked up at him, dazed, the look on her face utterly blissful. "Holy...shit." Damn. He loved her expression, the way she hid nothing. Dane let her legs slide down to his sides, and he leaned forward and kissed her deeply, liking that her mouth was so soft and giving under his own. She was boneless with her orgasms, soft and sated and warm. He'd give her one more so she'd know he wasn't full of shit...and because he wanted to hear that soft, throaty cry when she came a third time. He could listen to that sound forever. He quickly divested himself of his shorts and underwear, releasing his cock, aching hard under the need to come deep inside her. Pre-cum covered the head, and it throbbed with need. His entire _body_ throbbed with need. Miranda's gaze went to his cock and she reached for it, her fingertips brushing against the damp crown. "Can I—" He hissed, dangerously close to spilling himself, and pulled her hand away. Icing. Icing. Had to think about icing. "Give me two seconds, Miranda, and you can have all of me that you want." He pulled a condom out of his pocket and rolled it down his shaft quickly, then took a deep breath. He needed to pace himself if he was going to make her come again, and not bust as soon as he sank deep into her. Normally he had no trouble with control, but Miranda did something to his insides, where he lost all his macho swagger and became this addicted fool who lived to see her smile, to see that expression when she came... Slow was the key here. Slow and steady. He spread her legs wide, his hands caressing her soft, pale thighs and pushing them forward so that her feet were up in the air, her hips tilted at the perfect angle. With one hand on her thigh, the other guiding his cock, he fed the head of his cock into her tight, wet heat. Miranda let out a cry, her hips bucking, trying to pull him in deeper. "Oh God." "Just hold still," he told her, teeth gritting. She was clamping around him so tight. So hot and wet and so very tight. Sweat beaded on his brow, and he sank another inch into her, slowly. Carefully. One massive thrust and he'd lose it. Spill inside her and never give her that third orgasm. With gritted teeth, he ignored Miranda's breathless response and fed her several more inches, moving excruciatingly slowly. Icing. Icing. New Jersey Devils. Atlanta Thrashers. Wait, Atlanta was sold— "So full," she said, interrupting his wild internal monologue. "God, it feels—" "Shh," he told her, a hair trigger away from losing control. "Miranda, just hold on." She moaned underneath him and he felt her pussy flutter tight around his cock again. His control snapped. He rocked forward and impaled himself in her to the hilt. Her gasp as she arched underneath him was fucking beautiful. She gave a soft little whimper, lifting her knees and tucking her hands tight behind them to pull her legs close to her chest. Her posture pulled him deeper, her pussy tight and wet around him. He held still, on the verge of losing it, doing his damnedest to maintain control. After a moment, he pulled out, slowly, then buried himself to the hilt inside her again, the long sweeping stroke of his cock plunging deep inside her. Her gasp turned into a throaty moan, and her fingers clenched on his shoulders, nails digging in. Fuck, where had the Atlanta team moved to... "Oh God," she whispered, her eyes closed with the intensity of sensation. "So deep..." Oh, fuck Atlanta. He pulled out and drove into her again, noticing that this time, she raised her hips to meet his thrust, and gave another trilling moan as he sank into the hot, tight depths. He drove into her again, and then again. Each time, she rose to meet him. Her hips slammed against his, intense and powerful, as if he couldn't get deep enough to satisfy her. He moved hard against her, pounding deep into her core, his hands moving to pin her hips to just the right place. He thrust hard, then circled his hips in a long, languid motion, hoping to hit the right spot. Her eyes flew open and she gave a stuttering gasp, staring up at him. He felt her calves tense on his shoulders. Bingo. G-spot. He thrust again, repeating the motion, rolling his hips until he hit just the right spot and she clenched up against him again, trying desperately to raise her hips. His hands kept them firmly pinned in place, and he gave another sweeping, circling thrust. "Do you like that?" Miranda's lip quivered, and she began to say something, but at his next thrust, her words dissolved into a choked scream. "What was that?" he teased between gritted teeth, about to come just from her reaction. So intense on her beautiful face. "I couldn't hear you." Her hands suddenly clenched on his, her ankles digging into his shoulders. "Harder," she gritted. "Please, Dane." He didn't need to be asked twice. He thrust again, rocking her backward on the crinkling blanket, and was rewarded with another stuttering gasp of delight. Again, and again, and then he was pumping into her repeatedly, her hips pinned against his thrusts, her choked cries echoing in the forest as he thrust into her over and over again. A full-body tremble started and as he drove deep, he watched her arch again, her shoulders rising as she tensed, her mouth working in a silent scream of pleasure. Her pussy clenched and fluttered around him, hard. Miranda was coming again, and she was fucking beautiful in her abandon. He watched her, thrusting again—once, twice—her pussy spasming around him, milking him—until he couldn't stand it any longer. And then he went over the edge behind her, his own orgasm coming over him so hard and fierce that he growled, digging his fingers into her hips, his entire body tightening as if one single motion would shatter him. He came with a shout. She gasped for breath repeatedly, as if there was not enough air in the forest to fill her lungs, her legs still in the air, ankles on his shoulders. He leaned heavily on her, still sunk deep into her body, and stared down at her with a panting grin of his own. "That's three," he said smugly, and didn't care how pleased his voice sounded. Miranda laughed, soft and breathy. "You win." # NINE They folded the crinkling plastic blanket, but it was obvious that it had been used roughly and wasn't going back into the tiny pouch it had come from. "Maybe we could bury it," Miranda suggested. They did at the base of a tree, and put a stone over it to mark the spot. Dane made a mental note to come back and get rid of the evidence once the trip was done. After it was buried and hidden, they walked back to the campsite, fingers linked as if they were both reluctant to lose the intimacy they'd formed. Dane knew he wasn't ready for it to go away just yet. When they could see the coals of the fire in the distance, Miranda tugged at his hand and gave him a wicked, satisfied smile. "We going to do this again tomorrow night?" Sounded like a good idea to him. He was just about to respond when he heard the sound of a heavy log being dropped on the fire. It crackled in the distance and he froze. Someone was awake? Panic swept over him. They'd see him with Miranda and figure out that they'd been on a rendezvous in the woods. And if the clients found out, it'd get back to Colt and Grant. He'd fuck up the business, and then his friends would hate him for dicking over the team. All over again. Just because he couldn't keep his cock in his pants. He dropped her hand, regret washing over him. "Miranda, we can't do this again. Ever." She gave him an offended look. "What the f—" He clapped a hand over her mouth and stared down at her, face grim. "There's someone awake back at camp. I want you to wait ten minutes and then come back like you've just gone to the bathroom. Understand?" She glared from under his hand, but nodded. His own heart was thudding madly as he tucked in his shirt, pretending he'd just been on a bio break of his own, and reapproached camp. Steve was up, the older man giving him a proud look. He pointed at the fire. "Got it going again." "So you did," Dane said, his voice forced with casualness. "Thanks, man." "You seen Miranda anywhere?" Steve asked, his expression puzzled as he scanned the woods. "She was here when I left," Dane lied. "Dinner did a number on me, though. Maybe it made her sick, too." As if that were her cue, Miranda came to the edge of camp, holding her stomach. She looked the part—her hair was disheveled, her face flushed, her clothing askew, breasts loose under her T-shirt. Of course, he knew the real reason why she looked so rough, but it suited his needs. She glared at the two of them. "Good night." "Night," Dane said softly and moved back to the bunk. His heart was still hammering. Would Steve suspect something? But no, the older man puttered at the fire for a moment longer, then returned to the shelter. After a few tense moments, he began to snore. They'd been so close to being found out. His entire life, fucked up in one glorious, well...fuck. That couldn't happen. He couldn't touch Miranda again. Not until the class was over and done and no one thought of her as his student anymore. Playing around with her this week was just too dangerous. But there was always after. Next time she planned on a survival trip, she'd bring a comb, she thought grumpily as she dragged her long, tangled hair into a ponytail. And shampoo, she decided, and thought longingly of her shower back home. Maybe she'd invite Dane to come shower with her. She wouldn't have minded soaping him up and exploring his body with her hands...great. She was still horny, even hours after the best night of her life, ever. It seemed that the more she had sex with Dane, the more she wanted to have sex with him. Not exactly conducive to a revenge plot. She was the one who was supposed to hook him and make him dance to _her_ tune, not the other way around. "Let's break camp," Dane said, looking alert and utterly scrumptious this morning. There were no circles under _his_ eyes, she noted sourly. "Miranda, put out the fire. Will, go refill the canteens. Pete, you help Steve take apart the shelters." Miranda knelt next to the fire and began to put together her fire bundle for the day—a task that she'd shown George how to do last night as part of their cross-training. "Are we going to switch camps every day until we leave?" asked Steve. "Not every day," Dane said. "But today, we have our first team-against-team challenge. The winners get a special treat and a special camp." "And the losers?" Pete said, always the pessimist. She hated to admit it, but she was really starting to dislike Pete. "The losers get to find their own camp. I'll leave that up to you guys, if you lose." His voice sounded cheerfully confident. "But we're not going to." "What sort of challenge?" Miranda asked, using a twig to roll one of the coals out onto the tinder bundle. "You'll see," was all Dane said to them. "And you're all competitive people, so I think you'll enjoy it." Miranda didn't say anything. Was she competitive? She didn't really fit the mold of the average camper on this trip—she wasn't looking to learn skills that would bring her ahead in the corporate world, or learn about teamwork. She was here...well, she was here to bag the trainer. She gave Dane a thoughtful look. He was ignoring her this morning. It seemed a little off, given the mind-blowing interlude from last night. Hell, she couldn't stop thinking about it...or the fact that he'd turned her down flat afterward. She'd known he was skittish but his rejection had still come as a surprise. More than that, it made her determined. She was not going to go home with her tail tucked between her legs, wondering what might have been. Taking his rejection as the final word on things. Dane hadn't been trying to hurt her last night. She knew that; she wasn't being silly or emotional about this week. He was protecting his ass. She just needed to convince him that a week with her was far more enticing than a week without her. It was time to change up her game. Perhaps she wasn't being competitive enough. She began to think, devising a new game plan. The team hiked for a short time once camp was broken—Miranda falling to the rear again. She didn't mind. Not being in the midst of things gave her time to think, and she had a lot of thinking to do. Pete was chatty but she wasn't all that interested, and her responses to him were short and noncommittal. After a while, he got the picture and stopped talking. They entered a particularly rocky area, with a shale cliff off to one side. On the other side, atop the cliff, she could see an ATV trail, and she heard the sound of an engine in the distance. A flash of blue caught her eye as they ascended the ridge and as she watched, the blue team emerged from the woods a short distance away, led by Colt Waggoner. For some reason, she wasn't enthused to see them. Seeing Colt reminded her of what Dane had thought—that they'd set her up to flirt with him. That she'd been some sort of grand test that he'd failed. She frowned in his direction. If they didn't trust that Dane was on the straight and narrow, they were going to ruin her plans to ruin him. If anyone was going to destroy his life, she wanted to be the one to do it. Dane was going to be hers to make or break. Strange how that thought left a bit of a sour taste in her mouth this morning. She blamed it on the camping. A purring engine revved and a bright red four-wheel ATV sped up the trail, dragging a wheeled cart behind it. A woman was perched on the ATV, her sunglasses masking her expression and her shoulder-length curls sprouting from underneath a cheerful, decal-covered helmet. Brenna the assistant, Miranda remembered, and disliked it even more when both Colt and Dane's faces lit up at the sight of her. "Got your gear, boys," Brenna called out. "One gun for each person on your team and a hundred rounds apiece." Guns? Rounds? This did not sound like a good idea. Forgetting their team lineups, the groups splintered and gathered around the ATV as Dane and Colt began to dig through the equipment cart that Brenna had lugged up the trail. Sure enough, there was a stack of what looked like paint-splattered rifles in the back, along with some boxes. "What's all this?" one man asked before Miranda could. "Paintball competition," Colt said, his expression cold and unfriendly. Brenna chimed in for Colt, casting beaming smiles at everyone. "That's right. Today is paintball day. Doesn't have much to do with survival, but everything to do with teamwork. And it sure is fun." "Greaaat," Miranda said unenthusiastically. A bunch of men running around in the woods shooting at one another did not sound like her idea of a good time. She eyed the others on her team—clearly she was the only one with reservations. The others looked positively giddy at the thought. She took the helmet handed to her and gave it a wary look—it was a full face mask with goggles built into the faceplate. "Are we playing paintball or heading to a _Star Wars_ convention? Because I call dibs on the Boba Fett costume." Pete laughed uproariously at her joke, but the others just stared at her. Well, so much for that. Miranda put the mask on to hide the blush on her cheeks, took the dark red jumpsuit they handed to her, and zipped it over her clothes. This was apparently so she wouldn't get paint on her own clothing. Thoughtful. There were enough jumpsuits for both the red team and the blue team. Brenna also passed out protective cups for the men. "Can I have two of those?" Miranda asked, and Brenna's eyebrows shot up. Miranda patted her breasts, and the other woman laughed, but didn't hand her a pair. Oh well. She finished dressing as Colt read off the rules of the game. It was a capture the flag game. Each time a team's flag was captured, they would score a point. The team with the most points at the end of the day would win the special campsite. The others would just have to go to whatever campsite was the reject campsite. Clearly the reward was the game just as much as it was the campsite. At least, it was for everyone but her. Alone time with Dane was the true reward, as far as she was concerned. As Brenna distributed gear and counted out bullets to the teams, Colt gestured that Dane should join him off to the side. Dane swore under his breath. Great. Colt wanted to have a chat, and Dane didn't feel like talking. He knew what this chat was going to be about. _How are you handling your team? Miranda giving you trouble?_ At least, he hoped that was how the conversation would go. He jogged over to the tree line and nodded at Colt. "How's your team doing?" Colt shrugged, his tall frame relaxed as he leaned up against a tree. "It's doing." He looked at Dane expectantly. "You?" Dane shrugged and glanced back at his small group. They were crowded around the ATV, laughing and teasing one another as they were handed equipment. Miranda seemed to be in high spirits like the rest of them, but she kept glancing over at him, as if curious as to what he was doing. Damn. He hoped Colt didn't notice her watching him. He turned back to Colt. "They're all right. They're learning. It's not quite what I expected, but it's not bad either." Colt crossed his arms over his chest, the muscles on his forearms flexing. With his lean build, ever-present dog tags, and high-and-tight haircut, he wore his U.S. Marine background like a badge of honor. It always made Dane slightly uneasy. He'd had a very different path than Colt to get where they were today. And while he tried very hard to rid his life of everything hockey, Colt still lived his life as if he were in the marines every day. He grunted now. "They're soft." "They are civilians," Dane said dryly. "If they were efficient killing machines, I doubt they'd need our class." Colt grunted again, scowling as he stared out at the team. The friendliest of instructors, he was not. He wondered how Colt's team was faring with the monosyllabic trainer. He knew Colt well—he'd take a bullet for his brother—and the man knew survival like the back of his hand. Teaching and conversation, however, were not two of his strong points. Colt nodded at the teams gathered. "Yours are worse than mine." "I don't doubt that," he said with a half laugh. "I had one try to pass off a dead fish as his catch." Colt's glare swiveled and turned to Dane. He scowled at the ex-marine. "What?" Colt's eyes narrowed. "Why are you _happy_?" _Oh, here we go_. The interrogation begins. "I'm not. Fuck off." "You just giggled." "Fuck you, bro. I did _not_ giggle." Colt snorted. "Like a schoolgirl." Dane elbowed Colt, hard, and was rewarded with his grunt. "Not fucking giggling. I'm just enjoying myself." When Colt still looked skeptical, he added, "I like being out here in the wild again. Reminds me of Alaska. Sometimes I miss Alaska. We didn't have a care in the world up there, you know?" It was a half lie. While he enjoyed being out in the wild, the reason he felt so relaxed and calm lately was largely due to Miranda. He liked being around her, liked seeing her smile. Liked feeling her hips rising under his. Liked that flushed, startled-with-pleasure look she got on her face when he licked her. But if Colt knew that, he'd kick Dane's ass for fucking up their new business. Colt liked a rigid adherence to rules. He wouldn't understand a deviation. So Dane watched and waited, hoping that Colt would shut up and go back to his team so he could go back to his. But Colt only grunted. "Alaska was cold. Hated that. Made my dick shrivel." Dane grinned and clapped his friend on the back, then leaned in. "I don't give a shit about your dick." Colt punched him, and Dane punched back, back to being friends and ribbing each other like nothing had ever changed. The bullhorn clicked on and Brenna's chirpy voice screeched over the trees. "If you two are done beating the crap out of each other, can we please play some paintball?" Bags full of paintball ammo, bottled water, and PowerBars, as well as other miscellany she didn't recognize, were handed out, and the teams shouldered their packs and readied to go into the woods. Through her lashes, she watched as Dane stripped his shirt off before putting on his jumpsuit. Watching his naked back gave her all kinds of ideas, and she smiled to herself as he zipped up and then turned to his team. Oh, she had lots and _lots_ of ideas suddenly. The blue team set off into the woods as Brenna called instructions after them. "Stay on this side of the stream," she warned. "Don't rove out too far. You all have a fifteen-minute start before I sound the air horn that tells you to begin. No stealing ammo...and no nut shots!" With that command ringing in their ears, the red team set off as well. George was elected the leader—despite his gray hair and a MFA in business administration, George was also a passionate paintball player in his spare time, so he'd been elected the team captain. Miranda had thought Dane would assume the position of leader, but he was happy enough to let the others take charge. Perhaps this was part of his training plan as well—letting others take the initiative. Clever _and_ subtle. She had to admit, she liked that in a man. Too bad it was Dane Croft who housed those qualities. George led them over a few hills and through the trees. They ran across a hill—heavily treed and with a sandy ridge that was sloughing away due to erosion—that George declared to be the perfect spot to plant their flag. They did so, and then he began to direct orders. "All right—we need one person to stay at camp at all times to guard the flag. Miranda, that's you." She snapped out of her musings and glared at George. "Why is it me?" He looked as if he was going to comment that it was because she was female, and then thought better of it. "All right, Pete, you can stay at base this round. We'll take turns. Plenty of time to score points. All we need to do is get to ten first." That seemed fair enough. Mollified, she listened as George laid out a battle plan—but she was only half paying attention. Evil Miranda needed to make a play today to win back Dane's attention, she had decided. She had tried walking next to him as they'd marched out to plant the flag, but he'd slowed down, acting as if he were re-lacing his boots. Painfully obvious. Was he afraid that just being seen with her meant that someone was onto their little midnight trysts? Or was he regretting their rendezvous? Or simply done with her? He didn't get to be done with _her_ , she thought with a scowl. She got to decide who was done with whom, not him. And today, she'd just have to up the game a little. No problem. And if her plans failed, well, she could always resort to one of the nut shots they'd been warned about. That stirred a thought in her mind. A dirty, naughty little thought that made her a bit scandalized...and a lot excited. She shouldn't... But she really, really wanted to. She peeked over at Dane, his paintball gun resting on his shoulder. He slouched, leaning one shoulder against a nearby tree, and avoided looking in her direction. That decided her. Dane Croft wouldn't stand a chance against Evil Miranda. After a few minutes, the team split up to take positions for the game. Some of the team would be sniping from secluded positions, and others would attempt to take out the camp. Miranda had volunteered to snipe from a vantage point. She grasped her gun, put on her helmet, and began to stalk through the woods. Once she was out of sight, she cut through the trees. She was supposed to guard a lookout point a good quarter mile away from the flag, but screw that. Instead, she cut across the forest back to the trail that she'd passed on her way out of camp and headed down it, and after Dane. He was a decent clip ahead of her, and she lagged behind to make sure that he didn't see her. If he did, he might warn her away. She trailed him for a bit, and when he arrived at his location, she ducked behind the nearest bush and unzipped her paintball jumpsuit. She stripped off her clothes—all of them—and then put the jumpsuit back on again. The air was definitely crisper without that second layer of clothing, and she felt her breasts bob and jiggle with every step she took. She put a gloved hand on her breasts and held them still against her chest. She hadn't thought about the bouncing. Hopefully the others wouldn't notice if they should run across her. She buried her clothes in a pile of leaves and arranged a branch over it so it would be easily recognizable when she came back. Then she went to find Dane. She found him a short distance away, crouched behind one of the taller trees in the area and scanning the distance. His helmet was on, but she remembered that he had the only one that was a solid black. Plus, nobody else out here had those broad shoulders or that ridiculously tight ass. She moved toward him from behind. He hadn't noticed her yet, which was good. She liked the element of surprise. Smiling to herself, she crept quietly behind him and pulled off her helmet, carefully placing it on the grass nearby. Her gaze dropped to the ground and she noticed a nearby berry bush. A mental image filled her mind of nibbling berries off of his naked body, and she pulled a few of the berries off of the bush. What would be sexier than slowly licking and eating the berry in front of Dane? Just another gun in Evil Miranda's holster, she decided. She slid up behind Dane and when he was close enough to tap, she reached out to run her hand along his nape. He turned so violently that she stumbled backward in surprise. With a lightning-fast motion, he raised his paintball gun, then lowered it almost as quickly. "Miranda! What are you doing here?" Dane checked behind her. "Are you being followed by someone on the other team?" "Nope," she said in a sultry voice. She pulled out one of the berries and began to toy with it, playing with the small red fruit along her lips. "Guess what?" His gaze became glued to that berry, just as she anticipated. Nice. "Uh, Miranda," he began. "I couldn't wait to see you," she purred in her best Evil Miranda voice, then tapped the berry against her lip playfully. "You've been very naughty, Dane Croft. Why are you being so cruel and ignoring me?" "You know why." He swallowed hard. "Can I tell you something?" Her smile curved and she leaned forward to flick the berry against his mouth. "Of course." He caught her hand and forced her to drop the berry. "That's poisonous." "Oh." She stared at him blankly for a second, and then added, "I knew that." He raised an eyebrow. She furtively scrubbed her mouth. "Poisonous...by contact?" "Digestion," he told her, his look very serious. "How many did you eat?" "None." Thank God. He nodded at her and then turned back to his paintball gun. "You'd better go back to your post, then." She frowned. This was not how she'd envisioned their rendezvous going. Now he thought she was stupid. Rubbing her lip one last time, she slid over next to him and tossed her gun aside. "Was that all you wanted to tell me?" He glanced over at her briefly, then went back to scanning the woods. "That was it. What else did you have in mind?" She shrugged. "I don't know. I guess we could talk about what happened last night." She kept her voice light and sultry. "You've been avoiding me all morning." "That's because the others are going to find out about us." His gaze flicked back to her. "You know I can't let that happen. I made a vow to my partners that I'd be on my best behavior and I lied. I'm not proud of myself right now, Miranda." That didn't sound like it would fit her plans at all. If he was going to be this cold with her for the rest of the week, she doubted he could turn it back on like a switch as soon as they were out of here. Her window of opportunity had to remain open, and she had to keep Dane hooked and insatiable. So she bit her lip and thought for a moment more, then scuffed her hiking boot on the ground. "Hey, Dane." He didn't turn back around, his voice short. "What?" She unzipped the top of her jumpsuit an inch or two—not enough to expose skin, but just enough for him to notice the sound. "Guess who's not wearing anything under their jumpsuit?" Crouching at her feet, Dane froze. Slowly, he turned back toward her. "Miranda, what..." And she pulled the zipper down the rest of the way. # TEN Dane stared at Miranda as she slowly unzipped the jumpsuit. Sure enough, she exposed inch by sultry inch of her perfect skin until the zipper paused below her navel, and she looked up and gave him a naughty look. "What do you think?" _I think you're trying to kill me_. "I think we need to talk, Miranda—" She reached up and placed a finger over his mouth to shush him. "You've talked enough, and I'm not liking what you're saying." "Miranda—" She shook her head at him. "I like you, okay?" Her eyes were big and wide as she said that, as if she were trying to convince herself of that as much as him. "And I can't stop thinking about what happened last night. And you know, I liked the three-peat." She blushed at the words, trailing her finger down his front, as if fascinated by him. "And I thought you liked it, too." He had. He'd liked it too much. That was exactly the problem. He was losing his head where Miranda was concerned, because all he could think about was bending her over and sinking into the wet, slick satin of her pussy again. Just at the thought, his cock started to rise in his jeans. His mouth tightened and he glanced down at Miranda's body, the tantalizing narrow strip of flesh made visible by the gaping zipper. "I like you, too, Miranda, but—" Again, she put her finger over his mouth. "So many buts," she said softly. She took a step forward and then she was pressing those amazing, soft breasts against his chest and his cock grew hard as a rock. "You're not going chicken on me, are you?" she said in a soft voice, and trailed a finger up the zipper of his jumpsuit. "Because I thought a big man like you wouldn't be afraid of anything." She'd deliberately sighed over the word _big_ , and that made him all the harder. It was getting difficult to think straight. All he kept seeing was that curve of her breast exposed by the zipper, that sexy dip of her belly button. "We don't have any privacy here in the woods." But that was the wrong thing to say. Her smile grew brilliant. "We have privacy right now," she said, and her hand tugged the jumpsuit over to one side, revealing the perfect globe of one pretty breast, the tip taut and rosy. Underneath her breast, he saw that small triangular-shaped mole that seemed to almost mock him. "And I really want you to touch me again," she said, her voice almost a sigh. "I keep thinking about your hands all over me, and your mouth, and your cock deep inside me..." Her fingers skimmed over the tip of her breast, as if just the memory made her want to touch herself right in front of him. "And I keep thinking about how much I enjoy touching you, too." He was having a hard time swallowing. His throat was so fucking dry all of a sudden. Dane's Adam's apple bobbed as he fought to swallow hard, but it was useless. He couldn't see anything but that perfect breast. "Oh? So you've been thinking about touching me?" "I have," she agreed, and when she reached for the zipper of his jumpsuit, he didn't stop her. She was a sultry enchantress, so sexy and confident and utterly sensual. Her gaze followed the zipper and it exposed his bare chest underneath. He let his hand brush over her covered breast, pushing the fabric back and exposing the other to the air. He loved Miranda the seductress in that moment. "Fuck," he said in a ragged voice. Despite his resolve, here he was giving in to her demands. All it took was for her to show him one lovely breast, a pink nipple, and a teasing mole and he was a goner. "Would love to," she said with a sultry smile. "But right now I'm interested in a little something else." His hand went to cover her breast and he laid his palm over the peak, his tanned, callused hand looking foreign against her smooth skin. "And you want me to touch you?" "Oh yes," she breathed, then almost looked embarrassed at how it had come out. A hint of a flush touched her face and crept down to her breasts. "I lose my mind whenever you touch me. I like it. It allows me to...lose myself in the moment." "And lets you come?" He whispered in a husky voice, leaning in close to her. Even after several days in the woods, Miranda still smelled good. Clean and fresh and warm. She bathed in the stream every morning, and he had to admit that he had picked campsites by streams for that very reason. "Every time," she admitted, biting her lip as if she felt scandalized bringing this up. He let his hand slide to her belly button, then slid it into the V of the zipper and felt the curls of her sex. They were completely soaked and he hissed at that. "You are so wet." Her hands gripped him, her body locking as if it was too much to process, and she gave a shuddering gasp. Her hips flexed against his hand. "Oh, Dane. Touch me, please." As if he could resist when she begged so very sweetly. With one hand gripping her around her waist, he pulled her closer and dug one blunt finger between the damp curls. She gasped as he grazed her clit, then sank the finger into her hot, silky warmth. "Mmm, I like that," she said, and her arms wrapped around his neck, dragging his face down so she could kiss it. Naked and panting, he twisted so her breasts pushed against his chest, and her hips worked over his hand. He slid the finger in and out slowly, enjoying the way she moaned and twisted with each small motion. Her mouth frantically searched along his jaw as he pressed the heel of his palm against her clit and sank his finger in again. Miranda gave a muffled gasp that echoed in his ear right before she bit down on his earlobe, and he thrust his finger into her hot core again. Once, twice, and then she shivered all over, released his ear and buried her face in his neck to muffle her low moan. Warmth spilled over his palm as she gave a deep gasp and came against his hand, unable to hold out longer. It was incredibly erotic. He continued rubbing her pussy, enjoying the little quivers and involuntary jerking pulses she made as she came down from her orgasm. Her eyes were closed and the flush had returned to her cheeks and breasts, and she looked gorgeous in the dappled sunlight. He didn't care if this was a mistake. Seeing that open, raw look on Miranda's face and feeling her come all over his hand? He'd do it again in a heartbeat. Her brown eyes opened slowly. She smiled up at him, the expression languid and satisfied, as if she'd never felt so good. "We're getting better and better at that, aren't we? Soon I'm going to come just from you looking at me." Hearing that made him even harder. If he was any harder, he'd start losing blood to his brain. "Where's the fun in that?" She gave a throaty, sexy little laugh and slipped out of his arms. "It's fun for _me_." His hands slid from her body and he watched her move away with a bit of regret. She had the most beautiful breasts. "But you've made me all distracted. That isn't what I came here for," she said in a coy voice. He raised an eyebrow. "Oh, no? What are you here for, then?" The tiny, sly smile remained on her mouth and she reached for his zipper. She pulled it down below his waist, and then her hands went to the top button of his jeans, causing his straining cock to jump. "Want to guess?" Dane swallowed hard. Without even a second of hesitation, he reached out and brushed her long brown hair back over her shoulder, where it had fallen over her breast. "Sure." "How about I just show you?" She wiggled her eyebrows at him in a look that managed to be playful and sexy all at the same time. And then she dropped to her knees in front of him. He groaned under his breath as her hands slid to the hard bulge in his jeans where his cock rose, desperate to be free. "Miranda," he said softly. "What are you doing?" She gave him a playful look and undid the buttons of his fly. "I'm touching you." His hand went to her hair, tangling in the silky locks. "You are driving me crazy. Are you usually this naughty?" Her hands slid to the front of his boxers. "Actually, I'm normally just a buttoned-up librarian," she said and winked up at him. She tugged at the sleeves of his jumpsuit, and he helped her, pulling his arms out. She pushed his clothes down until they bunched at his knees, then followed suit with his jeans and boxers, until all his clothes were pooling at his knees and his cock was free. The breath of surprise that Miranda sucked in was immensely gratifying. "Oh my," she breathed, her hands going to cup his balls. Her other hand slid to the base of his cock. "I never get tired of seeing this." She gave him a pleased look and wrapped her fingers around the base, as if testing the girth. The squeeze was almost enough to send him over the edge, and he leaned his head back, his hands moving to grip the bark of the tree. He dug in with his fingers. Miranda was amazingly sexy, kneeling before him and handling his cock, and he did not want to blow his wad before he had a chance to thoroughly enjoy himself with what she was offering. "You don't have to do this," he mentioned, because he felt like he had to. Just because he'd touched her didn't mean she had to go down on him. She rolled her eyes at him. "Like I'm doing this for you." She leaned in and her breath touched the head of his cock, hot and silky. "I'm doing this for me. Understand?" He did. "In that case," he said, digging his fingers into the bark harder so he wouldn't reach out and touch her and somehow ruin this moment, "be my guest." From below his waist, Miranda grinned. She leaned in and gave his cock another squeeze, then slid forward and swiped at the head of it with her tongue. Her tongue flicked against the crown, tasting the drop of liquid beaded there. He groaned. Goddamn. He held his breath, unwilling to do anything to spoil the moment. His entire body tensed, waiting for her to lean forward again and take more of him into her mouth. And she did. She moved in and the head slid past her lips, into the wet, sucking cavern of her mouth. He could feel her tongue tickle the vein on the underside of his cock, and she pumped the base as she pulled him farther into her mouth. Dane panted, his gaze glued to her intent face. She looked as if she was concentrating hard, as if she wanted to get it just right. For him. The expression on her face was driving him almost as crazy as the feel of her tongue against his cock. He felt her suck him all the way into her mouth, felt the head of his cock butt against the back of her throat, and then she released him, pulling away before pumping the base and sliding him deep into her mouth again in one long, sinuous motion. She was amazing. That playful look was back on her face again as she let his cock fall from her mouth, then rubbed the head of it on her lips. He wanted to fuck her mouth, to fuck the playfulness off of her face and have her return to that intense look that he liked better than anything. The intense look that told him she was concentrating so hard because she was so very distracted. When her lips curved around the tip of his cock again, he put a hand to her hair and tangled it there, guiding her head. Would she like this? Some women didn't. She gave a little moan of pleasure around his cock and let him lead as he fucked her mouth, stroking into it over and over again. He noticed that as she worked his cock, she pushed her breasts against his thighs, making sure that her nipples scraped against his skin repeatedly. He reached down to help her with that, his other hand moving to tease the curve of one breast, rubbing against a nipple. She arched her back and leaned in to his touch, and he felt the low groan in the back of her throat against his cock, still buried deep in her mouth. He loved touching her. The way she felt, the way she responded—there was something about Miranda that just drove him crazy with lust. He wanted to see her come again, to see her eyes flutter as she rocked into another orgasm. "Touch yourself, Miranda," he murmured, his hand tangling in her hair again, guiding her back and forth as her mouth stroked his cock. "I want to see you touch yourself." Her hand had been teasing his balls and it fluttered on his thigh. She looked up at him with those brown eyes, and then her hand slid into her jumpsuit. She rolled her hips and gave another whimper, and he tweaked a nipple as he drove into her mouth again. "You are just about the hottest thing I have ever seen," he murmured, unable to stop touching her. One hand anchored in her hair, guiding her head, and the other skimmed along her breast again, then her arm, her cheek, anywhere he could touch her, to let her know how much he liked her mouth on his cock. Her hand began to work frantically between her legs, her hips giving that delicious shiver he was starting to recognize as Miranda being close to an orgasm, and it made his entire body surge. She was about to come again, just from taking him in her mouth, pleasuring him. The thought drove him wild, and he began to go over the edge. His balls tightened and he nearly lost it, but froze, breathing hard. "Miranda...I'm about to come. Do you want me to..." In response, her hand stole out from between her legs and she cupped his balls, toying with them with her fingers. Her mouth sucked down on his cock and she pulled him deeper into her throat. That was enough for him. Hand tangled hard in her hair, he pumped into her mouth and came, groaning her name as he did so. Cum shot down her throat but she didn't tear away from him, and he finished with a heavy breath, his cock sliding away from her lips as she closed her mouth and swallowed all that he'd given her. Her hand fluttered between her legs again, working her sex as she tried to come again. He pulled her up to her feet and she stopped, but he guided her hand back between her legs. Her eyes met his and he leaned in and kissed her neck as she stroked her flesh. When he bit her, hard, on the neck, she came in a soft little scream. She collapsed in his arms, as if the orgasm had made her totally boneless, and he cradled her and helped her slip to the ground. He expected her to sit up, brush off, something. Instead, she lay back in the grass and the leaves and gave the most sultry, satisfied giggle he had ever heard. He moved to sit next to her, a hint of a smile touching his mouth at her satisfied expression. "What was that laugh for?" The look on Miranda's face was dreamy. "I was just thinking I should have done that days ago." That had not been what he expected to hear from her. He watched her, her breasts pointed up to the skies, her hair pooled back behind her, and in that moment, Dane was truly and utterly caught. Hell. When he got back to the base camp, he was going to have to tell Colt and Grant that he'd been messing around with one of their new clients. Because there was no way he was letting Miranda Hill get away from him again. He grinned at the thought. He'd tell them when he got back. They'd understand...eventually. The little minx glanced over at him, and smiled. "Your pants are still down around your ankles." "You're still naked under your jumpsuit," he rebutted. "Mmm, true." She looked as if she didn't care. She didn't move to cover her breasts. "You know, I've decided I like the woods," she said softly. "And why is that?" Dane hitched his boxers back up around his waist, and then his pants. "No prying eyes," she said with a light sigh, as if it were the most blissful thing in the world. "It's just you and nature. No nosy townspeople." Her fingers played with a leaf near her head. He chuckled at the thought. Now she sounded like him. It was the very reason he'd gone off the grid in the wilds of Alaska—he just wanted to get away from everyone. To never see another city again. People hadn't understood his decision at the time—they'd assumed he'd go to Hollywood and cash in on his notoriety. It was just another part of Miranda's appeal that she felt the same way he did. Though it was odd that she had a reason to. "Someone in town giving you a hard time?" She glanced over at him and the smile slid off of her face, as if she had realized what she just said. Her hands stole to the zipper of her jumpsuit and she quickly zipped it shut, then stood. Her hands pulled leaves out of her tangled hair and she shrugged. "Nah. Just thinking aloud." With that, she glanced at the trees in the distance. "I think I'm going to go find my clothes. See you later, Dane." With a casual wave, she turned and left him, stopping only to swoop over and pick up her paintball gun from where she'd dropped it. He was oddly tempted to follow her. But instead, he turned and picked up his paintball gun, put on his helmet, and hoped it covered the satisfied smile on his face. Miranda flopped to the ground once she found her clothes and just breathed for a few minutes, determined not to let panic sink in. Dear lord, what the hell was she doing? It had all seemed like such a brilliant plan. Find Dane, flirt with him. Flash him a little cleavage. Taste that big cock—to please herself—and then leave him wanting more. But she'd been an idiot and started yapping about town and privacy of all things, and hell, why didn't she just confess her entire plan to him right then and there? She put a hand to her forehead and rubbed. Stupid. Stupid, stupid. She'd let things get carried away, and then she'd been so hot and bothered and content afterward that she'd just started spouting all kinds of crap. Good Miranda needed to back the hell off and let Evil Miranda stay in charge. Evil Miranda would have just sucked his dick, winked at him, and left. Good Miranda had to have her cuddle time and let him see the vulnerable side of her. Good Miranda was an idiot. She didn't regret pleasuring Dane—it had been far too enjoyable. What she regretted now was that she was losing control of herself. This was supposed to be _her_ revenge. And while she was twisting Dane around her finger...she found herself enjoying his kisses and his touch far too much. She could still taste him in her mouth, still feel his hands on her body. When had she started to like spending time with him? When did the revenge lose its focus in exchange for the next orgasm? Shit. Shit, shit, shit. Good Miranda kept going on about her _feeeeelings_. Feelings only got in the way of things. Frowning to herself, Miranda pulled out her panties and resolved that from now on, she was going to stay in control. Emotionless flirting and a teasing fuck or two in the woods. She needed to keep him off guard. Most of all, she needed to keep in control. She wouldn't let herself lose control of the situation again. Because apparently she couldn't be trusted around the man. He touched her and she just lost all sense of reality. And that couldn't keep happening. * * * The afternoon of paintball did a lot to distract Miranda from her worries. There was something incredibly cathartic about shooting a man, which probably said a lot about more about her mind-set than it should have. To her surprise, their team won the afternoon—despite the fact that both she and Dane had been distracted for a good while. George had been a superb leader and a terrific paintball player, and his skills and enthusiasm had pushed them over the top, though it had been close. The special prize had indeed been a nice campsite. There was a tiny cabin in the woods with two sets of bunk beds and extra blankets. It was situated next to a well, and the only roughing that was required was cooking and eating the steaks that had been left for them. The team was in high spirits, and no one even pointed out that this wasn't much of a survival skill. No one cared. They had a nice, quiet evening—it was Steve's turn to set up the fire and she helped the others collect wood while George and Pete went to set up a new trapline. Dane had gone back to treating her like one of the guys, but it was Miranda who put a little distance between them. She'd been opening up to him too much. He didn't want to know what she was thinking or how she felt—he just wanted his next roll in the hay. It was time to take a breather on the chatty stuff and just seduce the man when she needed to. So she stayed on the far side of camp and kept to herself. Dane didn't seem too thrilled at that, but it was nice to have the tables turned at least. That evening, they drew straws for cots, and Will, Jamie, and Dane ended up sleeping on the ground. Miranda got a top bunk and slept warm and cozy for the first time that week. So the bunk smelled a little stale—didn't matter. She was happy to have her own bed, even though she was unable to sleep. In the middle of the night, she leaned over the edge and glanced below. Sure enough, Dane was down there, looking at her. He grinned and pointed for the door, a wordless question. Her body quivered with need at the thought, but she forced herself to shake her head and roll back on the mattress and go back to sleep. She was going to call the shots around here, even if she wanted to give in. And right now, she really, _really_ wanted to give in. Morning dawned, and that day was Miranda's turn to learn foraging from the designated subject matter expert. Jamie was their forager for the week, and that suited her just fine. She wanted nothing more than to get away from camp for a few hours and clear her head. But Jamie had left early that morning to learn traplines, and now she sat at camp drumming her fingers. Nearby, Dane sat making more knots in some rope for Pete. He'd shown the poor gamer CEO the same knots every day that week, and every day, Pete needed to be shown again. Dane didn't lose his temper, but she was pretty sure he was getting sick of Pete. The man was lazy around camp, whined, and took every opportunity available to tell them how much money he was making at his company. No one liked him. "You know, Miranda, I can show you how to forage," Dane offered with a casual sideways glance. "I'd be more than happy to take you out and show you if the others aren't going to be back for a while." She could just bet how that would go. As soon as Dane left the campsite, she'd become entranced by the flex of his muscles as he bent over to pick up something and then she'd throw him down on the ground and hump him with abandon. Flushing at the thought, she shook her head. "That's okay, thanks." Pete seemed to wake up at that. He stood up and brushed off his shorts, drawing attention to his pale, bony legs. "Miranda, if you want, I can show you fishing. You still need to learn that, right?" This made Dane frown, and he gave Miranda an imperceptible shake of his head. Well, she loved driving the man crazy by doing the unexpected. She stood. "That sounds lovely, Pete. Thank you. Let me just change into some shorts." She did, and they set off into the woods, Dane giving her an odd look as they left. He looked as if he wanted to talk to her. She decided she'd let him stew for a bit instead. After all, she'd been stewing all night and she was horny. Angry and horny—it was a terrible combination. She would be strong, though, and not give in to his sexy smile and tight ass. Damn Dane for thinking he could snap his fingers and she'd be on all fours, ready and willing. Of course, that mental image didn't help her libido, and she gave a sigh. The stream wasn't too far away—thanks to the better campsite—and she took the line that Pete offered her. He showed her how to set up the hook he'd created from crudely carved wood and how to string a dead grasshopper on it. There wasn't much more to it than tossing the line in and watching the piece of wood (tied to the string to form a makeshift float) bob on the surface. It was a soothing sort of task, with nothing more to do than occasionally tug on her line to see how things were coming. Pete was chatty but she wasn't all that interested. It was a pleasant way to spend an afternoon—peaceful and quiet after living in close quarters with six men for the past few days. Best of all, it allowed her to get away from Dane for a few hours and clear her mind. It definitely could use some clearing. Right now she was feeling so lost about him that she didn't know what to think. "How's the fishing coming?" Pete said a while later, wading over in the water to get closer to her. His question broke her nice quiet reverie. So much for relaxing. "It's fine. No bites just yet." And there wouldn't be if he kept wading around close to where her bobber floated. "Let's just give it some time." He moved to stand next to her and threw his line out again, crossing over hers. She flashed him an irritated look, but said nothing, simply moved out of the way and dragged her line over a few feet. Was the man not even good at fishing? Wasn't fishing hard to screw up? "So...Miranda," Pete said when she'd gotten her line resettled. "How are you enjoying being out here in the wild with nothing but a bunch of big, sweaty men?" Her hackles went up. It could have been a perfectly innocent question, but the way he phrased it made her think otherwise. "Trip's going just fine," she said tersely. He nodded. "Good, good." There was a moment of silence, and then he looked over at her again, the smarmy grin on his face. "Any particular man you have your eye on?" She immediately rounded on him in irritation. "Where is this going, Pete?" The look he gave her was innocent. "I'm not sure what you mean, Miranda. I was just asking if you had a favorite guy you liked hanging out with this week. That's all." What did he think he knew about her and Dane? Miranda's eyes narrowed at the gamer CEO. "What makes you think I have a favorite out here?" "Oh, I don't know." He said casually, trailing his fingers through the water and making little circles that were certain to scare away any remaining fish. "You seem like the type who wouldn't mind a strong, attractive man at your side to make you feel beautiful and shower you with attention. To see to your every need..." She stiffened. Had he seen her and Dane in the woods? "... and I think you're the type that might appreciate the perks that come with dating a CEO." He paused and gave her an expectant look. Blank, Miranda stared at him, trying to piece together his thoughts. Was Dane a CEO of something? The only CEO this week was...Oh. She began to laugh with relief as she realized just who he was referring to. At her laugh, Pete scowled. "What's so funny about this?" She tried to smother the laugh behind her hand, but a bubble still escaped her. "Nothing. Sorry. It's not funny. It just wasn't what I expected, that's all." His look became cunning. "Maybe you expected me to mention Dane? I see the way you're slobbering after him." That quieted her laughter. "I'm not slobbering after anyone, you jackass." "Oh, come on. Don't think I haven't seen you snuggling up to Dane every chance you get. You follow him around like a puppy and hang on his every word. It's pretty obvious that you have a thing for him." That was sobering. Did everyone think she had a thing for him? And here she thought she'd been so very sly and clever this week, so very careful to only let Dane see what she was offering. Her hand went to her collar anxiously. Now that Pete had his feelings hurt, he seemed to be on a bit of a rampage. "If you think you have any hope of scoring time with our instructor just because he's some big shot hockey player, I'm here to disabuse you of that notion," he said, curling his lip at her. "I asked Dane flat out if he was interested in you. If he was going to pursue you. And do you know what he said?" Her entire body froze. Somehow, it had become very important for her to hear the words. "What did he say?" Her voice sounded incredibly calm despite the hammering of her heart. "He said that I was welcome to you. That he wasn't interested in you. So there." Miranda turned her gaze back to her fishing line, watching the stick bob back and forth, and not really caring if anything was biting on the other end. Of course Dane had said that, if confronted. Why wouldn't he? And yet...had he really said that Pete was welcome to her? Seriously? That stung. As if realizing that he'd gone too far over the line, Pete moved forward and put his hand on her shoulder in an effort to comfort her. "Hey, Miranda, I'm not trying to hurt you. I'm just trying to make you realize that you could do better than that cocky dickhead." She swallowed the knot in her throat and gave him a skeptical smile. "Like you?" "Yeah, like me," he said, offended that he had to point out the fact. "I'm a multimillionaire, you know. I—" "You're the CEO of Hazardous Waste Games. Yes, I know," she interrupted, then turned and held her fishing pole in his direction. "And while the offer is very sweet, Pete, I'm really not looking to hook up with anyone on this trip. I'm starting a new life in two weeks, and I don't plan on leaving any baggage behind." "That's fine," Pete was quick to offer. "I'm totally game for a one-night stand." She rolled her eyes and turned to leave. "Take my fishing pole, please. I need some time alone." She shoved it into his hand. "I think it has a bite." Ignoring his yelps of surprise, Miranda ducked past a tree and moved deeper into the woods. She needed time to breathe. Time to think. Was Dane just using _her_? She'd thought she'd seen real interest in his eyes, but was she being blind? Was this just round two of Dane's twisted games, nine years after he'd ruined her life the first time? The thought made her sick to her stomach, because she couldn't tell what Dane was up to. # ELEVEN Something was amiss. Dane scowled at Pete as he returned to camp, a fat fish strung through the gills on a large stick. The man looked utterly proud of himself for finally catching a fish, but he also looked...guilty. He wouldn't meet Dane's eyes as he returned to camp. So he approached Pete. "Caught something, I see?" "I did," Pete agreed, and before Dane could even ask him to, he sat down and began to dress the fish. Well, that was an improvement. Pete never did anything voluntarily. Dane glanced at the woods, looking for Miranda. It was getting dark and she should have been with Pete, but there was no sign of her. "Where's your partner?" Pete's skinny shoulders raised in a shrug. "She went for a walk." When Dane continued to loom over Pete with a frown, Pete added, "She was in a bad mood. I didn't ask. You know how women get." Not only was that sexist, but that was unfair to Miranda. She'd had good reason for getting mad at Pete the times that she had. He crossed his arms over his chest, deciding to wait it out. It could have been that something had upset her stomach and she was ill, and she wanted to be sick in the woods away from nosy Pete. That he could understand. As he fed another branch to the cozy fire, he turned and looked at the others in the camp. Steve was showing Jamie how to lash the shelter properly, but it was finished for the evening, with even a cozy side shelter for Miranda. George and Will had caught a squirrel in one of the traplines and Will was busy cleaning the kill. Their little camp was busy and productive. Except for Miranda, who had run off into the woods. He decided to wait a few more minutes, to see if she would come back. So he gathered wood and answered Steve's questions—Steve always had more questions—and loitered around camp as they cooked dinner. Pete whistled an annoying, off-key tune as he finished scaling the fish and trussed it up through a stick to roast over the fire next to the squirrel. They had some pecans and dandelion greens to go with the small meal, and spirits were high. When the food was ready and dark had set on the camp, the men looked at him expectantly. "Should we wait for Miranda before eating?" "The portions are awfully small for seven people," Pete helpfully pointed out. "Maybe she's already eaten something and we should go ahead without her?" That little shit. Part of him wanted to tell Pete to fuck off, and that if he touched Miranda's share of the food, he'd skewer him over the fire next. But that wouldn't make him a great teacher, would it? He frowned and moved to his backpack, pulling out the GPS tracker. He turned it on and punched in the code for Miranda's bracelet, and she began to beep on his radar. "You guys eat without us. I'm going to go check on Miranda and make sure everything is all right." Gray-haired George immediately stood and moved to Dane's side. "I'll come with you, just in case she's hurt herself and you need help." "No," Dane said quickly, thinking of his last few rendezvous with Miranda. She'd surprised him from out of nowhere repeatedly and pounced on him. Maybe this was another one of her sexy little games. The last thing he wanted was for George to come upon Miranda trying to seduce him in the woods again. "I'm sure I can find her. Don't worry about us. If I need help, I'll radio back to the base. You guys go ahead and eat. Don't worry about Miranda." George nodded and moved back to the fire, and Dane set off in the woods, tracking Miranda. He found her a short while later, a mile or so from camp, sitting alone in the woods on a fallen log. Though the sky was darkening and the stars were out, there was plenty of light to see by thanks to the rising moon that leaked in through the trees and the flashlight in his hand. He could have found her without the flashlight, but it would give her a chance to see him coming and to not be scared. Miranda was anything but scared, though. Instead, when he came up to her, she looked pissed. "Miranda, are you all right?" "Fine," she said in an oddly flat voice. She didn't sound fine. "Did you hurt something? Did you get bitten?" He gestured with the flashlight back the way he came. "The others were waiting for you at camp." "I needed to clear my head," she said, and didn't offer more. Dane frowned and clicked off the flashlight. "Clear your head? Why?" He watched her tilt her head to the side, as if considering his answer and how she would respond. After a moment, she gave a long, weary sigh. "It's complicated, Dane. Very, very complicated." She sounded so dejected that he felt a ridiculous urge to go and comfort her. Miranda didn't need comforting, did she? "So complicated that you can't talk to me about it?" For some reason, he didn't like that. Miranda didn't need protecting, but while she was on this trip, she was his responsibility, and he didn't like seeing her troubled. He liked seeing that playful smile on her face. It wasn't there right now, and he hadn't realized how much he'd enjoyed seeing it. So he moved toward her and tilted a finger under her chin, forcing her to lift her face. "Hey," he said softly. "You can talk to me." She gave a small, brittle laugh at that. "Can I really, Dane?" "Of course," he said, a little stung. "You can trust me." Another mirthless laugh escaped her with that. "What's so funny?" "Nothing," she said with a sigh, standing up and brushing off her shorts. "Just laughing at myself, really." He moved forward and put a hand on her arm. "Miranda," he said, brushing his thumb across the bare skin of her upper arm. "No one is...approaching you incorrectly, are they? They're not harassing you?" He remembered Pete's "accidental" grab of her breasts and his jaw clenched, hard. Client or no client, if that little shit had tried anything... "No," she said in a soft voice. "It's nothing. I'm just mentally not all here today." Her arm patted his. "Really. It's nothing." He decided to make her smile, instead. "It's because you missed me, right?" His hand reached up and touched her chin. "Couldn't stand the thought of being away from me for a few hours, and spending it with Pete was enough to make you sad?" She laughed at that, and he felt her mouth pull into a genuine smile. His thumb grazed over her lip, and a long, quiet moment passed between the two of them. "See? It's not so bad being with me," he teased. That seemed to sober her again. He frowned. Was she mad at him for some strange reason? The last time they'd had a moment alone together, she'd flopped back on the grass after giving him the best blow job of his life. She'd looked content and just a little pleased with herself. Miranda had him turned inside out lately. What had brought around this newest change of heart? Before he could question her further, she reached up, grasped his ears, and pulled him down. Her mouth devoured his in a wild kiss, her tongue thrusting into his mouth. Was Miranda fired up, then? Practically every time she saw him alone, she attacked him. The woman had a killer libido. Still, he wasn't looking this gift horse in the mouth. His arms wrapped around her and his hands settled on her waist, then he grasped her ass and pulled her in closer. His tongue thrust back against hers, a willing partner. Her kiss broke off with a low growl, and she stared up at him in the moonlight for a long moment. Then she said, "I started this, Dane. And I'm going to be the one to finish it. Understand me?" What the hell was she talking about? "Not really," he said with a half grin. "Care to explain?" "No," she said. "We're going to do this by my rules." And she pulled him in for another searing kiss. Her hands went to the front of his shorts. His cock jumped. "You want to do this right now?" She gave him another wicked smile. "Are you chicken?" "Hell no," he said, and pulled her in for another searing kiss. He wanted to kiss that smirky little smile off her face, get her expression to change from the almost-angry intensity to softness. To watch her mouth part and hear those soft little noises she made in her throat when he touched her. So Dane continued to kiss her, his tongue licking into her mouth. Her hand slid over his cock in his shorts, rubbing the length of him in a way that made his entire body flare with need. He groaned against her mouth, trapping her hand just as she brushed her fingertips over his sac. "You're a little wild today, aren't you?" She grinned against his mouth, and then ran her tongue over his lower lip. "I've heard you have a thing for wild, Mr. Croft." Did he ever. Miranda's flirty freight train of lust was driving him insane. When her fingers squeezed his sac again, he groaned and pulled her against him, turning so her back was cradled against his stomach. His hands pinned her hips against his, and he leaned in and whispered against her ear. "How would you like it if I turned the tables on you, Miss Wild?" "Why, I thought you'd never ask," she purred. Dane's hand skimmed over her breasts. Her nipples were hard little pricks under her T-shirt, and he coaxed each one with a gentle brush of his fingers. She inhaled sharply, and he leaned in and kissed the side of her neck, which was slightly damp with sweat. He loved it when she was sweaty, her skin slightly dewy with exertion. When she arched into his hand, he teased her nipple a moment more before skimming lower. His hand went to the waistband of her shorts, and then he skimmed his fingers along the edge of the fabric, teasing her. Miranda wriggled in his arms, as if anxious for his touch. Her hand curved backward, stroking the short burr of his hair. "Are you going to touch me or not?" He nipped at the side of her neck. "Depends on how badly you need it." Her other hand slid between them and he felt it brush against the hard length of his cock again. "Seems like I'm not the only one who needs it bad," she teased. "You're as hard as iron back here." Dane groaned at her touch. "I am." She slithered out of his arms and turned toward him with a mischievous look on her face. "I can help you with that." "Can you, now?" She dropped to her knees, tugging down his pants and freeing his cock from his boxers. A second later, she was taking him in her mouth and her hot, wet tongue stroked the underside of his cock, along the thick vein there. He shuddered hard, his hand going to her hair. He should tell her to stop, should take control again, pull her back in his arms, and tease the ever-loving hell out of those pretty nipples. But then Miranda's hand stroked his sac as she lapped at the head of his cock, looking up at him with a teasing glint in her eyes. And he pretty much forgot everything. When she took him deep in her throat and began to hum, he groaned and began to thrust into her mouth. He was building fast and hard, and he wasn't going to last long. Miranda seemed to know this, encouraging it with her fingers dancing across his sac. "Miranda," he gritted out. "Might want to pull away if—" She bore down on him, pulling him deep into her throat and humming louder. The vibration of her vocal cords made his entire cock feel like it was bathed in sensation. He came, shooting hot cum down the back of her throat. She continued to stroke his sac, fondling him as he came down from the short, violent orgasm. When he finished, she looked up at him with a very naughty expression and wiped the corners of her mouth. "Not much staying power, I'm afraid." Minx. He hitched his shorts back up and, when she stood, snagged her in his arms. "Not so fast. Turnabout is fair play." His hands went to her shorts and he began to unbutton them. She stiffened in his arms for a moment, then relaxed and helped him slide them down her legs. Her panties quickly followed after them, he tugged her against him and slid to his knees, burying his mouth in her wet pussy. She gasped in shock and then her fingers curled in his hair, her body shuddering as his tongue darted into the slit of her pussy and stroked her clit. He wanted to make her come as fast and as hard as he had. One hand stole between her buttocks and he ran his fingers along the seam of her sex, then plunged a finger into her wet core. His tongue continued to flick and suck against her clit, relentless and determined. Miranda shivered against him, and then he heard her breath choke in her throat, and her fingernails dug into his scalp, rough, and she came in a wet, salty rush against his tongue. He gave her clit one last, satisfying lick, savoring the taste, and then looked up at her and grinned. "Not much staying power?" She gave him a dazed laugh, the tension in her face gone. "Guess not." For the next two days, they didn't have a chance to sneak in any alone time. It was starting to make Miranda antsy—each time they'd sneak off into the woods, someone would follow them. They hadn't been able to do more than steal a kiss or two, and Miranda was starting to feel anxious. It was going to be tricky to keep Dane interested if they couldn't have sex, after all. So she did her best to cast him longing glances and let her fingers linger on him when she had the chance. Not that she had to try hard to look at him longingly—she couldn't wait to have sex again. Day six was designated as team-building and team exercises. Brenna showed up again on her ATV and brought more props, and they had obstacle races and puzzles that couldn't be solved without the help of all team members. There were challenges and the prize of the reward camp again. Their team lost. By the time they got back to their last designated camp and restarted their fire, they were exhausted and curled up in their respective shelters. Miranda didn't approach Dane that night at all, remembering what Pete had said about her snuggling up to Dane over and over again. Yet another night of enforced celibacy. Miranda went to bed hoping that Dane was starting to feel just as sexually frustrated as she was. The final day dawned rainy and wet, and with it came an odd sense of anxiousness on Miranda's part. This was the last chance—her chance to "hook" Dane Croft. She wasn't ready, she thought as they broke down the camp as the rain trickled over them, the skies gray and the ground underneath their feet slushy and wet. Once the campsite was pristine once more, Dane gathered them all together. "You've all learned a lot this week, and I'm proud of you. You have one more task to go—I'm going to send you out in the wilderness on your own until tomorrow, and we'll see how you do." Miranda crossed her arms, watching Dane as he gave them a pep talk about the final day. She knew what it would be about—they'd go out, make a shelter, build a fire, catch some dinner, show off their skills, and then go home with a blue ribbon. She'd had different plans for this day earlier in the week—seduce Dane at her little campsite and take pictures of him, naked. But her camera had been confiscated and she'd been sleeping with the object of her revenge all week. And she liked it. And him. And she didn't know what to think about that. She could still implement the revenge plot, of course. Tease him, invite him to her camp, and then she'd find out if she'd hooked Dane enough for her to put the next phase of her revenge into action. Otherwise, this was all for nothing. Not nothing, she amended with a flush, thinking of all the times she'd gone boneless with ecstasy in his arms this week. If anything, this crazy little revenge-slash-experiment had proven something very important to her: that she was sexual after all, and the problem wasn't on her end. That made her feel immense relief. If she could have an orgasm with her nemesis, she could surely have one with a regular boyfriend. Strangely, the thought of archenemies and revenge was making her uncomfortable. She shifted on her feet and hugged her arms close, only half paying attention to the lecture Dane was giving them. The others were eating it up, but she didn't plan on showing off her skills. She planned on getting Dane alone and seducing him. He'd shown her knots two days ago and as he'd demonstrated the appropriate knots to use for trapping, her nipples had gone hard for reasons that had nothing to do with trapping. Something about the way he'd pulled the rope taut and held it out to her had made her instantly wet. He'd noticed it, too, the heated look returning to his own eyes. But all he'd done was pass her the rope and say nothing more. She knew he was conflicted—that Colt and Grant would eat him alive if they'd found out what the two of them were up to. The thought left her with uneasy guilt, and she pushed it aside. Guilt was for the weak, and she was finally going to take what she wanted even if she had to step all over the man she was sleeping with to do so. Too bad he was so distracting. She thought of his naked body, gleaming in the moonlight, all hard muscles and that amazing little dip at his hip that she always wanted to run her tongue along— "Here you go, Miranda," Dane said, appearing before her. She blinked up and automatically took the items he handed to her. It was a tiny bundle with a tightly wrapped piece of plastic-covered paper and what looked like a wristband of some kind. She raised an eyebrow at him. "Were you listening to what I was saying?" he said patiently. "Um, sort of," she said with a half smile. "Refresher?" He chuckled and took the wristband from her, pulling her arm out and snapping it around her wrist. "That's so I can find you if you get lost." He handed her the plastic-covered piece of paper. "That's your map. You're going to be at camp six tonight. You need to find it and set up. I'll swing by later to check on you." "Oh, of course," she said hastily. "Sorry. I thought you were talking about something else. Camp six. Yep. Got it." God, she was babbling. "Thanks." They stared at each other for a moment, and then he gestured at the forest with a lazy smile. "You going to go or stick around here all day?" Oh! She looked around but sure enough, almost everyone else had shouldered their packs and was heading into the woods. She looked back at Dane, then the woods. "Are you going to, um...find me?" He looked at her very calmly, then seemed to scan the woods, as if she'd asked him about directions instead of a rendezvous. A quick glance showed that Pete was hanging around camp, obviously waiting to talk to her. Sigh. Steve immediately broke off from the others and began to jog into the forest, eager to start out on his own. The others were just as eager. Not Pete. He wanted to wait for his new BFF Miranda, it seemed. So much for inviting Dane to her campsite tonight. She glanced over at him. "You know where I'll be," she said in a soft voice that she hoped was inviting. "I'll skip the panties." And with that, she turned and walked out of camp, heading for the trees, armed with her tiny map. Camp six was across two creeks and over a hill—quite a hike for her. That was okay; in the last week she'd found she enjoyed hiking quite a bit. Pete hung behind for a minute, and when Miranda moved forward, he showed her his map. "I'm at camp three, Miranda. Where are you at?" God, he was annoying. In the last week of spending every day with the man, she'd learned to appreciate not having him in her life. He was bothersome as hell and didn't seem to realize it. He was also clingy and tried to go with her everywhere. "I'm at six," she replied after a moment, thanking the powers that be for the extra campsites between theirs. She didn't want to be his neighbor. He looked crestfallen at the thought. "Camp six is all the way on the far edge of the map." "That sucks," she lied. It really didn't suck much at all. Had Dane given her the most private campsite on purpose? Was he going to meet her tonight? Pete gestured at the woods. "You want to walk the same way until we hit the creek?" This would be the last time she'd have to deal with his aggravating self, at least. So she plastered a cheerful smile on her face and didn't even look back at Dane. "Sure thing, Pete." # TWELVE She ditched Pete on the far side of the stream. He'd offered to follow her to her campsite and help her with setting up, but she'd been a little affronted that he thought she needed his unasked-for help and had chased him off with a few cheerful encouragements that he should set up his own camp. She was glad he was gone, though. Pete was just underfoot too much, was too eager, too chatty, too everything. He was a nice guy—cute in a geeky sort of way and well off, but her mind was laser-focused on Dane. The silence left behind by Pete's departure was pleasing. Without him in her ear making small talk, she was able to relax and enjoy the day. She'd given up on stressing over Dane. If he showed up tonight, great. If he didn't, well. If she couldn't keep him interested long enough for phase two of her revenge plan, then it wasn't meant to be. Picking up a piece of firewood, she frowned to herself at the thought. Jeez, where had the laissez-faire attitude come from? She'd come out here to destroy a man in the most cutthroat manner. She was blatantly using him for her own ends. When had she gone from "Destroy Dane and everything he touches" to "Oh well, it is what it is"? That wouldn't do at all. She stopped for a minute, picturing the photos on that horrible cheap website with "Casanova" flashing in some poorly animated gif. The looks she got in town. The snickers. The awkward conversations at the town get-togethers. Her mother's total mental breakdown. The old familiar pain began to burn in her belly, filling her with an angry reminder of what she was doing here. She was here to destroy a man. So she'd ended up sleeping with him. It didn't mean that she had to change course. And it didn't mean that she had to have feelings for the man. She didn't. She wouldn't. She couldn't. Scowling to herself at the traitorous thoughts, she scooped up a few likely branches as she walked. Her camp was easy to find—there was a nice clear spot with a small red flag stuck into the ground. The creek was a short distance away, the trees were tall, and the area was secluded. Nice. She set down her wood, her pack, and set to work. The first task for the day would be a fire. Once she had it going, she could begin the next task—a shelter. That would be a bigger project, as she needed to make it big enough for two. Just in case. Food was last on the priority list, since it would probably involve leaving camp, and she wanted to stick around, just in case Dane showed up. _When_ he showed up, she amended. He'd said he'd come by to check how her "survival day" was coming along. No sign of him yet, she thought as she laid the wood in a pile for her fire, but it was early. After an hour's hard work she'd produced a fire with a spindle and bow and was oddly pleased she was able to do so. You couldn't make a fire every time with rubbing sticks, but she'd been able to do it today, and that was a nice feat. Wouldn't Dane be impressed? She fed it more kindling to build the flame, and when it was nice and healthy, she added a few small logs to the fire. Then she set to work on her lean-to. She took her time, the task requiring a lot of work, a lot of trekking back and forth, and tending the fire. First she had to build the A-frame and lash it together. When it was solid, she made a lattice of small branches on one side to form the windbreak, and then continued to stack tree branches on it, shoving dirt high against the bottom edge to ensure that nothing could crawl under. Then she worked on laying some soft pine branches as a bed, and spread her spare hooded sweatshirt down over the branches. After that was done, she stood up and wiped her brow, exhausted and surprised at how much the tasks had taken out of her. There was still so much to do—she had to catch dinner or go foraging, boil water for her canteen, gather more wood for the fire...She eyed the creek, then eyed the sun, low in the sky. She'd do that stuff after she had a nice rinse off. If Dane showed up— _when_ , she corrected herself—she didn't want to be sweaty and exhausted. She wanted to be fresh and sexy. She stripped and took a quick dip in the stream, letting the water refresh her spirits and ease her aching muscles. She quickly dressed in her last pair of cute, clean panties and her last bra—a delicate pink set edged with black lace, just enough to make it girlish with a naughty side. A quick glance around camp told her that food wasn't going to magically spring forth, so she sighed and grabbed a stick that would make a likely fishing pole. The creek had had a few deeper, slower-moving areas with overhanging branches, which were the perfect spots for fish to hide. She would have the best luck there. She fished for two hours (with frequent trips back to her campsite to check on her banked fire) but by the time the sun was going down, she'd caught nothing. Well, she'd eat tomorrow. It didn't matter, really. She could survive on her own—she had fire, a shelter, and she could eat some grasses and nuts. She had water to drink, and a fire to boil it over. She was set. She wondered if the other students had had as much success as she had. After a moment, she gave a rueful smile and decided they probably had. Dane was a good teacher. When she returned to her campsite, she knelt next to the fire and adjusted the logs. The bushes behind her rustled. Miranda whirled, startled. Was Dane finally— But no. To her surprise, Pete emerged from the woods, sweaty and unkempt. A smear of dirt bisected one cheek, and he held a fish strung through a small branch, carrying it toward her camp. Well, hell. She glanced around uneasily—was Dane not coming? Was she going to be stuck entertaining Pete all night? "Hey, Miranda," Pete said cheerfully. "I brought you a fish." "Pete, what are you doing here?" He looked confused that she wasn't greeting him more happily. "I brought you a fish." And he raised it in the air, as if it weren't obvious. "Just in case you weren't able to catch your own." Irritation flashed through Miranda, but she quickly tamped it down again. He meant well, even if he managed to insult her with everything he did. "Thanks, Pete, but I'm good. I don't want your fish. We're supposed to be surviving on our own, remember? No help from the others." He looked surprised at her rebuke, as if it had never occurred to him that he'd need to do stuff on his own—or that she'd be capable of handling herself. "Oh. I see. Well, I just thought I would help." He gave her a wounded-puppy look. "Sorry, Miranda." She sighed and forced a smile to her mouth. "It's a sweet thought, Pete, but don't you think you should get back to your own campsite with that fish before Dane drops by? You want to pass the course." "Oh, he's already dropped by," said Pete casually. "I'm guessing he's already stopped by the others, too." "Oh?" He gave her a knowing look. "Miranda, I'm not stupid. I know what's going on between you two. I think we both know where Dane's last stop is going to be tonight." Her heart pounded hard in her chest, her breath disappearing. "What do you mean?" He gave her a wry look. "Come on, Miranda. I've seen you and Dane sneaking off to be together this week, and I've seen the way you've been looking at him. I just want to tell you that you're going to get your heart broken. He's not interested in a relationship." The constriction in her chest relaxed a little, and she felt absurdly like laughing. Was that what he was worried about? That she might be used by Dane and would need a bit of rescuing by a white knight? Generous of him, but totally incorrect. She was being the user in this relationship. "That's very nice of you, Pete, but I'm a big girl. I can handle myself." "It's not right," Pete said, a hint of peevishness in his voice now. "He shouldn't be sleeping with clients. I've half a mind to go and tell his partners what he's been up to." Alarm shot through her body. He couldn't do that. If anyone was going to ruin Dane's career, it was going to be her, dammit. And just the thought of Pete messing things up made her nerves fray. "Pete, please," she said softly. "I would prefer if no one knows but us." He looked unconvinced. "For my sake?" she said, turning on the charm and moving forward to touch his arm. She swung her hair a little, mentally wishing she could punch him in the face for even suggesting such a thing. He looked at her and licked his lips, then sighed. "I won't say anything, Miranda. I just...I wish you were interested in me, not him." She smiled and leaned forward, impulsively kissing him on the cheek. "I wish I was, too." He turned his head into her kiss. She recoiled. So much for her goodwill toward him. She pulled away and smiled tightly. "You should go back." He had the grace to look embarrassed. "I'm going." With arms crossed, she watched him leave her campsite, still carrying his fish. She wouldn't miss him when this week was over. The only one she'd miss was Dane. The thought made her breath whoosh out of her lungs. _Miss_ Dane? Ridiculous. She was here to destroy the man. _Really?_ Good Miranda said inside her mind. _Because you seem more interested in sleeping with him than actually getting revenge._ Damn. She hated Good Miranda. Especially when that sounded closer to the truth than she liked. Confused by her own feelings, she moved toward the fire and tossed another log on. Did she like Dane? Really? Or did she only like playing with him? Where had her hate gone? She had been brimming with it earlier this week, and yet now she couldn't seem to muster it. There was no question that the two of them were compatible together—every time he touched her, her entire body exploded into feeling, every nerve ending singing with delight. No question that he knew how to touch her and what she wanted in bed. But relationships weren't built on that, and she didn't want a relationship with the man. She wanted to make his ass sorry for what he'd done to her nine years ago, and then shake the dust of this small, annoying town off her boots. She wanted to leave the Boobs of Bluebonnet behind. She wanted a life that didn't involve Dane Croft. Didn't she? And yet...she stared down at the camp around her. She'd worked hard this afternoon to make her camp perfect. Her fire was roaring, her shelter done, and he'd find no fault with it. She'd tried extra hard this week so he wouldn't think of her as lazy. She'd never complained, even when she was dripping with sweat. She wanted him to like her, too. _Not like,_ she corrected with a wince. _Respect, not like!_ If he respected her, the revenge would ache all the more. A warm arm wrapped around her from behind, and a hot mouth pressed a kiss to the side of her neck. She yelped in fright. "You're lost in thought," Dane said between nibbles. A shiver of delight charged through her body at the affectionate touch, and she had to stop herself from leaning in to his embrace. "Just thinking of this week," she said lamely. "Sorry I wasn't paying attention." "Here I thought you were thinking up new ways to ambush me again," he said, his eyes gleaming with the firelight. "But obviously not, since I scared you." "I thought you were Pete," she admitted. He stiffened against her and turned her to face him. Gone was the playful expression, replaced with a possessive, angry glare. "Miranda, has he been harassing you again? Say the word and I'll go over there and wring his little pencil neck—" "What? No, I'm fine." He growled low in his throat, his hand clenching possessively in her hair. "I've seen the way he's been looking at you. Say the word and I'll make him regret it." Funny, Pete had been at her campsite a short while ago saying the same thing about Dane. She gave him a weak smile. "It's fine. I can handle myself." She tried to ignore the thrill that his jealous protectiveness shot through her. She didn't care, remember? Didn't care. This was all part of the master plan. "Besides, you can't go beat him up. He's a client." "I don't care," Dane gritted. "You need me to beat him up, and I will." That was...sweet. She smiled and turned her face up to his for a short, hot kiss. "I'm fine," she repeated. "So, are you here to check out my stuff?" His hand slipped to her ass, cupping it and drawing her body against his. "Exactly what stuff did you have in mind?" Her arms twined around his neck and she gave him a flirty smile. "My fire, of course. I worked hard to give it just the right spark." "Looks like it's burning hard to me," he murmured, not glancing away from her face. "Shall I check?" "What?" she said, turning in his arms and giving him a playful look. "You're not going to pass me simply because we're sleeping together?" He grinned. "Nope. Your campsite needs to make the grade. And that means you need to show me your stuff." He slapped her ass. "Hot stuff." A girlish giggle escaped her throat and Miranda sidled away. Now that she'd slipped out of his arms, he moved forward to her shelter, inspecting the frame of it. "Not bad, not bad." "There are boughs on the bottom of the shelter to make it more comfortable," she blurted. "And I packed the earth around the bottom so there's no breeze going through the cracks." For some reason, she was anxious to please him—to show him that she'd been paying attention this week. That she wasn't like Pete. That she was someone he could respect. "Very nice," he said. "I have a nice fire going, too," she added. "No fish, though. A few nuts and some dandelions for greens." She gestured at the food that she'd set aside on a hastily made plate of a few long strips of bark that she'd washed clean. "Hungry?" "Starving," he said, grinning, and his hands moved to her waist again, pulling her close. "For some reason, I'm thinking we're both not talking about food," she said softly, her gaze going to the curve of his mouth, the scars that gave him that rakish look. As if sensing her thoughts, he tugged her against him and pulled her in for a kiss, his tongue stroking deep between her parted lips. It made her think of sex, and she gasped at the pulsing response of her body. His tongue thrust into her mouth again and then stroked against her own tongue, the feather-light touches tickling and making her quiver with desire. No one kissed quite like Dane, as if he had all the time in the world to do nothing but kiss her and his entire goal was to devour her body. That was one of the things she liked the most about him—his devoted attention to her pleasure, and how he took charge, making sure that she would get her orgasm before he got his. She'd let other boyfriends take the lead in bed before, but they'd hesitated, asking what she wanted. She didn't know what she wanted most of the time, but Dane seemed to instinctively know, and he used that to play her body like a violin. And she liked it. A lot. His tongue stroked into her mouth again, and she brushed hers against it, returning his caress with one of her own. She could feel each stroke, and each one reminded her of his mouth on her pussy, his cock thrusting deep inside her, and she grew wetter and wetter with each thrust of his tongue, until her hips were squirming against his own. Her arms wrapped around his neck and she lifted one knee. He grasped it and pulled it tight against him, pulling the cradle of her hips against the thick ridge of his erection. Just the feel of it against her sex made her moan with desire. She needed it inside her. Deep, hard, plunging... With one last teasing lick, Dane broke the kiss and gazed down at her, brushing a strand of hair away from her cheek with his free hand as she clung to him. "Miranda." "Yes?" Her voice was breathless, soft. "Show me your knot work." She blinked for a moment. "My what?" "Your knot work," he said in a husky voice, as if it were the sexiest thing imaginable. "I need to check your traplines and your knots to make sure you've passed that portion of the course." Miranda frowned up at him. Who cared about knots at a time like this? But he only gave her ass a friendly pat and released her leg, leaving her throbbing and wanting, her pulse pounding through her veins. Confused and utterly turned on all at once, she stood there for a moment, watching him. When he didn't move, she gave him a bewildered look and gestured at her pack. "I have my rope over there." "Good," he said. "Show me a square knot." The man was odd. Couldn't they do this later if he did want her? Grumbling mentally to herself, she moved to her pack at the base of a nearby tree and pulled her assigned length of rope out. Though Dane had showed them all how to create a rope from dried reeds and grasses, it was a task that would require more time than they had left in the training and they'd been parceled out a length of rope instead. There hadn't been time to practice her trap-making. He moved to stand over her, looming and blocking out the rest of her fading light just as she began to tie the ropes in the knots that he'd shown them this week. His attention made her flustered, and she dropped one end of the rope, her knot falling to pieces. "Tsk," Dane said in an oddly pleased voice above her. "Looks like someone needs a lesson on ropes after all. Stand up." "I can do it if you're not looming over me," she grumbled, but stood and handed him the rope. He quickly began to make a complex series of knots, the rope forming a loop on one side. Then he extended a hand to her and waited. Miranda frowned down at that outstretched hand. "I don't have anything else." Dane's look was utterly serious. "Give me your hand." A small, naughty thrill shot through her, and she stared at him. Then, ever so slowly, she put her hand in his. He grinned and kissed her mouth once—quickly, hard. Then he took her hand, slid it through the loop he'd created, and raised it over her head. The rope slithered over a branch just above her head and came down on the other side, and Dane extended his hand to her again. "Do you trust me?" he asked. Alarm bells shot through her body, and she immediately thought of the camera in the closet, long ago. The pictures of her breasts. "I..." He leaned in and kissed her forehead, an oddly tender gesture. It was a small caress, but a reassuring one. "If you don't want to, it's all right." She thought quickly. Odds were that there weren't cameras in the woods. Even though Dane had picked out this campsite for her, she hadn't seen any evidence of equipment when she'd foraged and gathered nearby. And if she didn't trust him, their little games would come to a screeching halt. And for some reason, she didn't want that. Not at all. "I trust you," she said softly, a slight waver in her voice. She hoped she didn't regret that trust soon. Dane didn't know her plans. There was no way this could be a setup. "But if there's another emu lurking around here and I can't run away, I'm going to kick your ass." "No emu. I promise." He pressed a kiss to her nose and then quickly fashioned a knotted loop in the rope to hold it in place, hiking her other arm above her head. Now she stood, both arms pinned taut above her head, the rope looped over a high branch. She gave each arm a tug and realized that the ropes were knotted tighter than she'd imagined. She couldn't slide free. Miranda shivered, alarm and excitement pounding through her. Dane grinned at the sight of her, his hands running slowly up and down her sides, stroking her body. "Someone should have learned her knots," he said in a husky voice. "Someone didn't have the chance to show what she knew," she retorted, twisting one arm in protest. "You're cheating." "I am," he agreed, his fingers sliding to the waistband of her shorts and tugging her shirt free. "I admit that I've been thinking about this all week." A shiver crossed her skin as he slowly hiked her shirt up, past her breasts. The material bunched under her arms and he tugged it over her head, until it lay trapped behind her head. Her body was exposed, her bra stark against her skin. His gaze was rapt as he stared down at her, eyes hot with need. Dane brushed the back of his hand over one lacy cup and her breath sucked in. "So lovely," he murmured. "Delicate and rosy. The only thing prettier than these breasts are your nipples," he said, and brushed the backs of his fingers against those very spots. She gasped at the jolt of sensation that rocketed through her. Her hands clenched against the ropes. Her body was exposed, helpless to do anything against his touch. Not that she wanted to escape. Dane's eyes gleamed and he pulled a length of fabric out of his pocket. "Let's raise the stakes a little, shall we?" And then he blindfolded her. The bandana cut off her vision, and her tremors of excitement were mixed with apprehension. She couldn't see where he was. His hand brushed along her arm and felt her quivers, and then his mouth brushed against hers. "I'm here, Miranda. Anytime you want to call this off, just tell me to stop and I'll take you down. All right? Let me know if it gets too intense for you." She gave a jerky nod. "That's my girl," he said in a pleased voice, and rewarded her with a kiss along her jaw. She hadn't expected his mouth there and trembled at the feeling. With her vision blocked and her hands unable to touch him, her senses were narrowed down to passively feeling, to smelling, to hearing. His lips were soft against her neck, and she felt the flick of his tongue against her pulse, the strum of it matching the spike of pleasure that shot to her pussy. The stubble of his beard rasping against her skin was an oddly pleasurable sensation, and she barely felt his fingertips gliding along her side in light, ticklish motions as he explored her body. "And that, my sweet Miranda, is how you tie a square knot." A nervous giggle escaped her throat. "Very funny." "Now let me think," he said softly. "Since you're completely at my mercy, where shall I touch you first?" She quivered at the thought, her body tingling with anticipation. "My breasts?" she offered. "Hush," he said, and gave one outthrust breast a teasing slap that made her body jolt. "I get to decide, and it's much more fun if you're surprised." "Then why'd you ask?" "It was a rhetorical question," he said with a chuckle. "You're all laid out and delicious, and I need to concentrate." "Fine, I'll be quiet." She liked this playful side of Dane. It gave her heart a funny little flip to hear his soft chuckle. That's just desire, she told herself. Nothing more. His hands—rough with callus—skimmed along the soft flesh of her outstretched arms. She shivered at the feather-light touch moving along the inside of her arm and grazing back down until his fingers skimmed her collarbones. "So pretty," he said huskily. "Do you mind if I play with you tonight? Have all the control?" Her nipples tightened at the thought. She swallowed hard. "I don't mind." A hand fisted in her hair, tugging her head backward, and the breath caught in her throat. "Do you think I'm going to hurt you?" Her breath coming in anxious little pants, she forced out a light, "I wouldn't let you tie me to a tree if I did." Dane laughed at that, and his mouth brushed hers. Her tongue slid out to caress his, but he was gone an instant later. "I won't hurt you, I promise. If you tell me to stop, I'll stop. But I want to give you pleasure tonight. To drive you out of your mind with it." Her entire body tightened in anticipation at the thought. "I've noticed a little something about you, Miranda. Whenever that brain of yours gets going, that body of yours stops enjoying. And I noticed that mind of yours working overtime this morning." His finger brushed against a rock-hard nipple and she gasped in response, her entire body pulsing with need. "What were you thinking about?" She knew immediately what she'd been thinking about all day—Pete's words to her about Dane and how he was using her. Of course, she couldn't confess that. "I...I was thinking about what I needed to do to make sure that I could get the camp set up properly." "Liar," he whispered, and tweaked her nipple again. Desire shot through her and she groaned, her pussy clenching with need. "Please," she panted. "Not until you tell me what's going on in that head of yours." His knuckles slid to the curve of her breast, rubbed the fabric covering them. A small cry of frustration escaped her throat and she twisted in the bonds, trying to angle her breast so his touch would graze her nipple. As soon as she did, though, he pulled away. "Bad girl," he said in a husky voice that thrilled her to her core. "Tell me." She licked her lips and was gratified by the sudden intake of his breath. "I was thinking about you," she admitted, since that was part of the truth. "Thinking about how you had touched me and how it's been three long days since we've been able to have sex. I was wondering if you'd come back to me tonight, or if I'd have to spend all night touching myself." He gave a low groan of desire. Feeling bold at his reaction, even though she couldn't see it, she licked her lips again and continued. "I was thinking about the hard, thick length of you deep inside me, pumping into me so hard that I can feel you slamming through my body—" He groaned and the hands were in her hair again, his mouth angling over hers in a thrusting, hard kiss of possession. Greedily, she sucked at his tongue. His hands stroked up and down her back and her hands fisted in the bonds, her core so wet she could feel the slickness between her clenched thighs. His mouth broke from hers and she gave a small whimper of distress, then felt his tongue graze along the column of her throat. She tilted her head back, enjoying the caress as his tongue trailed along the collarbones and back down to her bra. His hands plumped her breasts together, forming a valley that his tongue slid between, his thumbs grazing her nipples. Miranda groaned, arching her back into his touch. "My bra," she panted. "Take it off. Please." "You're not the one that gets to decide," he said, and his hands slid to the waistband of her shorts, tugging them down her legs. They fell to the ground, pooling around her ankles, and at his light touch, she lifted a foot, then the other, so he could remove them. She wiggled in place, longing for him to remove her thong next. She'd saved her sexiest one for that evening—lace with a saucy bow just above the cleft of her ass, the front a mere satin strip that covered nothing and teased everything. She'd worn them for him, saved them for this night. Miranda gave her hips a little wiggle. Did he like? His hands moved to her ass, clenching the rounded cheeks of her buttocks, and she quivered, waiting for him to spank her. Something. Instead, she felt his teeth graze one of her pebbled nipples and then bite it through the bra. A shuddering bolt of desire blasted through her, and she whimpered. "More." "Does sexy little Miranda like having her breasts bitten?" he teased in a husky voice, and she felt him rub his chin against her breast, his stubble catching on the fabric. Then his fingers pushed her bra away, exposing her nipple, and his mouth was on it again, sucking hard at the tip and then giving it another tiny bite. At her quiver of pleasure, he flicked the other nipple with his fingers. Dane tugged her bra down, exposing both breasts to the air—and his touch. He teased one with his tongue, then licked it, over and over, as if he were a cat lapping at cream. His other hand brushed against the other nipple, teasing it to a hard, aching point. Each lick sent an erotic thrill straight to her pussy, and her hips flexed involuntarily with each touch. "Such pretty breasts," Dane told her, then gave one a harder nip. She could tell he enjoyed seeing her gasp, because his laugh was a low, husky rumble. "Your breasts bounce when I startle you. Fucking love that." She arched against his touch, her breath coming in hard, quick bursts when his mouth moved away and his fingers slowed. "Then keep touching me, if you like it so much." He plumped her breasts again, his fingers working them, then bit at the nipple of one breast, then the other. "Do you like being tied up, Miranda? Like giving all your control to me?" She tugged at the ropes again, but he was right—she had no control. He could walk away and she'd be left here, topless, helpless—wet with need. The thought both excited her and terrified her. "I—I like it," she said when he nipped at the peak of her breast again. "It's just different. Scary. Exciting." "You're thinking again," he said, and his mouth dipped to her belly button. "Time to put a stop to that." And then she felt his hands slide to her hips and give a nudge that she should part her legs, and she got even wetter in anticipation. Tension coiled through her body as she waited for him to put his hands on her, his mouth on her. Prickles of anticipation made her nipples harden. Then she felt him. Fingers tugged her scrap of panties down her thighs and she wiggled to help them along. Then his hands—his thumbs—parted the slick heat of her pussy and she felt his tongue dip in and tease the wetness. Her breath sucked in. Again, the tiny insistent flick—no more than the tip of his tongue. But it drove her wild with need, every nerve on her body springing to life and crying out. His fingers gripped her hips and then dug into her buttocks as his tongue plunged between the damp folds of her pussy again, a long smooth stroke from her core all the way to her clit. When he reached that small hard button, he circled it with his tongue, hard and wet. Tiny gasps erupted from her throat. She needed...she needed... One hand lifted from her ass and slid between her legs. She felt the brush of fingers between her thighs a mere moment before a hard, thick finger glided into her heat. She whimpered. God, that felt so good. And God, it felt like not nearly enough. She bucked against his finger, crying out when he thrust it deep into her again, his tongue flicking against her clit in soft, teasing motions. Not fast or frantic, just slow and steady and making her pulse race with need and want. As if he had all the time in the world. As if he could sit there on the forest floor and lick her pussy for days on end, every stroke of his tongue slow and sensual as if the taste of her were a treat all its own. A fresh whimper rose in her throat at the mental image. "Oh, Dane," she moaned when he gave her a particularly long, sensuous lick. The words came out as a breathless sigh. "I need...I...I need..." The words slipped from her brain with every stroke of his tongue, as if she couldn't think while he tasted her. Her hips circled, trying to move his head to just the right place. She needed something. She was so close. The finger thrusting deep into her core suddenly felt thicker, harder, and she realized he'd slipped a second finger into her slick warmth. His tongue began to flick against her clit faster, the same tiny stroke of his tongue over and over again. It was maddening and she felt the hot spiral of her orgasm begin to slip over her again, and her cries became more urgent, her arms pulling hard at the bonds over her head. The tree branch rattled and shook, raining leaves. She didn't care. She was so close, her breath hot, panting gasps that ran into one another. Suddenly Dane's mouth was gone, and his fingers slid from her pussy. A sound of dismay escaped her throat, and then she felt his mouth kiss a breast, her shoulder, and his fingers were on the rope knots. Her hands fell free, and she tugged the blindfold off, staring up at him in distress, her orgasm ebbing away as if it had never been. "I don't understand. Why did you—" He leaned in and kissed her tenderly, then took her hand in his. "You were hurting yourself. I don't want that." She stared down at the rope burns on her wrists. They were reddened and chafed. "Oh. I didn't even realize." "I know," he said with chagrin. "That's one of the things I love about touching you, Miranda. It's that you lose yourself when you're in my arms. I just need to remember that. No more rope play for you." His thumb brushed her cheek in a soft caress. "My fault." She stared at him, uncomprehending, then slid a hand to his cock, hard and thick. Her finger glided over the pre-cum coating the crown. "We're not stopping, are we?" "God no," he groaned, and swept her up in his arms. Dane carried her the short distance to the shelter she'd made, and he laid her down on the freshly cut boughs she'd placed on the ground as a makeshift bed. No sooner had he laid her on her back than he was over her, and she heard the rip of the condom package. Then Dane's weight was on her, pushing her thighs forward, and he was sinking hard and deep into her, a swift stroke that took her by surprise. Her delight emerged in a muffled shriek as her body surged back to life, remembering the orgasm that had been so close. Her calves tensed as he pushed down on her, her knees pressing against her breasts as he pulled back and stroked deep again and again. "So tight and hot," he gritted, slamming deep into her again. "Dreamed of doing this to you for years." Thrust. "Taking you in my arms and fucking the living daylights out of you." Thrust. "Better than I ever thought it would be." Her pussy clenched hard with every word, her moans turning into a soft, continuous cry. Every time he stroked, a hard pulse of pleasure washed through her, her entire body tightening until she felt as if she'd explode. It was like she was trapped in an endless orgasm—coming and coming—and yet with each hard thrust, he pushed her just a little higher. Then something shattered in her, and a broken little cry escaped her throat as her entire body pulsed, hard, and he bit out a curse at the same time. "Fuck yeah," he growled. "Come for me, Miranda." She did. Hard. And when her cries died down, he bit out another oath and got his own release. Then he collapsed on top of her, panting, his forehead damp with sweat. Her legs eased down to his sides and she wrapped them around him, easing her arms over his shoulders and clinging to him as her body quivered in aftershocks. That had been...intense. What he'd made her feel...there were no words. Or if there were, she didn't know them. The way he'd been so intent on her pleasure made her blush just to think about. And she thought of the words he'd said as he'd fucked her deep and hard. _Dreamed of doing this to you for years. Better than I ever thought it would be._ He'd been thinking about her? Daydreaming about her? For years? And all this time, she'd hated his guts so bad she'd fuck a man just to ruin his life. Miranda didn't like herself very much in that moment. Dane's hand cupped her head and he turned her face toward him, kissing her lips softly. "Thank you," he said in a husky voice. A knot formed in her throat and she closed her eyes, pretending to yawn. "For what?" He toyed with a lock of hair—damp with sweat—on her forehead. "For this. For this week. It was pretty much perfect. I...have to tell you something." She stiffened under him. Oh God. What was he going to say? She couldn't open her eyes, couldn't bear the thought of looking at him and seeing the truth. "Dane—" "Shh. I want to tell you." His fingertips tucked the strand of hair in place and then glided along her cheekbone. "I had a...rough time when I left the NHL. A woman...she was responsible for me being fired. I turned her down and she made up a bunch of stories about me to the press." A knot formed in her throat. So he'd been used and publicly humiliated? She didn't know what to say. Did he want her to speak? Or was there more to tell? Hesitantly, she brushed her hand across his nape, stroking the soft skin there. It was a touch to comfort and encourage. To let him know she was there, and she was listening. "I hated her," he said after a long moment, as if warring with himself. "I hated her so much, and felt so betrayed at the moment when I was the most vulnerable, and I thought...well. I thought I was going to give up women for a long, long time. Maybe forever. Because I couldn't look into a woman's face and not see that bitch glaring back at me." She opened her eyes and continued to stroke his neck, waiting. He turned back to her, his eyes hooded. She met his gaze, and he searched her face, as if looking for something there. "But that was before this week. Before you. I haven't touched a woman in over three years. Didn't want to...until I saw you again, and realized what I wanted. And I wanted to say thank you. Like I said, this week was perfect." He kissed the side of her mouth. She twitched under him, not saying anything. After a moment, he chuckled and rolled off her, then pulled her close, cuddling her. Miranda said nothing, simply closed her eyes and waited for more. He snuggled close, his breath in the curve of her neck, and his breathing grew deep and even. Sleeping. She couldn't sleep. Her mind was freaking out. Dane had just confessed why he'd been kicked out of hockey. Someone else had done it to him. Casanova Croft was a fraud. He wasn't a ladies' man or a poon hound. Underneath the sexy, ultraconfident exterior was a man who'd apparently been thinking about nailing her for nine years, and who had been so hurt by a woman that he'd not had sex since being betrayed. The jerk had a soft side. A really big soft side. A vulnerable one that he'd launched straight in front of her bull's-eye. And in that awful, wonderful, tender, horrific moment, Miranda realized two things. One—that she wasn't going to be coldhearted enough to get her revenge on Dane Croft after all. And two—that she was still terribly, horribly, head over heels in love with the man and likely had been since high school. Well...fuck. # THIRTEEN The next morning dawned crisp and cool, though Miranda had been warm curled up next to Dane all night. They'd made love several more times before she'd fallen into an exhausted slumber in his arms. Part of her hadn't expected him to stay at her side all night, but when she'd rolled over and yawned, he'd woken her up with a kiss. "Hey, gorgeous," he'd said with a grin and a light smack to her ass. She given him a flustered smile back, but her mind was racing a million miles a minute. He hadn't left her last night. They'd slept in each other's arms. That felt like relationship material. Under no circumstances could she entertain a relationship with Dane. None. Zero. Preoccupied, she hadn't minded when he'd kissed her forehead, dressed, and went about breaking down camp. She dressed just as quickly. "So...shouldn't we be heading back?" A grin broke across his handsome face, and she felt her breath catch. Lord, he was easy on the eyes. She'd grown too used to seeing that face when she woke up. That'd change soon enough. "Right." He belted his shorts and pulled a piece of paper out of it and handed it to her. It was a small map with instructions. "When I met you yesterday, I was supposed to give you this." He gave her a look that was part chagrin, part pleased with himself. "Looks like we forgot. You can follow this back to the main camp. I need to swing by and check on the others to make sure they broke down their campsites." "Great," she said with a bright smile. "I guess once I have things taken care of here, I'll see you back at base camp." His look immediately became troubled. "Miranda..." She moved toward him and couldn't stop herself from plucking a pine needle off of his shirt and brushing it clean. Her hands lingered on his chest, thinking of last night and how good it had felt to be in his arms. "I'm not going to say anything to anyone," she said softly, knowing that was his unspoken question. "It's your job, and I know that if we exposed our secret, it could ruin you." "Wrong," he said, and tugged her closer, pulling her hips against his as if he wanted to drag her back to the remnants of camp and lie in bed for a few hours longer. He smiled down at her. "I've been thinking..." "Oh?" She forced a light smile to her face. Nothing good ever accompanied the words _I've been thinking_. "When I get back, I'm going to talk to the guys. Let them know about us. We shouldn't have to hide what we did. I'm not ashamed." She stared up at him. "What?" "I'm going to tell them about us," he repeated patiently, and tugged at her hips as if it could drag her attention back to the conversation. "You and me...I want them to know about us. I just need some time to talk to the guys. Ease them into it. Leave it to me. I want them to know you're mine, and we're together." Miranda smiled up at him, the pit of her stomach sick. "If you're sure..." "I'm sure. Just leave it to me." He reached down and touched the side of her neck, then pulled her in for a long, hard kiss. "See you back at camp." Miranda broke down her campsite and headed back to the base camp. It took about half the morning, but along the way, she ran into Steve and they walked back together in cheerful companionship. Though she forced herself to answer his conversation with calm, happy responses, her mind was wild with uncertainty. Dane wanted to continue their relationship. She was leaving for Houston far too soon, starting a new life. There was no room for him there. What could she do? Tell him the truth? That she'd been out for revenge due to a high school prank but he was so amazing in bed she'd changed her mind, and they should make a go for it until she had to bail out and move to Houston? Say nothing and just disappear? Confess the truth? She was torn. Following the coordinates on the tiny map, they were able to find a finish line tape set up between two stout trees. Brenna and Grant waited there, excited to see the students trickle in from the woods. Nearby, a few other students had already returned. They still had their backpacks on, and stood chatting, clearly not ready to leave yet. In the distance, Miranda could see the ranch house that was the business headquarters. Brenna wore a party hat. She blew a paper horn at the sight of them and whirled a noisemaker as Miranda and Steve stepped through the ribbon at the same time. "Congratulations!" she called. "You both passed with flying colors! Come over here so I can give you your certificates." Miranda was suddenly surrounded by other well-wishers—people from her team, people from the other team, Grant, Brenna—everyone wanted to shake her hand and chat with her about how the week had gone. Brenna handed her a certificate. "Thank you for being a part of Wilderness Survival this past week." Dazed, Miranda took the certificate and glanced around. No Dane, no Colt. No Pete, but that was a good thing. "Is...everyone here?" "Not yet," explained Brenna. "I think we had one or two get lost in the woods. Dane went to track them down." She grinned at Miranda. "Still working the kinks out in everything with it being the first class. Glad you made it, though!" Miranda gave her a weak smile. Grant stepped in front of her, camera in hand. Oh. "Hey, Miranda," he said with a friendly smile. "Good to see you again. I heard you'd signed up. You're just in time for me to get your picture for our graduation board." Miranda froze, her skin crawling at the sight of the camera. Suddenly, she did not want her picture taken. She didn't want to stand here and awkwardly wait for Dane. She didn't want the others to smile and hug her and chat. She wanted to run very far away. She wanted to leave this week behind and forget it had ever happened. She was sorry she'd ever gotten in the closet with Dane Croft nine years ago. She was sorry about the pictures, and about her revenge that had gone so very, very wrong. Houston and her new job was her future. Bluebonnet was her past. And that past now included a very torrid week with Dane Croft. She held up a hand in front of her face, blocking the camera. "Can I talk to you, Brenna?" The assistant cocked her head and studied Miranda with piercing green eyes. "Sure." She moved to the edge of the trees, away from the others, and waited for Brenna to follow. When the assistant did, Miranda pitched her story, careful to place a hand on her lower abdomen and look pained. Her excuse? Girl problems. Brenna looked sympathetic, and when Miranda said she wanted to leave early, even escorted her out to her car. She had to sign some paperwork certifying that she'd finished the class, but within a few minutes of arriving back, she pulled her car out of the gravel parking lot and was turning onto the highway, her mind whirling. Okay, so she'd just run away from her problems. Cowardly, yes. But it was for the best. A nice, clean break with Dane would be easiest. After all, it had been a nice clean break nine years ago, hadn't it? Sort of? "Here we go," Dane said, forcing a cheerful note to his voice as he clapped George on the back. Brenna had set up the finish line again and tooted her celebratory horn as he led the older man back to the finish line. Others stood around and clapped, laughing and smiling. They looked happy. Dane was glad. Right now, he was just tired. It had been a long week and he wanted to crawl into a shower, and then crawl into bed. Preferably both with Miranda at his side. She'd been quiet that morning, no doubt wondering how their relationship was going to last now that the class was over. She probably thought they were just fuck buddies, and he'd seen a hint of something in her eyes last night. Something had been bothering her. And he knew, after seeing that unease and unhappiness in her eyes, that he wanted to take care of it for her. Wanted to be there for her. And it seemed he'd never really gotten Miranda out of his system, had he? Even now, they'd spent a few hours apart and he craved seeing her, scanned the crowd for her pretty, flushed face and that long sweep of silky brown hair that made him hard as a rock when it brushed against him. Nine years and it had felt like it was just yesterday that he was holding hands with Miranda after graduation, lusting after her. Being with her had reset something cold and hard in his system. Something that he hadn't liked in himself. The part of him that had withered when he'd quit hockey. It was back now. Damned inconvenient timing, but you didn't get to choose when you felt yourself stirring back to life again. Sometimes life just happened. So Dane shook hands and smiled and posed for photos with his students for a time, but he didn't see Miranda. Bathroom break? Had she run off to freshen up? He kept glancing around, looking for her, waiting to hear her sultry laugh. A big hand clapped him on the shoulder, and he turned to see Colt grinning at him. "Good week." "Good enough," said Dane evasively. "How'd it go on your end?" "Uneventfully," Colt said. He crossed his arms over his chest and nodded at the group milling around. "Everyone passed, though there were one or two that had no sense of direction and needed some help. Thought we were gonna starve on day two, but they figured it out after a while." He eyed Dane. "You?" "One fool," he said, thinking of Pete. "Other than that, no complaints." "So how was Miranda?" Colt asked. "She whine the whole time about getting her hands dirty or something?" He forced himself not to stiffen or act evasive. Why was Colt asking about Miranda specifically? "She was a real trouper," he said. "No complaints." "Huh," Colt shrugged. "I remember her being friends with Beth Ann, is all. That blonde is way high maintenance. Thought Miranda'd be a little more prissy and scared of the woods. So what made her sign up?" "I didn't ask," he said. Was Colt fishing for information? What did he think he knew? Dane wanted to talk to him privately—Grant, too—but with all the clients around, now was not the time to have a discussion about the client he'd been sleeping with. He knew Grant was not going to react to the news well. They needed quiet, and a bit of time to wind down from the class before he let them know about Miranda and him. And if they didn't like it, well, it wasn't any of their business. Plus, he really just wanted to find Miranda at the moment. "Listen, I thought she'd be able to find it back on her own, but I might need to go rescue her." "She's already come and gone," Colt said with a shrug. His eyes narrowed and focused on the other man. "What?" "Like I said, gone." Colt turned away, done with the conversation. Frustrated, Dane scanned the small crowd and saw Grant's tall form in the distance. He plowed through the crowd and approached his friend, who was messing with a tripod. "Where'd Miranda Hill go?" Grant shrugged, double-checking the settings on his camera. "Saw her chatting with Brenna and then she hightailed it out of here fast. Shame she's gonna miss the team photo." Had to be a mistake. Miranda had come in his arms so sweetly last night. She'd liked him. Trusted him enough to let him tie her up. Hell, trusted him enough to fuck him like her life depended on it. Surely she wouldn't have left without giving him her phone number. Something. He stalked off after Brenna. "Good to see you, too," Grant said drily as he walked away. Brenna was busy at her little table, filling out certificates and chatting with the clients. She gave him a cool sideways glance under her long lashes. "'Sup, Dane?" "Where'd Miranda Hill go? I don't see her here with the rest." She looked unconcerned, and returned to filling out the latest certificate. "She left already." Disbelief flared. "What do you mean, she left already?" "I mean she left already," Brenna said slowly, as if she were speaking to someone mentally incompetent. "She got in her car and left. Said she was done here anyhow." What the fuck? Was she cutting and running? Why? "Un-fucking-believable." "Oh, don't get your panties in a bunch," Brenna said, misunderstanding his reaction. "We can take the team picture without her. One person isn't going to make a difference." When he said nothing, she added, "It wasn't because she was unhappy with the class or anything. Said she was real pleased. I think she was sick." Sick? He shot Brenna a look of disbelief. "She was sick and you let her go off on her own?" Brenna gave him a look of disbelief, lifting her pen from the endless pile of paperwork. "Are you serious? What was I supposed to do? Cling to her leg as she tried to get into her car? You want me to do that to everyone that tries to leave? I hate to break it to you, Dane, but every single one of these people is going home today." He ran a hand down his face and sighed. "Never mind. Just give me her contact information." Brenna pulled one folder out from the stack on her folding table, grumbling about how she preferred it when he was out in the field. "Here," she said finally, flipping through the waivers and handing him one. Miranda's curly handwriting stared up at him. He remembered it from high school, from the notes she'd passed him. Seeing it now brought back a surge of memories. Without asking, he grabbed Brenna's sat phone off the table and dialed the number Miranda had given. It picked up on the second ring. "Bluebonnet Library," said a sour voice. Okay, that was unexpected. "Miranda there?" "Ms. Hill no longer works here." So why'd she give a bogus number? He murmured his thanks and hung up, then stared at the paper to make sure he hadn't misread it. The address caught his eye. 1 Honeycomb Drive. He knew that address—it was the high school, named after the school mascot of the Bluebonnet Bees. "You don't go into the city much, do you, Brenna?" "Should I?" she asked, wrinkling her freckled nose. "Do I need to be familiar with the city, too?" Dane sighed and handed her back the paper and the phone. Brenna wasn't local. She didn't know what anyone in Bluebonnet would have immediately picked up on. "Never mind." Why had Miranda given bad information at the beginning of the week? Why so secretive? It didn't make sense. She wasn't the type to come up with fake addresses just to be a jackass about it. She'd genuinely not wanted anyone to contact her when they were done. Fuck that. He was heading into town as soon as they were done here, because he wanted to know what the hell was going on and why she'd run off. He was starting to think she'd lied. Maybe she was married after all. If she was...hell. He didn't know what he was going to do. The thought made him want to punch something. Miranda should have headed home first. She was tired and hungry, and she needed a shower. Most of all, she needed to have a good cry and figure out her head. Still, instead of heading home, she found herself turning down Main Street and parking in front of California Dreamin'. There were two cars already parked there, so Beth Ann was busy, but Miranda didn't care. Grabbing her keys, she headed inside. Beth Ann's tiny salon had one chair in the waiting area, and it was occupied. In the waiting area, a teenager with orange-dyed hair and blue bangs flipped through a hairstyle magazine. Across the room in the barber chair, a white-haired elderly woman had her curls teased into a bouffant by Beth Ann. Beth Ann glanced up and her eyes widened at the sight of Miranda. "You're back," she exclaimed, her lovely face breaking into a smile. "How'd it go?" Miranda leaned against the wall and closed her eyes. "Not...well." "Hold on just a sec," Beth Ann said, and finished brushing the last stiff curl into place in old Mrs. Porter's hair. "There you go, Janey. All good for this week." The old woman put on her glasses and paid, departing in a cloud of hairspray and powdery perfume. The teenager stood and Beth Ann turned to her. "Can I get you to reschedule, Laini?" The girl rolled her eyes. "You serious?" Beth Ann opened the front door and scooped up a piece of paper, holding it out. "I'll give you a free mani if you come back tomorrow." "See you then," she drawled, grinning, and snatched the ticket from Beth Ann. Beth Ann flipped her OUT TO LUNCH sign and then shut the door, turning to Miranda with wide eyes. "Tell me everything." Miranda dropped into the barber chair Mrs. Porter had vacated. It still smelled of powder. "I don't even know where to begin," she said wearily. Beth Ann automatically reached for her hair and then recoiled. "God, Miranda. I don't mean to be mean, but you stink like smoke and dirt." "Do I?" She sniffed her shirt, but really couldn't tell. Dane hadn't seemed to mind her smell at all, but maybe he'd smelled the same and she'd been around it so long she couldn't tell. The scent of campfire would always remind her of Dane after this point. She sighed. "Oh, Beth Ann, I totally messed this one up." Her friend's eyes widened and Beth Ann turned the chair to look Miranda in the eye. "What happened? Did you see him? Talk to him? Get the pictures?" Miranda hung her head, unable to meet Beth Ann's gaze. "What?" Beth Ann said, horrified. "What's so awful? Were you not able to get pictures of him after all? Did he find you out?" Miranda sighed. "I saw him. And I slept with him." Beth Ann blinked. "Okay. I didn't realize that was in the plan." "A lot," Miranda added. "I slept with him a lot." "Oh." She appeared to digest this for a moment, then asked, "So this was part of the revenge scheme? Lots of sex?" "That's the worst part about it," Miranda said with a wail. "It was supposed to be a meaningless hookup. I was supposed to have sex with him and just toss him aside when I was done. Use him like men use women. Get my revenge pictures and then move on. Except...now I like him. And the sex." Beth Ann pursed her perfectly made-up lips and then grabbed Miranda by the shoulders. "You're going to sit down over here so I can do your nails, and you're going to tell me everything." Miranda sniffed, and nodded. Beth Ann steered her friend to the manicure table and while Miranda explained what had happened in the past week, Beth Ann filed her nails and cleaned a week's worth of grime out of her nail beds. She listened without a word as Miranda spoke, not judging. Miranda avoided the part about her inability to have an orgasm prior to Dane. That was a little too personal and open even for her best friend, who wouldn't understand. Beth Ann had always had a steady relationship up until this year, when she was taking time off from her relationship with Allan, her high school sweetheart and on-again, off-again fiancé. "So that's what happened," Miranda said softly as Beth Ann put a glossy coat of clear polish over her short nails. "I went into the woods knowing I wouldn't be able to get the photos, and I did it anyhow. And I figured that okay, I'd just sleep with him and then get the pictures after the class was over. But last night, when we were sleeping together, I...I couldn't do it." She squeezed her eyes shut in anticipation of Beth Ann's response. "You think I'm an idiot, don't you?" "Honey, no," Beth Ann soothed. "Not at all." "But you don't approve." Beth Ann's pink lips pursed. "No, I don't. He's always been the guy that dicked you over in high school and left you out to dry. I don't care if he has puppy dog eyes now and a particularly fine ass. He's always going to be that jerk who hurt my best friend, even if you don't want your revenge." Miranda managed a miserable smile. "Thanks, Bethy." She patted Miranda's hand. "I can't judge you for sleeping with the wrong guy. Heck, look at me. I've had a relationship with a man who can't keep it in his pants, and yet I somehow keep forgiving him, right?" She gave Miranda a sad smile. "So who am I to judge?" "You guys have been split for a year now, Beth Ann. You stood up for yourself," Miranda said encouragingly. Beth Ann gave her a weak smile and wiped away a stray smear on Miranda's cuticles. "At least you believe in me. Everyone else seems to be waiting for me to 'come to my senses.'" Miranda snorted, and Beth Ann grinned. "Well," Beth Ann said after a moment. "One thing's for certain." "What's that?" "Next time you go on a camping trip, I should probably give you a bikini wax." Miranda smacked her best friend on the arm and laughed. # FOURTEEN Dane pulled his jeep up on Main Street, looking for a familiar building. Several things had changed in Bluebonnet since he'd last lived here, and while he hadn't been into town much since he'd returned, he knew there were a few things that had stayed the same. One of them was Hill Country Antiques, the little shop window just as cluttered as ever, the wooden sign hanging crookedly. And Miranda had mentioned that her mother, Tanya, still ran the place. He stepped inside the shop, a cowbell clanking against the glass door to signify his arrival. "Just a minute," a warbling voice called from the back. He didn't answer, just waited, looking around. The entire place needed a good dusting—it reminded him of something from the show _Hoarders_ , always had. Like all kinds of a yard sale, Hill Country Antiques was stuffed wall to floor with old junk. A massive glass case along the back wall locked up the really "valuable" stuff, and he could see a few Elvis plates on one shelf. An old rocking horse and some wooden furniture were scattered on the floor to his left. Shelves listed under the heavy weight of their items and needed obvious repair. There seemed to be a fine coat of dust on everything, and he brushed a finger under his nose, anticipating a sneeze. This place hadn't changed in nine years, he decided, remembering how embarrassed Miranda had been as a teenager that her mom was the crazy junk lady. But if anyone knew where Miranda was, Tanya Hill would. He knew Tanya didn't like him—when he'd called Miranda's house, right after he'd joined the NHL, she'd screamed and screamed at him as if he'd gotten her daughter pregnant or some shit, and then had never let him talk to her. But he'd tried his other options already—no one at the library would say where she lived, and she was unlisted in the phone book. Tanya Hill was his best option. Two minutes later, he wasn't so sure. The woman popped out of the back room, clutching a stack of old LPs. She still wore her hair in a feathered fringe of bangs, but it had all gone gray and the ponytail down her back was shorter than he remembered. Her face was heavily lined, and her eyes widened behind a pair of glasses at the sight of him. "You!" she screeched. "Get out of my store!" Well, he'd known she'd hated him, but he hadn't realized how much. "Mrs. Hill," he began. "I just want—" The woman picked up a cast-iron frying pan from behind the counter and hefted it with both hands, as if she were going to swing at him. "Get out of my store, you bastard, or I'm calling the cops!" He raised his hands, brows going up. "I just need to know where Miranda is." "You need to get the hell outta my store, you two-bit trash!" "Look, I'll buy something if—" "Get out!" she screeched again, then raced for the phone. "I'm calling the cops!" Great, just what he needed. He put his hands up higher in surrender. "Don't call, I'm leaving." As soon as he left the store, he heard her feet clomp across the wooden floors, and the door locked behind him. The OPEN sign in the window winked out. Well. Not the reception he was used to getting. Dane scratched his face ruefully. Damn. He probably smelled like ass and was all unshaven. Maybe her mother thought he was a wino or something? The woman had always been a little off. Frustrated, he glanced across the street. Kurt's Koffee was new, and had a few people in it. Maybe an Internet search... As soon as he entered, the man behind the counter broke into a wide grin. "Well, shut my eyes and call me a blind man," the stoner drawled. "If it isn't the star of the Las Vegas Flush, Mr. Dane Croft, come to pay us another visit." "Hey, Jimmy," he said casually, though his mind was racing. Damn. So much for keeping his presence quiet. "I'm looking for Miranda Hill." "I'll just bet you are," the stoner said with a smirk and raised his hand in a high five. Dane ignored it. "So you know where she lives?" "Small town," Jimmy said, lowering his hand and nudging his sad tip jar down the coffee bar at him. "I know where everyone lives." He scowled at the barista, but pulled a few bills out of his pocket and shoved them into the empty tip jar. "This is a coffeehouse, not a bar, Jimmy." "Barista, bartender, it's all the same. We're just a couple of dudes slinging drinks for a few bucks, man. Tip's a tip." He leaned forward. "So. You remember where Old Johnson Lane is?" Miranda's house was just as empty and small as she'd left it. Boxes were scattered through her living room, but she hadn't had a chance to pack much. She set down her backpack on the end of the couch and felt the overwhelming urge to collapse. She sat on the edge of the couch and then stood up. First, a shower. Someone knocked at the door. Miranda groaned. Not today. Not now. Her mother had called seven times in the past week and she'd been furious that Miranda hadn't answered. She'd soothed her mother with a cover story about scoping out her apartment in Houston, and she'd managed to deflect the worst of her anger. Miranda had avoided going over, but her mother still called. In fact, she'd called three times in a row just now, and Miranda had avoided all three calls. She didn't want to talk to her again. Not while she felt so utterly lousy and unhappy and lonely. Miranda hesitated, staring at the door with frustration. Her mother wouldn't go away. She'd just keep knocking, even if Miranda pretended not to be home. With a heavy sigh, she moved to the front door and pulled it open. "Mom, I'm just not—" A big, male form stood in her doorway. Broad shoulders and a gorgeous body lounged just inside her screen door, and Dane gave her a slow, pleased smile. "Surprise." The look of unhappy surprise on her face wasn't a pleasant welcome. Miranda stared up at Dane with her mouth hanging slightly open, her pretty brown eyes fuzzy, as if she wasn't quite able to piece together exactly how he'd managed to show up on her doorstep. That just made his stomach sink all the way down to his work boots and confirmed his suspicions. Miranda was married and he'd been nothing but a cheap fling on the side. His mouth tightened and he shoved his hands into his pockets, doing his best not to crane his head and see who sat in the living room of the tiny house. Still, he'd gone to all that trouble—he wanted confirmation at least. "Should I go? Is your husband home?" Her astonished expression grew even more confused and she opened her mouth wider, then closed it, then tilted her head in a way that made her hair spill over her shoulder and drove him absolutely wild. "Husband? I—I'm not married." "Good," he growled low in his throat, feeling pleased. "Can I come in, then? I think we should talk." He half expected her to put up a fuss or make excuses, but she only pushed her hair back over her shoulder and then stepped aside, swinging the door wider so he could enter. "Sorry, the place is a bit of a mess," she mumbled. His gaze moved to the boxes scattering the room. "You just move in?" She gave him an odd smile. "Yep. Still haven't unpacked." And then she darted past him, picking up shoes and the bra she'd apparently discarded as soon as she'd come in the door. She scooped up the items and tossed them into her bedroom, then shut the door. "Have a seat on the couch." He didn't want to sit on the couch like some uninvited guest. Dane wanted to pull her into his arms and kiss her now that they didn't have to hide it from prying clients. He wanted to hold her body against his and feel every curve, soft and naked, and pull her down to the floor and make love to her. To kiss and tease and coax that vague, worried look out of her eyes that told him she was thinking entirely too much right now. "Miranda, come here." "Oh, no," she protested with a half smile, retreating a step even as he advanced. "I smell like I've been living in the woods for a week." "You smell good to me," he murmured, snaking one arm around her waist and drawing her close. "But then again, I've been living in the woods, too." A girlish giggle escaped her throat and her gaze went to his face, and she smiled, her body melting against his. His cock grew instantly hard. "You and I," he said slowly, "need to talk about why you ran away earlier today." Her smile disappeared and she tried to slide out of his arms. "I wasn't feeling well. Girl troubles." He didn't buy it. "So did you have girl troubles before or after you wrote down a fake address and phone number on your documentation?" Her eyes flew open. "How did you—" "Because I tried calling you, dammit. Once I found out you disappeared this morning, I wanted to know where you'd run away to." His voice dropped and his hand slid down her back; he sensed she was escaping him, trying to flee even if she stood stock-still in his arms. He needed to anchor her or cut loose for good. So he told her the truth. "I thought we'd had something. It wasn't just a fling in the woods for me." Miranda had never been "just a fling" for him. She'd been the one that had gotten away. The one he'd dreamed about for years. The one that he was going to risk his job for when he told his friends they'd been sleeping together. But he didn't tell her that. He simply said, "I want to keep seeing you, if you want it." She seemed to hesitate, then she slowly melted against him, as if all the things that had been bothering her had dissolved and left her boneless. One hand slid over his shoulders, touched the hair at the nape of his neck, and her gaze roamed over him in a gesture that was both shy and possessive. "You do?" "Of course I do. Did you think I was just fucking you in the woods and jeopardizing our first class simply because I'd nail any hot piece of ass that walked past?" "The thought occurred to me," she said meekly. He winced. "Sorry," she added. "You were just...flirty, back when we were teenagers." "Flirty with you," he said. "And half a dozen starlets afterward," she added in. His jaw set, aching with tension. "So my past is going to be a problem?" "If it was," she said in a low, trembling voice, "I wouldn't be in your arms right now. I just don't want to get hurt, Dane." Miranda's eyes met his and he saw stark terror in them. "I'm terrified of being used." She seemed intensely vulnerable in that moment, and he didn't know what to make of it. Like she was offering him everything she was—and was completely terrified to do so. He brushed her cheek with his fingers and leaned in and gave her a feather-light kiss, sweeping his mouth over hers. "How about you just use me instead? I thought that was how our relationship worked, remember?" A soft chuckle escaped her, the sound going straight to his cock. Damn. He loved to hear her happy. "Very well," she said in a playful voice. "You are mine to use and abuse." "Sounds good," he agreed. She reached for his shirt and then wrinkled her nose up at him. "I smell." "I do, too," he said with a grin. "I came to find you before I showered. Hope that's okay." She smiled, a wide, lovely smile that covered her entire face. "I had no idea I was such an urgent matter." "To be honest, I was afraid you were going to waltz back out of my life again, and it scared the shit out of me." She looked pleased. "Come on," she said, giving his shirt a tug. "You can scrub my back." "Yes, ma'am," he drawled. "You wanna wash me, too?" She gave him a sultry look over her shoulder. "Absolutely." As she entered the bathroom and began to run the water, he moved into her small bedroom. Though it felt like an invasion of privacy, he hadn't brought any condoms with him, and he needed to find some. He wasn't leaving this house until he'd made love to Miranda again. Guessing, he pulled open the drawer of her nightstand. A magazine lay inside, a scatter of condoms, and a bright blue vibrator. Now, that gave him ideas. Grinning, he grabbed a condom—close to expiration. It was pretty obvious Miranda didn't buy them often. He'd fix that. Tucking the condom into his pocket, he headed into the bathroom after her. Like her bedroom, Miranda's bathroom was neat and clean, the counters shiny and white. A cheerful yellow shower curtain matched a plush bath rug, and she sat on the edge of the tub, peeling off her socks. "I can't wait to shower," she admitted with a tiny smile at him. "I'm not exactly feeling sexy at the moment." "That's fine," he said, trying not to think too hard about the condom in his pocket. "We'll clean up first, and then have sex." She laughed and finished stripping her clothes off. "All right." This, Dane decided, was going to be the shortest shower ever. He quickly stripped out of his own clothes and tossed them on the rack while Miranda stepped into the streaming water. Just the thought of her naked body all wet and gleaming made him hard, and her moan of pleasure made his balls tighten. Damn. Get in, wash himself, wash her, then back into the bedroom. Five minutes, max. He could do this. Steeling himself, he stepped into the shower. She stood in front of the spray, the water only grazing his body as she soaped up. Her long hair trailed rivulets of water down her back and he sighed, hard, thinking of how he'd like to take her in this shower, bend her over, and— "You want the soap?" He took the pink bath pouf she offered him, accepted the squirt of fruity shower gel, and began to rub it on himself with grim, quick intensity, concentrating on getting himself clean rather than on the warm, soapy woman who stood less than a foot away, her face blissful as she washed her hair. "Do my back?" She turned and presented it to him. Dane set his jaw. He began to methodically scrub her back, swiping the pouf over her in quick, rapid strokes. Miranda yelped in surprise and jerked away. "Are you trying to scrub my skin off?" "Sorry," he said, averting his eyes. Damn, he'd looked over and her breasts were dripping water, the globes of them slick and inviting. He wanted to shove his cock between them and come all over those pretty tits. Five minutes, he reminded himself. She'd asked him to wait five minutes. Surely he could do that. "Your turn to rinse off," she said, and parted the curtain, stepping out. "I'm done." Thank Christ. This was the longest shower in all eternity. He quickly rinsed his body off, staring through the small gap in the shower that showed pink buttocks being rubbed dry by a fluffy towel— Fuck it, he was done with this shower. He turned the water off and shoved the curtain aside, reaching for Miranda as soon as he stepped out of the tub. She squealed in surprise as he pulled her into his wet embrace, and he kissed the surprise out of her voice. She was lovely and soft and smelled like fruit, and she was driving him utterly insane just by being here. His tongue slid into her mouth and he gave her a long, sensual lick that told her exactly what he wanted to do to her. She shuddered in his embrace and wrapped her arms around his slick shoulders. The hallway would do. His hands on her, he dragged her a few feet out of the bathroom onto the rag rug that ran down the length of the hall and dropped to his knees, pulling her down with him. Her throaty giggle just made him harder. "Right here?" "First time right here," he agreed, separating from her for just a brief moment, long enough to reach for the condom he'd stolen from her drawer. His other hand continued to roam over her body, his mouth kissing her pretty jaw and throat. "Next time, in the bed. Time after that, we'll wing it." "Mmm," she said in response, and he knew she approved of his plan. Her fingertips slid over his abdomen, feeling the wet muscles and sliding lower to grasp his cock. Goddamn. He closed his eyes and groaned, bracing himself. He'd nearly lost it then and there. "Need a condom." He tore the packet open with his teeth. "Got one." Dane slid between her knees and she wiggled on the rug below him, her breasts jiggling with that small movement. Beautiful sight. He leaned down and kissed one tip as he quickly rolled the condom on. Her breath caught in a sexy little gasp. Fuck, he wanted to hear that all over again. Condom in place, he hauled one of her legs up around his waist and slid a finger down through the heat of her pussy, seeking her entrance. Was she wet? Was he moving too fast? Her gasps turned into soft cries and she pushed against his finger, raising her hips. Not only was she wet, she was hot and slippery with need. Beautiful. He let his fingers graze her clit once before removing his hand, enjoying the little jump her body gave in response. Then he took his cock in hand, guided it to her opening, and slammed home. Miranda gave a breathy little shriek, her eyes widening. Her hands found his shoulders and her nails dug in. "Oooh, that was good." "You like that?" he gritted out, doing his best not to fuck her right across this floor and spend himself in two seconds flat. He needed to make sure she came, or else he'd be as bad as those other pricks she'd dated. "I did," she said in a soft, breathy voice, lifting her other knee so her hips tilted up. He circled his hips against her own, rocking deep inside her, and she moaned. "Like that?" he murmured again, watching her head fall back with pleasure. "Want me to fuck you slow, baby, or fast and hard?" "Fast and hard," she whispered, her nails digging like claws into his back, her hips twitching under him. He didn't need any more encouragement. He drew back until he was almost out of her, then slammed home again, and was rewarded with her calves tensing against him, a slight flutter in her pussy in response. The only sound she made was another sharp gasp. "Like that?" She nodded. He thrust again. And then again. And again, until he was pumping her hard, his fingers digging into her hips to keep her anchored in place. Each thrust slammed into her, and he knew he wasn't going to last long. She was covered in droplets falling from his skin, her body wet and slippery like his, and those breasts gleamed and bounced with each hard drive of his cock, and it just made him wilder with need. She gave a small moaning breath with each thrust, her eyes closed with pleasure, and she raised her hips to meet his thrusts almost violently, until he was afraid he was going to hurt her. She wasn't hurting, though; her moans grew louder with every lift of her hips. He felt his balls tighten, knew he was close, but she hadn't come yet. Though it killed him, he slowed, circling his hips gently again, still embedded deep into her hot, tight passage. He needed to think about something else to pace himself. As always, when he needed to slow his orgasm, his thoughts went back to hockey. Drills. That was what he needed to think about. Think about passing. Better yet, passing in the offensive zone. That was what he needed to do. Pass to her—get her to come first. He slid a hand between them, searching for her clit. When he found it, she nearly came off the floor. "Dane!" _That's right, baby._ Now he was on the offense. He brushed her clit with his thumb, circling the wetness over it as she shuddered under him, crying out. Her nails were scratching the shit out of him, but he didn't care. He wanted her to come just as hard as he was about to. Patiently, he continued to circle it with the pad of his thumb, waiting for her to fall apart and then he'd finish claiming her. But first, she needed to score. A tiny, keening whimper rose in her throat and she pushed against his thumb, harder and harder, and then froze. He felt her pussy flutter and clench around him, hard, as she began to come, and he continued rubbing, extending her orgasm. She continued to clench around him, her voice calling out his name in a broken, rasping half sob, and he lost his control. He shuddered, trying to think about hockey. His mind was full of visuals of sliding the puck home, like he was sliding his cock home inside her. With one final, hard thrust, he came, gritting his teeth against the yell of pleasure that threatened to erupt, emptying himself deep inside her even as she quivered and her pussy clenched around him in multiple aftershocks. He continued to move in her, slowly thrusting even as he came down from his orgasm, cock throbbing, blood pounding in his ears, and then he collapsed to the side of her, pulling her close in his arms. They lay there for a long minute, neither one moving. Dane felt his heartbeat slowing, felt Miranda's breathing returning to normal. Then she gave a low, throaty chuckle. "Are we going to cuddle in the hallway?" He leaned in and kissed her shoulder. "Would you rather cuddle in the kitchen?" She smiled, her eyes still closed, the blissfully dreamy expression on her face. "I was thinking more along the lines of the bedroom, but hey." Dane grinned at that, thinking of the bright blue vibrator she'd had in there. "We can go to the bedroom." She untangled herself from him and stood, moving back into the bathroom and grabbing fresh towels. "I should probably clean this water up first." "Leave it," he said, then turned away and disposed of his condom. "Come on." Taking her hand in his, he tugged her back into the bedroom and down onto the bed. They were still wet from the shower, but he didn't care and he suspected she didn't either. Her dark, glossy wet hair flew across the blankets and she grinned up at him. "You don't look like you're in a cuddling mood." "No? How do I look?" She ran a finger over one pectoral, outlining it. "Predatory," she said, the sound a sigh of delight. "Like you want to capture me and eat me." "While that sounds delicious, I have other plans for you tonight," he said, and waggled his eyebrows at her. She gave him a curious look, a smile curving her lips. "Oh?" He loved looking down at her, soft and warm and curvy under him. Dane grinned. Was it possible to fall in love in a week? He wasn't a romantic, but there was something about being with Miranda that made him feel whole, centered. He liked it. And he liked her. And while the timing wasn't ideal, he loved being with her, and he was going to keep being with her. Colt and Grant would just have to adjust. He wouldn't tell her that just yet, though. Getting all sappy and yakking about feelings with a chick after a week would probably make any sensible woman run like the wind. He teased her belly button with one finger, toying with the dip. "How much do you trust me?" The look that Miranda gave him was immediately wary. "Why?" Well, damn. His playful mood deflated in an instant. She looked almost frantic with worry, her body tensing under him. What was going through her head right now? "In bed," he clarified. "How much do you trust me in bed?" "Oh." Her fear dissipated so abruptly that it stunned him. "Of course I trust you in bed." Well, what the fuck was all that about? Why was she so terrified of trusting him? Had she been burned in the past and that was why she flinched every time trust came up? A fierce, possessive surge swept through him. Whatever bastard had hurt her, he was going to find the man and take him apart, piece by piece. Miranda was a funny, loving, sexual, incredible woman, even if she didn't think she was. All he had to do was prove it to her. And if that took three weeks or three years, he was up to it. "So you trust me?" he repeated. "Trust me to bring you pleasure?" She nodded at him, her brown eyes soft and sexy. "Of course." "Trust me to make you come so hard your toes will curl?" She grinned and wiggled her toes at him in response. "Every time." He reached over her and opened the nightstand drawer, then pulled out the bright blue vibrator. "Enough to let me use this on you?" Her playful look froze on her face, and then she blushed—not the delicate, charming usual blush—fiery red. "Dane!" Her outraged expression was delightful to see. "You're not supposed to know about that." "Not supposed to know about what?" he teased, pulling it out of the drawer and examining it with great thought. She reached for it, but he continued to hold it out of her reach. "I had to dig for condoms earlier, and I saw this. Hope that's all right." "It's just embarrassing, that's all," she said in a hushed voice, and then reached for it again. "Give it back. I don't need it when I have you anyhow." "My sweet, darlin' Miz Hill," he drawled, and winked down at her. "Now that's where you're wrong. I think there's plenty of room in this bed for the three of us." Her blush gave way to confusion. "You want to use my vibrator?" Dane grinned. "Maybe some other night you can use it on me. I'm more interested in using it on you and watching you come again." The crimson returned to her cheeks. He moved up on the bed and kissed her lips. She was pliant underneath him, and he detected a slight tremble of her body—in excitement or uncertainty? "I won't do it if you don't want me to, Miranda," he murmured against her lips. "Not if you're embarrassed." "A little," she said. "What embarrasses you about it?" She thought for a moment, then shrugged. "I guess when we have sex, we're in it together so I'm not thinking too much about you staring at me or watching my reactions. But with this..." He chuckled and slid a knuckle over the soft swell of one breast, enjoying how her nipple tightened under his touch. "You think I'm not going to participate here? That it won't turn me on?" "Kind of silly, huh?" "Very silly," he agreed. "Hell, I got turned on just watching you make fire when we were on the trip." "Must have been because I was handling all that wood," she teased, lightly skimming her fingers over his arms. Then she looked over at him and bit her lip. "It's okay," she said after a moment. "I want to do this. I trust you, remember?" He felt like he'd been given a gift. Miranda trusted him to take her out of her comfort zone, and he wouldn't disappoint her. Dane kissed her again, nibbling and sucking at her lower lip. When she gave a low moan in her throat and her arms slid around his neck, he slipped from her grasp and began to move down her body, kissing a path. Her chin. Her collarbones. Her breastbone. He lingered at the twin mounds of her breasts, so lovely and full. He nuzzled each nipple and made them stand erect, leaving the tips wet and gleaming and Miranda squirming and panting under him. Farther down, he kissed her belly button and noticed that her body was beginning to quiver with tension. A good sort of tension, he hoped. When he got to her mound and pressed a kiss there, she didn't make a sound. He glanced up and saw her dark eyes watching him, biting her lip as if she were afraid to show him her reactions. Well now, that wouldn't do at all. She was thinking too much again. Dane sat up and slapped her thigh lightly. "Pull up your knees, baby." She frowned at him, and slowly did as she was told, pulling her knees close to her breasts. "Hug them to you," he said. "Lock your hands behind them and don't release until I tell you to." He watched the blush return, and she pulled her knees in tighter, her head tilted to the side. Now she could no longer see what he was up to, and he'd have the freedom to work her over like he wanted to. She was lovely like this, her legs pulled back, displaying long, lean thighs pressed together tightly. Between them, the dark strip of her pussy and the wet gleam of the folds there. She looked so delicious that he leaned in and brushed his tongue over the wetness, tasting the salty flavor of her. She shuddered underneath him, her breath catching in her throat. Now, that was more like it. With one finger, he parted her pussy lips and slid his finger back and forth, making sure she was juicy and wet. She was—her core was hot and slick, and he rubbed the slippery wetness up and down through her labia. She whimpered every time he touched her clit, but he didn't linger there. "Soft and pretty. And so very wet. Are you excited about me using the vibrator on you?" The tremble swept over her again, and her pussy clenched when he dipped a finger deep into her core. He teased it deep, then slid it back out again, and waited for her response. "Y-yes," she finally allowed. "It feels...naughty." "Well, I happen to love a naughty girl," he said, and rewarded her with another thrust of his finger. She jerked her hips in response, but she seemed tense, waiting for something. He knew what it was. He clicked on the vibrator, and she seemed to almost vibrate with need herself. He didn't insert it—not just yet. He wanted her to anticipate it for a minute more. Instead, he continued to let his finger circle the heat of her core, then glided it through the wetness back up to her clit, as if he had all the time in the world. All the while, she made soft whimpering noises and her legs tightened and flexed, over and over again, her thigh muscles working repeatedly. It was beautiful to watch. Miranda's vibrator was a long, smooth blue column, the head slightly flared and curved to brush against her G-spot. It shivered in his hand, and he spread her pussy lips and laid the vibrating head of it against her clit. Miranda nearly came off the bed. A cry escaped her throat and she clenched, hard. "Keep your legs up," he reminded her, and placed his own arm over hers to keep her stationary. "Oh God," she moaned. "Dane, please—" "Please what?" he said in a husky voice. "Please tease it against your clit?" He rubbed the vibrating head against the hard, swollen little nub. She jerked against his grasp but he held her pinned there, using the head of the vibrator with great precision, rubbing it back and forth against her clit. He was as hard as a rock watching her reaction. Her pleasure was so overwhelming that her eyes were closed, her mouth working in soft cry after cry. And then she shattered, her legs clenching hard. He continued to rub the head of the vibrator against her clit, milking her reaction until her muscles loosened under him and she began to pant. "Oh, Dane," she said in a husky, wondering voice that made his balls tighten with need. "Oh God." He rubbed the head of it through her labia, getting it good and slick, and then he sank the tip of it into her pussy. He felt her tense under his arm, felt the shiver building through her legs. Her moan rose again. When he sank the vibrating length in to the hilt, she cried out his name again. "Dane!" "Do you like it, baby?" He twisted the vibrator inside her, rotating it in a circle like he would his cock if he were deep inside her. "Like it when I make you come?" "Yes!" He pulled it out, then thrust the slick length of it deep again, enjoying the way her body jerked in response to the thrust. He repeated the motion, then began to piston it slowly, letting it glide in and out between her wet pussy lips, and leaned in to flick his tongue against her clit. She was so wet that she was soaking, and he lapped at the taste of her and was rewarded with another hard clench of her pussy. Then another. "I'm coming again," she cried out, and her pussy clenched hard against the vibrator. He continued to work it and she cried out, over and over. He kept thrusting it into her, and her cries turned into a shriek of his name as she spasmed hard with her orgasm. Dane turned the vibrator off, breathing hard. He'd made her come so freaking hard and his own breathing was shallow and panting, his dick hard as a rock. He wanted to be deep inside her, wanted to be the one fucking her and making her shriek. Wanted to be the one she clenched around. He slid the vibrator out of her still-clenching pussy and pulled his arm off of the backs of her thighs, trying to compose himself. He'd give her a moment, and then he'd finish what they'd started. She was on him in a flash, rolling up on the bed and kissing him hard. Surprised, he kissed her back, and then hissed when she reached for his cock. "Let me do you," she said against his mouth, then tugged at his lower lip with her teeth. "With the vibrator?" She shook her head and gave him a naughty look, her face gleaming with a sheen of sweat. "The old-fashioned way." And she slid a hand down his chest and pushed him backward. He went, his cock standing straight up in the air. As he watched, Miranda straddled him and went straight for his cock, her mouth suddenly sliding over the head and encasing it in warmth. Her warm hands grasped the shaft and then she sucked, hard, on the head. Oh fuck, he was going to come if she did that again. "Miranda, baby," he groaned. She only wiggled her ass and continued to swipe her tongue against his cock, licking the head and tasting the pre-cum that beaded there, then taking him deep into her throat and pumping the base with her hands. His fingers were wrapped in her still-wet hair and he held her as she worked his cock, using her tongue in wicked little licks that made him want to come all over her face. Then she got the naughtiest look of all on her face and pulled him deep into her throat, sucking hard. It was so fucking good he nearly saw stars. He almost missed when she switched on the vibrator and held it against his balls. A raw shock wave of pleasure coursed through his body. Then he was exploding, shooting hot jets of cum down her throat and yelling, fucking her mouth even as he came, and she continued to rub the vibrator against his sac, working her mouth over his cock and getting every last ounce of cum from him. When he collapsed back on the bed again, she sat up and licked her lips, giving him the most satisfied expression. "Damn," he panted, grinning up at her. "I think I need another shower." She smiled and moved up to kiss him. "In a minute," she said. "I just want to touch you for a while without my head exploding in another orgasm." "Is that such a bad thing?" "Hell no." He grinned, pulled her into his arms, and tucked her close, trapping her leg in between his. No way was he letting Miranda Hill get away again. She was amazing. She was wild in bed, and hot as hell. And she was all fucking his. # FIFTEEN One Week Later Beth Ann dropped by Miranda's place after she closed her salon. Usually when Miranda had a day off, she'd stop in to chitchat, and Beth Ann would trim her ends, give her a manicure. But the salon had been strangely quiet this week, and Beth Ann's suspicions were roused. She'd called Miranda a few times, and her friend had seemed cheery but distracted. "You busy moving?" Beth Ann had asked her, and Miranda had said she was. But Miranda had the week off, and she hadn't stopped in to say hello or hang out. _Fine way to treat a friend when you were moving away for good_. So she brought a roll of contact paper with her to work that day, and when Miranda didn't drop by the salon, she went to Miranda instead, toting her present as an excuse. Miranda had a small, neat little cottage on a quiet street. Tall pecan trees littered the yard, and her tiny rental house was older, but charming. Beth Ann had half a mind to take the lease off of Miranda's hands when she left—anything would beat another month living at home with her parents until Allan got his act together... She sighed. She needed to stop thinking that way. Allan wasn't getting his act together, and she wasn't getting together with him ever again. That was her mother planting ideas in her head. She swung the screen door open and knocked on the wooden door. Silence. Beth Ann glanced in the window—lights were on. She leaned in close to the door. A murmur of voices, and then a scramble to get to the door. Annoyed, Beth Ann hit the doorbell. The door swung open quickly, and a flushed Miranda answered, pushing strands of her hair out of her face. "Hey, girl," she exclaimed in greeting. "What are you doing here?" "Thought I'd help you pack," Beth Ann drawled, not fooled for a second. Miranda's shirt was untucked, the zipper on her jeans was down. Her feet were bare. Yeah. Beth Ann wasn't dumb. She shoved the contact paper roll into Miranda's hands and pushed her way inside. "Since you're so busy packing, I thought I'd come and help you finish," she said. "Brought you some contact paper for the new place." The house was just as she suspected—boxes lay scattered in the room but nothing seemed to be put in them. In fact, if she looked hard, it almost seemed as if there was less stuff in them than the last time she'd been over. Beth Ann whirled, tapping one pink fingernail on her chin. "You didn't have to do this, Beth Ann," Miranda said awkwardly. "I'm almost done with the packing." Beth Ann turned to look back at her friend, hurt. "You are a terrible liar." There was a sound in the bedroom, and Miranda stiffened. A dreadful feeling began to rise in the pit of Beth Ann's stomach. Before Miranda could stop her, Beth Ann moved to the bedroom door and pushed it open. A large man sat on the edge of the bed, sliding on his shoes. He wore no shirt over his bronzed, rippling muscles, and he looked up at the sight of her in the bedroom door. "Hi," said her best friend's worst enemy. "Beth Ann, you haven't changed." "Neither have you," she said through clenched teeth, and shut the door in his face. She turned and gave Miranda a look of disappointment, and then walked right back out of the house. Miranda followed her out. "Beth Ann! It's not what you think—" "Really?" she snapped, angry and afraid for her friend all at once. "Because I'm thinking he's moved in." "Don't be silly," Miranda scoffed. "We've only been seeing each other for the last week." Beth Ann crossed her arms over her chest. "And how many times has he slept at his place in the last week?" Silence. Beth Ann gave her friend an exasperated look. "Really?" To her credit, Miranda blushed. "I know what you're thinking—" Beth Ann whipped out her cell phone. "That I should just take the picture of your tits and post it on the Internet right now so we can get this over with?" Miranda flinched, and immediately Beth Ann felt like a jerk. She sighed and moved to hug Miranda. "I'm sorry, girl," Beth Ann said. "You're just worried about me," Miranda said in a soft voice. "I know." "He hurt you so badly," Beth Ann said, and hated the knot rising in her throat. "Humiliated you in front of everyone and broke your heart. I know what that feels like, too." "Of course you do," Miranda said soothingly, and patted Beth Ann on the back. "But this is...this is different, Beth Ann." Her face flushed with pleasure, and her pretty brown eyes gleamed. "He's not the guy I thought he was. He's different. You just have to trust me." "Oh, honey," Beth Ann said, and gave her best friend another squeeze on the arm. "Of course I trust you." It was that low-life fink Dane Croft that she didn't trust. She and Miranda chatted for a moment longer on the porch, and then Beth Ann made up an excuse about having to go back to the salon to make sure she'd unplugged everything. Miranda looked uncomfortable. "You sure you don't want to stay for a few? I made some sweet tea." She shook her head and managed a cheerful smile. "Gotta run, but thanks for asking. I'll stop by tomorrow and help you pack for real." She gave her friend a stern look. "No excuses." "No excuses," Miranda said with a smile. Beth Ann moved back out to her small, sea green Volkswagen Bug and started the car. But instead of turning back toward Main Street, she got on the highway and headed outside of town, toward the Daughtry Ranch. She'd find out on her own if Dane Croft was playing games this time. If Miranda wasn't worried...Beth Ann would be worried for her. One of them needed to be ready, and Beth Ann wanted to be prepared for the worst. Beth Ann parked her car in front of the Daughtry Ranch. There was a gravel parking lot and a scatter of cabins, but other than that, it really didn't look like much. In the distance, she could see a long barn, but it looked deserted. Beth Ann got out of the car, gripped her keys, and headed for the big ranch house. A sign hung above the door, proudly proclaiming WILDERNESS SURVIVAL EXPEDITIONS, and a plastic pamphlet case nailed to the porch wall was stuffed full of brochures. She considered knocking, but it was a business, right? She'd treat it like one. No one had to knock before entering her salon. She opened the door and stepped in. No one looked up as she entered. A woman she didn't recognize had a phone to her ear and was writing furiously in a steno pad. Grant Markham—a total blast from her high school past—didn't even look up from his computer. Gee. He'd gotten friendly in the last nine years. One person stood and moved to the door. "Can I help you?" he said in a low, almost raspy voice. She stared at him in surprise. Colt Waggoner—she remembered him from high school, too. She shouldn't have been surprised—she'd seen his picture in the brochure. He was...different. He'd been silent in high school, and she guessed that much hadn't changed, but there was something hard and lean about him now. Something slightly dangerous. He'd filled out from the rangy form she remembered, too—this man was all ropy muscles and coiled strength. And he was devouring her with his eyes. Taken aback, she stifled the surge of pleasure that his appreciation brought. It was nice to have someone make her feel pretty again, but that wasn't why she was here tonight. "Thought I'd come by and ask you if you know where your friend Dane is," she said, keeping her voice mild. Colt crossed his arms over his chest. "Ain't his keeper." "No, you ain't," she said, emphasizing his slang. "But you are trying to run a business here, aren't you? How's that going to look if your instructor is having a relationship with one of his students while on one of your retreats? You'll never have another woman sign up ever again." Colt's appreciative look turned to a scowl as he glared at her. When the woman put down her phone and Grant looked up from his computer, Colt took her by the elbow and dragged her back outside. "Beth Ann Williamson? Is that you?" Grant said, getting up from his desk. "Wow, long time no see. Did you say something about students—" "She doesn't have time to talk," Colt gritted, pushing outside and dragging Beth Ann with him. She glared as he pulled her along, and when they were back in the parking lot, she jerked away from him. "That's far enough, thank you." "There a problem?" Colt's tone was abrupt. "Darn right there's a problem," Beth Ann said, crossing her arms over her chest. She refused to be intimidated by his raw, physical power and that sexy rasp in his voice. Funny, but that wasn't how she remembered him at all. He'd been a silent, aloof jerk in high school, and nothing had changed, it seemed, except the package. "The issue is your friend. He's ruining the life of mine. Again." Colt stared at her a long, long minute, his eyes piercing. When she thought he was ignoring her, he finally said, "Don't know what you're talking about." "No?" She eyed his lean body, the casual set of his shoulders. "Where's your buddy Dane? Not at work, I see." Colt was silent. Ah yes, she remembered why he'd infuriated her so much in school. She'd always been lively and chatty, and he was anything but. "Well?" "Heard you didn't marry Allan." She stiffened. "That's none of your business, and I'll thank you to keep your nose out of it." He grunted. Flustered, she brushed her bangs behind her ear. "Look. All I'm saying is that Dane and Miranda are hooking up, and I think it's mighty unpleasant that one of your instructors is hooking up with one of his students." Colt continued to watch her in that scrutinizing way she found so unnerving. "Not true," he said finally. "Dane's not seeing anyone." The man was unreal. "Are you blind? Should I have taken a snapshot to savor the moment and show to all my friends? Oh wait, that's y'all's job, isn't it?" He scowled. "Look. I came up here to warn you." "Oh, you did, did you?" His cool tone suddenly got thirty shades cooler. "Came to warn me that we're not welcome here?" "I don't care if you pitch a tent in the middle of town," she declared. "But if Dane hurts Miranda one more time, I'm going to geld him with my haircutting scissors, understand me?" "Miranda's a former student. No more. She's done with the class and he's done with her." Beth Ann could have sworn his mouth turned up in a hint of a smile. "All I'm saying is that he needs to back off and leave her alone. All of you. Understand me?" The interested glint in his eyes died and was replaced with ice. "Yes, ma'am," he drawled, then nodded at her car. "I'll see you out." "We're already out." "Then I guess I'll see you gone." Jerk. The town jackass count was higher by one with him back. She turned on her heel, hair flaring over her shoulder as she stomped away. Grant looked up as Colt reentered the office. "What was that about?" "Town shit," Colt drawled, and picked up his Xbox controller. "Nothing worth repeating." "She came all the way out here to talk to you about town shit?" Grant asked. "Sounds like bull to me." Colt shrugged. "She's just imagining shit. Or making it up." "Why would she drive out here over something she made up?" "Maybe she's hot for me." Grant snorted. "I must have mistaken that look of desire on her face for rage." He grabbed the stack of paperwork on his desk and began to pick through it, oblivious to Colt's silence. Then again, Grant always was preoccupied with one project or another. The man focused—okay, obsessed—so much that he lost track of reality. After a few minutes, Grant spoke again. "Is it about a girl?" "Huh?" Colt looked up from the TV. "A girl," Grant said, shoving a stack of ledgers aside to glance at his friend. "Dane hasn't been himself lately. Distracted. She mentioned a girl. Think he's seeing someone in town?" "Nailing, maybe," Colt said with a shrug. "If he was really seeing someone, he'd bring her around." "Because we're both so cuddly and lovable? We read him the riot act about keeping his dick in his pants. The guy probably won't bring a date within twenty miles of this place because he thinks we'll flip out on him." Colt grunted. "She's just being hysterical and overreacting. Dane promised he'd keep his hands off the clients." "True." Grant stared at the door, then looked back to Colt. "You keep your hands off all your clients?" Colt scowled. "Fuck off. That's not even funny." Grant just grinned, ignoring Colt's anger. "You never know, I think any woman would run screaming if she realized she'd have to spend time with you and your sunny personality." "I'm all charm, no harm." "That doesn't even make any sense." "To you, maybe." Colt nodded at the enormous stack of paperwork on Grant's desk. "Anyhow, it ain't a girl. He hasn't said a thing about anyone and he would have said something to me. He never misses a class and he never shows up smelling like cheap perfume or with lipstick on his collar." "I asked if he was dating a girl, not your mom," Grant said dryly. Colt shot him the bird. "Funny. You done with that shit so we can go get a beer in town?" Fingers drummed on the stack of paperwork as Grant weighed the options. Then he stood and shoved his chair out from the desk. "I guess those press releases can wait another day or two." "Damn straight," Colt said, getting to his feet. He gestured for Brenna to take her headphones off. "We're going into town to get a beer," he yelled. She brightened and pulled the headphones off, her freckled face lighting up. "Beer? Can I come?" "No," Grant said sharply. "You still haven't finished the supplies inventory I gave you two days ago." She scowled at him, slipped the headphones back on her head. "Haters." Grant looked as if he wanted to take the headphones off the girl and choke her with them. Colt nudged his friend on the shoulder. "Come on. What's that place in town with the bar?" "Maya Loco," Grant said, finally turning away from glaring at Brenna to move toward the door. "You sure it wasn't a girl?" "If it was, he'd say something," Colt drawled. "Huh. True." Miranda played with Dane's fingers, locking her own with them as she rested her head on his chest. "Do you have work today?" "Classes," he said. "School camping trip. After that, I'm free until the weekend. What about you?" She stiffened in bed. She should be packing. Should call her job to confirm that everything was lined up. Instead, she'd spent the past week lazing in bed with Dane, and when they weren't in bed, they'd spent every waking moment together. Being with him made mundane trips—like going to the grocery store—a pleasant experience. It was depressing how much she was enjoying his company, even more because she knew it couldn't last. The thought made her unreasonably sad. "No work today." He locked his fingers around hers and wiggled their linked hands. "You haven't been to work in more than two weeks. You on vacation?" "Yeah." It was easier than telling him the truth. That she was leaving in less than a week, and that she didn't plan on seeing him ever again. "My trip's an overnighter," he said, then tapped her arm with their linked hands. "You want to come with? I'm sure we could come up with a reasonable excuse." "No, that's okay," she said, feigning a yawn. "I'll clean up around here." "You should probably finish unpacking," he agreed, though he didn't get up from the bed. His free hand played with her hair. "It's my fault you haven't had a chance. Want me to help when I get back?" "Nah," she said, her heart thudding. "I can handle it. I've taken up enough of your time lately." To her surprise, he lifted her hand and kissed it. "I don't mind," he said huskily. "I'll miss you when I'm gone." _Me, too,_ she thought, but said nothing. His arms went around her and he rolled over in the bed, dragging her until she was pinned underneath him. Dane gave her a wicked grin. "Class won't be there for at least another hour or two." She smiled back at him and wiggled her hips suggestively. "Plenty of time for you to make me breakfast, then." He tugged at her clothing. She only wore a tank top and panties to bed, and Dane seemed to appreciate her clothes—or lack thereof. His hands slid under the hem of her tank top and his mouth descended on her belly button, licking and sucking at the indention. "I had something else in mind." Miranda's laughter died in her throat, emerging as a low moan, her hips rising when his fingers brushed along the edge of her panties. Then he was sliding them down her thighs, and a moment later, he tossed them to the floor. His breath was warm on her stomach, and he leaned in to kiss her navel again, his tongue dipping in. Her hands flexed over his shoulders, her nails digging into his skin. God, she loved the feel of him over her, all hard muscles and delicious masculine scent. She'd miss wrapping her legs around him and feeling him sink deep inside her, miss the taste of his skin when he was slick with sweat from fucking her for so long... A needy sound escaped her throat, and she tugged at his shoulders. "I want you inside me, Dane. Please." He moved up her body and loomed over her, leaning down to press a quick, hard kiss against her mouth. She pushed up against him, making the kiss urgent with need. She wanted to forget that this was all going to be over soon. That this might be one of her last moments with him. Her hands went to his cheeks and her tongue stroked deep in his mouth, her hips rising, and she felt the hard, hot length of him against her thigh. He chuckled at her eagerness, moving to press hot kisses along her face, her nose, her chin. "I'm only going to be gone for a few days, Miranda. Then we can spend the next two in bed together." She shook her head, still pulling him against her with need. She couldn't tell him why she felt this intense urge to have him deep inside her. He wouldn't understand. "It's going to be a long day," she said. "Can't I just miss you?" Dane reached over her and grabbed a foil packet from the nightstand, then ripped open the package and rolled the condom on as she clung to him, kissing every inch of toned flesh that she could reach. A brief moment later, his hips settled between hers and she raised her legs, locking her ankles behind his back just as he surged deep into her. Miranda cried out. Her hands tugged at him wildly. "God yes, like that." He gave a low growl deep in his throat and began to fuck her hard, each thrust pumping deep. "Miss you, too," he breathed, leaning in for another hard, possessive kiss. He ended it with a slick thrust that seemed to almost reach her core. His tongue teased along the seam of her parted mouth, thrusting when he drove deep again. "Think about you the whole time. You in this bed, your dark hair all wild, in nothing but your tank top and panties, waiting for me to come home. Thinking about that vibrator and then thinking about you using it while thinking of me." Her pussy clenched at his words. "You like that?" He surged deep inside her again, then began to pump slowly, punctuating his harder thrusts with another searing kiss. She clenched again, and was rewarded with a hiss of his breath—he'd felt the contraction of her muscles deep inside. "I love it when you touch me," she said softly, gazing up into his green eyes, cloudy with lust. "Love it when you're deep inside me, so hard." He stroked deep inside her again, and his mouth descended on hers, swallowing her soft words into another intense kiss. Her hips rose to meet his and they began to move in rhythm, her hips rising to meet each thrust, rocking together. Miranda's nails dug into his skin and she began to whimper with each thrust, the intensity overtaking all rational thought. Her movements became jerky, jagged, and she lost the rhythm as her legs began to tense. He stroked deep and then circled his hips, and her breath choked as he grazed her G-spot deep within. Then he pushed deep, surging hard inside her, and held. His green eyes stared into her own, and she memorized his face in that moment, gleaming with sweat, the intense gaze framed with spiky, dark lashes. The stubble on his cheeks and chin, the curve of his mouth. He was beautiful. He rocked inside her again and she came, a low moan erupting from her throat. Her muscles quivered as the orgasm swept through her. She gasped with relief, curling her toes as he surged deep again. Then he was coming too, the cords in his neck standing out as his entire body shook with the force of his orgasm. Miranda watched him with fascination, memorizing the lines of his face in that vulnerable moment. She could watch him forever. Dane leaned heavily over her, panting, and pulled her close for one last kiss. "Thank you." She gave him a breathy laugh and wrapped her arms around his shoulders, pulling him on top of her fully. The heavy weight of him atop her was a delicious feeling. "Why are you thanking me?" He leaned in and kissed her nose. "Morning sex always takes the edge off the day. Now I don't have to worry about those kids getting on my nerves. I can just go through the day in a daze of endorphins." Miranda laughed. When he tried to roll off of her, she shook her head and tugged him close again. "Not yet," she said softly. "You have a few minutes before you have to leave, and I kind of want to hold on to you for a bit longer." He grinned down at her. "You do realize I'm only going overnight, right?" A day, true. But she didn't have many more days here. She avoided his gaze and ran a light finger over the muscles in his arms, tracing the veins. "Even that can seem like forever," she said lightly. "Is it such a crime to like the feel of you over me?" Dane gave her an intense look, all teasing leaving his gaze. The grin slid from his mouth, and she recognized the flare of desire in his eyes. "Not in my book." And with that, he rocked his hips, just a little. Enough to remind her that he was still seated deep inside her. And he was already hard again. The breath caught in her throat and her gaze flew back to him. "You said we had a few minutes yet, right?" He leaned in and kissed her, then tugged her lower lip into his mouth, sucking on it lightly for a moment before releasing it. "I think we should make the most of it." "I'm game," she said lightly and pulled him closer. Sometimes, she thought she could stay in his arms forever. After Dane left, she picked through some of the stuff in her house. She was restless, something vaguely nagging her. Guilt for her relationship with Dane? Beth Ann hadn't been happy when they'd talked last, and she hated the thought of disappointing her best friend. She picked up her phone to call and noticed her mother had called again, twice. No messages. Probably just checking in. She should stop by and visit her anyhow. She had a box of stuff to drop off, and it'd give her a chance to swing by the salon and visit Beth Ann. Talk for a bit, maybe explain that things weren't the way they seemed. Because really, they didn't seem like wise decisions at the moment. Miranda drove into town, parking her truck in front of her mother's antiques store. The crooked sign wobbled in the breeze, and she sighed at the sight of it. Someday she'd have to pay to get that fixed. The inside of the store was quiet, and she found her mother sitting on a trunk, crying quietly in the back storage room. "Mom?" Her mother looked up as Miranda entered, and only cried harder. Oh jeez. Her mother had always been a little fragile of spirit, and her crying jags could mean anything. Lately they had meant that she was sad Miranda was leaving for Houston, but they'd talked about this already. She tamped down the feeling of annoyance and sat next to her mom on the trunk, patting her back. "You okay, Mom? What's bothering you?" "Oh, Miranda," her mom wept. "I don't know how to tell you this. It's just like before." "What's like before?" Her mother waved a dramatic hand in the air. "The rumors, the laughing. The staring. The people that come by to tell me what he's doing to you." A cold pit started in Miranda's stomach. Her mother was overreacting again. "You mean my dating Dane, Mom?" Tanya hissed, as if the thought caused her pain. "Not dating. He's using you," she accused. "Just like before. He's going to use you for sex and then post pictures of it all over the place to inflate his own sorry ego." Miranda flinched. "It's not like that." "It's not?" her mother said bitterly. "I looked. The pictures are still up. If he liked you so much, why are they still on the Internet for everyone to see?" Miranda pursed her lips, hard, and continued to stroke her mother's back. She'd always had to be the adult in their relationship, the one who assured her mom that everything was okay. Even still...her question made twisted sense, and Miranda didn't like that. "It's not like that—" "No?" Her mother turned tear-bright eyes to Miranda, and her chin wobbled as she mopped at her cheeks with wadded-up Kleenex. "Then tell me how it is, Miranda Jane Hill. Does he take you out on the town? Buy you nice dinners?" Well, no, they'd stayed at Miranda's tiny house the entire time. "Mom—" "Does he introduce you to his friends? Take you to his house?" An uncomfortable feeling prickled through Miranda. She said nothing. "Oh, baby," her mother said in a sad, sad voice. "Did you forget how he treated you all those years ago?" A knot burned in her throat and she stood. "I haven't forgotten." The conversation with her mother had been disturbing in its accuracy—and at the same time, inaccuracy. Her mind was full of questions that she had no answers for. Was Dane using her? But she'd approached him, she'd been the one to declare their relationship a secret. It had been her idea for them to use each other for sex. It had been Dane that had said he was going to break the news to Colt and Grant, let them know he and Miranda were together. They'd taken it surprisingly well, given that Dane had been sleeping with a client. There'd been no fallout at all. That bothered her a little. Her head was full of questions that she had no answers to, and she needed to talk to someone. Instead of getting back in her car, she headed down the street to Beth Ann's salon. As she entered, another woman was leaving—Mary Ellen Greenwood. She gave Miranda a dismissive look as she entered, clutching her purse more tightly under her arm, as if Miranda's presence might somehow contaminate her. Miranda put a hand to the collar of her shirt and tugged it upward. Beth Ann was sweeping hair in the main salon room and looked up as Miranda entered. "Hey, honey," she said, her voice uncertain. "My mom is crying," she gestured down the street to the antiques store. "She thinks Dane's using me." Beth Ann sighed and shook her head, continuing to sweep. "Well?" "Well what?" Beth Ann looked up, her expression grim. "Do you want me to tell you the truth, or do you want me to tell you what you want to hear?" Ouch. Miranda stared at her. "I want you to tell me the truth, Beth Ann. You're my best friend. If I can't count on you, who can I count on?" Beth Ann patted the barber's chair. "Sit. I'll tell you what I know." As Miranda sat down, Beth Ann swept a hot pink leopard cape over Miranda's shoulders. "I went and visited Colt the other day..." # SIXTEEN Her eyebrows waxed, nails manicured, and heart aching, Miranda got back into the car and stared out the windshield without seeing a thing. Dane hadn't told the others he was dating her. He'd told her he would, and that everything was okay. Why was he keeping her a secret from his friends, unless he'd never planned to take things seriously between them at all? Worst of all, everyone in town knew. She'd stopped by the hardware store and had heard the "Boobs of Bluebonnet" whispered again, and her heart sank. In a small town, it was impossible to keep a secret, but damn. If everyone already knew they were sleeping together, why hadn't he told his friends—the ones that counted? It was just like before, except worse. Because she should have known better this time. Her phone rang and Miranda jumped, digging her cell phone out of her purse. Was it Dane calling? Had his class been canceled and and he'd decided to swing by? She hated the little thrilling flip her heart had given at the thought. But that was silly—he didn't even have her number. The caller ID showed a phone number she didn't recognize. "Hello?" "Miranda Hill?" The chirpy voice said on the other end. "This is Kacee Youngblood with HGI Incorporated. I wanted to give you the details about your drug test. We'll need you to complete it before you begin work here at HGI." "Oh, of course," she mumbled, and halfheartedly listened as the woman chirped instructions in her ear. She hung up after agreeing to head into Houston tomorrow to get the tests taken care of. Her mind was still stuck on Dane, and her heart felt like it was shattering into a million pieces. When had this turnabout happened? When had she gone from user to the one being used? Was she truly that stupid? He'd promised her he'd tell them. Promised. And he'd been so sweet and charming about it. He'd been the one to suggest it, not her. Someone was lying to her. Maybe Beth Ann had heard wrong. Maybe Colt was lying. Instead of going home, she turned her car onto the highway, heading for the Daughtry Ranch, her heart aching with every mile. When she turned in, the parking lot had a scatter of cars. They had a class today. She stepped over the rope railing that served as a partition for the parking lot and began to stride through the grass, heading for one of the trails. This class would be students. He couldn't be taking them far— "Can I help you?" A man jogged up behind her, and Miranda wasn't thrilled to see it was Grant. "Hey, Miranda," Grant said with an easy smile. "We didn't get a chance to talk the other day. It's nice to see you again—" "I'm looking for Dane," she said, continuing to stomp forward. "He's out in a class right now. He won't be back until tomorrow." Grant parked himself in front of her, effectively blocking the path. "Is there something that I can help you with?" "I really need to talk to Dane, actually—" Grant didn't budge, but he gave her a friendly smile and gestured back at the main cabin. "You want to come in and hang out for a bit since you're here? Colt's out with students, too, so it's just me and Brenna. We can catch up—I hear you've been here in Bluebonnet for the past nine years." Oh, lovely. Like she wanted to talk about the town right now. "I just..." Grant smiled at her. "This is perfect, actually. We were hoping you'd come back. We need your picture for the student hall of fame board." She gestured at the woods. "But Dane—" "He doesn't have to be involved for the picture," Grant said with an easy smile. "It's fine." She stared at him for a long, long moment, not moving. Waiting for him to break character, to show her that he'd been teasing and he knew perfectly well that she was in a relationship with Dane. That he was just giving her a hard time. But he said nothing, continued to patiently wait for her. "Did you know I fucked Dane?" He stiffened, his smiling expression growing dark. "Excuse me?" "I was one of Dane's students," she pointed out, hurt by his shocked surprise and wanting to hurt back. "I approached him that first night and made a pass at him. Then we met in the woods and he fucked me. And then we fucked almost every night out there." Grant's face grew stiff with anger, his mouth thinning. "Are you serious?" "Yes, actually. He loves fucking me. In fact, we're still fucking," she said with a bitter laugh. "He disappears at my house every night and we fuck like bunnies until morning." Fury and hurt exploded through her. "He told me he was going to tell you and Colt that we're sleeping together, but it looks like that was a lie. I'm guessing that isn't the only thing he's lied about lately, huh?" The words choked in her throat, and she whirled to go, stomping away on the gravel path back to the parking lot. She waited for him to try and stop her, but he didn't, and when she looked back, Grant was staring at the woods with a thoughtful expression. Miranda slammed the car door and drove home. Time to put the plan in action. "Just remember that we do all kinds of survival classes—from school-age to adult, beginners to experts." Dane said as he shook hands and offered pamphlets. The class had gone well—it was their first school group, and he'd been a little nervous, but the kids had taken to the classes with enthusiasm. There'd been no problems on the overnight, and they'd all had a good time telling stories around the campfire and bunking out. The class should have wrapped earlier today, but the kids had been so enthusiastic that he'd dragged the lessons out for a bit longer and then treated them to a round of paintball, and they'd been more than happy to take him up on it. It had been fun. He liked kids. He wouldn't mind doing more classes with children, he decided, and made a mental note to ask Brenna to schedule more of them. Maybe they could do a summer survival camp for kids. Brenna jogged up next to him. "Oh, good. You're back. Grant's been looking for you since yesterday." He tried not to groan. Grant was not his favorite person to see at the end of a class. The micromanager in his friend wanted a complete detail of how the class had gone, from what tasks they'd performed to what each person had worn. Grant wanted to take all the details that he could and record them, and then study and analyze in the hopes of improving the client experience. Of course, Colt thought he did it just to keep himself occupied. Dane wasn't sure. Either way, a long, intensive, detailed narration was not what he had in mind right now. He missed Miranda and wanted to go home and curl up in bed with her. "Grant? What about?" "Dunno," Brenna said, shrugging her shoulders and wrinkling her nose, as if the thought of Grant made her queasy. "He was upset about something. Insisted on talking to you as soon as you got back. If you want to run out, now's the time." He grinned at Brenna. She was like a kid sister to him and Colt. To Grant, she was a messy, disorganized, lackadaisical nemesis. She didn't care for Grant's micromanagement either, and would have been fired a long time ago if it wasn't for Colt and Dane. "All right," he said, and squinted up at the setting sun. "It's getting late. I'm going to head into town for the night. If he's looking for me, tell him I'll be back tomorrow morning for the next class." She winked at him, her green eyes laughing. "Got it. I will thoroughly enjoy telling him that." "I imagine you will." Dane laughed. Miranda's street was dark, but the lights were on in her bedroom, and he knocked at the front door softly. The wildflowers in his hand had seemed vibrant when he'd picked them back at the ranch, but they had wilted in the car on the drive over. Hopefully that wasn't a sign of how the evening would go. "It's open," she called. He stepped inside. The living room was dark and tidied—the boxes that had scattered the room were stacked neatly in the corner of the room. He couldn't tell where she'd unpacked more, however. Lights flickered in the bedroom, and he headed in that direction. The room was lit with dozens of flickering candles, the bed stripped of everything but a sheet. Miranda lay on the bed, completely naked except for a pair of red fuck-me pumps and a slick of lip gloss. His cock went immediately hard at the sight. "Welcome back," she said in a soft, throaty voice. "I missed you." Damn, he'd missed her, too. He watched her sit up, dark brown hair spilling over her shoulder, and felt as if he was going to lose control right then and there. The sultry way she was eyeing him did crazy things to his mind. "This is a nice surprise," he said, grinning, and presented the flowers. "Makes mine seem kind of pathetic in comparison." "That's okay," she said lightly, getting up from the bed and taking the flowers out of his hands. "I'll make up for it." Her hands were immediately on his shirt, tugging it over his head and then tossing it to the side. He leaned in to kiss her, but she gave him a teasing, coy smile. "My turn to lead tonight." He raised an eyebrow. "Oh?" Dane sucked in a breath when her hand went to his cock and stroked it through the fabric of his jeans. "Hard already," she purred. "Guess you did miss me." "Like crazy," he said, leaning in to kiss her. Again, she moved away, giving him a coquettish little frown. "My games tonight." "You want to be in charge?" At her sexy little nod, he felt a thrill flow through his body. "All right." She ran her hands over his skin as she undressed him, ripping his clothes away from his body. It was an incredible turn-on, and yet...there was something about Miranda's single-mindedness that was bothering him. When she ripped his boxers away, he grasped her hands and pulled her naked body against his. "Slow down. We have all night." She relaxed against him, her hands stroking over his skin. "Sorry," she said with a breathless smile. "I was getting carried away." "Oh?" "I have a surprise for you." There was a fierce glint in her eyes. His hands slid down, cupped her beautiful ass and pulled her against him. "I'm all ears." She wriggled out of his grasp and crossed the room to the side of the bed. Miranda reached into her nightstand and pulled out a pair of handcuffs. And she gave him an expectant look. Well, damn. His sweet little Miranda was taking control of her wilder side, and he liked it. Dane grinned. "Are those for me to use on you?" "After I get to play with you," she said coyly. "Volunteers get a special treat." "Then I'm definitely volunteering," he murmured huskily. With that, he extended a wrist out to her. She linked the handcuffs over one wrist, and trembled. So turned on she was shivering? The thought made him even harder, and he groaned, nuzzling into her thick, silky hair. She shuddered at his touch and led him to the bed, then handcuffed one wrist to one of the wooden posts of her bed. He lay on his back, testing his left wrist, which she'd cuffed to the left post on her bed; his arm stretched over his head, though not uncomfortably so. To his surprise, she took out a second set of handcuffs and reached for his other wrist, her eyes hot as she gazed at him. Did she expect him to flinch away? He offered her his wrist. "I'm all yours." She gave him a wry smile and locked his other cuffed wrist to the opposite post on the bed, then crawled over him. The brushing of her ass and breasts against his body was driving him wild, and he groaned low in his throat, his balls tight with pleasure. To his surprise, though, she continued off the bed and moved to the far side of the room. Another surprise? He strained against the cuffs as she rummaged in a drawer, trying to see what she'd pull out. A feather? Chocolate sauce? A dildo? Damn, he hoped not. He swung a lot of ways, but a chick using a dildo on him wasn't really his thing— Suddenly, she turned and he was blinded by a flash. "What the—" _Click. Click click click._ He stared up at Miranda. She held a camera, and she was taking pictures of him. Naked. And turned on. Cuffed to her bed. He jerked. He hadn't given permission for this, and this felt...invasive. "Miranda, what the fuck? Put the camera down." "No." Her voice was hard and cold. _Click click._ He stared at her. What the hell was her problem? "This isn't funny. Put the camera down." "You're right, it's not funny." She took a few more snaps and then changed the settings on the camera, glancing through the photos she'd just taken. "And now you know how it feels, don't you?" "What the fuck are you talking about?" He was starting to get mad now, jerking at the cuffs. Damn it, he'd trusted her to tie him up, and this was what she did? "Miranda, untie me—now. Now." "You know what I'm talking about," she cried, setting the camera aside and shrugging on a sundress and panties. Dread crawled through his stomach as he watched her dress. She...she wasn't leaving, was she? Why was she getting dressed? The uneasy feeling in his mind intensified. Even more upsetting was the look of anguish on her face. "Miranda—" "I thought you were different! I thought you had changed, and then I find out _nothing_ has changed." "Miranda, I don't understand—" "Why are the pictures still up? After all this time? Do you hate me? Is that it? Is that why you didn't tell your friends about me? Just having a laugh with slutty Miranda Hill, is that it?" Pictures? What pictures? The ones she'd just taken? She wasn't making sense. She knew he hadn't said anything? And wait—Slutty? "No! I—What are you talking about?" She straightened her dress and picked up the camera again, breasts heaving as if she couldn't get enough air. "You and I are through tonight, Dane Croft. Through once and for all. I was an idiot, because after we started talking again, I thought maybe you'd changed. You weren't the asshole that I thought you were, but I suppose it's my fault for being stupid enough to fall for a pretty face. Well, it's my fault no longer." She shook the camera. "I have these pictures of you, Dane. That's why I went on that camping trip. It wasn't to learn survival skills. It was to get compromising pictures of you and ruin your life the way you ruined mine. I got what I wanted, and you can expect these pictures to show up on the Internet in the next few days." She tapped her chin. "Or maybe I'll sell them to a tabloid. I haven't decided." "Miranda!" "Good-bye, Dane. You're never going to see me again." With that, she exited the room. He stared at her in shock. What the fuck was all that? Why was she taking pictures of him—naked, for fuck's sake—and declaring that she was never going to see him again? He was in her goddamn house. She had to be coming back. She _was_ coming back, wasn't she? Suddenly paranoid, he jerked at the cuffs, twisting his hands. Tight and unyielding. He couldn't slip his wrists through. Damn. Maybe if he jerked hard, he could break the bed. He didn't know what the fuck was going on—or why Miranda had suddenly gone psycho—but he intended to find out. _Juuuuust_ as soon as he got free. For the next few minutes, he strained hard, clenching his fists and jerking at the bedpost. It was no good—he couldn't get the right angle. Dammit. He heard the door in the living room and stiffened. Had she called in someone else? Was she inviting her friends to come and gawk? They wouldn't have much to see—his cock had pretty much shriveled at this point. But no—Miranda stumbled in a moment later, clutching the camera and crying as if her heart were breaking. It made his own gut give a miserable twist—God, why was she so very unhappy? His own anger at her disappeared at the sight of her misery. "I can't do it," she sobbed. "I can't do it. I know what it's like, and I can't do it to another person. Especially not you." "Do what?" he said, jerking at the cuffs again. He needed to get free. Then he could touch her, comfort her, figure out what in the hell was going on in her head. "Miranda, get me out of here—" "No," she said, wiping her eyes. "I'm not going to do it. With the pictures. Put them on the Internet." Her eyes were wild and glassy. "Even a jerk like you doesn't deserve that." "Deserve what? I don't even know what you're talking about." "Yes you do," she yelled at him, and looked as if she wanted to throw the camera at his head. "Quit playing dumb, Dane. You already ruined my life once. I won't let you ruin it again because I was stupid enough to fall in love with you. And I won't ruin yours either." She flung the camera down at the end of the bed. "I'll call someone at the ranch to come get you in a half hour. And don't come looking for me or call me and try to explain, because I don't want to hear it. We're done, you and I. This time for good." She wiped her cheeks and left. He stared after her, but she didn't come back. Maybe she wouldn't this time. What had she been going on about? Something with pictures and him ruining her life. It made no sense—he hadn't seen her since he last left Bluebonnet right after high school, and she'd been the one to turn him away, not the other way around. He didn't understand, and Miranda had been incomprehensible. She'd been totally heartbroken, too, and it clenched at his heart. What was so wrong that she wanted to hurt him to try and fix it? She said she'd fallen in love with him. He'd fallen for her, too—hell, he'd been in love with her for the past nine years and was just too damn dumb to see it. Whatever she'd felt, she needed to destroy it, he thought, and kicked the camera. It shot against the wall and shattered into several pieces. He wanted to go after her. Have her explain herself. Shake some sense into her. Hold her and stroke her hair until the tears went away, and fix her problems for her. He wasn't even mad anymore, just downright confused. Dane couldn't stay mad—not when she cried as if her heart was breaking. Hell, her sobs were breaking his heart. All he wanted was to comfort her. But he couldn't do anything, because she'd chosen to cut him out of her life in this bizarre manner. He was stuck until someone came to rescue him. And so he sat, and waited, and seethed with worry for Miranda. Someone knocked at the door a short time later. "Come in," he shouted, wishing he had something to cover up with. "I'm in the bedroom." Footsteps, and then a tall, lean figure leaned in the doorway of the bedroom. Cool eyes narrowed at him, and Colt rubbed his head. "'Preciate the offer, bro, but you ain't my type." "Very funny," Dane said, jerking at the handcuffs again. "Just get me down from here so I can find Miranda and paddle her ass until she tells me what's going on." The keys had been left in the nightstand, and it took mere seconds for Dane to be freed. He rubbed his wrists and then dressed without a word. When he put his shoes back on, he went over to the camera, and stomped it to bits, taking out some of his aggression on it. "Pics of your bad side?" He turned and scowled at his friend. "You know anything about Miranda Hill and something that ruined her life?" "Nope," Colt said. "You fuckin' her?" He narrowed his eyes at Colt. "She's my girlfriend. Was my girlfriend." Colt scowled. "Fucking a _client_?" "It's not like that. And it's not fucking. Quit saying that or I'm going to punch it out of your mouth. Hear me?" Colt scowled for a moment longer, then nodded. "Her prissy friend knows something 'bout this. She came by to yell at me a couple days ago." "Then let's pay her a visit," Dane said grimly. * * * "Why am I not surprised it's a beauty parlor?" Colt said with disgust as they pulled up in front of the bright pink sign and shop window painted with garishly bright flowers. "Shoulda known." Dane vaulted from the jeep as soon as it stopped, barely pausing to clear the curb. The sign in the window was off, but there was a light on inside. He banged on the door repeatedly. She had to be inside. Had to be. After two minutes of straight banging, he heard someone inside and then the rest of the lights flicked on. A familiar blonde glared out at him from the other side of the window. "I'm closed. Come back tomorrow." "I'm looking for Miranda," he shouted through the glass. "Not here. And even if she was, I wouldn't let her talk to the likes of you." She gave him a cool look and flicked the light off again, clearly intending to leave him standing out there. He banged on the door again, harder. After a moment, Beth Ann flicked the lights on again. "Don't make me call the cops." "We just want to talk," Colt said gruffly at his side, his gaze hot on the blonde. "'Bout Miranda." She scowled at both of them but hesitated. "I'm not leaving my salon open late if no one's getting their hair cut, understand?" "Fine," said Colt, and gave a crisp nod. Beth Ann opened the door and heaved an exasperated sigh. "Don't you two make me regret this." Dane walked into the shop after Colt. It was cute and bright and covered in bottles of all kinds of girly stuff. It smelled faintly floral, and he recognized the scent—Miranda's shampoo. Just the scent sent a bolt of unhappiness through him. "Where's Miranda?" "Probably halfway to Houston by now," Beth Ann said, and picked up the pink leopard cape. "Who am I cutting?" Colt removed his hat and sat in the chair, and Beth Ann gave a delicate snort. Colt's hair was already cut razor short and close to his scalp. She must have been a sadist, because she put that ugly pink cape on Colt and pumped the foot bar on the chair to adjust the height. Colt glared into the mirror. "Houston?" Dane said, leaning against the wall and crossing his arms tight over his chest. "Why the hell is she going to Houston?" Beth Ann started the clippers and began to run them over the back of Colt's neck. "Because she's moving there? You need to listen when a girl tells you something, Dane Croft. You're not God's gift to women." Dane frowned at her. "What are you talking about? Why is Miranda moving? She just moved in." In the chair, wearing the pink cape, Colt sat stone-faced, watching his reflection in the mirror. His eyes shifted a fraction, and Dane realized that it wasn't his reflection he was watching—it was the perfect blonde his gaze was focused on. Watching her like an eagle sights prey. He wondered if Beth Ann had any idea that Colt was watching her so carefully. But he didn't have time for this shit, and Beth Ann wasn't volunteering information about Miranda, which was why he was here, not to see Colt staring at the blonde with possessive eyes. "Well? Why is Miranda moving if she just bought a house?" "Oh, honey," Beth Ann said in a tone that was a little sorry and a lot mocking. "Bless your heart. She wasn't moving in, she was moving out. She couldn't wait to leave this darn town. Been waiting nine years to get out." "Nine years? Why?" "Because of those damn photos—" she snapped, then pursed her glossy lips. "Sorry. Language. But you know what I'm talking about." "No," he gritted. "I don't." She narrowed her blue eyes at him, then moved across the tiny salon to a laptop on a small desk. As she bent over, he watched Colt's expression narrow a bit more, as if his world had suddenly focused in on a soft pair of hips in a jean skirt. "This," Beth Ann said, moving to the side and showing him the screen. "This is why she couldn't wait to leave." He moved forward and stared. It was a horrible website, with an ugly background and noisy graphics on it. The URL read "Boobs of Bluebonnet" and he stared at a picture of Miranda's perky breasts, that beauty mark under her left one staring back at him. Some asshole had his hand down her pants and his other underneath her breasts, as if plumping them for the camera. Even worse, her head was tilted back in ecstasy. "Who's the dick?" he growled, the urge to beat the shit out of someone rising. His fists clenched, hard. Beth Ann frowned at him and scrolled the website down to the next photo. This one clearly showed the face of the man as Miranda knelt before him, with his hand twisted in her hair as if she were about to suck him off. "You are, you stupid fool. And you ruined her life." * * *** When they left the salon, Dane sat in the car, numb. Colt drove, every once in a while scratching at his neck for phantom hairs left from the shave. Dane didn't know what to think anymore. Miranda thought he'd ruined her life. She thought he'd taken the photos. Thought he was getting some sort of psycho revenge on her when he'd left all those years ago. Beth Ann had spilled the whole story, though she'd clearly been reluctant to divulge her best friend's secrets. She was only telling him, she said, what anyone in town would tell him. How Miranda's mom had had a nervous breakdown and Miranda had had to run the store until she recovered. The rumors. The nickname. The photos had followed her for the last nine years. And all this time he'd never known. No wonder her mom had freaked when he'd entered the store. No wonder Miranda had taken his picture and said she'd wanted to ruin his life. Hell, he didn't blame her. He knew what it was like in small towns, and Bluebonnet was one of the smallest. You knew everyone, and everyone knew _everything_ about you. And everyone knew Miranda's breasts intimately. God, poor Miranda. She'd been so strong to quietly suffer all these years and put up with shit for her mother's sake. Beth Ann had explained that she'd gotten her master's from nearby Sam Houston State University and had applied for jobs, eventually landing a plush one at a big corporation in downtown Houston. Beth Ann wouldn't say where. He didn't blame her...and at the same time, he wanted to shake the news out of her. "Will she be back?" he'd asked, feeling like the world had just fallen out from under his feet. "Don't know," said Beth Ann. "I have her number, but she's turned her phone off for the move. She said she'd call me in a few weeks, when she's settled. Wants to get a fresh start first." Her hurt was obvious, and she gave Dane an accusatory look. She'd been closed out of Miranda's life, too, and it was no thanks to him. Judging from the look on her face, Beth Ann wouldn't be forgetting that anytime soon. He didn't care. All he could see was Miranda's tearful face, sobbing as she left the camera with him. Even though she'd wanted revenge...she hadn't been able to do it. She'd said she'd fallen in love with him. It was like a knife twisting in his gut. He'd loved her all these years and she'd thought the worst of him. He thought back—the pictures must have happened at the after-graduation party at Chad's house. He'd never known there was a camera in the closet—he'd been too caught up in the beautiful girl in his arms and the fact that he was finally, finally getting to touch her. He'd had a call that day from the NHL, and between that and Miranda, he just wasn't thinking about anything else. His head had been full of hockey hopes and dreams and he'd been cocky and arrogant. And he hadn't realized. Rage pulsed through him and he slammed his fist into the passenger side of the car. "Fuck!" "Tryin' to deploy my air bag?" Colt asked casually. "I want to punch the fuck out of that asshole who did this to her," Dane snarled. "I want to slam his face into the ground and make him realize how much he hurt her." He'd hurt her...and Dane couldn't fix it. He wanted to fix it, and didn't know how. Colt gave him a long look, and then turned the car back around. "Where are we going?" "To the bar. We're gonna ask some friends how our high school buddies are doing." Dane nodded, rubbing his knuckles, contemplating another jab to the panel, or maybe putting his fist through the glass. Her pain ate away at him, gnawed in his belly and he couldn't do a damn thing about it. He'd somehow caused this. Some jackass had ruined her life, and he'd been completely clueless about the entire thing. Colt glanced over. "Why didn't you say something? 'Bout you and Miranda?" His jaw tensed. Dane stared out the window, his mood black. "I was going to tell Grant first. Get the bad shit over with. Then I was going to let you know." There was a long moment of silence in the cab. Then Colt spoke again. "You know I don't care who you're fucking around with as long you keep it on the DL." Unusually chatty for Colt. Dane knew his friend was pissed that he hadn't been looped in. When he got mad, he got talky. "Yeah, and as soon as Grant finds out I've been sleeping with a client, he'll blow his lid." "He's concerned about the business, that's all. You planning on fucking all our clients?" Dane scowled at his friend. "Exactly my point. So this ain't a big deal. Just tell the man and be done with it." He should have. Of course, now Miranda was gone, and it was too late. Bluebonnet boasted exactly one place that served alcohol. It was a Tex-Mex restaurant in a converted house, but it had a bar, and that was good enough for most of the residents of Bluebonnet. After hours, the men in town showed up to drink a few beers, watch sports on the TV on the wall, or rack up a few balls in the town's only pool table. Dane walked up to the bar, ordered a longneck. He chatted with the bartender for a few minutes. The man—who'd likely been tending the same bar since Dane was last in town nine years ago—was all too eager to hear stories of Dane's time in the NHL. He told a few stories, had the men at the bar smiling, and then eased into other topics. "Seems like everyone still lives here in town." "Yup. Seems like." "Chad Mickleson still live around here?" "Yup." Dane nodded, took a sip of his beer, tried to act casual. He had a guess as to who had taken those pictures, and he wanted to talk to the guy. "You know how I can get ahold of him?" "Sure do." He gave vague directions to a nearby car garage and Dane made a mental note to visit there in the morning. "What about Miranda Hill?" He asked casually, almost afraid of what he'd hear. "You know anything about her?" "Ol' boobs? Yeah, She's legendary around here," the man said, grinning. "Turned into a hot little librarian. Why? You planning on tapping that?" "That's my girlfriend," he growled. The conversation ceased. "You know who took those pictures of her?" "Well," the bartender said slowly. "Kinda thought you did." Minutes passed like hours, and Dane tossed and turned in his bunk. His own house was a small cabin on the edge of the Daughtry Ranch, and he normally liked it just fine, but tonight it was too quiet. He missed Miranda, her warm breath tickling his chest as she slept, the soft curve of her body against his. How quickly he'd gotten used to having her in his life. How hollow he felt right now since she'd run away from him. He was filled with the same helpless rage he always felt when thinking about it. When the sun came up, he was in his jeep and heading to the garage, his mind full of grim determination and Miranda's sad hopelessness. The directions the bartender had given him were dead on, and he pulled in. A mechanic came out to greet him, wiping his hands. "Need an oil change?" "I'm looking for Chad Mickleson," Dane said. "He work here?" "Yup, he's just inside," the man said, then broke into a wide grin. "Hey, aren't you—" Shit. "Yes." "I'll be damned," the man said, following him in. "Hey, Len! You'll never guess who just drove up! The local legend himself." Dane ignored him, striding into the garage, looking for a face he only vaguely remembered. Sandy brown hair and big eyebrows—that was all he remembered of the guy. One of the mechanics turned around and boom, there he was. Dane's hands instantly clenched to fists—if he'd have had his hockey gloves on, he'd have dropped them. The other man's eyes lit up. "Holy shit. Dane Croft. How are you, man?" Dane punched him square in the jaw. The man went down like a light and dropped to the floor of the garage. Someone yelled. "You and I have a lot of talking to do," Dane said in a low, dangerous voice. "Now get up." # SEVENTEEN One Month Later Miranda stared at her Outlook calendar in dismay. She clicked on the meeting, then buzzed her secretary. "Shirley, could you come in please?" The woman—easily three decades older than Miranda—hefted in and gave her a patient smile. "What can I help you with?" Miranda pointed at her computer monitor. "Why do I have a three-hour conference call on Saturday?" "Oh, that." Shirley picked a piece of lint from her black cardigan. "You have a meeting with the CFO of the fabrication division." "On a Saturday?" Shirley blinked. "Your Friday and your Monday are booked solid." They were? Miranda clicked on the calendar again and swore under her breath. Sure enough. They hadn't even left her enough time to run to the restroom or catch lunch. She'd seen a few women wear their headsets into the bathroom stalls and had thought they were absentminded. Perhaps they'd simply had too many meetings. "Thanks, Shirley," Miranda said, feeling a little bit sour. Life at HGI was definitely...fast-paced. They loved projects, and collaborations, and meetings. They _loved_ meetings. She'd half joked with someone that they needed meetings just to determine if they needed meetings or not, but no one had laughed. Probably too close to the truth. She stared out the window of her tiny corner office and down the busy streets of downtown Houston. It was just a big change, she told herself for the hundredth time. Once she settled in, she'd start to like her job. Maybe she'd even appreciate the constant stream of meetings. Eventually. With a sigh, she turned and swiveled in her ergonomic chair. A meeting invite flashed onto her screen and she ignored it, feeling the sudden urge to rebel. Loosening one of the buttons of her severe black-jacketed pantsuit, she stared at the only picture on her desk—of her and Beth Ann holding up beers while boating at the lake. She picked up the picture and stared at it for a good long minute. The setting sun on the water made her think of that week in the woods. And that week in the woods reminded her of Dane, and camping, and that last night at her campsite, where it had been just the two of them in her tiny lean-to. That evening had been so perfect; they'd had sex, laughed, cuddled, and they hadn't worried about others discovering them. It had been just the two of them in a small slice of paradise. She'd felt so at home, like she'd waited nine years for a missing part of her to come back. Which was stupid, really. But a sudden pang of homesickness shot through her and she picked up the phone and began to dial. "California Dreamin'," a cheerful voice answered. "This is Beth Ann." "Hey," Miranda said softly. "It's me." A pause, then a girlish squeal on the other end. "Oh my God! Miranda! Hang on, let me set Bessie Roberson under the dryer." The phone clanged against something on the other end of the line and Miranda heard a soft murmur of conversation, and then the buzz of the dryer in the background. The phone picked up again and Beth Ann returned. "Dang, girl! I've missed you. How are you?" "I'm good." "How's the new job? Your mom and I have been wondering how it's going over there." "It's good," she said, then sighed. "Just really different." "Uh-oh," Beth Ann said in a teasing voice. "Different as in good, or different as in 'Oh mercy, what have I gotten myself into?'" "It's hard to say," Miranda confessed. "It's everything I thought it would be—corner office, amazing benefits, and everyone here is so driven..." "But?" "But they're driving me crazy," Miranda confessed. "I have four more meetings today. Seven tomorrow and one on Saturday. Then I start over again on Monday. I'm starting to see PowerPoint slides when I close my eyes at night." Beth Ann laughed. "I'd laugh, too, except it's too pathetic," Miranda said caustically. "It's just...I don't know. It's not what I expected on that point. I thought I'd do more stuff myself. Instead, I'm just advising everyone else and making sure projects stay on task, and then turning around and reporting to the big guys." "You miss your library?" "I do," Miranda said softly, thinking of the slightly dusty smell, the silence-but-not-silence, the rows and rows of books. "I didn't realize how much control over things I had there until I got this big fancy job...and now it feels like I have no control." After a minute, she blurted, "And, well, it's not the only thing I miss." "You miss me, too, right?" Beth Ann said with a grin. "And your mom? We miss you like crazy, honey, but we're proud of you." "I miss you guys," Miranda admitted. Then she added, "And—" "You miss Dane, don'tcha?" Miranda swallowed, the knot in her throat making it hard to speak. "You think I'm stupid, don't you?" Beth Ann chuckled. "Let me tell you a little story, Miranda Jane. Seems I had a couple of visitors that night you left town." "Oh?" "Yep. One big pissed-off ex-hockey player and his buddy showed up, demanding to know where you went. And when I told him you'd gone and skipped out of town, well...you could have knocked him over with a feather." Miranda smiled at the visual. "He couldn't imagine why you'd lied to him, so I tried explaining to him that he'd ruined your life, and you know what he told me?" Her throat ached, it was so tight. "What?" "The boy didn't know what I was talking about," she said. "I had to pull up the pictures to show him because he didn't believe me. And when I did show him, he asked me who the guy was in the photos!" A cold chill shivered over Miranda. "What do you mean? How did he not know?" "I know! That's what I said. When I told him that it was him, I thought he was going to fall over in shock. And then he got real mad, Miranda. Real mad. I've never seen a man get so angry. He didn't put his fist in the wall or anything, but he looked like someone had just shot his dog or something." Dane had been...angry? And confused? He'd seemed so confused at her anger the last time she'd seen him, too, when he'd been handcuffed to her bed and her heart had been breaking with every moment. She'd thought he'd been playing stupid to defuse her anger, but...what if he was telling the truth? It didn't make sense...did it? "So you aren't going to believe what happens next, right? Tommi Jo told me that he went to Maya Loco asking how he could find Chad Mickleson. And when someone brought up your name and the boobs thing, he shut them down, real polite-like, and told them you were his girlfriend and not to be talking about you." Her heart gave a funny tingle. "He...did?" "Oh, it gets better, girl," Beth Ann said, pleasure in her voice. "So...the next morning, he shows up at the garage where Chad's working, walks up to the man, and punches him flat." She gasped. "What?" "It's true! Said they needed to have some kind of talk. I don't know what about, but I hear from Tommi Jo that Dane was plenty, plenty mad at Chad and had to be talked down a few times." "Did someone call the cops?" "Not from what I heard. Tommi Jo said that the guys talked for a long time, and then Dane came out and signed some autographs and gave some of the guys invites to a free class or two, and that was that." So strange. "I don't understand." "Me either," Beth Ann said sweetly. "But I do know your pictures aren't on the Internet anymore. I checked. It's gone." "Gone?" Unbelievable. She grabbed her smart phone and punched in the URL. Sure enough, it didn't come up. Was it possible? Hope and joy surged in her chest. Dane had realized it was hurting her and had taken the pictures down? "They're not there anymore?" "Nope. Tommi Jo thinks Dane was defending your reputation. She said your name came up a few times, though she couldn't hear everything that was said." Damn Tommi Jo. Why hadn't she gotten more than just pieces of the conversation? Her curiosity burned in her chest. "I wonder what they were saying." "Don't know. But I know what I do know. Someone mentioned that you had a big collection of teen books at your library that you'd bought out of your own pocket, because the city wouldn't give you the money for that vampire stuff, right? Well, Dane overheard that at your mom's store, and I heard he went down to the library the next day and donated ten thousand dollars of his own money. Isn't that something?" It was dizzying. Why had he done that? Why was he so generous on a pet project of hers at an old job that she wouldn't be returning to? She thought of all the books that would buy. The teens in town would flock to the library for more than just the Internet. Oh, she'd love that. The big-city libraries carried so many books that she couldn't possibly compete, but with that...she could start a reading club, just for teens, maybe get a manga section... "Wait," she said, something registering in her brain. "Did you say he heard that at my mom's store?" She cringed at the thought. "Mom hates Dane. Even the mention of his name will send her into hysterics." "You should ask your mom about that," Beth Ann said slyly. "I hear Dane's been at her store almost every day lately." They chatted for a few more minutes about inane things, and then promised to check in with each other again very soon. As soon as Miranda hung up the phone, Shirley was on the line, buzzing in. "Miranda? They're waiting for you in the Indigo Conference Room." "Be right there," Miranda said. Then she added, "And I need you to clear my Saturday. I'm going out of town." "Out of town?" Shirley repeated disapprovingly. "But the meeting with—" "Can wait," she said firmly. "I'm going back to Bluebonnet for some unfinished business." When she drove into town, nothing looked different. She'd been away for a whole month and it was like she'd never left. Well, one thing had changed—there was a big banner on the gazebo in the town square for the upcoming Hill Country Spring Festival, but they trotted that thing out every year. So why was it that when she looked at the too-small, overly intrusive, Podunk town she grew up in, it no longer filled her with helpless anger? Why did it fill her with nostalgia instead? Surely she didn't miss Bluebonnet, of all places. She drove down Main Street—her mom's shop had several cars in front of it, which meant she was busy. Miranda opted to head to her library instead, see how things were going. Because Bluebonnet was a small town, the library was sandwiched into City Hall, squished between the Water Department and the police station. When she walked in, the smell of the place made her heart flutter with longing. The faint scent of dust and old paper made her senses tingle, and a possessive surge came over her. This was _her_ library. She'd missed it. She moved to the new releases section and ran a hand over the spines, looking for new books purchased with Dane's donation. Nothing—they were all books she'd purchased. She glanced over at the checkout counter—it was stacked high with books waiting to be returned to the shelves. Perhaps old Mrs. Murellen, her replacement, was running a bit behind. Well, she had a little time to kill. Miranda picked up a few books and began to shelve them. As she did, she noticed that when she put them on the shelves, even more books seemed out of place, and she continued to put books in their proper homes, frowning as she did. She'd never let her library get so sloppy when it had been hers. _You didn't want it, though, did you?_ she reminded herself. _You wanted to be a corporate big shot, and now you are._ Right. She shelved the last book and resisted the urge to dust and straighten. That was someone else's job now. She turned the corner and nearly ran into a student. "Oh, I'm sorry," Miranda said with a smile, recognizing Trisha Ellis. "I didn't see you there." The girl's face widened into a smile, and for a moment, Miranda thought she was going to hug her. "Ms. Hill—I am so glad to see you. I can't find the teen books!" Inwardly, Miranda groaned. Not again. "Did you check in our normal hiding place under the nine hundreds?" "They're not there, and the fake slipcovers you made are gone," she said, her expression crushed. "I think they took them off the shelves again." It had been an ongoing battle with the city council, who thought the books that teenagers were reading were trash. They didn't seem to understand how wonderful it was that they were reading at all, so Miranda had purchased her own small library of popular teen novels and shelved them with fake jackets that a few of the students had helped her create. They were the most popular section in the library. "I'll check with Mrs. Murellen," Miranda said, heading toward the counter. Trisha trotted on her heels close behind. There was no one at the counter, books stacked everywhere, the return bin overflowing. Trisha immediately started to pick through the return bin, looking for missed favorites. Miranda slipped behind the counter and went to the back office, knocking softly. No response. She opened the door. Mrs. Murellen sat behind her desk, chin propped on a hand, snoring. "Mrs. Murellen," Miranda said, her voice sharp. "Wake up." The older woman snorted awake, and peered at Miranda. "Oh my goodness. Did you come back for your job?" "No—" Mrs. Murellen looked sad. "Oh." "Someone here is looking for the teen reading books. Where did you move them to?" "I took them off the shelves," Mrs. Murellen said, adjusting her glasses as she stood up. "Did you know that they were about vampires? Sexy vampires? Terrible stuff." "They're perfectly fine," Miranda explained, going through the shelf of books in the tiny office. Sure enough, it was crammed full of P. C. Cast, Richelle Mead, and Stephenie Meyer, as well as anything that had a teenager on the cover. She sighed and grabbed several of them off the shelf. "Those are going to be removed from circulation—" "No they're not," Miranda said firmly, and handed them to an excitedly waiting Trisha. "I think we should talk." Miranda spent the next hour straightening up at the library, doing her best not to lecture Mrs. Murellen, and reshelving the teen literature. She couldn't be mad—it was obvious Mrs. Murellen didn't want to be Bluebonnet's only librarian; she'd offered Miranda her job back three times in a half hour. When that didn't work, she tried to get Miranda's advice on what to spend Dane's donation on—she had no idea what books to buy, and didn't know where to start. Miranda came up with a list of bestsellers that she'd had her eye on and wrote Mrs. Murellen a lengthy shopping detail, as well as a to-do list of chores that she was neglecting. Miranda shouldn't have come back. Her mind was now filled with treacherous ideas. Ideas of returning and running things with a firmer hand. Ideas of how to spend the money Dane had donated. Ideas of driving over to the Daughtry Ranch and throwing Dane down on the floor, apologizing, and then making love to him until the sun came up. But she'd burned that bridge. Across town, Beth Ann picked up her phone and dialed. "Wilderness Survival Expeditions," a gruff voice said. Great, she'd gotten Colt. "This is Beth Ann. Let me talk to Dane." "He's out." Short, abrupt. Why was she not surprised? The man acted like it was a crime to string more than two words together. "Why is he out?" she persisted. "Class." "Yeah, well, you'd better go get him," she said irritably. The man crawled under her skin way too fast. "Because you'll never guess who just drove back into town. And I think he'll want to see her." Colt found Dane surrounded by a group of students. They crouched in a clearing, Dane at the center of the group. Ignoring their surprised looks and Dane's puzzled one, he quickly told Dane the details of Beth Ann's phone call. "She's here?" Dane stared at Colt, disbelieving. He dropped the fire-making implements in his hands. "You're sure?" "No, I lied." Colt turned around to leave. Dane lurched forward over the group and grabbed Colt by the shoulders. He turned and looked his buddy in the eyes. "Miranda's back?" "That's what Beth Ann said," Colt drawled. "And that's why my ass is out here and not playing Xbox." Dane ran a hand down his face and then looked over at Colt. He glanced at the waiting students, then back at Colt. "I'm gonna need your help." "All ears, buddy." He couldn't resist the grin that crossed his face at Dane's hopeful expression. He began to pace. Colt crossed his arms and leaned on a nearby tree, watching him. "I need a way to say that I'm sorry." "Didn't do anything wrong," Colt pointed out. "No, I know. I'm not apologizing for me. I'm apologizing for the situation." He paused, put his hands on his hips and stared up at the sky, thoughtfully. "Needs to be a big gesture." "She won't care if it's big or not," Colt felt he had to point out. "Yeah, but I care," Dane said. He paced for a moment, then snapped his fingers and dashed off into the woods. Colt looked back at the confused students, then at his retreating friend. "Class dismissed. Brenna will be out here in a bit to lead you guys home," he said, then dashed off after Dane with a grin. Whatever he had in mind, it ought to be interesting. # EIGHTEEN Miranda walked into her mother's store, and for the second time since returning to Bluebonnet that day, she got that weird feeling of déjà vu. It was like she'd left and returned to a town that was the same...and yet not. Her mother's cluttered store, exactly the same for all twenty-seven years of Miranda's life, was different. Not much, but enough that it bothered her as she looked around. Miranda stared at the overflowing shelves and tried to figure out what it was. After a moment, it hit her. None of the heavy shelves were leaning anymore. They had all been repaired, the warped wood of each slat replaced, the contents dusted. She glanced outside and sure enough, the crooked sign she'd been so used to seeing had been repaired as well. Her mother finished ringing up some customers, and when they left, Miranda stepped forward. Tanya Hill gasped with delight at her daughter and flung her arms open, and they hugged for a long, long minute. "How's it going, Mom?" she asked with a smile. Her mother looked happy—healthy. Smiling. God, that was so good to see. She'd been so worried that her mother was going to have another one of her depressive episodes while Miranda was gone, and then she'd have to return to town. A little sad part of her twinged at that. Surely it wasn't disappointment? That was so wrong of her. She immediately shut down the thought. Her mother didn't need her here, babysitting her in town, and she was finally free. Miranda should have been thrilled, and she'd hated that the first thing that came to mind was hurt. Didn't her mother miss her? "I'm good, I'm good," her mother gushed. "I went to an estate sale and picked up all kinds of things for cheap, Miranda. You wouldn't believe the deals I got!" As her mother went on about the sale, showing her new items in the store, Miranda couldn't help but make a mental note of all the small changes. "Did you hire a carpenter, Mama? I saw the sign was fixed." Her mother beamed. "No. That nice Dane Croft came by and fixed it for me." The world spun. She never thought she'd hear her mother say _nice_ and _Dane_ in the same sentence. "What?" she said weakly. "You hate him." "I did hate him," her mother said proudly. "But that was before he got that horrible Chad Mickleson to take down those pictures of you." "What?" "Yes," her mother said solemnly. "He punched Chad right in the face in front of everyone, and then made him go home and take down those pictures while he watched." "But...I thought..." She felt weak, and collapsed onto an antique wooden stool nearby. Beth Ann had told her that the pictures were down, but she hadn't realized that it was because Dane had threatened someone _else_ and forced them to take it down. She thought he'd finally had a change of heart. "I thought Dane put those pictures up." "So did I," her mother said with a sniff. "All this time it was that horrible Chad. You know I never liked him. Shifty eyes. Anyway, after Dane had the pictures taken down, he came over here and apologized to me. Said he'd been unfair when we'd talked on the phone all those years ago, and he wanted to make it up to me. And he offered to fix my sign." Her head wouldn't stop spinning. Miranda pressed a hand to her forehead, unable to comprehend. "I'm sorry, what? What calls are you talking about?" "Back when he left for the NHL," her mother said patiently. "He kept calling for you and I wouldn't let him talk to you. We got into a nasty argument. That was when I had my nervous breakdown." "Dane called for me back then?" she said weakly, surprised. She'd never known. "Why didn't you tell me?" "Why, darling, I didn't want to upset you. He was an awful boy back then, but he's turned into a nice man now." Oh _God._ Not only had Dane been innocent of the picture-taking, but he'd called and asked for her after he'd left for the NHL? He wasn't the bastard she'd made him out to be? Miranda remembered his confusion that evening a month ago, as he'd stared up at her, handcuffed to her bed. Totally betrayed. He hadn't understood why she'd been so upset. Because he'd _truly_ had no idea. God, she was going to throw up. She clutched her stomach, horrified. Dane...hadn't been the one all those years? Her revenge? Her burning hatred? Directed at the wrong person? And...oh God. Dane had really liked her? Oh my _God_. And she'd ruined it. Acted like a crazy woman, screaming and crying at him. Handcuffed him to the bed and left him there. Taken naked pictures of him. Taken what they'd had and stomped it into the ground. "Mom," she said softly. "Are you all right, honey?" "I think I'm going to be sick." She was going to pass out. She really was. She bent over and put her head between her knees, breathing hard. Horror swirled in her stomach, a hard knot that threatened to make her puke. Her mother patted her back. "I'll get you some water, dear." Miranda didn't move. Maybe she could curl up right here and die. She'd ruined one of the best things that had ever happened to her. For nine years she'd obsessed over Dane Croft, and when she'd gone and fallen in love with the man and he'd seemed to care for her back...she'd destroyed it all just to get revenge. And why? Just because he hadn't told his buddies about her? She'd told him _not_ to when they'd gotten together. Miranda buried her face in her hands and moaned. She'd been so stupid. A horn honked outside. Once, twice, three times. She glanced up, but didn't see anything outside the cluttered shop windows. Her mother went to the window, and then covered her mouth, smothering a laugh. "You had better come see this," she declared. "I don't care, Mama," she said weakly, lost in thought. God, how could she have been so cruel to him? "I really think you should, Miranda Jane Hill." Curious despite her nauseated misery, Miranda pulled herself from the chair and followed her mother over to the front door of the shop. There, coming down the street, was a naked man. A very, very naked man. He wore nothing—even his feet were bare, and his body was corded with muscle. She could see tan lines on his arms and collarbone from a shirt, and she automatically looked south for more tan lines. His hands were in front of his privates, and he was holding something white and round there. She squinted, but the window was dirty and cluttered, and it was hard to see. Two people stood behind the man, and as he strode down the sidewalk, she noticed that cars were stopping to honk. The man was oblivious, striding forward with purpose in his step, ignoring the photographer that hovered a few feet away, rapidly taking pictures with an oversized camera and keeping just a few paces in front of the man as he walked. _What an idiot,_ she thought, her hand going to the high collar of her shirt even as she admired his body. Whatever this stupid gesture was for, he was never going to be able to live down the pictures. She knew that full well... Then she recognized what the white thing in front of his privates was—a hockey helmet. She swallowed hard, and moved to the front door, scarcely able to breathe. And she went out in the street and stared. And _stared_. From behind Dane, Beth Ann giggled. The man standing next to her didn't crack a smile, but it didn't matter—Miranda barely glanced at him. Her gaze went to the bronzed body of Dane Croft, completely naked and heading in her direction. Crowds of people had started to come out of the shops, flooding onto Main Street and whispering. A few women catcalled at his bare ass. "Dane," Miranda hissed as he walked down the sidewalk and moved to stand in front of her. "What the hell are you doing?" He grinned at her, her heart flipping at the sight of those flashing white teeth. The photographer clicked away, but Dane seemed oblivious. "I wanted to talk to you before you run away again, and I figured this might be the best way to do it." She stared around nervously as even more people flooded into the street to watch. Even the cars nearby slowed down and parked to watch the show. And God, he was _really_ naked. "Dane—" "Let me speak, Miranda," he said softly. "I have a lot to say." She swallowed the knot in her throat. "All right." Dane's face grew serious and he stared down at her, his green eyes thoughtful. "Miranda, I came out here today, naked," he said, pausing to look at the audience, and then shifted on his feet and turned back to her, "to tell you that I was an idiot in high school." "I don't—" "I'm not done," he said with a grin. "Just getting started." She raised a brow at him and crossed her arms over her chest, waiting. "I," he repeated loudly, "was an idiot. When I got the offer to play for the NHL, I forgot about everything else. I'd trained for the past ten years to get there, driving back and forth to Houston every morning before dawn to get a few hours of practice in before school, and practice after school. I never imagined that when I was in high school, I'd meet a girl with pretty brown hair and smiling eyes who would make me think that there was anything in my life other than hockey. But when the NHL called, I went." He tilted his head at her. "And it never occurred to me until just now that I never said good-bye to that pretty girl. I just up and left." She said nothing, stone-faced. "I do remember that I tried calling her over and over again when the initial excitement of the NHL wore off and I missed her," Dane said slowly, his eyes on her. "But by then, her mama didn't like me much and I couldn't figure out why she wouldn't let her come to the phone. So I stopped calling and went on with my life. And, well, like I said, I was an idiot. I let myself get carried away with my own celebrity, because I was a legend in my own mind. I played hard, and partied hard, and didn't care about anything until I woke up one day and everything was gone." He gave a small shrug of his shoulders. "Just like that, I had nothing left, all because I didn't listen to anyone. But it was a good thing, because it made me strong. When I lost everything, I had to rebuild myself into a different person. One who could stand on his own. One who didn't have to have hockey or celebrity to have meaning in his life. And it brought me back here to the pretty girl with the long brown hair that I'd never stopped thinking about." His green gaze grew tender and he looked as if he'd like to reach out and touch her, but stopped himself. Her breath caught in her throat. "And when I saw that girl again, I knew I had to have her. So when she flirted with me, I flirted back. And before I knew it, we were seeing a lot of each other, and I couldn't figure out why this lovely, gorgeous girl was keeping secrets. And then, one day, I found out. And we broke up." Her cheeks flushed, remembering that ugly night. How she'd screamed at him and sobbed, while he'd stared up at her, betrayed and uncomprehending. It wasn't her proudest moment. "And I didn't understand at first," he said softly, so low that only she could hear it. "How one thing that was so small could ruin someone's life. And when I did understand it, I thought to myself, how is this girl ever going to forgive me?" "But it wasn't you, was it?" she said softly, hanging her head in shame. "I was wrong. I should have asked—" "Hush. You're ruining my story," he said with a louder drawl. "So anyhow, I thought to myself, what do I want most in this world?" She looked back up at him sharply. "Is it a business with my friends, or hockey, or something else? And I realized it was something else." His gaze softened as he looked at her, a curve of a smile playing at his mouth. "A something else with pretty brown eyes and a killer pair of"—he coughed—"heels." Behind him, the crowd laughed. "And I had to figure out how to show this smart, funny, wonderful, strong woman that she's the most important thing in my life. So I quit my job and abandoned a bunch of clients back in the woods." His grin grew sheepish. "Pretty sure we're going to have a bunch of angry students asking for refunds once they find their way back to the ranch." She gasped. "And I took off all my clothes and went to the paper," he said, grinning. "And told them that I was going to march down the street naked so I could know what it felt like to be so exposed to everyone." Tears pricked in her eyes. This stupid, beautiful man. "And I'm pretty sure there's a cop somewhere back there, waiting to arrest my ass," Dane said wryly. "But I want to finish my apology first." "What are you apologizing for? It wasn't you who took the photos—" To her surprise, he handed her the helmet and dropped to his knee, stark naked, in the middle of the street. She blushed as she realized just how naked he was, and barely noticed that Beth Ann stepped forward and handed him something. "Miss Miranda Jane Hill," he said loudly. "I am very sorry that I fell in love with you back when we were both eighteen years old and it took me nine years of being an idiot to figure it out. I still love you and want to be with you. Will you marry me?" With that, he flipped open a ring box and showed her the diamond inside. Her eyes widened and she stared down at it, half horrified, half shocked. The entire street was deadly silent. "Mommy, I can see that man's peepee," said a child nearby. A strangled giggle erupted from Miranda's throat, and she smothered it with her free hand. He knelt there, staring earnestly up at her, so beautiful and serious. He understood. He understood what it was like—of course he did. He'd been a tabloid mainstay back when he was famous. Why had she been too wrapped up in herself to realize that? And now he knelt here naked, in front of her, in front of the whole town, to put them on an equal playing field. He couldn't make it right for her, but he could put himself in an equally embarrassing situation, just so she knew that he knew how it felt. A tear slid down her cheek. Dane looked utterly crushed at the sight. "Damn, Miranda. I didn't mean to make you cry." He began to get up, and she rushed forward, flinging her arms around his neck. "You're such an idiot," she sobbed. "I know," he said, patting her hair. "I'm an idiot, too." "I know," he said, and she could hear the chuckle in his voice. "Is that a yes?" "Yes," she said, and the town square erupted in cheers. # NINETEEN In the spirit of things, the cops let Dane off with a warning, a towel was quickly located, and Beth Ann fussed over Miranda and Dane as Colt glowered at her. Miranda's mother hugged them both, whispered into Miranda's ear that she and Dane needed some alone time, and took the opportunity to usher onlookers into her shop. "My car's nearby," Miranda told Dane breathlessly, clutching his hand in hers. "Want to go somewhere private?" He gave her a smoldering look. "Absolutely. But my car's nearby, too, and my clothes are in it. I can drive." "Your car, then. We can't go to my place, though. My house is being rented out," she said with a grimace. "We'll go to my place." She looked at him in surprise. "At the ranch?" He'd never invited her back before. "Are you sure?" "One hundred percent," he said and leaned in and gave her a hard, quick kiss. "I'm quitting anyhow. Grant is going to have a fit when he finds out I just abandoned a class in the woods to come here. If I can't be with you, I'm going to find something else to do. Maybe I'll teach hockey." She stared at him in surprise. "You hate hockey now." "It reminds me of a part of my life when I was too stupid and full of myself to think about anyone else. But it's something I need to face at some point." They turned down the highway and sat in silence. Miranda twisted the ring on her finger. It felt so alien, but in a good way. She glanced over at Dane and wondered if he was regretting his impulsive proposal. He'd grown so silent. She chewed on her lip, thinking. He glanced over at her and pulled in down a gravel path into the woods. "You still have boxes?" Her brows furrowed. "I guess so. Why?" "We'll need them if I'm moving to Houston to be with you." Her jaw dropped as he got out of the car and then jogged around it to open her door. As she got out of the car, she stared up at him. "You want to move to be with me?" He gave her an odd look. "Miranda, I just told you that I love you and I want to marry you. One of us is going to have to move, and I wouldn't ask you to give up your career." A knot formed in her throat. "You wouldn't?" "Of course not." She gave him a weak smile. "What if I kind of hate my career?" "What do you mean?" He grabbed her and swung her into his arms, and she clung to his neck. "Don't you like your job?" She gave a wry laugh. "I think I like the concept of it more than the actual job. I bet you think that's stupid, right?" His mouth tugged up on one side. "Actually, that sounds a lot like me back when I was in the NHL." They approached his small cabin and she saw a note stuck to the door, pinned there by a hunting knife. She leaned forward in his arms and grabbed the note, tearing it down. "'Dane,'" she read, "'if you think you can quit on us, you're a bigger jackass than we thought. See you tomorrow.' It's signed by Grant." He grunted, and then a slow smile spread over his face. "Guess I still have a job." She was happy for him. He loved what he did, and he was good at it. Miranda stared at the paper, then back up at him, and smiled. "Mrs. Murellen asked me if I wanted my job back at the library." His eyes burned into hers. "Miranda, I don't want you to give up anything—" "It's what I want," she said, then brushed her fingers across his stubbled cheek. "Are we going to go inside or not? I'm afraid if we stand out here for much longer, we're going to be attacked by a rabid emu." Heat flared in his eyes and he hastily opened the door, kicking it wide and staggering in. As soon as they were inside, he set her on her feet and whirled to shut the door. She glanced around—a plain sofa sat in the corner of the room and there was a queen-sized bed, but no TV or electronics. A stack of books lay next to the wood-burning stove. "Wow. You live pretty simply." He shrugged and put his hands to the sides of her face, cupping her cheeks as he tilted her face toward his. His mouth licked at hers and she felt the shiver of delight all the way down to her toes. "God, Miranda, the last month has been hell," his voice was ragged with need, and he closed his eyes, pressing his forehead against hers. "I thought I'd lost you again just when I'd found you." "I'm so sorry, Dane," she whispered. "You must think I'm crazy." "I don't," he said vehemently. "I think you were hurt and you thought I caused it. I'm surprised you ever wanted to touch me again." He leaned down and kissed her nose, then her cheeks, kiss after kiss, so delicate and heartfelt that her own heart ached in her chest. "Was it for the revenge?" "At first," she admitted, and hated when he flinched. "But then, after we slept together, it was for _me_ and I just kept telling myself that it was for revenge." He kissed her mouth again, slowly, sweetly. "I'm glad." She was, too. So, so glad. Impulsively, her hand slid to the front of his pants and caressed his cock. He was hard and straining, the length of him making her pussy slick with anticipation. "Are you going to fuck me or are we going to stand here and talk all day?" "You want to be fucked?" he said in a low, dangerous voice, his hand sliding to her skirt and pushing it up until his hand rested on the damp V of her pussy. She whimpered and clutched his shirt. "We need to make it over to that bed, and fast." He hauled her up against him and her legs went around his waist, her sex cradled against his cock. She moaned again as he carried her to the bed, his mouth slanting over hers in hard kiss after kiss. "How important are those panties to you?" he growled against her mouth. "Totally unimportant," she breathed and slicked her tongue into his mouth, darting and flicking. "Good." He dropped her on the bed, then reached under her skirt again, pushing it up around her waist. He tugged her panties off and tossed them aside. One finger slid across her pussy, then delved deep into the well of her sex, and she gasped at the surge of pleasure. "I want to take you right now. Fast and hard until you're screaming my name." "So do it," she breathed with excitement, and her pussy clenched against his finger. "What's stopping you?" "Condoms," he said. "They're out in the jeep—" "Fuck the condoms," she said and locked one leg around his back, trying to pull him forward. "I'm on the pill. Get those pants off and get inside me." His groan was swallowed by a hard kiss, this time from her, and both sets of hands fumbled at his belt. Miranda ran her fingers over the length of him under the pants, so excited she could hardly stand it. She was so wet and needed him so badly. His belt flew to the ground, then his pants, and he slid out of his boxers an instant later, and then his hot, warm length was probing at the aching core of her. "You sure, Miranda?" "God yes," she breathed. "Please. I need you." He surged deep, and her moan caught in the back of her throat as a broken little gasp at the burn of him. It had been a month since they'd had sex, and she stretched and tensed around him, the sensation of being filled so tightly making her toes curl and her pussy throb. "I can feel you all around me," he gritted out. "So fucking hot." His hands locked around her hips and he pulled out, then thrust deep again, causing her toes to curl once more. She dug her heels into his buttocks, urging him forward. "Again," she breathed. "Please." He surged deep again, and then again, the next thrust causing her to moan his name. He groaned. "Not going to last." "Then touch me," she said, her breath coming out in short, excited pants. He began to thrust again, his hips rocking against hers. His hand moved between them and slid into the wet heat of her folds, seeking her clit. When he found it, her entire body tensed and she shrieked her pleasure. He began to stroke it in time with his thrusts, the rhythm hard and fast, the touches on her clit feather light. She came mere seconds later, a scream building in her throat, her fingernails digging into his skin. "Dane!" He growled low in his throat as she clenched around his hard cock, and then suddenly he was leaning on her hard, his thrusts coming so rapid and rough that she thought they'd fall off the edge of the bed where she perched. But he stared down at her with wild green eyes, thrusting deep. Their eyes connected and when he pumped into her, it was even more intense because they were connected. "I love you," she whispered on his next thrust. He came with a yell, and she felt him deep inside her, coming hard. He tensed and rocked against her a few moments more, still staring into her eyes with his beautiful green ones, and then he collapsed on top of her. She hugged him close, and caught the glint of her ring in a beam of sunlight. A ring. She was engaged to Dane Croft. Holy shit. "Are you sure you want to do this?" "At least twice a day for the next thirty years," he said in a husky voice, and she felt him kiss her collarbone. She laughed. "No, dummy. The ring. Marriage." He rolled over on the bed and dragged her over him until they'd switched places and she lay sprawled atop of him. Then he went very still, his eyes serious. "Don't you want to marry me?" "Of course," she said with a smile. "But don't you think we're moving too fast?" "I like fast women," he said, reaching down and playing with a lock of her hair. "I seem to recall someone who seduced me in the woods the very first night of class." "Are you complaining?" she said with a grin. He pulled her down for another kiss. "Never."	{ "redpajama_set_name": "RedPajamaBook" }	5,466
Q: Unir tablas SQL a partir de códigos distintos según los últimos dígitos del id Necesito obtener los datos de una tabla, a partir de otra, pero la clave en común no tiene el mismo código, son distintos, ejemplo: Tengo en una tabla FACULTAD: facu_codigo facu_nombre 1 VICERRECTORIA 2 FACULTAD CS SOCIALES 3 . 4 . . . . . 10 . y otra tabla UNIDADES ACADÉMICAS: uaca_codigo 10 2001 2002 2003 . . 2010 Entonces necesito obtener nombre_facu a partir de la unión entre facu_codigo y uaca_codigo, donde los últimos 2 dígitos de uaca_codigo da referencia al facu_codigo y de esa manera podría hacer la relación. ¿Cómo puedo hacer para que uaca_codigo se convierta en los últimos dígitos para igualarlo a facu_codigo y que me dé el nombre? Ejemplo: 2001 ----> 1 2005 ----> 5 2010 ----> 10 A: Así de pronto se me ocurre que le restes 2000 al código de uaca_codigo, por ejemplo: SELECT * FROM FACULTAD F INNER JOIN UNIDADES_ACADEMICAS U ON F.FACU_CODIGO = (U.UACA_CODIGO -2000); A: SELECT [uaca_codigo], CONVERT(INT, RIGHT([uaca_codigo], 2)) FROM [dbo].[UNIDADES ACADEMICAS] Así cortarías por la derecha 2 caracteres y convertido en numérico para quitar el 0 de su izquierda. Un saludo	{ "redpajama_set_name": "RedPajamaStackExchange" }	3,160
\section{Introduction and statement of results} Let $\Fscr$ be a collection of sets. A {\em transversal} of $\Fscr$ is a set $A$ that intersects every member of $\Fscr$ (that is, $A \cap F \neq \emptyset$ for all $F \in\Fscr$). The {\em transversal number} of {\em piercing number} $\tau(\Fscr)$ of $\Fscr$ is the smallest size of a transversal, i.e. \[ \tau(\Fscr) := \min_{A \text{ transversal of } \Fscr } \|A\|. \] \noindent (Note that $\tau(\Fscr)=\infty$ if no finite transversal exists.) A collection of sets $\Fscr$ has the $(p,q)$-property if out of every $p$ sets of $\Fscr$ there are $q$ that have a point in common. In 1957, Hadwiger and Debrunner~\cite{HadwigerDebrunner1957} conjectured that for every $d$ and every $p \geq q \geq d+1$ there is a universal constant $c = c(d;p,q)$ such that every finite collection $\Fscr$ of convex sets in $\eR^d$ with the $(p,q)$-property satisfies $\tau(\Fscr) \leq c$. (By considering hyperplanes in general position it is easily seen that for $q\leq d$ no such universal constant $c$ can exist.) Many years later, in 1992, Alon and Kleitman~\cite{AlonKleitman1992} finally proved the conjecture of Hadwiger and Debrunner by cleverly combining various pre-existing tools from the literature. In the special case when $p=q=d+1$ the Hadwiger-Debrunner conjecture reduces to the classical theorem of Helly~\cite{Helly1923} which states that if $\Fscr$ is a finite family of convex sets in $\eR^d$ such that every $d+1$ members of $\Fscr$ have a point in common then $\tau(\Fscr) = 1$. A variant of Helly's theorem states that if $\Fscr$ is an infinite collection of closed, convex sets in $\eR^d$ and at least one member of $\Fscr$ is compact then $\tau(\Fscr) = 1$. Erd\H{o}s conjectured that in the first nontrivial case of the Hadwiger-Debrunner problem, a similar variant would be true. That is, he conjectured that if $\Fscr$ is a collection of closed, convex sets in the plane with the $(4,3)$-property and one of the members of $\Fscr$ is compact, then $\tau(\Fscr) \leq c$ for some universal constant $c$. Boltyanski and Soifer included this conjecture in the first edition of their book ``Geometric Etudes in Combinatorial Mathematics'' and they offered a prize of \$25 for its solution. Eighteen years later, Gr\"unbaum found a simple counterexample while proofreading the second edition, earning the reward. Gr\"unbaum also made a conjecture of his own, stating that if $\Fscr$ is as above, and {\em two} members of $\Fscr$ are compact then $\tau(\Fscr)$ is finite. (See~\cite{SoiferBoek}, pages 198-199.) Here we show that Gr\"unbaum's conjecture fails as well: \begin{theorem} There exists a collection $\Fscr$ of closed, convex subsets of the plane such that \begin{enumerate} \item $\Fscr$ has the $(4,3)$-property, and; \item Two of the elements of $\Fscr$ are compact, and; \item $\tau(\Fscr) = \infty$. \end{enumerate} \end{theorem} \noindent On the positive side, we show that any collection $\Fscr$ of closed, convex sets in the plane that contains two disjoint compacta and satifies the $(4,3)$-property does have universally bounded transversal number: \begin{theorem}\label{thm:disjoint} If $\Fscr$ is a collection of closed, convex sets in the plane such that \begin{enumerate} \item $\Fscr$ has the $(4,3)$-property, and; \item $\Fscr$ contains two disjoint compacta, \end{enumerate} then $\tau(\Fscr) \leq 13$. \end{theorem} \section{The counterexample} Let us set \[ F_1 := [-1,1]\times\{0\}, \quad F_2 := [0,2] \times \{0\}. \] \noindent Let $t_1 < t_2 < t_3 < \dots$ be a strictly increasing sequence of numbers between $0$ and $1$, and let $s_1 > s_2 > \dots$ be a strictly decreasing sequence of negative numbers that tends to $-\infty$. (For instance $t_n := 1 - \frac{1}{n}, s_n := -n$ would be a valid choice.) Set $p_n := (t_n, 0)$; let $\ell_n$ denote the vertical line through $p_n$; and let $\ell_n'$ denote the line through $p_n$ of slope $s_n$. For $n \geq 3$ we now let $F_n$ be the set of all points either on or to the left of $\ell_n$ and either on or above $\ell_n'$. See figure~\ref{fig:counterex}. \begin{figure}[h!] \begin{center} \input{Fn4.pspdftex} \end{center} \caption{The construction of $F_n$ for $n \geq 3$ (left) and part of the collection $\Fscr$ (right).\label{fig:counterex}} \end{figure} \noindent Observe that by construction $F_n$ contains all sufficiently high points on the $y$-axis for all $n\geq 3$: \begin{equation}\label{eq:yn} \text{For each $n\geq 3$ there exists a $y_n > 0$ such that $\{ (0,y) : y \geq y_n \} \subseteq F_n$.} \end{equation} \noindent Let $\Fscr := \{ F_1, F_2, \dots \}$ be the resulting infinite collection of closed convex sets. We first establish that $\Fscr$ has the $(4,3)$-property. \begin{lemma} $\Fscr$ has the $(4,3)$-property. \end{lemma} \begin{proof} Let us pick four abitrary distinct indices $i_1 < i_2 < i_3 < i_4$ and consider the quadruple $F_{i_1}, F_{i_2}, F_{i_3}, F_{i_4} \in \Fscr$. If $i_1=1$ and $i_2=2$ then clearly $F_{i_1} \cap F_{i_2} \cap F_{i_3} = \{p_{i_3}\}$, so that $F_{i_1}, F_{i_2}, F_{i_3}$ is an intersecting triple. We can thus assume that $i_2, i_3, i_4 > 2$. In this case $F_{i_2} \cap F_{i_3} \cap F_{i_4} \neq \emptyset$ by the observation~\eqref{eq:yn}. \end{proof} \noindent It remains to show that $\Fscr$ does not have a finite transversal. \begin{lemma} $\tau(\Fscr) = \infty$. \end{lemma} \begin{proof} It suffices to show that every point of the plane is in finitely many elements of $\Fscr$. Let $a = (a_x,a_y) \in \eR^2$ be arbitrary. If $a_y \leq 0$ then $a$ is in at most three elements of $\Fscr$. Let us therefore assume $a_y > 0$. In this case, if $a_x \geq t_n$ for all $n \in \eN$ then $a$ is in no element of $\Fscr$. Let us therefore assume that there is at least one $n\in\eN$ such that $a_x < t_n$. Let us fix an $n_0$ such that $a_x < t_{n_0}$, and set \[ s := - \frac{a_y}{t_{n_0} - a_x}. \ (Note that $s$ is exactly the slope of the line through $a$ and $p_{n_0}$.) \\ Since $s_n \to -\infty$, there is an $m_0$ such that $s_n < s$ for all $n \geq m_0$. Observe that for all $n \geq \max(n_0,m_0)$ the point $a$ is below the line $\ell_n'$ (as the point $p_n$ is to the right of $p_{n_0}$ and $\ell_n'$ has a steeper slope than $s$). This shows that $a \not\in F_n$ for all $n \geq \max(n_0,m_0)$. Hence $a$ is in finitely many elements of $\Fscr$ as required. \end{proof} \noindent {\bf Remark:} By adding additional compact sets to $\Fscr$ that each contain $[0,1]\times\{0\}$ we can obtain a collection $\Fscr'$ that contains an arbitrary number of compacta, and still has the $(4,3)$-property and $\tau(\Fscr') = \infty$. \section{The proof of Theorem~\ref{thm:disjoint}} The proof of the Hadwiger-Debrunner conjecture by Alon and Kleitman~\cite{AlonKleitman1992} does not give a good bound on the universal constant $c$. A better bound on this constant for the special case when $p=4, q=3$ was later given by Kleitman, Gyarfas and T\'oth~\cite{KleitmanEtal2001}. \begin{theorem}[Kleitman et al.~\cite{KleitmanEtal2001}] If $\Fscr$ is a finite collection of convex sets in the plane with the $(4,3)$-property then $\tau(\Fscr) \leq 13$. \end{theorem} \noindent A standard compactness argument (which we do not repeat here) shows that the same also holds if $\Fscr$ is an infinite collection of convex compacta with the $(4,3)$-property. \begin{corollary}\label{cor:Kleitman} If $\Fscr$ is an infinite collection of convex, compact sets in the plane and $\Fscr$ has the $(4,3)$-property then $\tau(\Fscr) \leq 13$. \end{corollary} \begin{proofof}{Theorem~\ref{thm:disjoint}} Let $\Fscr$ be an arbitrary infinite collection of closed, convex sets with the $(4,3)$-property with two sets $A, B \in \Fscr$ that are disjoint and compact. Let us set \[ F_0 := \conv(A\cup B). \ Let us first observe that \begin{equation}\label{eq:obs1} F \cap F_0 \neq \emptyset \text{ for all } F \in\Fscr. \end{equation} \noindent To see this, suppose that some $F \in\Fscr$ is disjoint from $F_0$, and let $F' \in \Fscr$ be an arbitrary element distinct from $F, F_1, F_2$ and $F_0$. Then the quadruple $F_1, F_2, F, F'$ does not have an intersecting triple as every triple contains a pair of disjoint sets. But this contradicts the $(4,3)$-property! Hence~\eqref{eq:obs1} holds as claimed. Next, we claim that \begin{equation}\label{eq:obs2} \text{If $F_1, F_2, F_3 \in \Fscr$ are such that $F_1\cap F_2\cap F_3 \neq \emptyset$ then also $F_0 \cap F_1\cap F_2\cap F_3 \neq \emptyset$.} \end{equation} To see that the claim~\eqref{eq:obs2} holds, consider an arbitrary triple $F_1, F_2, F_3 \in \Fscr$ such that $F_1\cap F_2\cap F_3 \neq \emptyset$. Let us assume $F_1\cap F_2\cap F_3 \not\subseteq F_0$ (otherwise we are done), and fix a $q \in (F_1\cap F_2\cap F_3)\setminus F_0$. By considering the quadruple $A, B, F_1, F_2$ we see that we either have $A \cap F_1 \cap F_2 \neq \emptyset$ or $B \cap F_1 \cap F_2 \neq \emptyset$. In either case, there is a point $p_{12} \in F_0 \cap F_1 \cap F_2$. Similarly there are points $p_{13} \in F_0 \cap F_1 \cap F_2, p_{23} \in F_0 \cap F_2 \cap F_3$. By Radon's lemma the set $\{q, p_{12}, p_{13}, p_{23} \}$ can be partitioned into two sets whose convex hulls intersect. Note that we cannot have that $q \in \conv(\{p_{12},p_{13},p_{23}\})$ since $q \not\in F_0$ and $p_{12},p_{13},p_{23} \in F_0$ and $F_0$ is convex. Hence, up to relabelling of the indices we have either $p_{23} \in \conv( \{q, p_{12}, p_{13}\} )$ or $[q,p_{23}] \cap [p_{12},p_{13}] \neq \emptyset$. In the first case we have that $p_{23} \in F_0 \cap F_1 \cap F_2 \cap F_3$ since we have chosen $p_{23} \in F_0 \cap F_2 \cap F_3$ and $\conv( \{q, p_{12}, p_{13}\} ) \subseteq F_1$ as all three of $q,p_{12},p_{13} \in F_1$ and $F_1$ is convex. In the second case we have that the intersection point of $[q,p_{23}]$ and $[p_{12},p_{13}]$ is in $F_0 \cap F_1 \cap F_2 \cap F_3$. This is because $[q,p_{23}] \subseteq F_2 \cap F_3$ and $[p_{12},p_{13}] \subseteq F_0 \cap F_1$. Thus,~\eqref{eq:obs2} holds as claimed. We now define a new collection of sets by setting: \[ \Fscr' := \{ F \cap F_0 : F \in \Fscr \}. \] \noindent Since the members of $\Fscr$ are closed and convex and $F_0$ is compact and convex, each element of $\Fscr'$ is compact and convex. By~\eqref{eq:obs1} each set of $\Fscr'$ is nonempty (this is needed since otherwise there cannot be any transversal of $\Fscr'$), and by~\eqref{eq:obs2} together with the fact that $\Fscr$ satisfies the $(4,3)$-property, the collection $\Fscr'$ also satisfies the $(4,3)$-property. The theorem now follows from Corollary~\ref{cor:Kleitman} as every transversal of $\Fscr'$ is also a transversal of $\Fscr$. \end{proofof} \section*{Acknowledgement} I thank Bart de Keijzer and Branko Gr\"unbaum for helpful discussions. \bibliographystyle{plain}	{ "redpajama_set_name": "RedPajamaArXiv" }	3,635
{"url":"https:\/\/marckhoury.github.io\/on-computable-functions\/","text":"# On Computable Functions\n\nWhat does it mean to be computable? A function is computable if for a given input its output can be calculated by a finite mechanical procedure. But can we pin this idea down with rigorous mathematics?\n\nIn 1928, David Hilbert (see [4]) proposed his famous Entscheidungsproblem, which asks if there is a general procedure for showing that a statement is provable from a given set of axioms. To solve this problem mathematicians first needed to define what it meant to be computable. The first attempt was through primitive recursive functions and was a combined effort by many researchers, including Kurt G\u00f6del, Alonzo Church, Stephen Kleene, Wilhelm Ackermann, John Rosser, and R\u00f3zsa P\u00e9ter.\n\n### Recursive Functions\n\nPrimitive recursive functions are defined as a recursive type, starting with a few functions that we assume are computable, called founders, and operators that construct new functions from the founders, called constructors. The founders are the following three functions:\n\n\u2022 The constant zero function: a function that always returns zero\n\u2022 The successor function: $S(n) = n+1$\n\u2022 The projection function: $\\text{proj}_{n}^m$ is an $m$-ary function that returns the $n$th argument\n\nComputability theory wasn\u2019t going to get very far if these functions weren\u2019t computable. Next, we have two operations for constructing new functions from old: composition and primitive recursion.\n\n\u2022 Composition: Given a primitive recursive $m$-ary function $h$ and $m$ $n$-ary functions $g_1,\\ldots, g_m$, the function $f(\\textbf{x}) = h(g_1(\\textbf{x}),\\ldots, g_m(\\textbf{x}))$ is primitive recursive.\n\u2022 Primitive Recursion: Given primitive recursive functions $g,h$ the function % is primitive recursive.\n\nThe set of primitive recursive functions is the set of functions constructed from our three initial functions and closed under composition and primitive recursion. Many familiar functions are primitive recursive: addition, multiplication, exponentiation, primes, max, min, and the logarithm function all fit the bill.\n\nSo are we done? Is every computable function also primitive recursive? Sadly, no: the Ackermann function would be proven in 1928 to be a counterexample.\n\nThe Ackermann function is a total (defined for all inputs) function that is clearly computable but not primitive recursive. Indeed, in 1928 Ackermann (see [1]) showed that his function bounds every primitive recursive function: it grows too fast to be primitive recursive.\n\nSomething was clearly wrong, but early computability theorists didn\u2019t want to abandon primitive recursive functions entirely. What came next was a rather surprising idea at the time: perhaps computable functions need not be total! This was the key that unlocked computability theory: focusing on partial functions, those that may not be defined on all possible inputs.\n\nThe reason for focusing on partial functions is to allow an unbounded search operator. That is, we want to be able to search for the least input value that satisfies a condition and simply be undefined if no such input value exists. This operation is captured by Kleene\u2019s $\\mu$-operator.\n\n\u2022 $\\mu$-recursion: $f(x) = (\\mu y)(g(x,y) = 0)$ returns the least $y$ such that $g(x,y) = 0$ and is undefined if no such $y$ exists. The function $g(x,y')$ must be defined for all $% $.\n\nTaking the closure of the $\\mu$-operator with all primitive recursive functions gives a class of $\\mu$-recursive functions. In 1943, Kleene (see [5]) used his $\\mu$-operator to provide an alternative, but equivalent, definition of general recursive functions. The original definition was given by G\u00f6del in 1934 (see [3]), based on an observation by Jacques Herbrand. It would later be shown that $\\mu$-recursive functions are the exact same class of functions defined by two competing approaches (see [6]).\n\n### $\\lambda$-Calculus\n\nSimultaneously, from 1931-1934, Church and Kleene were developing $\\lambda$-calculus as an approach to computable functions. The syntax of $\\lambda$-calculus defines certain expressions as valid statements, which are called $\\lambda$-terms. A $\\lambda$-term is built up from a collection of variables and two operators: abstraction and application.\n\nLet\u2019s start with a collection of variables $x,y,z,\\ldots$ and suppose $M, N$ are valid $\\lambda$-terms. The abstraction operator creates the term $\\lambda x. M$, which is a function taking an argument $x$ and returning $M$ with each occurrence of $x$ replaced with the argument. The application operator creates the term $M N$, which represents the application of a function $M$ on input $N$.\n\nThe $\\lambda$-term $\\lambda x.M$ represents a function $f(x) = M$ and - like recursive functions - many familiar functions are $\\lambda$-definable. The $\\alpha$-conversion and $\\beta$-reduction are classic examples of \\emph{reductions}, which describe how $\\lambda$-terms are evaluated. An $\\alpha$-conversion captures the notion that the name of an argument is usually immaterial. For instance $\\lambda x.x$ and $\\lambda y.y$ both represent the identity function and are $\\alpha$-equivalent. A $\\beta$-reduction applies a function to its arguments. Take, as an example, the $\\lambda$-term $(\\lambda x.x)y$, which represents the identity function $(\\lambda x.x)$ applied to the input $y$. Substituting the argument $y$ for the parameter $x$, the result of the function is $y$. So we say $(\\lambda x.x)y$ $\\beta$-reduces to $y$.\n\nIn 1934 Church proposed that the term \u201ceffectively calculable\u201d be identified with $\\lambda$-definable. While Church\u2019s formalization of computability would later be shown to be equivalent to Turing\u2019s, G\u00f6del was dissatisfied with Church\u2019s work. In fairness, G\u00f6del also was dissatisfied with his own work! Church would go on to advocate that \u201ceffectively calculable\u201d should be identified with general recursive functions (which G\u00f6del still rejected). In 1936 Church (see [2]) published his work proving that that the Entscheidungsproblem was undecidable: there is no general procedure for determining if a statement is provable from a given set of axioms.\n\n### Turing Machines\n\nMeanwhile, after hearing about Hilbert\u2019s Entscheidungsproblem, a 22 year old Cambridge student named Alan Turing began working on his own solution to the problem. Turing was unaware of Church\u2019s work at the time, so his approach wasn\u2019t influenced by $\\lambda$-expressions (this wasn\u2019t the first time Turing failed to perform a literature review). Instead, he envisioned an idealized human agent performing a computation, which he called a \u201ccomputer\u201d. To avoid confusion with the modern definition of computer, we\u2019ll adopt the terminology of Robin Gandy and Wilfried Sieg and use the term \u201ccomputor\u201d to refer to an idealized human agent. The computor had infinite available memory called a tape, essentially an infinite strip of paper, that was divided into squares. The computor could read and write to a square, as well as move from one square to another.\n\nTuring put several conditions on the computation that the computor could perform. The computor could only have finitely many states (of mind) and the tape could only hold symbols from a finite alphabet. Only a finite number of squares could be observed at a time and the computor could only move to a new square that was at most some finite distance away from an observed square. He also required that any operation must depend only on the current state and the observed symbols, and that there was at most one operation that could be performed per action (his machines were deterministic).\n\nFrom this, Turing would go on to define his automatic machines - which would later come to be known as Turing machines - and show the equivalence of the two formalizations. He\u2019d then show that \u201ceffectively calculable\u201d implied computable by his idealized human agent, which in turn implied computable by such a machine. Turing then went on to show that the Entscheidungsproblem was undecidable. Shortly before publishing his work, he learned that Church had already shown that the Entscheidungsproblem was undecidable using $\\lambda$-calculus. Turing quickly submitted his work in 1936 (see [7]) - six months after Church - along with a proof demonstrating the equivalence between his machines and $\\lambda$-calculus.\n\nAfter reading Turing\u2019s seminal paper, G\u00f6del was finally convinced that the correct notion of computability had been determined. It would later be shown that all three formalizations - Turing machines, $\\mu$-recursion, and $\\lambda$-calculus - actually define the same class of functions. That these three approaches all yielded the same class of functions suggested that mathematicians had captured the correct notion of computation, and supported what would come to be known as the Church-Turing Thesis.\n\nThree years later, in 1939, Turing completed his Ph.D. at Princeton under the supervision of Church. In his thesis he\u2019d state the following (see [8]): \u201cWe shall use the expression \u2018computable function\u2019 to mean a function calculable by a machine, and let \u2018effectively calculable\u2019 refer to the intuitive idea without particular identification with any one of these definitions.\u201d\n\nChurch-Turing Thesis: Every effectively calculable function is a computable function.\n\nChurch intended for his original thesis to be taken as a definition of what is computable. Likewise, even though he never stated it, Turing had the same intention. In fact, the term \u201cChurch\u2019s Thesis\u201d was coined by Kleene many years after Church had published his work. These days, many people take the Church-Turing Thesis as a definition of what is computable; less formally stating that a function is computable if and only if it can be computed by a Turing machine.\n\nIt\u2019s important to stress that the Church-Turing Thesis is not a definition as many believe. It does not refer to any particular formalization that we\u2019ve discussed and is not a statement that can be formally proven. It is a statement about the nature of computation. Everything that is \u201ceffectively calculable\u201d, in the vague and intuitive sense, is a computable function.\n\n### References\n\n1. Wilhelm Ackermann; 1928; Zum hilbertschen aufbau der reellen zahlen; Mathematische Annalen, 99(1): 118\u2013133.\n2. Alonzo Church; 1936; An unsolvable problem of elementary number theory; American Journal of Mathematics; 58(2): 345\u2013363.\n3. Kurt G\u00f6del; 1934; On Undecidable Propositions of Formal Mathematics Systems; Institute for Advanced Study.\n4. David Hilbert; 1900; Mathematical problems; International Congress of Mathematicians.\n5. Stephen C. Kleene; 1943; Recursive predicates and quantifiers; AMS; 53(1): 41-73; http:\/\/www.jstor.org\/stable\/1990131.\n6. Stephen C. Kleene; 1952; Introduction to metamathematics; North-Holland Publishing Company.\n7. Alan M. Turing; On computable numbers, with an application to the entscheidungsproblem; Proceedings of the London Mathematical Society; 2(42), 1936.\n8. Alan M. Turing; Systems of logic based on ordinals; Proceedings of the London Mathematical Society; 2(1):161\u2013228, 1939.\n\nTags:\n\nUpdated:","date":"2018-11-19 08:57:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 67, \"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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8241946697235107, \"perplexity\": 752.6714566097098}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-47\/segments\/1542039745522.86\/warc\/CC-MAIN-20181119084944-20181119110944-00504.warc.gz\"}"}	null	null
Q: Unable to convert classes into dex format Unity Using the google play games services and Admob plugin I cannot seem to get my back to build into an APK and I am receiving the following error message. Any help would be appreciated as I have removed some duplicate files but I am still failing to find why I cannot build my app. I am using Unity Unity 5.6.1f1 (64-bit). game services: https://github.com/playgameservices/play-games-plugin-for-unity Ads: https://github.com/unity-plugins/Firebase-Admob-Unity CommandInvokationFailure: Unable to convert classes into dex format. C:/Program Files/Java/jdk1.8.0_121\bin\java.exe -Xmx2048M -Dcom.android.sdkmanager.toolsdir="C:/Users/Jack/AppData/Local/Android/sdk\tools" -Dfile.encoding=UTF8 -jar "C:\Program Files\Unity\Editor\Data\PlaybackEngines\AndroidPlayer/Tools\sdktools.jar" - stderr[ Uncaught translation error: java.lang.IllegalArgumentException: already added: Lcom/google/android/gms/internal/zzbyb; Uncaught translation error: java.lang.IllegalArgumentException: already added: Lcom/google/android/gms/internal/zzbyc; 2 errors; aborting ] A: Some Background When Unity builds your project for Android, it invokes several tools from the Android SDK. As part of that process, it converts all of your native (Java) code for Android into a file format called DEX (Dalvik executable). All of your Android plugins get built up into a single package using that tool. The problems start when a few plugins have the same compiled Java code (classes) in them. This will cause the DEX tool to fail with an error like the one you're seeing: Uncaught translation error: java.lang.IllegalArgumentException: already added: Lcom/google/android/gms/internal/zzbyb; Uncaught translation error: java.lang.IllegalArgumentException: already added: Lcom/google/android/gms/internal/zzbyc; This means that classes with those names were already included in another library, and so they cannot be added again. Possible Causes As explained above, this error occurs when you have duplicate plugins in your project; this can happen in different scenarios, here are some examples: * The same plugin is included more than once, under different folders of the project. The same plugin is included more than once with different versions. A plugin contains other dependencies "embedded" inside it, but these dependencies are already included in the project in some form. How To Fix You should look up duplicate Android plugins in your project and eliminate them (keep only 1 copy). From the error message you posted, the issue here is related to Google play services libraries. You should look into that (libraries named play-services-xxxx.aar). Paid Help (Shameless Plug) I provide a professional service for fixing this exact kind of issue. In case you (or anyone else) are not able to resolve such an issue themselves, feel free to contact me and get it solved. A: I was facing the same problem after add admob in my project, here are 2 possible solutions: Go to the file >> build settings and change the option internal to gradle Open libs folders separately of both Facebook SDK and Google Play services and delete the matching files It would be something like (support v4) A: Google ads 11.2 has errors. It reuses certain jar classes . On its own, it isnot an issue Once you add in another google class, ie firebase, your done for. There is a fix though. Google put out firebase 4.2 and ads 11.4	{ "redpajama_set_name": "RedPajamaStackExchange" }	4,420
What is Elbow Osteochondritis Dissecans (OCD) in Pets? Three bones make up the joint of a dog's elbow: the radius, the ulna, and the humerus. These three bones are supposed to grow together and fit perfectly to form the elbow joint. Osteochondritis dessicans is a condition in which a piece of cartilage comes loose or pulls away completely from the surface of the joint, resulting in inflammation and pain. After the inflammation or "itis" is gone, the condition is called osteochrondrosis dessicans. Most often the cause is rapid bone development, so OCD is usually found in puppies between four and eight months old. However, it can occasionally be found in older dogs, as well as smaller breeds. It affects male dogs about five times more often than females. What causes Elbow Osteochondritis Dissecans (OCD) in Dogs and are there certain breeds at risk? Conservative care for Elbow Osteochondritis Dissecans (OCD) include physical therapy, use of non-steroidal anti-inflammatory drugs (NSAIDs), rest from sport for 6-8 weeks, and bracing.	{ "redpajama_set_name": "RedPajamaC4" }	5,272
{"url":"https:\/\/cs.stackexchange.com\/questions\/23804\/showing-that-deciding-whether-a-given-tm-accepts-a-word-of-length-5-is-undecidab","text":"# Showing that deciding whether a given TM accepts a word of length 5 is undecidable\n\nI'm having trouble grasping this the concept of reductions. I found the solution and it looks like this:\n\nAssume that $M_5$ is a Turing Machine that can decide if a given Turing Machine $M$ accepts any string of length $5$, i.e., $L(M)$ contains a string of length $5$. The above figure shows how we can use this to construct a Turing Machine that can solve the Halting problem.\n\nThe output $M\u2032$ of our translator behaves as follows:\n\n1. $M\u2032$ erases its own input and replaces it with the string w.\n2. It them simulates $M$ on $w$.\n3. If $M$ halts on $w$, then it goes into a final state (accepts its input). It is clear that if $M$ halts on $w$, $M\u2032$ accepts all its inputs. So it accepts a string of length 5 as well. If $M$ does not halt on $w$, $M\u2032$ does not accept any string at all. So it does not accept any string of length 5 either. So $L(M\u2032)$ includes a string of length $5$ if and only of $M$ accepts $w$. So by running $M_5$ on $M\u2032$, we can decide if $M$ halts on $w$ or not. But we know that this is not possible since the halting problem is undecidable. Hence $M_5$ does not exist and the given problem is undecidable.\n\nWhat I am confused about is: \"if $M$ halts on $w$, $M\u2032$ accepts all its inputs\" and \"If $M$ does not halt on $w$, $M\u2032$ does not accept any string at all\". Can someone clarify why this is the case? I've been trying to work out the logic for so long. If any of you guys could help this would be great!\n\nSource\n\nTo decide if $M$ halts with input $w$, we use the Machine $M'$ that simulates $M$ with input $w$. If $M$ accepts $w$, $M'$ also accepts $w$ (from point 3 of your description). If however $M$ does not halt on input $w$, the simulation can not halt as well.\n\nSo far, nothing special has happened.\n\nNow consider point 1 of your description: $M'$ does take an input, but immediately discards it.\n\nSo to decide if $M$ halts on $w$, we use $M_5$ to decide if $M'$ accepts a word of length 5. But $M'$s behavior is completely independent of it's own input, since it just simulates $M$ with fixed $w$.\n\nNow: if $M$ halts on $w$, $M'$ will halt on every input (because it is discarded and $w$ is used instead), especially one of length 5. This will (magically\/hypothetically) be detected of $M_5$.\n\nIf however $M$ does not halt on $w$, $M'$ will not halt on any input (again, because $M'$s input is irrelevant), especially it will not halt on any input of length 5. This can also be detected by $M_5$, and we have solved the halting problem.\n\nSuppose that $M$ halts on input $w$ and consider the behaviour of $M'$ when it is given input\u00a0$w'\\!$. $M'$ first deletes its input $w'$ and replaces it with $w$, which means that the behaviour of $M'$ doesn't depend on what input it actually receives. Then, $M'$ simulates $M$ on input\u00a0$w$. So, if $M$ halts on $w$, $M'$ halts for every input, becaues $M'$ ignores its input and just pretends to be $M$ running with input\u00a0$w$. Likewise, if $M$ does not halt on\u00a0$w$, $M'$ doesn't halt for any input.\n\n\u2022 I understood the explanation of why this problem is undecidable. But my question is, we have finite strings of length 5 and if a TM really accepts string of length 5 then it should take maximum of 5 steps. So we have finite number of strings which we can run for finite number of steps . So shouldn't it be decidable ? \u2013\u00a0Zephyr Sep 9 '17 at 6:48\n\u2022 @Zephyr It's not true that a Turing machine that accepts a string of length five must do so in at most five steps. Indeed, there's no computable bound on the number of steps it might take. \u2013\u00a0David Richerby Sep 9 '17 at 18:26\n\u2022 every character in the input has to take minimum 1 step in standard Turing machine right ? \u2013\u00a0Zephyr Sep 10 '17 at 5:42\n\u2022 @Zephyr That would argue for a minimum of five steps, not a maximum. But note that a machine can halt before reading its whole input -- for example, the language \"the first character of the string is '$0$'\" can be decided in one step, regardless of the length of the input. \u2013\u00a0David Richerby Sep 10 '17 at 9:03\n\u2022 @Zephyr any machine that moves its head to the left at least once. \u2013\u00a0David Richerby Sep 10 '17 at 9:21","date":"2019-08-20 01:29:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5924944281578064, \"perplexity\": 201.12223011206493}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027315174.57\/warc\/CC-MAIN-20190820003509-20190820025509-00159.warc.gz\"}"}	null	null
Q: Нужно вставить изображения в черную область, возможно как-то через overflow-hidden Нужно вставить изображения в черную область, возможно как-то через overflow-hidden,тогда как сделать такую форму для div ?Нужно сделать на CSS.Хелпните, плиз A: Создание таких криволинейных контуров на CSS довольно проблематично. Для реализации таких форм идеально подходит SVG Предлагаю использовать для решения подобных задач инструменты SVG: Mask SVG, clipPath, pattern По сравнению с масками на css, маски svg являются более мощным инструментом, так как они появились на много раньше. Документация на русском: тут, тут и тут Вкратце смысл применения маски,- создание обтравочного контура и обрезка изображения с помощью этого контура. Что достигается при окраске маски в белый цвет. Обратите внимание на fill="white" в path маски. Техника создания обтравочного контура подробно дана в соседнем ответе. SVG Mask <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="85%" height="85%" viewBox="0 0 912 816" preserveAspectRatio="xMinYMin meet"> <defs> <mask id="msk"> <path fill="white" d="M420.5 150.5c33.8-6.1 71.9-2.8 101.5 11.8 15.1 7.5 23.4 26.6 27.8 42.8 16.4 61 41.8 56.8 47.3 57.5 54.4 7.6 78.2 16.7 101.1 45.7a164.3 164.3 0 0 1 35 98c-.2 26.8-20.9 54.4-45.6 65.2-24.6 10.6-46.8 15.4-68.5 26.9-43 22.9-64.9 81.5-104.4 115-23.2 19.6-52.6 26.4-81.2 29.4-34 3.4-65.6-4.3-97.8-16.8-10-3.8-16.4-6.9-24.1-21.6-8-15.1 0-23.5 1.6-35 4-28.3 13.3-41.6-14.7-84.9-12.7-19.7-20-7.6-54.2-46-42.7-48.2-46-141 11-192.2 34.8-31.2 64.7-43 98.7-62 21.2-11.7 42.7-29.5 66.5-33.8z" /> </mask> </defs> <image mask="url(#msk)" xlink:href="https://i.stack.imgur.com/rkQWc.jpg" width="100%" height="100%"/> </svg> Обтравочный контур создан и его можно использовать также для обрезки изображения с помощью clipPath или для заполнения контура изображением с помощью pattern clipPath <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="85%" height="85%" viewBox="0 0 912 816" preserveAspectRatio="xMinYMin meet"> <defs> <clipPath id="clip"> <path d="M420.5 150.5c33.8-6.1 71.9-2.8 101.5 11.8 15.1 7.5 23.4 26.6 27.8 42.8 16.4 61 41.8 56.8 47.3 57.5 54.4 7.6 78.2 16.7 101.1 45.7a164.3 164.3 0 0 1 35 98c-.2 26.8-20.9 54.4-45.6 65.2-24.6 10.6-46.8 15.4-68.5 26.9-43 22.9-64.9 81.5-104.4 115-23.2 19.6-52.6 26.4-81.2 29.4-34 3.4-65.6-4.3-97.8-16.8-10-3.8-16.4-6.9-24.1-21.6-8-15.1 0-23.5 1.6-35 4-28.3 13.3-41.6-14.7-84.9-12.7-19.7-20-7.6-54.2-46-42.7-48.2-46-141 11-192.2 34.8-31.2 64.7-43 98.7-62 21.2-11.7 42.7-29.5 66.5-33.8z" /> </mask> </defs> <image clip-path="url(#clip)" xlink:href="https://i.stack.imgur.com/rkQWc.jpg" width="100%" height="100%"/> </svg> pattern <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="85%" height="85%" viewBox="0 0 912 816" preserveAspectRatio="xMinYMin meet"> <defs> <pattern id="pat" patternUnits="userSpaceOnUse" width="1425" height="1267" viewbox="0 0 1425 1267" > <image xlink:href="https://i.stack.imgur.com/rkQWc.jpg" width="100%" height="100%"/> </pattern> </defs> <path fill="url(#pat)" d="M420.5 150.5c33.8-6.1 71.9-2.8 101.5 11.8 15.1 7.5 23.4 26.6 27.8 42.8 16.4 61 41.8 56.8 47.3 57.5 54.4 7.6 78.2 16.7 101.1 45.7a164.3 164.3 0 0 1 35 98c-.2 26.8-20.9 54.4-45.6 65.2-24.6 10.6-46.8 15.4-68.5 26.9-43 22.9-64.9 81.5-104.4 115-23.2 19.6-52.6 26.4-81.2 29.4-34 3.4-65.6-4.3-97.8-16.8-10-3.8-16.4-6.9-24.1-21.6-8-15.1 0-23.5 1.6-35 4-28.3 13.3-41.6-14.7-84.9-12.7-19.7-20-7.6-54.2-46-42.7-48.2-46-141 11-192.2 34.8-31.2 64.7-43 98.7-62 21.2-11.7 42.7-29.5 66.5-33.8z" /> </svg> A: В чем сложность? Где именно не получается? Создайте внутри обертки два блока, ипользуйте flex, чтобы расположить их рядом горизонтально. Ставьте внутри правого блока хоть фоном, хоть тегом нужное изображение/изображения. <flexWrap> <textContentWrap> <h1>Заголовок</h1> <p>Описание …</p> <button>Кнопка</button> etc. </textContentWrap> <imageContentWrap style="background: url(bolb.png);background-repeat: no-repeat;background-size: 300px 100px;"> </imageContentWrap> </flexWrrap>	{ "redpajama_set_name": "RedPajamaStackExchange" }	3,032
George Albert Wade Revision as of 15:00, 22 April 2016 by RozB (talk \| contribs) George Albert Wade (1891-1986) of Wade Ceramics 1891 July 19th. Born in Burslem, Stoke-on-Trent, to George Wade and his wife Marie Start Poxon - a family who ran a pottery business. 1901 Living at Watlands, Porthill, Wolstanton: Harriet Wade (age 72 born Whitechurch), Widow. With her five children; William Wade (age 43 born Tunstall), Manufacturer of Earthenware and Tiles and Employer; Eleanor Reade James (age 45 born Tunstall), Married. Also Harriet Annie James (age 22 born Norton in Moors), grand-daughter and Elementary School teacher; George Wade (age 37 born Tunstall), Manufacturer of Pottery used in Textile Industry and Employer. Also Marie Hart Wade (age 38 born Castle Donnington), (daughter-in-law); George Albert Wade (age 9 born Burslem), grandson; Albert Joseph Wade (age 35 born Tunstall), Manufacturer of Earthenware and Tiles and Employer; Mary Ann Wade (age 33 born Tunstall).Two servants.[1] Sir George was a pupil at Newcastle High School, which has been amalgamated into the Newcastle-under-Lyme School, and left at age 15 to work at the family's pottery factory. 1911 Living at The Mount, 1 Porthill, Longport, Stoke On Trent: George Wade (age 48 born Tunstall), Pottery Manufacturer and Employer. With his wife Marie Start Wade (age 49 born Castle Donnington) and their son George Albert Wade (age 19 born Burslem), Manager at Pottery Manufacturer. Two servants.[2] He was a soldier between 1914 and 1919 in World War I, first as a private with the North Staffordshire Regiment, and then as a lieutenant with the South Staffordshire Regiment in the Machine Gun Corps, and he served in France and Egypt. 1915 September 18th. Married Florrie Johnson Sir George became chairman of the family's pottery business, Wade Ceramics, a manufacturer of porcelain and earthenware, whose main factory was in Burslem, Stoke-on-Trent. In the 1950s, Sir George and the Wade potteries created and manufactured "Whimsies", small cheap solid porcelain animal figures, which became popular and collectable in Britain and America. Sir George never fully stopped working, but in the early 1980s he gave the routine running of the business to his son George Anthony (Tony) Johnson Wade. His hobbies included painting and ornithology. He was knighted in 1955 for political and public services. 1986 January 27th. Died Retrieved from "https://www.gracesguide.co.uk/index.php?title=George_Albert_Wade&oldid=839620"	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	5,050
Cnephasia zernyi is een vlinder uit de familie van de bladrollers (Tortricidae). De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1959 door Razowski. zernyi	{ "redpajama_set_name": "RedPajamaWikipedia" }	9,889
Q: Print in VB 2010 is working but page missing I'm trying to print my form which I created with Visual Studio. I'm using power pack printing. All goes really well but I'm missing 2cm on the right side of my printout. Page on printer is set to A4. This is what i try to do but no luck: PrintForm1.Print(Me, PowerPacks.Printing.PrintForm.PrintOption.ClientAreaOnly) Whatever I change, it is always printing the cropped page. I'm using xps printer at the moment and there is no zoom option. Can I set the form to zoom to fit the page? The borders are also too big. Thats why my printout gets cropped, but I have no idea how to change it. The size of my form is 711 x 1034 pixels, it's a desktop application. Thanks everybody. A: Ok finally find it :D probably too tired yesterday :D If anyone need to change margin on printer here it is!!! Code: PrintForm1.PrinterSettings.DefaultPageSettings.Margins = New System.Drawing.Printing.Margins(50, 50, 50, 50) PrintForm1.PrintAction = Printing.PrintAction.PrintToPrinter PrintForm1.Print()	{ "redpajama_set_name": "RedPajamaStackExchange" }	5,505
The Lilac Screen is crafted in 100% solid Moso Bamboo and combines a beautiful design and a green sustainability story. With a rich honey-toned caramelized finish, the Lilac Screen is both elegant and practical. Moso Bamboo is 20% harder than Red Oak! Absolutely no plywood, particle board or MDF is used in the construction of this room divider.	{ "redpajama_set_name": "RedPajamaC4" }	2,039
title: Blog permalink: /blog/ --- <!DOCTYPE html> <html> {% include head.html %} <body id="page-top" class="index"> {% include blog/blog_nav.html %} <div class="content"> {% include blog/blog_header.html %} {% include blog/blog_posts.html %} </div> {% include footer.html %} {% include js.html %} </body> </html>	{ "redpajama_set_name": "RedPajamaGithub" }	8,365
\section{Introduction} Integrated circuits (ICs) are a critical layer for security in modern electronic systems. However, there are security concerns due to third parties in the supply chain. As external design houses and foundries have full access to the IC intellectual property (IP) during production, attackers could reverse engineer the IP for malicious purposes, such as IP theft and hardware Trojan insertion~\cite{10.1145/3060403.3060495}. Design-for-trust methodologies aim to counteract such threats. Logic locking has been recognized as a premier technique to safeguard ICs throughout the supply chain~\cite{10.1145/3342099}. Logic locking builds on the concept of design obfuscation~\cite{yasin2020trustworthy, EPIC2008, 7362173, 8203496,tan2020benchmarking}, where designers insert key-driven logic to functionally and structurally alter ICs, thus concealing functional intent. Only the correct activation key unlocks the intended functionality of the IC. Recently, machine learning (ML) techniques have challenged the security of gate-level locking~\cite{sisejkovicVLSISoC21Survey}. ML-driven attacks exploit the predictable relation between the key value and the functional or structural aspects of locking. This has led to potent attacks that can either predict the correct key value or remove the locking circuitry from the netlist~\cite{OMLA2021,gnnunlock2021, sisejkovicJETCSnapShot2021, sailAttack2021}. While ML-driven attacks often lack output certainty, their applicability adds another requirement for logic locking success---\textit{prevention of key-related residue within the locking mechanics}. As long as the structural change is related to key values, it is possible to use ML to guess the keys. \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{motivation_small.eps} \caption{Machine learning vs. logic locking: impact on RTL?} \label{fig:motivation} \end{figure} Traditional gate-level locking schemes are limited to local changes and do not use the semantic information of the circuit, as logic synthesis and optimization disperse the semantics to a low granularity. Therefore, gate-level locking schemes operate ``blindly'' on the design without considering its functional traits. In response, RTL locking has emerged as a way to overcome this issue~\cite{pilatoTVLSI2020assure, 5401214}. At the RTL, locking can use the full spectrum of semantic information, including operations, constants, and control flow constructs. Hence, RTL locking is a promising basis to build ML-resilient locking. However, compared to gate-level locking, ML attacks on RTL locking remain unexplored as shown in Fig.~\ref{fig:motivation}. \textbf{Contributions:} Thus, this study explores ML resilience of RTL locking focusing on \textit{operation obfuscation}, where we: \begin{itemize} \item Introduce theoretical concepts to evaluate ML-resilience of RTL locking. \item Expose security faults in ASSURE RTL locking~\cite{pilatoTVLSI2020assure}. \item Define ML-resilience security metrics for RTL locking. \item Introduce two ML-resilient locking algorithms: (1) ERA{}: \textbf{E}xact ML-\textbf{R}esilient \textbf{A}lgorithm and (2) HRA{}: \textbf{H}euristic ML-\textbf{R}esilient \textbf{A}lgorithm. \item Evaluate the locking algorithms against an RTL adaptation of the ML-based SnapShot attack. \end{itemize} To the best of our knowledge, the presented concepts and locking procedure are the first to address the challenges of ML resilience on RTL. \textbf{The implementation of this study will be made available to the community once published.} \section{Background} \subsection{Threat Model} Since ML-driven attacks do not need a working chip (oracle) to succeed~\cite{sisejkovicJETCSnapShot2021}, our threat model includes the following assumptions. (1) The attacker has only access to the locked design in the form of a locked gate-level netlist or GDSII layout file. As a working copy of the locked chip is not available, this attack model is often referred to as \textit{oracle-less} (OL). Starting from the provided design level, the attacker can perform reverse engineering to recover the RTL design. To simulate the \textit{best-case scenario for the attacker}, we assume the attacker can retrieve an \textit{exact copy} of the initial, locked RTL design. (2) The attacker is aware of the algorithmic details of the applied locking scheme. (3) The location of the key inputs (pins) is known (i.e., \textit{distinct ambiguity}~\cite{10.1145/3342099}). In the rest of this study, we refer to the locked RTL design under attack as the \textit{target}. \subsection{ML-Driven Structural Attacks}\label{ml-driven-struc-attacks} In the OL model, an attacker has only access to the target design without I/O patterns. An ML-based attack has to exploit \textit{structural} key-related patterns to produce a (correct) key prediction. Thus, we selected the OL SnapShot attack~\cite{sisejkovicJETCSnapShot2021} as a basis for the evaluation. SnapShot was initially designed to attack locked gate-level netlists, following four major steps (Fig.~\ref{fig:snapshot}). First, the attack prepares a set of locked samples by \textit{relocking} (self-referencing) the target benchmarks with new keys. Second, a training set is assembled by extracting a netlist sub-graph for each single-bit key input from all data samples. The extracted sub-graphs are transformed into a vector of numbers, where each entry encodes a single gate from the derived sub-graphs. These vectors are referred to as \textit{localities}. In essence, \textit{a locality represents a key-affected portion of the netlist.} Next, the attack trains a dedicated ML model to associate localities with their respective key values. Finally, the trained ML model is deployed to predict the key of the target design. Since SnapShot has previously only been applied on gate level, we adjust the extraction and ML model of SnapShot to support RTL locking in this work. Alongside SnapShot, the most prominent OL, ML-based attacks on gate-level locking are OMLA~\cite{OMLA2021}, GNNUnlock~\cite{gnnunlock2021}, and SAIL~\cite{sailAttack2021}. OMLA and GNNUnlock use graph neural networks, thus relying on a graph representation of the input design that is natural to gate-level netlists. SAIL exploits the deterministic and local changes of gate-level locking by learning to reverse the transformations induced by logic synthesis. Since we operate on RTL and assume a perfect reconstruction of the locked RTL, SAIL is not considered. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{snapshot2.eps} \vspace{-0.28in} \caption{ML-driven SnapShot attack flow.} \label{fig:snapshot} \vspace{-0.1in} \end{figure} \subsection{The Concept of RTL Locking} To protect the design, we focus on the locking techniques proposed in ASSURE~\cite{pilatoTVLSI2020assure}---one of the latest RTL locking policies. ASSURE offers three locking techniques: constant, branch, and operation obfuscation. Constant obfuscation extracts constants into the activation key. For example, \texttt{a = 4'b1101} is locked as \texttt{a = $K$}, where \texttt{$K$} is the 4-bit constant stored as the key. Branch locking works by XOR-ing the condition of the branch with a key bit, thereby inverting the condition if the value is 1. For example, the condition \texttt{a $>$ b} is locked as \texttt{(a <= b)$\wedge{} K$}. Operation locking works by inserting a key-controlled multiplexer to choose between a real and dummy operation. For example, the expression \texttt{a = b + c} can be locked either as \texttt{a = $K$ ? (b + c) : (b - c)} or \texttt{a = $K$ ? (b - c) : (b + c)}, depending on the value of \texttt{$K$}. In terms of security, constant obfuscation does not offer any apparent attack vectors, as the secret is fully omitted from the attacker. Branch obfuscation only affects the existing control flow based on the key, without inserting additional logic. Operation obfuscation manipulates the design by inserting additional logic \textit{depending on the existing one}. This dependence offers the potential for an attack. Therefore, \textit{we focus on the security of operation obfuscation}. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{ASSURE_locking_example2.eps} \vspace{-0.28in} \caption{ASSURE operation locking and representation.} \label{fig:assure-example} \vspace{-0.2in} \end{figure} \begin{figure}[ht] \centering \includegraphics[width=0.95\linewidth]{ml_analysis2.eps} \vspace{-0.12in} \caption{Impact of operation selection on learning resilience in RTL locking.} \label{fig:ml-analysis} \vspace{-0.08in} \end{figure} \textbf{Operation obfuscation:}~The security of this locking concept lies in the assumption that the attacker cannot guess which operation of the observed pair is the correct one. The paired real and dummy operations are called \textit{locking pairs}. In the general case, locking pairs are defined as $(T, T')$, where $T$ and $T'$ are the real and dummy operations, respectively. On RTL, a locking pair is typically implemented in the form of a \textit{ternary operator}. For instance, as depicted in Fig.~\ref{fig:assure-example}a, the real operation $+$ can be locked in the form of $(+, -)$ for the correct key value 1, or in the form of $(-, +)$ for the correct key value 0. Hence, an addition is always locked in pair with a subtraction, and vice versa. Note that all other operations have predefined locking pairs~\cite{pilatoTVLSI2020assure}. In case an already locked pair is relocked, both $T$ and $T'$ are locked separately. As shown in Fig.~\ref{fig:assure-example}b, relocking results in a tree of multiplexers, i.e., nested ternary operators. The compact notation of locked pairs from Fig.~\ref{fig:assure-example} is used in the rest of this study for visualization purposes. \section{Learning Resilience for RTL Locking}\label{learning-res-for-rtl} Learning attacks make predictions about the key by studying the locked design. A scheme that is secure against learning attacks is considered \textit{learning-resilient}~\cite{sisejkovicTCADDeceptive2021}. As discussed in~\cite{sisejkovicTCADDeceptive2021}, netlists that exhibit regular, repetitive structures \textit{maximize} the exposure of a locking scheme's mechanics, making it is easy to identify potential leakage points. This is because key-related structural changes are more likely to be identified within repetitive constructs. To evaluate RTL locking for structural leakage, let us consider its workings on a structurally regular design that only contains connected $+$ operations. The following challenge arises: \textit{how do we lock it without suggesting anything about the correctness of the key?} Let us consider the two methods of operation selection in ASSURE: \textit{serial} and \textit{random}. Furthermore, we need to take two data sets into account: test and training. The test set consists of locking samples for which the key is \textit{unknown} and represents the design under attack. The training set comprises locking samples that are added in additional relocking rounds of the target with \textit{known} keys. This process is also known as self-referencing~\cite{sailAttack2021, sisejkovicJETCSnapShot2021}. The attacker uses the training set to collect observations about \textit{the relation between the locking pairs and the key}. The ensuing discussion follows visualizations in Fig.~\ref{fig:ml-analysis}, where we consider different locking scenarios using locking pairs and symbols from Fig.~\ref{fig:ml-analysis}a, and the $+$ operation network. \textbf{Serial selection}~is the standard selection in ASSURE. As shown in Fig.~\ref{fig:ml-analysis}b, the initial locking (test set) selects $+$ operations for locking to create locking pairs in the form of $(+, -)$ and $(-, +)$ (encoded with a single symbol for simplicity). This "serial" selection always selects the operations in a serial manner w.r.t. the design topology. Due to the serial selection, the subsequent locking rounds (training set) select the \textit{same} operations as the test set for relocking; already-locked $+$ operations are extended with additional locking pairs. In the example, the left operation is selected for key value 1 and the right for key value 0, according to the rule of ternary expressions. As portrayed in Fig.~\ref{fig:ml-analysis}e, both the $+$ and $-$ operations \textit{are equally related to the key value 0 and 1}, resulting in confusing observations. This suggests that ASSURE is, in principle, learning-resilient. However, this case arises only due to the deterministic order of selecting operations---it can easily be broken by either using a longer training key to ensure locking untouched operations during training or by randomizing the order of selection. Hence, the standard ASSURE procedure \textit{is not secure} w.r.t. data-driven attacks. \textbf{Random selection} is depicted in Fig.~\ref{fig:ml-analysis}c. Here, the samples from the test and training set are likely to overlap only to some extent. Hence, the observations of the training set are contaminated by some \textit{contradictory} observations. By analyzing the training set (Fig.~\ref{fig:ml-analysis}f), one can learn that the $+$ operation \textit{is more likely to be the correct one}. The random selection might result in a favorable outcome for the attacker when training and test samples do not overlap (Fig.~\ref{fig:ml-analysis}d). In this scenario, all observations from the training set (Fig.~\ref{fig:ml-analysis}g) suggest that the $+$ operation \textit{is always the correct one}. This knowledge can be used to infer a correct key. \subsection{Observations} $(1)$ Learning resilience on RTL can be achieved if the likelihood of any operation in a locking pair is equally related to key value 0 and 1. $(2)$ Operation selection impacts learning resilience. $(3)$ The initial distribution of operation types determines if learning-resilience is achievable. Evidently, the effectiveness of learning-resilient locking \textit{should not depend on circuit features}. Even if real-world designs are not represented by the $+$ network, focusing on this biased case ensures that the scheme offers security even in general cases. Based on the above observations, we can conclude that \textit{learning resilience on RTL is achievable if the occurrence frequency of every operation within a locking pair is equal for all operations in the pairings}. This is the case if the design has the same number of $+$ and $-$ operations after locking. In that case, any selection procedure for training results in an equal number of contradictory observations. In the next section, we introduce two locking algorithms that use this rule for learning resilience on RTL. \subsection{ASSURE Leakage Points} We analyzed the serial selection of ASSURE~\cite{pilatoTVLSI2020assure}, and the \textit{current pairing of operations is leaky} as operations are incorrectly paired. For example, ASSURE assumes these pairs: $(, +)$, $(+,-)$, and $(-,+)$. Here, is paired with a +, but + is also paired with -. Hence, if the locked pair $(, +)$ is encountered, the attacker can infer as the correct operation, as $(+, )$ does not exist. Similarly, leakage exists for modulo, xor, power, and division. Thus, \textit{currently ASSURE can be broken by analyzing operation pairs}. Hence, every operation must exist as a real and dummy operation with the same pair, e.g., $(, /)$ and $(/, )$. This fix applies to all evaluations in this study. \section{ML-resilient RTL Locking} Based on the discussion, we introduce the following definition: \begin{definition}\label{learning-resilience-def} An RTL design is \textbf{learning-resilient} w.r.t. operation locking if the number of operations of type $T$ is equal to the number of operations of type $T'$ for each locking pair for which at least one operation of type $T$ or $T'$ is locked. \end{definition} If neither $T$ nor $T'$ are involved in locking, the locked design is learning-resilient even if the number of $T$ and $T'$ operations is not balanced. The reason is that, during training, locking ``untouched'' $T$ and $T'$ operations does not provide feedback for the target samples. Henceforth, \textit{secure} refers to security in the context of Def.~\ref{learning-resilience-def}. Next, we introduce two ML-resilient locking algorithms built on top of ASSURE: ERA{}: \textbf{E}xact ML-\textbf{R}esilient \textbf{A}lgorithm and \textbf{H}euristic ML-\textbf{R}esilient \textbf{A}lgorithm. ERA{} guarantees security w.r.t. Def.~\ref{learning-resilience-def}, but requires a large key budget. HRA{} trades-off key length with security, yielding less secure solutions if the key budget is limited. \setlength{\textfloatsep}{8pt} \begin{algorithm}[t] \footnotesize \DontPrintSemicolon \KwIn{Locking type $T$, operation distribution table $ODT$, RTL design D, and pair mode P} \KwOut{Number of used bits} $n \gets 0$\tcp{Initialize used bits var} $T' \gets \Call{GetPairType}{T}$\; $o_{i} \gets \Call{RndSelect}{D.ops[T]}$\tcp{Select a T-type op.} $o_{j} \gets \Call{RndSelect}{D.ops[T']}$\; \uIf{$ODT[T] > 0$ \textbf{and} $!P$}{ $\Call{AddPair}{D, o_{i}, T'}$\tcp{Add T' node to $o_{i}$} $ODT[T] \gets ODT[T] - 1$\; $ODT[T'] \gets ODT[T'] + 1$\; $n \gets n + 1$\; } \uElseIf{$ODT[T] < 0$ \textbf{and} $!P$}{ $\Call{AddPair}{D, o_{j}, T}$\tcp{Add T node to $o_{j}$} $ODT[T] \gets ODT[T] + 1$\; $ODT[T'] \gets ODT[T'] - 1$\; $n \gets n + 1$\; } \Else{ $\Call{AddPair}{D, o_{i}, T'}$\tcp{Add T' node to $o_{i}$} $\Call{AddPair}{D, o_{j}, T}$\tcp{Add T node to $o_{j}$} $n \gets n + 2$\; } \Return{$n$} \; \caption{\sc{Lock}} \label{alg:lockingstep} \end{algorithm} \textbf{Operation distribution:}~The first step in ERA and HRA is to analyze operation distribution in the input RTL. We store this information in an \textit{operation distribution table} ($ODT$). For each $T$, the table stores a number representing the difference between the distribution of $T$-type and the locking-pair $T'$-type operations. Assuming the pair $(+,-)$, a design with 7 "$+$" and 5 "$-$" has the following $ODT$ entries: $ODT[+] = +2$ and $ODT[-] = -2$. A positive (negative) $ODT$ value indicates that the operation type has more (less) operations than its locking-pair type. The $ODT$ entries can inform a \textit{secure} design by balancing the number of $T$ and $T'$ operations. \textbf{The locking step:}~Algorithm~\ref{alg:lockingstep} outlines \textsc{Lock}, the common locking step for HRA{} and ERA{}. For a selected type $T$, the RTL $D$ is locked following three cases. If $ODT[T]$ is positive (lines 6-9), pair a new $T'$-type operation with an existing $T$-type to reduce the excess of $T$. If $ODT[T]$ is negative (lines 11-14), pair a new $T$-type operation with an existing $T'$-type to reduce the deficiency of $T$. Otherwise (lines 16-18), pair new $T$- and $T'$-type operations with existing operations. This is used by \textit{specific operation-selection algorithms} to derive HRA{} and ERA{}. Before describing the locking algorithms, we introduce a security metric for resilience w.r.t. Def.~\ref{learning-resilience-def}. \subsection{Security Metric for Learning Resilience}\label{metric} $ODT$ entries can be used as a vehicle to \textit{measure security} in the context of Def.~\ref{learning-resilience-def}. To design a metric that indicates how "far" a locked design is from the optimal distribution, let us consider the following notation. The content of $ODT$ in iteration $j$ of a selected locking algorithm can be represented as the vector $\bm{v_{j}} = [x_{0}, \ldots, x_{l-1}]$, where $l$ is the number of available locking pairs, and $x_{i}=\abs{ODT[T]}$. Note that $\abs{ODT[T]} \equiv \abs{ODT[T']}$. A secure solution is reached if all entries of $ODT=0$. Hence, the optimal distribution can be defined as $\bm{v_{o}}=[y_{0}, \ldots, y_{l-1}]$, where $y_{i}=0$ for $i\in{[0,l-1]}$. Using this notation, we can define the learning-resilience security metric as: \begin{equation} M_{sec}= 100 \cdot{\left(1 - \frac{d_{e}(\bm{v_{j}},\bm{v_{o}})}{d_{e}(\bm{v_{i}},\bm{v_{o}})}\right)}, \end{equation} where $d_{e}$ is a modified version of the Euclidean distance, $\bm{v_{i}}$ the initial distribution vector of the target design, $\bm{v_{o}}$ the optimal distribution vector, and $\bm{v_{j}}$ the distribution vector after the $j$-th locking iteration. Note that $M_{sec}\in{[0,100]}$, where the highest value indicates $\bm{v_{j}} \equiv \bm{v_{o}}$. In that case, all locking-pair operation types are equally represented within the locked design, disabling the ability of ML to learn from relocking (as discussed in Section~\ref{learning-res-for-rtl}). Furthermore, the formulation of the Euclidean distance was adjusted as presented in Algorithm~\ref{alg:e-distance}. For a selected $\bm{v_{o}}$, the algorithm allows the exclusion of selected $\abs{ODT}$ values from the calculation, enabling two metric variants: \textit{restricted} and \textit{global} learning resilience. \MakeRobust{\Call} \begin{algorithm}[t] \footnotesize \DontPrintSemicolon \KwIn{Current vector $\bm{v_{j}}$ and optimal vector $\bm{v_{o}}$} \KwOut{Distance} $s \gets 0$\tcp{Initialize sum var} \For{$i \gets 0; i < \abs{\bm{v_{o}}}; i++$}{ \tcc{Check if value should be considered} \uIf{$\bm{v_{o}} \neq $ 'x'}{ $s \gets s + (\bm{v_{o}}[i] - \bm{v_{j}}[i])^{2}$\; } } \Return{$\sqrt{s}$} \; \caption{$d_{e}$: Modified Euclidean Distance} \label{alg:e-distance} \end{algorithm} \textbf{Global security metric} ($M_{sec}^{g}$) considers all $ODT$ entries to determine $d_{e}$, regardless of whether operations from a selected locking pair are affected by locking or not. This metric is suitable to guide heuristics when it is not clear which operation types will be locked. Thus, $M_{sec}^{g}$ describes \textit{the potential for exploitation within a design.} Since $M_{sec}^{g}$ considers all $ODT$ values, $\bm{v_{o}}$ does not contain any 'x' values. Hence, $M_{sec}^{g}$ is monotonic. \textbf{Restricted security metric} ($M_{sec}^{r}$) only considers $ODT$ entries in which either $T$ or $T'$ are affected by locking. The reason is that an ML model cannot learn from operations from a selected locking pair if neither $T$ nor $T'$ operations are locked. In this sense, $M_{sec}^{r}$ captures the security of the design when only considering locked operations, i.e., \textit{the actual exploitability of the design}. If a selected locking pair is included during a locking procedure, certain 'x' values in $\bm{v_{o}}$ are set to 0. Thus, $M_{sec}^{r}$ is not monotonic. If all types in $ODT$ are affected by locking, $M_{sec}^{r}\equiv M_{sec}^{g}$. Furthermore, $M_{sec}^{r}=100$ does not imply $M_{sec}^{g} = 100$, since some operation types are not affected by locking. However, if $M_{sec}^{g} =100$ then $M_{sec}^{r} = 100$. These metrics are used by the locking algorithms. \MakeRobust{\Call} \begin{algorithm}[t] \footnotesize \DontPrintSemicolon \KwIn{Key budget $k_{b}$ and RTL design $D$} \KwOut{Locked RTL design} $\Call{LoadODT}{D}$\tcp{Populate ODT} $n \gets 0$\tcp{Initialize used bits var} $\Theta \gets \{(T_{1}, T'_{1}), \ldots, (T_{n}, T'_{n})\}$\tcp{Valid locking pairs} \While{$n < k_{b}$}{ $\vartheta \gets \Call{RndSelect}{\Theta}$\tcp{Select a pair} $T \gets \Call{RndSelect}{\vartheta}$\tcp{Select a type} \While{$\abs{ODT[T]} > 0$}{ $s\gets \Call{Lock}{T, ODT, D, False}$\tcp{Apply lock (Algorithm 1)} $n \gets n + s$\; } } \Return{$D$} \; \caption{ERA{}: Exact ML-Resilient Algorithm} \label{alg:exact-alg} \end{algorithm} \subsection{\textbf{E}xact ML-\textbf{R}esilient \textbf{A}lgorithm (ERA{})} ERA{} (Algorithm~\ref{alg:exact-alg}) ensures that locking always yields a secure design even if the key budget is exceeded. While the key budget is not exceeded (line 4), ERA{} randomly selects a type $T$ from a randomly selected pair $\vartheta$ from valid locking pairs $\Theta$ (lines 5-6). To ensure a secure solution after each selection, the algorithm repeats the locking for the selected type until $ODT[T]$ reaches 0 (lines 7-10). This way the selected operation pairs yield a balanced solution. Thus, $M_{sec}^{r}=100$ after each locking round even if the cost is more than allowed. Hence, ERA{} prioritizes security over cost. ERA{} always locks all selected pairs until $ODT[T]$ reaches 0---all affected pairs are guaranteed to be balanced. The security evaluation of an ERA{}-locked design will result in $M_{sec}^{r}=100\%$, but not necessarily in $M_{sec}^{g}=100\%$. The former states that all affected pairs are perfectly balanced. The latter indicates that other parts of the design are exploitable by ML \textit{if not locked properly} (if $M_{sec}^{g}<100\%$). \begin{algorithm}[t] \footnotesize \DontPrintSemicolon \KwIn{Key budget $k_{b}$ and RTL design $D$} \KwOut{Locked RTL design} $\Call{LoadODT}{D}$\tcp{Populate ODT} $n \gets 0$\tcp{Initialize used bits var} $\bm{v_{i}} \gets \Call{ExtractVector}{D.ODT}$\tcp{Initial vector} $\Theta \gets \{(T_{1}, T'_{1}), \ldots, (T_{n}, T'_{n})\}$\tcp{Valid locking pairs} \While{$n < k_{b}$}{ $M_{sec}^{g} \gets 0$\tcp{Track max metric value} $j \gets 0$\tcp{Track operation index} $P \gets \Call{RndBoolean}{\null}$\tcp{Include randomness} \uIf{$P$}{ $j \gets \Call{RndSelect}{\abs{\Theta}}$\; } \Else{ $\Call{Shuffle}{\Theta}$\; \For{$i \gets 0; i < \abs{\Theta}; i++$}{ $\Call{Lock}{\Theta[i][0], ODT, D, False}$\; $\bm{v_{j}} \gets \Call{ExtractVector}{D.ODT}$\; $M_{i} \gets \Call{EvalMetric}{\bm{v_{i}}, \bm{v_{j}}}$\; $\Call{UndoLock}{D}$\tcp{Undo last lock} \uIf{$M_{i} > M_{sec}^{g}$}{ $M_{sec}^{g} \gets M_{i}$\; $j \gets i$\; } } } $s\gets \Call{Lock}{\Theta[j][0], ODT, D, P}$\tcp{Apply lock (Algorithm 1)} $n \gets n + s$\; } \Return{$D$} \; \caption{HRA{}: Heuristic ML-Resilient Algorithm} \label{alg:heuristic-alg} \end{algorithm} \subsection{\textbf{H}euristic ML-\textbf{R}esilient \textbf{A}lgorithm (HRA{})} HRA (Algorithm~\ref{alg:heuristic-alg}) performs iterative fine-grained balancing of locking-pairs in the target design to get closer to the secure solution at every step without exceeding the key budget. While key bits are available (line 5), HRA{} randomly chooses (line 8). Either a random operation type is chosen (line 10) or the best type is chosen (lines 12-21). The latter case evaluates all locking pairs in $\Theta$ and checks which one yields the \textit{highest} increase in $M_{sec}^{g}$. In both cases, the selected pair is locked by the \textsc{Lock} function (line 23). As HRA{} performs fine-grained design adjustments, it uses the exact key budget and trades off against the guarantee to reach a secure solution. Since HRA{} ensures that every step increases security and decreases operation imbalances, it must be guided by the monotonic $M_{sec}^{g}$ metric. \begin{figure}[t!] \centering \includegraphics[width=\columnwidth]{metric_visualization.eps} \vspace{-0.25in} \caption{Security metrics: (a) search space and (b) evolution.} \label{fig:metric} \end{figure} \begin{figure}[ht] \centering \subfloat[KPA per benchmark]{ \begin{tikzpicture}[scale=0.49] \pgfplotstableread[row sep=\\,col sep=&]{ bench & v1 & v2 & v3 \\ 1 & 75.00 & 65.00 & 40.00 \\ 2 & 66.00 & 48 & 54.00 \\ 3 & 89.00 & 89.00 & 42.00\\ 4 & 66.00 & 53.00 & 49.00 \\ 5 & 83.00 & 85.00 & 57.00 \\ 6 & 68.00 & 58.00 & 51.00 \\ 7 & 53.00 & 54.00 & 59.00 \\ 8 & 99.00 & 100 & 47.00\\ 9 & 95.00 & 95.00 & 48.00 \\ 10 & 68.00 & 79.00 & 36.00 \\ 11 & 82.00 & 82.00 & 47.00 \\ 12 & 63.00 & 89.00 & 40.00\\ 13 & 90.00 0 & 100.00 & 51.00 \\ 14 & 50.00 0 & 50.00 & 50.00 \\ }\dataset \begin{axis}[ybar, ylabel={KPA (\%)}, xlabel={Benchmarks}, ylabel style={at={(-0.001,0.5)},anchor=north}, legend style={at={(0.1,0.96), font=\Large}, anchor=north,legend cell align={left}}, legend image post style={scale=1.4}, enlarge x limits={abs=0.8cm}, ymajorgrids = true, bar width=.38cm, width=30cm, height=4.3cm, ymin=0,ymax=140, ytick={0, 25, 50, 75, 100}, xtick align=inside, xtick={1,2,3,4,5, 6, 7,8, 9, 10, 11,12,13,14}, xticklabel style={align=center}, xticklabels = { {\tt DES3}, {\tt DFT}, {\tt FIR}, {\tt IDFT}, {\tt IIR}, {\tt MD5}, {\tt RSA}, {\tt SHA256}, {\tt SASC}, {\tt SIM\_SPI}, {\tt USB\_PHY}, {\tt I2C\_SL}, {\tt N\_2046}, {\tt N\_1023}, }, ylabel style ={font=\Large}, xlabel style ={font=\Large}, tick label style={font=\large}, legend entries={ASSURE, HRA, ERA}, legend columns=3, ] \addplot[blue(pigment),thick, sharp plot,dashed, update limits=false,forget plot] coordinates {(0,50) (15,50)} node[above] at (axis cs:13.7,50) {\Large Random guess}; \addplot[draw=black,fill=white] table[x=bench,y=v1] \dataset; \addplot[draw=black,fill=white, postaction={pattern=north east lines}] table[x=bench,y=v2] \dataset; \addplot[draw=black,fill=gray!80] table[x=bench,y=v3] \dataset; \end{axis} \label{fig:sim_ambient_temp} \end{tikzpicture} } \subfloat[Average KPA]{ \begin{tikzpicture}[scale=0.49] \begin{axis}[ybar, ylabel={KPA (\%)}, xlabel={RTL locking algorithms}, ylabel style={at={(-0.004,0.5)},anchor=north}, legend style={at={(0.5,0.96), font=\Large}, anchor=north,legend cell align={left}}, legend image post style={scale=1.4}, enlarge x limits={abs=1cm}, ymajorgrids = true, bar width=.38cm, width=6.3cm, height=4.3cm, ymin=0,ymax=140, ytick={0, 25, 50, 75, 100}, xtick align=inside, xtick={1,2,3}, xticklabel style={align=center}, xticklabels = {ASSURE, HRA, ERA}, ylabel style ={font=\Large}, xlabel style ={font=\Large}, tick label style={font=\large}, legend columns=3, every node near coord/.append style={ anchor=north, yshift=4ex, xshift=-2ex, font=\Large, rotate=90 }, every node near coord/.append style={ /pgf/number format/fixed, /pgf/number format/fixed zerofill, /pgf/number format/precision=2 }, ] \addplot[blue(pigment),thick, sharp plot,dashed, update limits=false,forget plot] coordinates {(0,50) (4,50)} node[above] at (axis cs:3,50) {}; \addplot[draw=black,fill=white,nodes near coords]coordinates {(1.2,74.78)}; \addplot[draw=black,fill=white,nodes near coords, postaction={pattern=north east lines}]coordinates {(2,74.26)}; \addplot[draw=black, nodes near coords,fill=gray!80]coordinates {(2.7,47.92)}; \end{axis} \label{fig:} \end{tikzpicture} } \vspace{-0.15in} \caption{Evaluation results for the ML-based SnapShot attack on RTL locking.} \label{fig:eval} \vspace{-0.1in} \end{figure*} \subsection{Metric-Guided Design} The proposed metrics can be used to design various locking algorithms targeting learning resilience. Let us consider a design with the following $ODT$ entries: $\abs{ODT[(+,-)]}=25$ and $\abs{ODT[(<<,>>)]}=10$. As depicted in Fig.~\ref{fig:metric}a, $M_{sec}^{g}$ represents a smooth, monotonic surface. Fig.~\ref{fig:metric}b presents the evolution of the metric in each step. The goal of locking is to move the target design from the initial point (bottom right) to the secure point (top left). The path between these two points represents different heuristic approaches. ERA{} forces selected $ODT$ values to 0, thus jumping in two steps to the secure solution alongside the edges of the surface. HRA{} travels in the steepest direction, taking small steps and remaining on the highest line across the surface. A greedy approach (same as HRA{} where $P$ in line 8 is always false) traverses the same points as HRA{}. Fig.~\ref{fig:metric}b suggests that a greedy approach is more efficient than HRA since it reaches full security (i.e., metric equal to 100) with fewer key bits. However, a greedy approach has a negative consequence: reversibility. An attacker can reverse the locking procedure alongside the steepest decreasing direction. Therefore, including random locking decisions within HRA{} (variable $P$) thwarts reversibility---even if it takes longer to get to the secure solution. Similar observations hold for $M_{sec}^{r}$ since $M_{sec}^{r}\equiv M_{sec}^{g}$ when all $ODT$ entries are affected. \section{Evaluation} We evaluate ASSURE-based locking policies against the state-of-the-art ML-based SnapShot attack on a subset of the benchmarks used in~\cite{pilatoTVLSI2020assure}. Some benchmarks were excluded due to the low number of operations. We also composed two synthetic benchmarks: \texttt{N\_2046} and \texttt{N\_1023}, representing a fully imbalanced (biased) design (a network of 2046 $+$ operations) and a fully balanced design (a network with 1023 $+$ and $-$ operations), respectively. We consider ASSURE (serial implementation), HRA{}, and ERA{}. Note that the cost of the proposed algorithms are in line with the original ASSURE, as the cost of a locking pair per key bit has not changed~\cite{pilatoTVLSI2020assure}. \textbf{SnapShot for RTL:} We adapted SnapShot (Fig.~\ref{fig:snapshot}) to learn RTL key leaks by extracting all key-controlled pairs $[K[i], C_{1}, C_{2}]$, where $K[i]$ is the key-bit value, and $C_{1}$, $C_{2}$ are encodings for an operation pair. We assign each type a unique integer. The extractor uses the Pyverilog library~\cite{Takamaeda:2015:ARC:Pyverilog}. Instead of one neural network type as in~\cite{sisejkovicJETCSnapShot2021}, we use auto-sklearn~\cite{feurer-neurips15a}, a library for automatic ML (auto-ml) model exploration. Auto-ml searches for an ML model among the implementations and optimizes the hyperparameters. We selected 600 seconds per attack iteration as this was enough for the attack to converge. \textbf{Attack setup:} The test set for each algorithm comprises every benchmark locked 10 times with different keys. We assembled the training set by \textit{relocking} each test sample 1,000 times with different keys. Relocking was performed with random ASSURE locking so that all parts of the design were used for learning; thus, simulating the most effective attack. Both test and training keys are set to 75\% of the operations for each benchmark. This was exceeded for the \texttt{N\_2046} benchmark, as its perfect imbalance requires a 100\% key budget for ERA{}. We assumed a best-case scenario for the attacker: a perfect reconstruction of the initial, locked RTL. \textbf{Accuracy metric:} Key Prediction Accuracy (KPA) is used to measure attack success~\cite{sisejkovicJETCSnapShot2021}. N\% KPA indicates that N\ of the key bits are correctly predicted. The baseline random guess has 50\% KPA. \subsection{Results and Discussion} \textbf{Results:} Fig.~\ref{fig:eval}a presents the KPA evaluation results per locking algorithm and benchmark, and Fig.~\ref{fig:eval}b presents the average KPA across all benchmarks. SnapShot correctly predicts 74.78\% key bits for the original ASSURE implementation, on average. The average KPA for HRA{} is slightly lower, 74.26\%. ERA{} averages $\sim$47.92\% KPA with consistent KPA values around (or lower) than a random guess. \textbf{Lessons learned:} SnapShot's success on HRA{} is at first surprising since it is supposed to have a higher level of security than non-ML-driven serial locking. However, since we use a key budget of 75\% of the available operations, parts of the design remain unaffected by locking. Hence, the training step \textit{can extract knowledge} about the design for an educated guess ($\sim$24 percentage points better than random). Once all operations are fully balanced---as guaranteed by ERA{}---the training fails to extract useful observations. The above leads to a significant conclusion: \textit{\textbf{when it comes to ML-driven attacks, half measures are not effective.}} Data-driven approaches can exploit even the slightest imbalance. In contrast, half-way measures can mitigate non-ML-driven attacks, e.g., slightly increasing the key length can deteriorate a brute-force attack. While HRA{} appears less promising, the heuristic is useful if multiple security objectives must be reached, such as learning-resilience, output corruptibility, and Boolean Satisfiability (SAT)-resistance~\cite{SMT2020}. Since ERA{} makes coarse-grained modifications, it might create radical changes in the design. HRA{} improves learning resilience of locked designs alongside other objectives \textit{in smaller and controlled locking steps} as it only decreases operation imbalance. \textbf{Limitations and opportunities:} This study exploits individual locking pairs---but is there a "global bias" among designs? If so, this bias could help determine the correct function of locked designs. The metric in Section~\ref{metric} can extract the initial distance for selected designs by considering the distance between the initial distribution and the optimal one. Are the locking algorithms resilient to oracle-guided attacks? Moreover, locking has recently been explored in combination with high-level synthesis~\cite{9586159, TAOLocking2018, 10.1145/3410337}. Future efforts should evaluate the problem of learning resilience on this abstraction level and address the mentioned challenges. \section{Conclusion} We introduced the first concepts on designing and evaluating RTL locking using ML-based attacks on operation obfuscation, and proposed two ML-resilient locking algorithms. The heuristic algorithm is a controlled procedure that decreases the imbalance of operations in an RTL design in small steps, adhering to the allowed key budget. The exact algorithm guarantees ML resilience but can exceed a key budget. We presented a security metric to assess resilience of RTL locking to ML attacks that can guide the design process of heuristic locking. Finally, we presented the first ML-based oracle-less attack on RTL locking by adapting the state-of-the-art SnapShot attack. \vspace{0.5em} \noindent \textbf{ACKNOWLEDGMENTS} \\ R. Karri was supported in part by ONR Award \# N00014-18-1-2058, NSF Grant \# 1526405, NYU Center for Cybersecurity, and NYUAD Center for Cybersecurity. \vspace{-0.5em} \bibliographystyle{ACM-Reference-Format}	{ "redpajama_set_name": "RedPajamaArXiv" }	3,455
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{"url":"https:\/\/zbmath.org\/?q=an:0809.20056&format=complete","text":"## Groupoids and the associative law. II: Groupoids with small semigroup distance.(English)Zbl\u00a00809.20056\n\nThis paper is a continuation of the first paper of the authors [ibid. 33, 69-86 (1992; Zbl\u00a00791.20084)]. Here, groupoids with small semigroup distance are investigated. The main results are Theorem: Let $$G$$ be a semigroup. Then $$G[a, b, c]$$ is associative for all $$a, b, c \\in G$$ iff $$\\text{card}(G) \\leq 2$$ and $$G$$ is a semilattice. Theorem: Let $$G$$ be a finite groupoid with $$n$$ elements and such that $$\\text{sdist} (G) = 1$$. Then $$1 \\leq \\text{ns} (G) \\leq 2n(n - 1)$$ and $$n^ 3 - 2n^ 2 + 2 \\leq \\text{as}(G) \\leq n^ 3 - 1$$. Moreover, if $$\\text{ns}(G) = 2n(n - 1)$$, then $$G$$ is isomorphic to one of the groupoids $$R_ n()$$, $$S_{n,1}()$$, $$S_{n,2}()$$, (to $$R_ 2()$$ if $$n = 2$$). This part seems to be new. Not much is known about the semigroup distance of (finite) groupoids and this topic would deserve a more detailed study.\n\n### MSC:\n\n 20N02 Sets with a single binary operation (groupoids)\n\n### Keywords:\n\ngroupoids; semigroup distance; semilattice; finite groupoid\n\nZbl 0791.20084\nFull Text:","date":"2022-08-14 12:51:07","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\": 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\": 0.9647941589355469, \"perplexity\": 428.8008737754382}, \"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-2022-33\/segments\/1659882572033.91\/warc\/CC-MAIN-20220814113403-20220814143403-00074.warc.gz\"}"}	null	null
Q: Node: 'Cast_1' Cast string to float is not supported [[{{node Cast_1}}]] [Op:__inference_train_function_24202] for Roberta I am getting this problem called " Node: 'Cast_1' Cast string to float is not supported [[{{node Cast_1}}]] [Op:__inference_train_function_24202] " enter image description here I wrote a code about IMDB sentiment analysis for 5000 data in Google Colab #importing the necessary libraries import pandas as pd import numpy as np from sklearn.model_selection import train_test_split from tensorflow.keras.preprocessing.sequence import pad_sequences from tensorflow.keras.preprocessing.text import Tokenizer #importing the data data = pd.read_excel('/content/drive/MyDrive/499A_Project/Dataset/IMDB5000.xlsx') #spliting the dataset into train and test train_data, test_data = train_test_split(data, test_size = 0.3, random_state = 42) #preprocessing the data #tokenizng the text tokenizer = Tokenizer() tokenizer.fit_on_texts(data['Review'].values) #converting the text into sequences train_sequences = tokenizer.texts_to_sequences(train_data['Review'].values) test_sequences = tokenizer.texts_to_sequences(test_data['Review'].values) #padding the sequences max_length = max([len(s.split()) for s in data['Review']]) train_padded = pad_sequences(train_sequences, maxlen = max_length) test_padded = pad_sequences(test_sequences, maxlen = max_length) #preparing the labels train_labels = train_data['Sentiment'].values test_labels = test_data['Sentiment'].values #importing Roberta model from transformers from transformers import TFBertForSequenceClassification #instantiating the Roberta model model = TFBertForSequenceClassification.from_pretrained('roberta-base') #compiling the model model.compile(loss = 'sparse_categorical_crossentropy', optimizer = 'adam', metrics = ['accuracy']) #training the model model.fit(train_padded, train_labels, batch_size = 32, epochs = 10, validation_data = (test_padded, test_labels)) This is the code I wrote for my dataset but it is not working and show the erros	{ "redpajama_set_name": "RedPajamaStackExchange" }	9,617
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\section{#1}} \linespread{1.2} \global\arraycolsep=1truept \newcommand\freccia[1]{\accentset{\leftrightarrow}{#1}} \begin{document} \bigskip \phantom{C} \vskip 2truecm \begin{center} {\huge \textbf{On The Nature}} \vskip.4truecm {\huge \textbf{Of The Higgs Boson}} \vskip1truecm \textsl{Damiano Anselmi} \vskip .1truecm \textit{Dipartimento di Fisica ``Enrico Fermi'', Universit\`{a} di Pisa, } \textit{Largo B. Pontecorvo 3, 56127 Pisa, Italy} \textit{and INFN, Sezione di Pisa,} \textit{Largo B. Pontecorvo 3, 56127 Pisa, Italy} damiano.anselmi@unipi.it \vskip1truecm \textbf{Abstract} \end{center} Several particles are not observed directly, but only through their decay products. We consider the possibility that they might be fakeons, i.e. fake particles, which mediate interactions but are not asymptotic states. A crucial role to determine the true nature of a particle is played by the imaginary parts of the one-loop radiative corrections, which are affected in nontrivial ways by the presence of fakeons in the loop. The knowledge we have today is sufficient to prove that most non directly observed particles are true physical particles. However, in the case of the Higgs boson the possibility that it might be a fakeon remains open. The issue can be resolved by means of precision measurements in existing and future accelerators. \vfill\eject The Higgs boson has unique features. For example, it is a scalar field, unlike every other field of the standard model. Its key role is to trigger a crucial mechanism that gives masses to the particles. While it solves many problems, it leaves other questions unanswered. In this paper we study the possibility that it might hide a little secret. Specifically, the Higgs boson might be a \textquotedblleft fake particle\textquotedblright , i.e. an entity that resembles a true particle in various physical processes, but cannot be observed directly. In quantum field theory, the poles of the free propagators are usually quantized by means of the Feynman prescription \cite{peskin}. In that case, they describe physical particles. An alternative quantization prescription is able to quantize them as fake particles \cite{LWgrav}, or \textquotedblleft fakeons\textquotedblright\ \cite{fakeons}. The fakeons are important in quantum gravity, because they allow us to build a consistent theory that is both unitary and renormalizable \cite{LWgrav} (see also \cit {UVQG,absograv}). A fakeon simulates a physical particle when it mediates interactions or decays into physical particles. However, it is not an asymptotic state, because unitarity requires to project the fakeons away from the physical spectrum. In other words, a fakeon cannot be detected directly. An important physical prediction due to the fakeons is the violation of microcausality, which occurs at energies larger than their masses. Quantum gravity predicts that at least one fakeon exists in nature \cit {LWgrav}. It has spin 2, it is described by a symmetric tensor $\chi _{\mu \nu }$ and its mass $m_{\chi }$ could be much smaller than the Planck mass. Its free propagator has a negative residue at the pole, so $\chi _{\mu \nu }$ is a \textquotedblleft fakeon minus\textquotedblright\ \cite{causalityQG} and its dynamically generated width $\Gamma _{\chi }$ is negative. The spin-2 gravifakeon is necessary to make the quantization of gravity consistent. In other sectors of high-energy physics, like the standard model in flat space, as well as its extensions, there might be no need of fake particles. However, if one fakeon exists in nature, it might not be the only one. Are there any other fakeons, maybe in the realm of the standard model? In this paper we provide enough arguments to exclude this possibility for most particles, but the cases of the Higgs boson and a few other particles remain unresolved. Standard model extensions can be built by adding physical and fake particles, as long as they satisfy the conditions for the cancellation of the gauge anomalies and their masses are large enough to avoid conflict with the data. We do not explore these possibilities here, although they might have interesting applications. Instead, we inquire whether the particles that have already been identified so far are physical or fake. Some particles, like the photon and the electron, are observed directly, so they are physical. Several other particles have not been observed directly, and probably will not be for a long time. We can mention the intermediate bosons, the Higgs boson, the quarks, the gluons and the neutrinos. All of these are potentially fakeons. Since the definition of direct observation of a particle is to some extent debatable, we prefer to determine the true nature of all particles, including the photon and the electron, by means of indirect, more objective methods. We show that, as of today, we have enough data to ensure that most particles are physical. However, we are unable to settle the matter in the cases of the Higgs boson,\ the top quark, the gluons and the right neutrinos. With the exception of the right neutrinos, the missing answers can be provided by precision measurements to be made in existing and/or future accelerators. Before proceeding, let us recall a few properties. Physical and fake particles are quantized by means of different prescriptions, which are the Feynman prescription and the fakeon prescription. Introducing infinitesimal widths $\epsilon $ and $\mathcal{E}$, the unprescribed propagator 1/(p^{2}-m^{2})$ is turned into \begin{equation} \frac{1}{p^{2}-m^{2}+i\epsilon },\qquad \frac{p^{2}-m^{2}}{(p^{2}-m^{2})^{2} \mathcal{E}^{4}}, \label{prescr} \end{equation respectively. Note that, by convention, $\epsilon $ and $\mathcal{E}$ have different dimensions. The fakeon propagator vanishes on shell (which means for $p^{2}=m^{2}$) for every $\mathcal{E}>0$. This is a sign that is does not propagate a physical particle. Instead, the Feynman propagator blows up on shell. Off-shell, for \ $\|p^{2}-m^{2}\|\gg \epsilon ,\mathcal{E}^{2}$, the two prescriptions are equivalent. The sign in front of the Feynman propagator must be positive, otherwise it propagates a ghost, instead of a physical particle. Instead, the sign in front of the fakeon propagator can be either positive or negative, which distinguishes the \textquotedblleft fakeon plus\textquotedblright\ from the \textquotedblleft fakeon minus\textquotedblright\ \cite{causalityQG}. The dynamically generated width $\Gamma $ of a fakeon plus (minus) is positive (negative). In the right expression of (\ref{prescr}) the poles of $1/(p^{2}-m^{2})$ are split into pairs of complex conjugates poles. Inside the Feynman diagrams, the loop energy $p^{0}$ must be integrated along a path that passes under the left pair and over the right pair. The fakeon prescription needs to be specified by a number of other instructions to compute the loop integrals, which we do not review here and have been recently summarized in ref. \cite{causalityQG}. We mention, in particular, the average continuation, which is a nonanalytic operation to circumvent the thresholds that involve fakeons. In the end, the most important property is that the fake particles can be consistently projected away from the physical spectrum. At the tree level the difference between the two options (physical or fake) is just the projection, which has no impact on the particles that are not detected directly, like the vector bosons and the Higgs boson. The true nature of these particles can be established by analyzing the radiative corrections to the scattering processes. Since the physical spectrum is defined with respect to the broken phase, $SU(2)$ invariance is not very helpful. This means that the quantization prescriptions of the $W^{\pm }$ bosons, the $Z$ boson, the photon $\gamma $ and the Higgs field $\eta $ are in principle unrelated to one another. \begin{figure}[t] \begin{center} \includegraphics[width=5truecm]{HiggsDecay.eps} \end{center} \caption{Higgs decay processes} \label{HiggsDecay} \end{figure} To begin with, consider a process like the one shown in fig. \ref{HiggsDecay . Since $H$ and $Z$ decay, both quantization prescriptions (\ref{prescr}) give the same prediction. This is a typical case where $H$ and $Z$ simulate real particles even if they are quantized as fakeons. Specifically, it is possible to show \cite{absograv,causalityQG} that if we take a fakeon $F$ and resum the powers of its dynamical width $\Gamma _{F}$ into the so-called dressed propagator, the imaginary part of (minus) the $F$ dressed propagator tends to \begin{equation} \pi Z_{F}\delta (p^{2}-m_{F}^{2}) \label{dress} \end{equation in the limit $\|\Gamma _{F}\|\rightarrow 0$, as if the fakeon $F$ \textit{were indeed a physical particle} of mass $m_{F}$, where $Z_{F}$ is the normalization factor. Applying these arguments to the final states of the process of fig. \ref{HiggsDecay}, we see no difference between true and fake $H$ and $Z$ particles. At one loop, the two prescriptions give significantly different results for the imaginary parts $\mathcal{I}$ of the radiative corrections to the transition amplitudes above suitable thresholds \cite{LWgrav,fakeons}, while the real parts coincide in the two cases. The relevant diagrams are self-energies, triangle diagrams and box diagrams. See fig. \ref{HiggsDiag} for examples and refs. \cite{hollik} for explicit formulas. We need to pay attention to the diagrams that contain at least one virtual fakeon and estimate the orders of magnitude of the various types of contributions. An important point is that the processes we are considering are far from the resonance peaks. In such conditions, the contributions of the three types of diagrams are of the same orders, so in many cases we can concentrate on the self-energies with no loss of generality. Consider a self-energy diagram $\mathcal{B}$ with internal legs of masses m_{1}$ and $m_{2}$. The imaginary part $\mathcal{I}$ of $-i\mathcal{B}$ is equal to zero if an internal leg is quantized as a fakeon \cit {LWgrav,fakeons,UVQG,absograv,causalityQG}, while it is proportional t \begin{equation} \theta \left( s-(m_{1}+m_{2})^{2}\right) \sqrt{1-\frac{(m_{1}+m_{2})^{2}}{s} \sqrt{1-\frac{(m_{1}-m_{2})^{2}}{s}} \label{ima} \end{equation if both internal legs are quantized as physical particles, where $s$ is the center-of-mass energy squared. The real part of $-i\mathcal{B}$ is the same with both quantization prescriptions. Typically, when neither of the particles circulating in the loop are fakeons, $\mathcal{I}$ and the real part of $-i\mathcal{B}$ are of the same order, when $s$ is larger than the physical threshold $(m_{1}+m_{2})^{2}$. Let $\mathcal{I}_{cd}^{ab}$ denote the imaginary part of $-i$ times the bubble diagram that has $a$, $b$ as external legs and $c$, $d$ as circulating particles. If $a=b$, we just write $\mathcal{I}_{cd}^{a}$. Consider $\mathcal{I}_{W^{+}W^{-}}^{\gamma }$, $\mathcal{I}_{W^{+}W^{-}}^{Z} , $\mathcal{I}_{W^{+}W^{-}}^{\gamma Z}$ and $\mathcal{I}_{ZH}^{Z}$. Since they have different thresholds, or depend on $s$ in different ways, it is possible to analyze their contributions separately in precision measurements. The imaginary parts $\mathcal{I}_{W^{+}W^{-}}^{\gamma }$, \mathcal{I}_{W^{+}W^{-}}^{Z}$ and $\mathcal{I}_{W^{+}W^{-}}^{\gamma Z}$ contribute to the cross section $\sigma (e^{+}e^{-}\rightarrow $ leptons, hadrons$)$ and their thresholds $2m_{W}$ are in the range of energies spanned for example by LEP\ II. Since no unexpected behavior has been noted (and the data of LEP\ II are precise enough), we infer that \mathcal{I}_{W^{+}W^{-}}^{ab}$ are nonvanishing, hence the $W$ bosons are physical and not fake. The same can be said of the left neutrinos $\nu $, from $\mathcal{I}_{\nu \bar{\nu}}^{Z}$, the charged leptons $\ell $, from \mathcal{I}_{\ell \bar{\ell}}^{\gamma }$, $\mathcal{I}_{\ell \bar{\ell}}^{Z} , $\mathcal{I}_{\ell \bar{\ell}}^{\gamma Z}$, and all the quarks $q$ but the top one, from $\mathcal{I}_{q\bar{q}}^{\gamma }$, $\mathcal{I}_{q\bar{q }^{Z} $, $\mathcal{I}_{q\bar{q}}^{\gamma Z}$. The case of the top quark $t$ remains unresolved. The self-energy contributions $\mathcal{I}_{t\bar{t}}^{\gamma }$, $\mathcal{I}_{t\bar{t }^{Z} $ and $\mathcal{I}_{t\bar{t}}^{\gamma Z}$ with circulating tops have been missed by LEP\ II, due to their thresholds $2m_{t}\sim 346$GeV. However, they can be studied in precision measurements with some effort of data analysis and background subtraction at LHC \cite{atlas} and HiLumi \cit {HiLumi}, or, more directly, at the International Linear Collider \cite{ILC , the Compact Linear Collider \cite{CLIC}, the Future Circular Collider \cit {FCC} and the Circular Electron Positron Collider \cite{CEPC}, if they will be eventually built \cite{Mangano}. \begin{figure}[t] \begin{center} \includegraphics[width=10truecm]{SelfHiggs.eps} \end{center} \caption{Relevant processes with $Z$ and $H$ bosons in loops} \label{HiggsDiag} \end{figure} The analysis just made, based on the self-energies, leaves out the photon \gamma $ and the $Z$ boson. We can prove that they are not fake by considering the box diagram of fig. \ref{HiggsDiag}. Here the imaginary part associated with the vertical cut has a threshold equal to $m_{Z}$, so it contributes to the processes studied at LEP II. Since no discrepancies with respect to the predictions of the usual quantization prescription have been reported, we infer that both $\gamma $ and $Z$ are physical. As far as the Z $ boson is concerned, we can also consider the box diagram with $\gamma $ replaced by a second $Z$, since the threshold $2m_{Z}$ has also been exceeded by LEP\ II. Let us now consider the Higgs boson. If the Higgs field were a fakeon, it would have to be a fakeon plus, since the possibility that it might be a fakeon minus is excluded. Indeed, to turn it into a fakeon minus, we would have to flip the sign of its kinetic term $(D_{\mu }H^{\dag })(D^{\mu }H)$. Then the squared masses of $Z$ and $W^{\pm }$ would also turn into their opposites. However, it is not possible to quantize takyons as fakeons [both prescriptions (\ref{prescr}) must have $m^{2}>0$], so we would have to flip the signs of the of $Z$ and $W^{\pm }$ kinetic terms as well. That would force us to quantize $Z$ and $W^{\pm }$ as fakeons (since the Feynman prescription would turn them into ghosts), which is contrary to the results obtained above. In the end, we remain with just two possibilities: the Higgs boson is a physical particle or a fakeon plus. To decide which it is, consider the cross sections $\sigma (e^{+}e^{-}\rightarrow \gamma \ell j)$, where $\gamma \ell j$ denotes any final state made of photons, leptons and/or jets. If $H$ is physical, the imaginary part $\mathcal{I}_{ZH}^{Z}$ of the $Z$ self-energy with a Higgs field (see the right diagram of fig. \ref{HiggsDiag}) starts contributing from $\sqrt{s}\gtrsim m_{Z}+m_{H}=216$GeV. Enough above the threshold (say, at $\sqrt{s}\sim 240$GeV) the contribution of $\mathcal{I}_{ZH}^{Z}$ is comparable to the one of the real part. It is also comparable to the contributions of the imaginary parts of the other main $Z$ self-energy diagrams, like $\mathcal{I}_{W^{+}W^{-}}^{Z}$. Thus, the difference between a physical Higgs boson and a fake one is important enough to be noted, whenever a self-energy diagram like the one of fig. \ref{HiggsDiag} contributes and the experiment is sensitive to it. If $\mathcal{I}_{ZH}^{Z}$ is found to be nonvanishing, then both $Z$ and $H$ are not fake. Instead, if $\mathcal{I}_{ZH}^{Z}$ is found to vanish, we conclude that $H$ must be fake, since we have already proved that $Z$ is not fake. If LEP II had not stopped right below the threshold $m_{Z}+m_{H}$, we would already know the answer to this question. At present, the only possibility to fill this gap is to perform precision measurements\ at LHC or wait for HiLumi, ILC, CLIC, FCC or CEPC. Other potentially relevant diagrams are the fermion self-energies that involve a virtual Higgs boson. The fermions $f$ must also be virtual, to turn on the imaginary part $\mathcal{I}_{fH}^{f}$, whose threshold is m_{f}+m_{H}$. These self-energy diagrams contribute for example to the Compton-like process of fig. \ref{compton}. However, the couplings of the fermions to the Higgs boson are suppressed by a factor $m_{f}/v$, where $v$ is the Higgs vev, and this ratio is squared in the diagram. The resulting contribution $\mathcal{I}_{fH}^{f}$ is too small to be observed in all cases apart from the one of the top quark, where the threshold raises to about 300 $GeV. The virtual top quark can be produced by top-gluon and top-photon interactions, as well as from the pairs $W^{+}b$ and $Zt$. These are processes that can be studied at LHC. Replacing $H$ with a gluon in fig. \ref{compton}, we obtain a contribution that allows us to test whether the gluons are physical or fake. In that case, it is enough to reach energies that are a bit larger than the quark mass and we can use any type of quark we want. We are not aware of data that can be immediately analyzed to obtain an answer in this case, but it is another problem that can be studied at LHC. \begin{figure}[t] \begin{center} \includegraphics[width=6truecm]{ComptonH.eps} \end{center} \caption{Compton-like process for fermion self-energy with Higgs boson} \label{compton} \end{figure} If we replace $Z$ or $\gamma $, or both, with $H$ in the box diagram of fig. \ref{HiggsDiag} we obtain another interesting diagram with a virtual Higgs boson circulating in a loop. Then the fermions can only be top quarks, since, for the reasons recalled before, the couplings of $H$ to the other fermions are too small. In conclusion, theoretical arguments and experimental evidence ensure that no particles of the standard model are fakeons, apart from possibly the Higgs boson and the top quark, the gluons and the right neutrinos. The top quark is related to other quarks by (approximate) family symmetries, which may suggest that it is probably physical. On the other hand, the possibility that the Higgs boson is a fakeon is more intriguing, given the peculiarities of this particle. All cases, apart from the one of the right neutrinos, can be settled in future experiments or by performing precision measurements at LHC. If one or more particles of the standard model turn out to be fakeons, it becomes interesting to devise specific experiments to search for the first signs of violations of microcausality. \vskip12truept \noindent {\large \textbf{Acknowledgments}} \nopagebreak\vski 2truept \nopagebreak We are grateful to U. Aglietti, M. Grazzini, M. Piva and A. Strumia for useful discussions.	{ "redpajama_set_name": "RedPajamaArXiv" }	500
{"url":"https:\/\/mattbaker.blog\/tag\/robert-coleman\/","text":"# Conference on p-adic methods in number\u00a0theory\n\nAfter somewhat of a hiatus, I\u2019m back to blogging again.\u00a0 The purpose of this post is to advertise the conference \u201cp-adic Methods in Number Theory\u201d, which will be held in Berkeley, CA from May 26-30, 2015. \u00a0The conference, which I am helping to organize, is in honor of the mathematical legacy of Robert Coleman.\u00a0 Please spread the word!\u00a0 Here is the current version of the conference poster, which will be mailed out soon to a math department near you:\n\nMany thanks to Janet Ziebell of the Georgia Tech College of Sciences for her help creating this poster, and to Ken McMurdy for designing the conference website.\n\nHere is a memorial article about Robert which I co-authored with Barry Mazur and Ken Ribet.\u00a0\u00a0 I encourage you to read it!\u00a0 It will be published in the new open access journal Research in the Mathematical Sciences, in a special volume dedicated to Robert.\n\nYou can find other interesting links related to Robert Coleman\u2019s life and work here, and in this older blog post of mine.\n\n# Newton polygons and Galois\u00a0groups\n\nIssai Schur\n\nA famous result of David Hilbert asserts that there exist irreducible polynomials of every degree $n$ over ${\\mathbf Q}$ having the largest possible Galois group $S_n$.\u00a0 However, Hilbert\u2019s proof, based on his famous irreducibility theorem, is non-constructive.\u00a0 Issai Schur proved a constructive (and explicit) version of this result: the $n^{\\rm th}$ Laguerre polynomial $L_n(x) = \\sum_{j=0}^n (-1)^j \\binom{n}{j} \\frac{x^j}{j!}$ is irreducible and has Galois group $S_n$ over ${\\mathbf Q}$.\n\nIn this post, we give a simple proof of Schur\u2019s result using the theory of Newton polygons.\u00a0 The ideas behind this proof are due to Robert Coleman and are taken from his elegant paper On the Galois Groups of the Exponential Taylor Polynomials.\u00a0 (Thanks to Farshid Hajir for pointing out to me that Coleman\u2019s method applies equally well to the Laguerre polynomials.) Before we begin, here is a quote from Ken Ribet taken from the comments section of this post:\n\n# Effective Chabauty\n\nOne of the deepest and most important results in number theory is the Mordell Conjecture, proved by Faltings (and independently by Vojta shortly thereafter). It asserts that if $X \/ {\\mathbf Q}$ is an algebraic curve of genus at least 2, then the set $X({\\mathbf Q})$ of rational points on $X$ is finite. At present, we do not know any effective algorithm (in theory or in practice) to compute the finite set $X({\\mathbf Q})$. The techniques of Faltings and Vojta lead in principle to an upper bound for the number of rational points on $X$, but the bound obtained is far from sharp and is difficult to write down explicitly. In his influential paper Effective Chabauty, Robert Coleman combined his theory of p-adic integration with an old idea of Chabauty and showed that it led to a simple explicit upper bound for the size of $X({\\mathbf Q})$ provided that the Mordell-Weil rank of the Jacobian of $X$ is not too large.\u00a0 (For a memorial tribute to Coleman, who passed away on March 24, 2014, see this blog post.)\n\n# Robert F. Coleman\u00a01954-2014\n\nI am very sad to report that my Ph.D. advisor, Robert Coleman, died last night in his sleep at the age of 59.\u00a0 His loving wife Tessa called me this afternoon with the heartbreaking news.\u00a0 Robert was a startlingly original and creative mathematician who has had a profound influence on modern number theory and arithmetic geometry.\u00a0 He was an inspiration to me and many others and will be dearly missed.\n\nRobert and Bishop in Paris","date":"2018-12-13 09:53:22","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\": 14, \"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\": 0.6496301293373108, \"perplexity\": 632.3498319940655}, \"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-2018-51\/segments\/1544376824601.32\/warc\/CC-MAIN-20181213080138-20181213101638-00508.warc.gz\"}"}	null	null
{"url":"http:\/\/physics.aps.org\/synopsis-for\/10.1103\/PhysRevLett.112.183902","text":"# Synopsis: Semiconductor Lasers Get Nervy\n\nSemiconductor lasers can exhibit characteristics of nerve cells, making them potentially useful for simulating biological neural networks.\n\nThe brain\u2019s information processing ability has inspired researchers to build neuromimetic computing systems from building blocks that behave like neurons. To accurately mimic neural behavior, these individual units should exhibit the characteristic of so-called excitable media, like axons\u2014appendages that transmit impulses away from the body of neuron cells: Axons respond to a stimulus below a threshold with small, linear responses; above threshold, they respond with a large nonlinear response. Essential to their operation is a refractory period following the nonlinear events, during which any response is temporarily inhibited.\n\nIn a paper in Physical Review Letters, Sylvain Barbay and colleagues at the CNRS Laboratoire de Photonique et de Nanostructures in France, demonstrate that semiconductor lasers can reproduce the refractory period of an excitable system and thus could form the basis of optical computing architectures. The researchers studied the response of a micropillar laser with an intracavity saturable absorber (a material that blocks low-intensity light but allows high-intensity light through). The micropillar laser is pumped continuously just below threshold while another laser provides pulses that push the excitation above threshold. The response stimulated by such pulses is measured via the microlaser emission.\n\nBy using pairs of pump pulses with varying delay, the researchers were able to map out the refractory period, which consists of an \u201cabsolute\u201d portion for delays less than $200$ picoseconds (no response at all to the pump pulse) and a \u201crelative\u201d portion between $200$ and $350$ picoseconds (the response begins to recover). The confirmation that the micropillar laser with saturable absorber is an excitable system with an absolute and relative refractory period may lead to its use in a variety of optical neural networks. \u2013 David Voss\n\n### Announcements\n\nMore Announcements \u00bb\n\nFluid Dynamics\n\nOptics\n\n## Related Articles\n\nQuantum Physics\n\n### Synopsis: Position Detector Approaches the Heisenberg Limit\n\nThe light field from a microcavity can be used to measure the displacement of a thin bar with an uncertainty that is close to the Heisenberg limit. Read More \u00bb\n\nAtomic and Molecular Physics\n\n### Viewpoint: Next Generation Clock Networks\n\nFree-space laser links have been used to synchronize optical clocks with an unprecedented uncertainty of femtoseconds. Read More \u00bb\n\nBiological Physics\n\n### Synopsis: Bacteria Create Own Swim Lane\n\nResearchers calculate the size of a low-resistance buffer zone created by microbial organisms as they swim through the mucus lining of the stomach. Read More \u00bb","date":"2016-05-31 10:04:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"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\": 0.41220661997795105, \"perplexity\": 3846.0205168367343}, \"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-2016-22\/segments\/1464051268601.70\/warc\/CC-MAIN-20160524005428-00215-ip-10-185-217-139.ec2.internal.warc.gz\"}"}	null	null
{"url":"http:\/\/preprints.sissa.it\/xmlui\/browse?type=author&value=Andreini%2C+Elena","text":"SISSA Preprints\n\nBrowsing by Author \"Andreini, Elena\"\n\nSort by: Order: Results:\n\n\u2022 (SISSA, 2011-01-31)\nLet $X$ be a smooth complex projective algebraic variety. Let $\\mathcal{G}$ be a $G$-banded gerbe with $G$ a finite abelian group. We prove an exact formula expressing genus $g$ orbifold Gromov-Witten invariants of ...\n\u2022 (SISSA, 2011-05-27)\nLet $\\clX$ a projective stack over an algebraically closed field $k$ of characteristic 0. Let $\\clE$ be a generating sheaf over $\\clX$ and $\\clO_X(1)$ a polarization of its coarse moduli space $X$. We define a notion of ...","date":"2019-08-22 05:09:04","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\": 0.8809916973114014, \"perplexity\": 393.05105185732344}, \"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-2019-35\/segments\/1566027316783.70\/warc\/CC-MAIN-20190822042502-20190822064502-00520.warc.gz\"}"}	null	null
La Jazz at Lincoln Center Orchestra es una big band y orquesta de jazz estadounidense dirigida por Wynton Marsalis desde 1991. La orquesta es parte del Jazz at Lincoln Center, una organización de artes escénicas de la ciudad de Nueva York. Historia En 1988, la Orquesta se formó como resultado de una serie de conciertos llamada Classical Jazz, con la dirección de David Berger. Cuando Wynton Marsalis se convirtió en su director artístico en 1991, hizo hincapié en la historia del jazz, particularmente en la herencia de la orquesta de Duke Ellington. El primer álbum fue Portraits by Ellington (1992), y siete años más tarde se honró el centenario de Ellington con el álbum Live in Swing City: Swingin' with the Duke (1999). Bajo el liderazgo de Marsalis, la banda se presenta en su sede en el Lincoln Center, realiza giras por los EE. UU. y por el extranjero, visita escuelas, aparece en televisión y actúa con orquestas sinfónicas. La Orquesta también respaldó a Wynton Marsalis en su álbum Blood on the Fields, que ganó el Premio Pulitzer. Desde 2015, los álbumes de la Orquesta se publican en su propio sello, Blue Engine Records. Personal Wynton Marsalis - trompeta Ryan Kisor - trompeta Marcus Printup - trompeta Kenny Rampton - trompeta Chris Crenshaw - trombón Vincent Gardner - trombón Elliot Mason - trombón Robert Stewart - saxofón Walter Blanding - saxofón Víctor Goines - saxofón Sherman Irby - saxofón Ted Nash - saxofón Paul Nedzela - saxofón barítono Dan Nimmer : piano Carlos Henríquez – contrabajo Obed Calvaire – batería Discografía Portraits by Ellington (1992) Jazz at Lincoln Center Presents: The Fire of the Fundamentals (1994) They Came to Swing (1994) Blood on the Fields (Columbia, 1997) Live in Swing City: Swingin' with Duke (1999) Big Train (Columbia, 1999) All Rise (Sony Classical, 2002) Lincoln Center Jazz Orchestra with Wynton Marsalis Plays the Music of Duke Ellington (Warner Bros., 2004) Cast of Cats (2004) A Love Supreme (2005) Don't Be Afraid: The Music of Charles Mingus (2005) Congo Square (2007) Portrait in Seven Shades (Jazz at Lincoln Center, 2010) Vitoria Suite (Universal, 2010) Wynton Marsalis and Eric Clapton Play the Blues (2011) Live in Cuba (Blue Engine, 2015) Big Band Holidays (Blue Engine, 2015) The Abyssinian Mass (Blue Engine, 2016) The Music of John Lewis (Blue Engine, 2017) All Jazz Is Modern: 30 Years of Jazz at Lincoln Center Vol. 1 (2017) Handful of Keys (Blue Engine, 2017) United We Swing: Best of the Jazz at Lincoln Center Galas (2018) Una Noche con Rubén Blades (2018) Swing Symphony (2019) Jazz and Art (2019) Jazz for Kids (2019) Big Band Holidays II (2019) Sherman Irby's Inferno (2020) The Music of Wayne Shorter (2020) Black, Brown, and Beige (2020) Rock Chalk Suite (2020) Christopher Crenshaw's The Fifties: A Prism (2020) Referencias Swing Grupos de música formados en 1988 Lincoln Center Big Bands	{ "redpajama_set_name": "RedPajamaWikipedia" }	6,922
Q: Validating Form Fields on button click in office fabric Ui control I have one TextField and PrimaryButton in my form. On button click i want to validate textfield.If there is no value then fire requiredfield validation on button click and display message same as display on onGetErrorMessage event. I know validation is fire onBlur event but i want to fire validation on button click. How can i do it? A: First you have to use refs to your TextField <TextField ref={(input) => { this.textInput = input; }} label='test' /> Now you can set onGetErrorMessage on click using refs onClick(e){ if(this.refs.input.value == undefined \|\| this.refs.input.value == null \|\| this.refs.input.value == '') this.refs.input.onGetErrorMessage = "you error message" } Update 1: Check out this code snippet class FormExample extends React.Component { constructor() { super(); this.state={ inputError:'' } this.ValidateText = this.ValidateText.bind(this) } ValidateText(e){ this.setState({ inputError:this.input.value?'':'testing' }) } render() { return ( <div> <div> <TextField ref={(input) => { this.input = input; }} label="Name" errorMessage={this.state.inputError} /> <input type='button' value='Submit' onClick={this.ValidateText} /> </div> </div> ); } }	{ "redpajama_set_name": "RedPajamaStackExchange" }	6,537
Q: JavaScript Loop not recognising global variable I am following a javascript book and trying to re-implement the books code examples in my own way. i keep getting "ReferenceError: test is not defined" in the Firefox debugger Thisis my code : <script> var test1 = "abcdef"; var test2 = 123; var test3 = true; var test4 = {}; var test5 = []; var test6; var test7 = { "abcdef": 123 }; var test8 = ["abcdef", 123]; function test9() { return "abcdef" }; for (var i = 0 ; i < 10 ; i++) { var probe = test[i]; alert((typeof probe).toString); } </script> A: test[1] is not at all the same as test1. What you want is var test = []; test[1] = "abcdef"; test[2] = 123; test[3] = true; test[4] = {}; test[5] = []; test[6]; test[7] = { "abcdef": 123 }; test[8] = ["abcdef", 123]; test[9] = function() { return "abcdef" }; for (var i = 0 ; i < 10 ; i++) { var probe = test[i]; alert(typeof probe); } A: When you write test[i], you approached the array named 'test', at the i index, which causes an error of course, since the test array does not exists. In order to implement your code correctly, you can either write: var test = []; test[1] = "abcdef"; test[2] = 123; and so on. This is fine too: var obj ={}; obj.test1 = "abcdef"; obj.test2= 123; And in your loop use: var probe = obj["test"+i]; (Your loop index should be initialized with 1 and not with 0) A: test is not really defined. so just change the test to some other variable name which you have declared already. var test1 = "abcdef"; var test2 = 123; var test3 = true; var test4 = {}; var test5 = []; var test7 = { "abcdef": 123 }; var test8 = ["abcdef", 123];	{ "redpajama_set_name": "RedPajamaStackExchange" }	9,604
The Last Podcast on the Left March 19, 2018 Mayra Leave a comment This weeks recommendation is "The Las Podcast on the Left" hosted by Ben Kissel, Marcus Parks, and Henry Zebrowski. These guys cover everything from murderers, cults, alien abductions, conspiracies, hauntings and more. You can listen from the beginning or pick and choose based on the topic you are interested in. They really dive deep into the subject of each episode with extensive research; while keeping their sense of dark humor with jokes and silly voices. So if you also have a taste for the weird and disturbing, give this podcast a try. AliensConspiracyCultsHauntingsMurderMysteryPodcastsSupernaturalThe Last Podcast on the LeftTrue Crime Crime Writes On… July 16, 2017 Mayra Leave a comment "A podcast about a podcast" is how they describe themselves. Hosted by Rebecca Lavoie and features Kevin Flynn, Toby Ball, and Lara Bricker as regulars. Though they got their start talking about Serial, now they have a broader spectrum on what they discuss. From other podcasts, true crime, pop culture, to Lara's cat. They cover anything they find interesting and always take suggestions from listeners. That is one of my favorite things about this podcast, they really read our tweets, emails, or comments. They even have a voicemail set up for listeners to leave messages and will sometimes play them during the podcast. Since they cover so many forms of entertainment, they usually let listeners know ahead of time what they will be talking about. That way we can all avoid the dreaded "spoiler." Rebecca and Kevin are true crime writers (hence the name) and also happen to be married. Toby Ball is a fiction author and Lara Bricker is a Licensed P.I. So as you can imagine, the conversations are very interesting/entertaining to listen to. Check them out! (Also, I'll be posting about some of their true crime books on a later date.) Crime writers onPodcastsSerialTrue Crime "Podcasts! There are a million of them and they're all amazing!" -Tom Haverford, Parks and Recreation. Parks and RecreationPodcasts I am starting off my long list of recommendations with what some might say is the most mainstream and influential true crime podcast in the history of podcasts. For those of you who have never heard of it, Serial is a 12 episode podcast hosted by Sarah Koenig and developed by This American Life. Its first season investigates the murder of Hae Min Lee. On January 13, 1999 Hae Min Lee; an 18-year-old high school senior at Woodlawn High School in Baltimore, was last seen around 3:00pm. Her body was then found in Leakin Park on February 9, 1999. Her ex-boyfriend, Adnan Syed, was arrested and charged for her murder later that month. Then he was convicted with a life sentence a year later. Adnan and his family have always maintained his innocence. A decade and a half later a family friend and attorney, Rabia Chaudry, reaches out to Sarah Koenig and asks her to take a look at his case. Koenig investigates for a year and is able to interview Adnan himself. What she finds is an infinite amount of unanswered questions. Why was there so little physical evidence? Why did the key witness, Jay Wilds, have such a conflicting and constantly changing story? Why did Jay know where Hae's car was? Why would Jay confess to taking part of a murder if it wasn't true? Why can't Adnan remember what he did that day? Where is Asia McClain, the girl who could potentially be Adnan's alibi, and why didn't she testify in his trail? If Adnan didn't kill Hae, then who did? Koenig is a great storyteller and keeps the listener in constant intrigue. She gives just enough information to keep you thinking and lets you build your own theories in your mind. There's ambiguity in every episode and the whole series leaves you wanting more answers. I loved this podcast and I highly recommend it to everyone I know. But having looked further into Adnan's case, I've realized how much information there is and how little Koenig touched on. I understand why she did this but at the same time it makes me feel like I was lied to in a way. So please listen and enjoy but dig deeper. This leads me to my next recommendation…Undisclosed. Adnan SyedHae Min LeePodcastsSerialTrue Crime	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	80
# Dedication For Marissa and Shane, and the new kid on the block. # Contents Cover Dedication Thank You Introduction Emotional People Buzzcocks The Cure Weezer The Classics Troublemaker Wordsworth Civilisation The French Revolution The Story is in the Soil A Motion and a Spirit Romantic Disenchanted Paint It Black and Take It Back Napoleon This Tragic Affair Passion Incapable of Being Converted into Action Sentimentalists Across the Sea Love Like Winter Alone and Palely Loitering A Forest Lemonade Anatomy of Mellon Collie Rock and Roll Suicide Screamin' Lord Byron Lord Byron Give Them Blood The Vampyre Goths Rocky Horror Vincent Frankenstein Edward Scissorhands The Dark Side of Human Things Mystery Utopia Utopiate The Degraded Present Blasphemous Rumours Paradise Lost The Disappearing God The Age of Simple Faith Faith World in My Eyes We Can Be Heroes Wagnerian Born to Run Pressure Schopenhauer Pinkerton Butterfly Satisfaction Boredom Notes from Underground How Soon Is Now? Why Bother? The Crystal Palace The Broken, the Beaten and the Damned Teenagers I've Gotta Get Out of the Basement! Myths of the Near Future Gustav Klimt Nietzsche A Night at the Opera The Wisdom of the Woods Personal Jesus Stronger Also Sprach Zarathustra Homo Superior Destroyer Such a Special Guy Expressionism The Pain Threshold Sprechstimme Everything Collapses Life Is a Cabaret Mother War Artists are Cleaners Distress Cries Aloud Rock Stars Are Fascists, Too The Black Parade Is Dead Insulation and Disaffection Leave Them Kids Alone. Notes Searchable Terms Copyright # Thank You Hey, Nietzsche! first began to resemble a book in November last year. At that point, my wife and I had been travelling for almost nine months, and I had accumulated six notebooks full of ideas and an ipod full of emo. In Thailand, we met an Australian girl called Chrissie, and one night I tried (for the first time) to explain my idea for this book to her. When I'd finished my spiel, she thought for a moment and said, 'You know, when you first told me you were writing a book, I thought, "What a waste of time". But now that you've explained it, I reckon it might actually be really good.' This was the second most valuable piece of encouragement I received while writing Hey! Nietzsche! — the first being something my father-in-law, Peter Lynch said to me twelve months later. I was, by this point, a little emo myself — the combination of the looming deadline and over a year's worth of exposure to angst-y poetry was taking its toll. Was I, I wondered out loud, attempting too much? Pete told me that I was, but that he believed this was the secret to success. 'Bite off more than you can chew', he said, 'and chew like fuck.' (I think this is more or less what Rainer Maria Rilke was getting at in his Letters to a Young Poet.) Along with Pete and Chrissie, I have a few other people to thank. Zan Rowe, Dan Buhagiar, Marc Fennell, Chris Scaddan and Linda Bracken at triple j; Wendy Were for having me to the Sydney Writer's Festival; Jenny Valentish at Jmag; Keith Hurst for legal business; Jase Harty and Michael Agzarian for their continuing support; and the amazing Brad Cook for his portraits of 'old dudes' and guys with names nobody can pronounce. I'd also like to thank everybody who helped to make the Culture Club book launch such a roaring success — The Devoted Few, Dr Lindsay McDougall, Sam Simmons, Brendan Doyle, Nina Las Vegas, and the Pemmell Pad crew; and everybody at ABC Commercial for their hard work on Hey! Nietzsche!, including Louise Cornege, Jacquie Kent, Megan Johnston, and especially Susan Morris-Yates — for her unwavering support and saint-like patience. Speaking of which, my friends and family — Michael 'Timmy' Rosenthal, Ben and Jess, the girls (and their boys), Pete and Lyn, Marissa and Shane and Mum and Dad — have been listening to me say 'it's nearly finished' for over a year now. As I type this, I think I can safely say that this time I really mean it. Thank you all for putting up with me. And thank you, Kirileigh Lynch. You have confounded my thesis by being the least emo person I know and the most romantic (in a good way). I promise to come out of the tower of doom and have some real fun. And why are you so firmly and triumphantly certain that only what is normal and positive — in short, only well-being — is good for man? Is reason mistaken about what is good? After all, perhaps prosperity isn't the only thing that pleases mankind, perhaps he is just as attracted to suffering...whether it's a good thing or a bad thing, smashing things is also sometimes very pleasant. I am not here standing up for suffering, or for well-being either. I am standing out for my own caprices and for having them guaranteed when necessary. Fyodor Dostoyevsky Notes From Underground # Introduction I. This book began with a song I heard on the radio about two years ago. It was a five-minute rock epic in three acts, a sincere denial of modern life, an affirmation of the power of dreams, and a conflation of corny melodrama and gut-wrenching personal confession the likes of which had not been heard since 'Bohemian Rhapsody'. I ranted and raved about this song and how great it was to everyone I knew, and quickly discovered that nobody liked it as much as I did. Or they liked it, but were unwilling to make any greater claim for it than that. I, on the other hand, was full of great claims. I believed this song had the power to change the world. I leaped to its defence. I argued, in all sincerity, that it was one of the Greatest Songs of All Time. I was becoming unreasonable — but I liked the feeling of being unreasonable. It was a feeling I remembered from when I was about fifteen years old, when I blundered into many similar arguments with my friends. I tried to convince them that Supertramp's 'The Logical Song' was the Greatest Song of All Time. My proclamations were met with embarassed silence or outright scorn. I was an outcast — a heretic. But my almost religious devotion to Supertramp allowed me to weather this isolation — my faith would sustain me. This was the irrational love I felt for my new favourite song, 'Welcome to the Black Parade' by New Jersey five-piece My Chemical Romance. And this, I soon discovered is the love that My Chemical Romance's fans feel for the band and their music. It's how the band themselves would like to be loved, as singer Gerard Way once explained to his audience: If for one minute you think you're better than a sixteen-year-old girl in a Green Day T-shirt, you are sorely mistaken. Remember the first time you went to a show and saw your favourite band. You wore their shirt and sang every word. You didn't know anything about scene politics, haircuts or what was cool. All you knew was that this music made you feel different from anyone you ever shared a locker with. Someone finally understood you. This is what music is about. This is one of the many Gerard Way quotes I began collecting around this time. I read every interview I could find. I listened to The Black Parade album over and over again, I studied the lyrics intently. I told people I was doing 'research' — but I wasn't fooling anyone. I was collecting pictures of the band and gluing them in my notebooks. I had crossed over to the other side. I was entering the advanced stages of pop obsession. 'Gerard Way and I really do have a lot in common,' I thought to myself. This is actually true. The bands Way talks about in interviews, the singers he says changed his life, are all the same ones I like. Queen, Bruce Springsteen, The Cure, The Smiths, and David Bowie. I realised that all of these artists share something in common — a quality that is present in 'Born to Run', 'Heroes', 'How Soon is Now' and 'Welcome to the Black Parade' — but that I was hard pressed to say exactly what that was. I started making graphs and charts, looking for the missing link in all of this music. My notebooks started to accumulate more collaged fragments — Bowie, Bruce, and Morrissey shared space with Gerard and the band. I was preparing a case — a passionate defence of My Chemical Romance and their music — and I knew I would need evidence to back up my claims. But I was also trying to explain something to myself. I wanted to understand what it was in these songs that moved me so much, and where that something came from. With my last book, I had attempted something similar. I had gone looking for the ideas that informed certain songs and albums of the last ten years, and had found them in the dimly lit corners of the twentieth-century avant-garde. I found Dada in Beck, Artaud in Gnarls Barkley, Walter Benjamin in The Scissor Sisters and Andre Breton in The Mars Volta. But I quickly realised that in this case, the twentieth century would be of no use to me. The ineffable something that connected the songs in my new lists could not be explained by any twentieth-century idea. So I took a leap back another hundred years — into the period bookended by the French Revolution in 1789 and the death of Friedrich Nietzsche in 1900. Here, I immediately found what I was looking for. Self-expression, the rejection of institutions, individualism, questing spirituality, the desire to escape society, the strong identification with criminals and madmen, the divinity of sin, the tragic view of nature, ideal love, dying young, solitude, melancholy, medievalism and an unhealthy obsession with death. These were the ideas that connected the songs in my list to one another. But they're also, I learned, the ideals and characteristics of romanticism, the artistic and philosophical movement that dominated much of the nineteenth century. As I learned more about the romantics, I began to realise how widespread their influence in rock and roll really was. Every poem I read reminded me of a song lyric, every letter of an interview with a musician. The collaged portraits in my notebooks became festooned with spidery lines. I began to see how Bowie was linked to Byron, what Freddie Mercury owed to Friedrich Nietzsche, and what put the romance in My Chemical Romance. For years I'd been reading that a singer's looks are 'Byronic', that an album is 'Wagnerian', that the singer's philosophy is 'Nietzschean', that the song's mood is 'gothic' and that the band are hopelessly 'romantic'. Behind these words I had always sensed a story that was not being told — now I realised I had an opportunity to tell it. My speech in defence of 'Welcome to the Black Parade' had grown to the point where no-one would have time to listen to it all. I imagined myself calling up a radio request show: 'And what do you like about this song, Craig?' 'I'm glad you asked. See, about two hundred years ago...' 2. One of the hurdles I knew I would have to face in presenting my argument was the Emo Factor. My Chemical Romance is considered emo. And emo is not considered acceptable. In fact, if you're into indie rock and over the age of twenty-five, there are few things less cool on the planet than emo. It is, almost by definition, un-cool. Cool implies a certain amount of detachment, an ironic attitude to life, a refusal to show too much feeling, a preference for playing with signs and surfaces without becoming too attached to their meanings. Emo has no truck with cool. Emo is about private passion, and its success is judged on how much passion is produced and how nakedly it's exposed. Emo wants avalanches of feeling, tragic romances, explosive rage, bottomless self-pity and gordian knots of self-absorption. Its symbols are fire, blood, churning thunderstorms, endless oceans and cold, dark earth. What do these symbols represent? Emotions — my emotions, not yours. Emo is first person — and for most of the songs the first person is the only person. Politics does not exist, society barely registers. In emo, the singer's emotional world is the whole world — nothing else is as big, nothing else is as important. Emo, I began to realise, is the most unashamedly romantic sub-culture in rock today. It represents the outer extreme of romanticism, its purest and most dangerous strain, the romanticism of Goethe's young Werther, of Frankenstein, of Byron and Nietzsche — a philosophy which rejects the idea of the greater good, which says that what is good is simply what's good for me. One of the extraordinary things about emo is the panic it's created in the media and society at large. It's been a long time since youth culture was this frightening to grown-ups. But behind the squawking over self-harm, 'suicidal messages' and school shootings is one, big dangerous idea — that emotions, my emotions, are the most important thing in the world, and the only justification I need for my actions, however extreme. Emo culture is a threat to society because it's irrational — and it knows this, which is why it looks a bit like goth. Emo, like goth, has a preference for horror imagery — hence the fondness for Nightmare Before Christmas merchandise and Misfits T-shirts. This too, I discovered, leads back to romanticism. The romantics' desire to escape from a de-valued, meaningless present frequently led them into the arms of the Gothic Revival, and the close kinship between these two early nineteenth-century ideas would eventually produce the founding work of the modern horror genre, Mary Shelley's Frankenstein. Shelley's monster represents many things — but chief among them is the threat of romanticism in society, the havoc that could result if a philosophy of feeling is followed to its conclusion. In Shelley's novel, the monster justifies his crimes by saying that he has been cast out of human society — why should he now be asked to obey its rules? This is exactly the threat implied by emo — that of a whole generation of kids so alienated from society that they no longer believe in society at all, and no longer care what happens to it. This, I've realised, is why people looked at me funny when I started speechifying about 'Welcome to the Black Parade'. They were scared. The relationship between My Chemical Romance and emo is complex, to say the least. The simplest way to put it would be to say that the band's fans are, but that the band itself is not. Gerard Way has repeatedly insisted that this is the case, and I tend to agree with him — for reasons I hope to make clear in this book. Emo is characterised by a hopelessness that is not compatible with Gerard Way's ambition. He is not content — as most emo bands are — to stir powerful emotions in his audience. He wants, as he has stated many times, to make a difference. He's taken it upon himself to defend the rights of his community — more than that, to lead them in a pitched battle against the world that has rejected their ideals. My Chemical Romance's nineteenth-century military jackets are not just for show. They've taken the romantics' obsession with Napoleon to its logical conclusion, by forming a liberating army in the name of emotion, dreams and solitude, and fighting for your right to be alone at the party. 'The Black Parade' represents romanticism on the march. 3. The fate of Gerard Way's quest to liberate the broken, the beaten and the damned forms one narrative that runs through the book, along with other strands following the adventures of Bowie, The Cure, and Weezer — whose career is crucial to the history of emo, and in many ways represents the polar opposite of My Chemical Romance's. But through it all, I've done my best to tell the story of the romantic movement in a fairly linear fashion — beginning with the last days of the Enlightenment and the formalism of Alexander Pope, moving through Rousseau, Goethe and Wordsworth and on to Byron, Keats, and Mary Shelley, then the Victorians — Matthew Arnold, the Pre-Raphaelites, and the late romanticism of Wagner, then Dostoyevsky, and finally, Nietzsche. I've then gone on to demonstrate how romantic ideas — Nietzsche's in particular — played themselves out in the early twentieth century, in Expressionist music, art and theatre. I've made no attempt to be definitive, but I have sought to bring history to life and, hopefully, to build a bridge (or several) between the history of the romantic movement and history of rock and roll. In the process I've learned new ways to love my record collection, thanks to (who knew?) poetry and philosophy. Having returned from my travels in the nineteenth century I've found (as travellers always do) that the world I thought I knew looks slightly different — and mostly better. I've heard new ideas and relationships jumping out of the speakers, a new sense of the history that informs a song like 'Heroes' or 'Born to Run' or 'Bohemian Rhapsody', a new sense of excitement at the demands being made in an album like Faith, The Queen Is Dead, Violator or Siamese Dream. I've discovered why I was so moved by the song I heard on the radio that day — it is, in a sense, a song two hundred years in the making, and I was feeling the full force of those two centuries. I've realised that my irrational, unreasonable devotion to 'Welcome to the Black Parade' was an entirely reasonable response; given that the song is a demand for the unreasonable and the irrational. In the following pages, I've tried to explain why the singer's demands deserve our attention — in as reasonable and rational a manner as my subject matter will allow. Gerard Way: 'Feel something!' # Emotional People BECAUSE THEY'RE YOUNG, American and wear a lot of black; because they play melodic punk rock with their hearts on their sleeves; and because they're fond of eyeliner and introspection, My Chemical Romance is, in the eyes of the world, an emo band. In fact, last year in the UK, they became the emo band — the only one people over the age of thirty would be able to identify in a line-up. Gerard Way's face, covered in make-up, screaming into a microphone, or just looking moody and mysterious, stared out from the cover of thousands of music weeklies and tabloid rags. His lyrics were the subject of urgent debate, the band's Black Parade tour was a news story. But this newfound notoriety did not please My Chemical Romance one little bit. The band's press coverage was up, but the quality of that coverage had plummeted out of all proportion. Way and the band rolled into town ready to talk redemption and rock and roll — what they got was a barrage of questions about things that, to them, had nothing to do with My Chemical Romance: things like Marilyn Manson, Mexican homophobia, teen suicide and — worst of all — emo. 'Emo', Gerard Way says, 'is a pile of shit'. He's not the only one who feels that way. Not only is the sentiment echoed by thousands of punks, goths, and indie-rock fans, it's a conviction shared by most of the bands people think of as emo — members of Panic at the Disco, Fall Out Boy, The Get Up Kids, Saves the Day, Weezer, and Jimmy Eat World have all, at one time or another, declared that they want nothing to do with it. It seems to be one of those genres that's only useful to music journalists and record store owners, along with other much maligned terms such as trip-hop, new rave and electroclash. Unlike these, however, emo has proved surprisingly durable — the history of the genre is long, and its fans are passionate about it. Whatever else it might be, emo — like punk — is important enough for people to keep arguing about it. Meanwhile, the sheer diversity of the music demonstrates just how unstable emo is as a concept. Sunny Day Real Estate's Diary sounds nothing like Weezer's Pinkerton, but both are emo landmarks. This diversity is the very reason why many insist that the term is useless. In the beginning emo referred to a very specific handful of Washington DC hardcore bands, beginning with Rites of Spring, who traded political rage for emotional angst and were thus labelled 'emo-core'. Now it seems any band whose singer is pale and pretty looking (or not) and sings about his feelings, accompanied by loud guitars (or not) is emo. Emo can sound like mall-punk, synthpop, goth, glam, country, classic rock, or Morrissey fronting a hardcore band. So it's impossible to work out whether My Chemical Romance is emo or not based on the sound of their music. Nor is it possible to peg them on the basis of their lineage. My Chemical Romance does occupy one of the outer branches of the emo family tree — they were supported early on by Geoff Rickley from Thursday, they've recorded a cover of David Bowie and Queen's 'Under Pressure' with The Used, and they've been touring and writing with James DeWees of the Get Up Kids. But by that logic both The Mars Volta and The Foo Fighters would also be emo — and nobody would argue that they are. Surely at this point, say the critics, it's time to either find a more precise definition for this thing, or forget about it entirely. In 2008 Tonie Joy, whose band Moss Icon was one of the first to be described as emo, had this to say: Over the years it's been diluted and shifted so much, it's pretty inaccurate compared to what the term was first pinned to. I think Ian MacKaye summed it up best when the term was first coined... He just thought it was stupid, saying that any music that's real is emotional, whatever the genre. William Goldsmith, formerly of Sunny Day Real Estate, agrees: Emo means emotional, right? Human emotions have been the driving force of all art since the beginning of the beginning. To say that...emotionally driven music is a new thing...it just doesn't make sense to me. Here, these two veterans of the scene have hit the nail on the head. When all the musical and genealogical arguments have been worn out, the only grounds for finding that My Chemical Romance is emo is that they're emotional artists who write emotional music. This they will admit to. 'We're all very sincere, emotional people', Way once told Spin magazine. Way's lyrics come from feelings, and his goal is to make you feel those feelings too. 'Feel something. That's what we've always been fighting for,' said the singer in 2007. But if that's your definition of emo, does that mean Kurt Cobain, Ian Curtis, Robert Smith, David Bowie, Bruce Springsteen, Mick Jagger, Richard Wagner and Beethoven are all emo too? For the term emo to exist, and for it to last as long as it has, implies that there is some kind of music which is less emotional, or not emotional at all. This is exactly how emo began, as an alternative to staunchly un-emotional music. In the late 1980s, American hardcore bands were singing about society and the world, and the kids at their shows, while enjoying the power of the music and the feeling of community it created, were virtually screaming, 'Sing about my feelings!' Over the next ten years, the history of emo would be written by the bands that answered this silent prayer. Rites of Spring, Mineral, Texas is the Reason, Sunny Day Real Estate and Drive Like Jehu turned the steely gaze of hardcore inward, the ruthless critique of the world became a ruthless critique of oneself and one's feelings. This rebellion against punk orthodoxy from within is as old as punk itself. # Buzzcocks IN 1977 MANCHESTER band the Buzzcocks released their first EP, Spiral Scratch. In punk's year zero it was the punkest thing you ever saw — recorded in a couple of hours using mostly first takes and one overdub, and released on the band's own label, using money borrowed from guitarist Pete Shelley's dad. On 'Boredom', singer Howard Devoto spat out his lyrics in a fake cockney accent at breakneck speed, pausing only to make room for Shelley's two-note guitar solo — a snot-nosed act of defiance toward prog-wankers and school music teachers alike. Music, Spiral Scratch seemed to say to its audience, is easy. All you need is a feeling and the will to express it. Blag a couple of quid off the old man, and you're away. But, coming as early in punk's history as it did, Spiral Scratch already contained an argument against punk — or against what it was becoming. The noise coming out of London, engineered in no small part by Sex Pistols impresario Malcolm McLaren and happily parroted by the music press and the papers, suggested a musical movement rising up from the streets; kids from council estates seizing the means of production and transforming everyday life. The Sex Pistols illustrated this with three singles that seemed to map out the road to revolution: 1. Discredit and destroy the existing political system. 2. Discredit and destroy the head of state. 3. Discredit and destroy rock and roll. By 1977 punk had a recognisable uniform, a mandate to critique the conditions of everyday life, and the momentum of a mass movement. But there was nothing in its charter about the importance of self-expression. With the possible exception of righteous rage, emotions were considered a luxury the punk singer could not afford to indulge in. And love, as Jon Savage notes in England's Dreaming, was the one topic punk bands (and punk critics) considered totally out of bounds. Love songs were stupid songs, chart songs, product. Love confined rock and roll to the teenage bedroom and the school dance. The love song was mere escapism; an easy way out. Punk, as Savage puts it, was 'determinedly in the world'. Punk saw myriad injustices and humiliations wreaked on the world by conservatives and capitalists — all of these needed to be exposed. Of course you like love songs, punk says, they let you off the hook. They want you to like love songs — as long as you're obsessing over your feelings, you won't notice what's really going on. But no amount of Marxist rhetoric or agit-prop sloganeering could convince teenagers, then or now, that their feelings are not important. This is the key to the success — and the continuing importance of — the Buzzcocks. Pete Shelley's songs are as punk as you like — fast, unpretentious and full of bile and snot. There're no clichés, no grandiose gestures, and very little ornament — but there's also none of the pious preaching or undergraduate politicking that characterises the work of agit-punk heroes like Tom Robinson Band or Sham 69. Shelley wrote about what he knew, his voice was the bitter voice of teenage experience. 'I just want a lover like any other,' he yelps on 1978's 'What Do I Get?': For you things seem to turn out right If it could only happen to me instead! What do I get? 'You didn't have to be a political rebel or riotous anarchist to relate to Buzzcocks' lyrics,' writes Annie Zaleski in Alternative Press, 'being a bored, disgruntled teenager or introverted social misfit was good enough'. Seventeen-year-old Robert Smith of Crawley was a little of both. Robert Smith: The melancholy man should make the best use of his moods. # The Cure ROBERT SMITH WAS galvanised by the energy of punk in 1976. But in the sprawling suburbs south of London where he lived, the call to riot in the streets seemed oddly useless. 'Living in Crawley you really didn't have to go out of your way to get beaten up,' says Smith. 'I couldn't really see the point in putting a safety pin through my nose.' The fury of the Sex Pistols was a much-needed shot in the arm, but in the Buzzcocks' nervy love songs for loners, Smith heard something of more lasting value, something he could use. The Buzzcocks had sent a signal out into the suburbs saying it was okay to be a punk and sing about your feelings, and Smith was about to take unprecedented liberties with this idea. Smith formed The Cure in 1976. The band released their first single 'Killing an Arab' two years later, and a debut album, Three Imaginary Boys, in 1979. The accompanying single 'Boys Don't Cry' had all the nervous adrenalin charge of punk, but the lyric was introspective and confessional. 'Boys Don't Cry' made perfect and immediate sense in the musical climate of 1979, and The Cure were regarded as promising. Then — as far as the critics were concerned — they blew it. From a band that seemed to have such a perfect grasp of the post-punk pop song, The Cure's next single, 'A Forest', was a baffling move. The song hardly seemed to be a song at all, more like a ghostly moan over three minutes of bleak Eurodisco. Smith's sense of humour and irony — the 'oh well, don't worry about me' — tone that had made 'Boys Don't Cry' so charming, had seemingly vanished along with the girl he was chasing in the gloomy lyric of 'A Forest'. And given that the song had no discernable hook or chorus, the only thing that seemed left was the singer's unhappiness. The faint twinge of despair that added spice to 'Boys Don't Cry' had become the entire musical world of 'A Forest'. Sadly, for those who still held out some hope for The Cure as the future of New Wave, 'A Forest' proved to be a sign of things to come. The Cure's second album, Seventeen Seconds was — like its single — a ghostly, impenetrable affair. Amazingly, it reached the Top 20 in the UK charts — but neither this, nor the critics' griping had much of an effect on Smith. From this point on, Smith's main goal was not to have hits or get good reviews, but to describe emotions. Over the course of three increasingly troubled records — Seventeen Seconds, Faith and Pornography — Smith would document his failing relationships and secret fears with unflinching honesty. These albums form one of the most important — and most misunderstood — bodies of work in the history of rock. They were hard to understand at the time of their release precisely because they were so personal, so singular, so much the product of one man's inner turmoil. There are musical reference points in early eighties Cure — Joy Division, Bowie, Eno, psychedelic rock — but in the end, Smith's most important influence seems to have been despair, of which he became an avid student. When Smith felt despair creeping up on him, he didn't do what the rest of us do — shrug it off, think positive thoughts, and try not to make too big a deal about it — instead he grabbed despair by the scruff of the neck and looked right into its pitch-black eyes. 'I was letting myself slip in order to write those songs,' he later recalled. 'I wasn't fighting it, whereas in everyday life you have to fight those feelings.' But the process of extracting and recording all this inner pain would take its toll, not just on Smith, but on the whole band. Recording an album like Faith was one thing, spending the next twelve months playing it in front of audiences was something else. Smith later admitted that Faith was the one record they shouldn't have toured with at that point. But they did, and as a result, Smith's gloom became self-perpetuating. By the time The Cure came to record Pornography, the singer was in the grip of a deep depression, further exacerbated by drugs and lack of sleep. Pornography opens with Smith howling, 'It doesn't matter if we all die'. Then it gets worse. Realising that he was driving the band into the ground and himself into an early grave, Smith called a halt in 1982, effectively declaring The Cure dead. He spent some time playing guitar on a Siouxsie and the Banshees tour. 'Fat Boy just does what he's told,' Siouxsie explained at the time. This was fine with Fat Boy, who was enjoying the feeling of playing music in which he had no particular emotional investment. Learning from this experience he began, in 1982, to write songs again — but of a very different kind from the ones on Pornography. Instead of recording his inner turmoil, he tried writing songs around themes — as though he were setting himself an assignment rather than writing a confession. Previously he'd aimed for authenticity, emotional sincerity; now he was trying to write a song that worked. He came up with a tune, gave it a solid, catchy beat, and loaded up the chorus with slinky 'doo doo doo doo's. He had a feeling 'Let's Go to Bed' would get played on the radio, and it did. He wrote another one called 'The Walk', and his mum told him, for the first time ever, that she liked it. Neither song had anything near the level of emotional honesty of those on Faith or Pornography. But Smith was done, for the time being, with ripping his guts out for his art. Form, not feeling, had become his goal. Rivers Cuomo: No feeling, no emotion. # Weezer WRITING ABOUT EMOTIONS is hard — it demands a level of self-absorption that even the most well-balanced individual would find it difficult to maintain. And of course, it's never the well-balanced individual who decides to pick up a guitar and pour the contents of his diary into a microphone — only a real lunatic would do that. So what's the alternative? Well, you could write about social life or politics or the state of your neighbourhood. But what if you don't care about any of that stuff? Is it possible to write a song that's just...a song? In 1983, Robert Smith proved you could; and in 1997, Weezer singer Rivers Cuomo — similarly exhausted by the effort of emotional music — decided to give it a go. He'd always written songs about himself, now he decided to see if he couldn't write a song about something that had never happened to him. Cuomo assembled a set of tried-and-true mythological images and set about describing them with words and music. He wrote a song about catching his sweetheart 'out in the eve, deep in the shady glen' in the arms of another man. The results surprised him. He'd always assumed great songs came directly from strong feelings triggered by personal experiences. But even though he'd never been in or near a shady glen, and the things in the song were entirely made up, the song still sounded great. To have discovered a method of songwriting by which he revealed nothing of himself was a huge relief for Cuomo. Not long before this, Weezer had released its second album, Pinkerton, in which Cuomo laid his heart completely bare. He'd spared nothing of himself — all his insecurities, his childhood anxieties, his sexual fantasies and his darkest thoughts went down on the tape. It was an extraordinarily brave thing to do, and it earned him, for his trouble, a pile of scathing reviews and a place in Rolling Stone's list of the Worst Albums of 1996. These days, Pinkerton is rightly seen as a classic — many believe it's Weezer's finest hour. But it's not impossible, listening to it today, to hear the reasons why it was so badly received in its day. Pinkerton is, in a word, embarrassing. It's embarrassing in the way that an unnecessarily maudlin twenty-first birthday speech can be, embarrassing in a reading-your-old-high-school-diary kind of way. It's the kind of embarrassment we feel for someone when they're over-sharing. Cuomo himself was not blind to this possibility. He'd first begun to experiment with this type of confessional songwriting in 1992, inspired by the example of New Radicals frontman, Gregg Alexander. In February of that year, he'd set about recording a cover of one of Alexander's songs 'The World We Love so Much'. He didn't want the guys in his band to hear it — he didn't want anyone to hear it. He didn't even hit record until he'd covered the walls of the room he was subletting with acoustic foam — not to enhance the sound quality, but to make absolutely sure no-one could hear him while he was 'emoting'. But the events of the following year made Cuomo bolder. The success of Weezer's insanely catchy — but surprisingly personal — debut album had given him reason to believe that there might be some level of interest out there in his emotions. So he decided to give them more emotions. Lots more. As a songwriter, he went into confessional mode — and he had plenty to confess. Cuomo was deeply uncomfortable with his newfound rock-star status. 'He hated himself for achieving it,' said music journalist Andy Greenwald, 'and he hated himself for loving it.' Stardom only increased his isolation and magnified the problems in his life, problems which went right back to his childhood. Rivers and his brother, Leaves, were raised on an ashram, an experience which left them totally unprepared for the brutal world of high school in America. The Cuomo brothers got the crap beaten out of them. But worse than that, they were outcasts, unable to connect with all the nice normal kids with their nice normal lives. No wonder, he thinks, he turned out weird. No wonder he can't communicate with people, except in the highly controlled form of songs like the one he's singing now. In 'Across the Sea' Cuomo dumps the blame for all of this squarely at his mother's feet. Then, disgusted by his own self-pity, and the entire song itself up to this point, he exclaims, 'goddamn, this business is really lame!'. This was more or less the mainstream music press's reaction to Pinkerton. Rolling Stone magazine, and the vast majority of the people who'd bought and loved the Blue Album, weren't ready for this kind of thing at all. Where were all the catchy little pop songs? Why is he screaming like that? The record-buying public stayed away in droves, reviewers were unkind, and Cuomo went into retreat. After we put out the first record, it seemed like a lot of the fans were really interested in me and were encouraging me to expose myself more, so that's what I did on the second record, and everybody hated it. I was really embarrassed. Pinkerton wasn't a disaster — it was an acquired taste. Grownup rock journals like Rolling Stone felt let down by Cuomo's failure to deliver on the promise of a bubblegum rock revival, and dismissed the album as morbid and self-indulgent. But younger fans loved it for exactly the same reasons. At this point, a parallel universe was created. In the world we know, grunge rose and fell, and rock-rap begat nu-metal. Meanwhile, in the other dimension, Pinkerton, not Nevermind, was the greatest album of the '90s, and emo began its crucial second phase. While the cult of Pinkerton was getting underway, Cuomo had begun writing non-autobiographical songs like 'Lover in the Snow', and judging their success not on their emotional authenticity, but on their formal qualities. Like Robert Smith (and for similar reasons) he was moving decisively away from emo toward what can only be described as 'formo'. Toward the end of the '90s, he began a study of rock structure in the form of his 'Encyclopedia of Pop', a ring binder full of hand-drawn charts in which Cuomo recorded the characteristics of hit songs by Green Day, Nirvana, Oasis and many other bands in an attempt to pinpoint the traits they share in common. This list has provided him with a set of models for songwriting, which he has been implementing ever since. The songs on the Green Album, he proudly told Rolling Stone's Chris Mundy in 2001, contain 'no feeling, no emotion', just music. Cuomo was not being entirely honest — 'Hash Pipe' and 'Island in the Sun' are emotional enough. But he was making the point that he would happily sacrifice feeling for form. This new direction irritated Pinkerton fans as surely as Pinkerton itself had annoyed the critics. One fan, writing as 'whatawierdo' on Songmeanings.com, said: I think Rivers has traded his personal touch of neurotic and clever songs for more standard, less emotional songs. Diehard fans gritted their teeth and put up with the 'horrible pop songs', scouring the albums for the rare flashes of Cuomo's old confessional mode that still showed up here and there. By the time of 2005's Make Believe, those who'd been seduced by Pinkerton's emotional authenticity had had enough of Cuomo's songs about nothin'. Pitchfork's reviewer wrote: Pinkerton triumphed by being an uncomfortably honest self-portrait of Cuomo. On Make Believe, his personality has vanished beneath layers of self-imposed universality, writing non-specific power ballads like he apprenticed with Diane Warren, and whoah-oh-ohing a whole lot in lieu of coming up with coherent or interesting thoughts. Cuomo would probably have taken the Diane Warren comparison as a compliment. The author of dozens of monster middle-of-the-road hits in the '80s and '90s, Warren's CV includes Starship's 'Nothing's Gonna Stop Us Now' and Chicago's 'Look Away', as well as co-writing credits with Bon Jovi and Cuomo's beloved Cheap Trick. The 'universality' and 'non-specificity' the pitchfork reviewer complains about on Make Believe is the key to the success of Warren's mega-hits. They're songs that (in theory) always work for everyone, because their structure is tried and true. That's why Diane Warren is a hit-maker. Hit-makers don't sit around waiting for painful personal experiences to happen to them — they sit down at the word processor and write hits according to the rules of hit-writing. By compiling his 'Encyclopedia of Pop' Rivers Cuomo was learning these rules for himself. Pretty soon, he was rhyming 'lady' with 'maybe' and telling his girl that 'you're the air that I breathe'. He sang this stuff on stage in front of a huge glowing 'W' and played solos like he was in Van Halen. He left emo in the dust, and embraced its opposite — classic rock. He'd proved it was possible to write powerful music whose goal was something other than the sharing of feelings. But in the process, he'd demonstrated that the modern indie rock singer does so at his peril. # The Classics THE KILLERS' DEBUT album Hot Fuss is full of lyrics about nothing. What is 'Somebody Told Me' about? Who cares? The important thing is that the chorus contains the words 'boyfriend' and 'girlfriend'. These are words that sound good in the choruses of new-wave rock songs, and Brandon Flowers has fifty years of pop history on his side as he yelps them out over the song's skipping beat and buzzing synths. What about 'All these things that I've done'? The glorious sing-along refrain, 'I've got soul but I'm just a soldier', works like a charm. But as comedian Bill Bailey has pointed out, Flowers might as well be singing, 'I've got ham but I'm not a hamster'. By the time it came to record the Killers' second album, Flowers was feeling guilty about having got away with these powerful, but un-emotional lyrics. He made amends by getting his diary out and writing an album about his childhood. Formalism in rock makes us uncomfortable. We're just as suspicious of Rivers Cuomo reducing rock to a series of lists and graphs as we are with the idea of songs being made-to-order by Diane Warren, or with Brandon Flowers writing lyrics by choosing words that sound good over music. Robert Smith was faintly disgusted with himself after he wrote 'Let's Go to Bed'. The song did the trick, but Smith felt like he'd got away with something, not like he'd made a great work of art. That's because in rock and roll, especially indie rock and roll, artists, critics and fans alike place an enormous premium on emotional authenticity. When an album is good, we say it's 'inspired', 'sincere', 'unflinchingly honest' or 'deeply personal'. If it misses the mark, it's 'formulaic', 'soulless' or 'unoriginal'. The idea of art or music as a form of self-expression is virtually taken for granted today. It's the artist's ultimate authority — 'I wrote it that way because that's how I felt,' says the artist. And this is no less than we expect of the artist — authentic self-expression, in defiance of fashion, sensible advice or the dictates of the marketplace. We accept the idea that music might have other goals — social commentary, political protest, making you dance, getting the singer laid. But in every case, what we're mostly interested in is the artist's feeling for these things. Emo is just an extreme and uncompromising variation on a theme which is universally accepted. From the rarefied air of The Wire magazine (where an almost unlistenable album will be lauded for the artist's refusal to compromise his vision), to the set of American Idol (where week in, week out, the judges advise the contestants to 'be yourself'), the mantra is the same. Authentic self-expression = good art. What we look for in music is passion, because passion, we feel, makes good poetry. This wasn't always the case. In eighteenth-century London, for example, nobody took much of an interest in poets' feelings, or how sincerely they were expressed. Back then, nobody would have cared very much about Robert Smith's depression, Gerard Way's rage, Rivers Cuomo's angst or Brandon Flowers' diary. In fact, most of the artists whose lyric sheets we pore over today would have been chased out of the coffee house for having too many feelings, for devoting too many stanzas to their emotions while forgetting all about the things that really make for quality poetry — a sense of balance and symmetry, a sound grasp of metrical composition, and the advancement of a useful moral theme or accurate social observation. Not to understand these things was, in eighteenth-century literary circles, as disastrous for one's career as to be seen about town in a poorly powdered wig. In the pursuit of this ideal, the poet's emotions could be of no particular use — in fact they were most likely to get in the way. What was needed to write great poetry was not passion, but careful study of the classics. Alexander Pope was a master in this regard — he studied Horace to the point where he could imitate his style perfectly. He became famous, and it became the ambition of all young poets to imitate Pope's imitation. And this was not impossible, since Pope took care to set out the rules of poetry he'd derived (and refined) from Horace in his Essay on Criticism. He even made them rhyme. Here, a poet could learn, if not how to write a great poem, at least how to avoid writing a bad one. These rules were more discussed than actually followed. 'No great writer,' literary historian Richard Barnard points out, 'allowed himself to be imprisoned in neo-classical theory.' And yet the fact that this theory existed and was seriously discussed offers a glimpse of an artistic climate completely different to our own, one where order and stability were the qualities most admired in a work of art, and originality — far from being the poet's goal — was something best avoided, since it meant you were more likely to screw things up. The good thing about formalism is that it usually works, but often that's about the nicest thing you can say about it. 'The classic,' wrote Walter Pater in his Essay on Style, 'comes to us out of the cool quiet of other times: as the measure of what a long experience has shown us will at least never displease us'. In certain periods of history, Pater says, the classics assert themselves — and this is what happened in Europe in the eighteenth century. But eventually, there will be — there has to be — a reaction to this insistence on order and symmetry. A demand for the wild, the quaint, the passionate, and the unreasonable will make itself felt. The pendulum will swing back. Rules will be broken, books thrown aside. # Troublemaker THE UN-EMOTIONAL, CLASSICAL phase that Rivers Cuomo had entered with 'Lover in the Snow' began winding down during the recording of 2005's Make Believe. But the real change came with the release of a compilation of his home demo recordings three years later. In collecting the material for this album, Cuomo went through tapes dating back to the very earliest days of Weezer. He listened, mesmerised, to the Rivers of fifteen years ago emoting in his carefully soundproofed isolation on 'The World We Love so Much'. He heard again the painful whimpering at the start of 'Crazy One', and the demo tapes of his wildly ambitious — and ultimately abandoned — space opera, 'The Black Hole'. He began to speak approvingly of Pinkerton for the first time since its traumatic birth. When Weezer finally released their new album later that same year, it quickly became apparent that something had been fundamentally altered in the singer's approach to his art. The Red Album's opening song, 'Troublemaker', is a manifesto for this new direction, in which Cuomo finally throws out the 'Encyclopedia of Pop', and asserts the value of originality and sincere personal expression. The singer insists that he is an original man with original thoughts. So instead of looking at books, he's looking inside himself: Who needs stupid books? They are for petty crooks I will learn by studying the lessons of my dreams Dreams crop up again on The Red Album, on a song Cuomo describes as 'a big symphonic art-type number', 'Dreamin''. The tug of war between emo and formo in Rivers' soul meant the song very nearly didn't make it onto the album. He wrote it, and then somehow lost his nerve. He scrapped it, and started working on a reassuringly classic-sounding verse-chorus-verse type song that became 'This is the Way', which the band, and the record company loved immediately. But by the time Weezer started recording the album, Cuomo was feeling adventurous again. He argued passionately for 'Dreamin'' to be included and 'This is the Way' to be left on the shelf. 'Dreamin'' is an ambitious ode to imagination and reverie, in which Cuomo expands on the idea contained in those lines from 'Troublemaker'. All his life, the singer explains, people have been trying to tell him there are rules. You have to go to school, you have to get a job, you have to learn to be responsible. And all his life, the singer has known in some profound way that this is a crock. How does he know? He just knows. 'Normal' life — school, job, etc, terrifies him to his soul. But when he's absorbed in his own imagination, he feels at home: Dreamin' in the morning Dreamin' all through the night and when I'm dreamin' I know that it's all right The song moves through several different movements that illustrate the dreamer's different moods. At the beginning, when he's just staring out the window, the backing has a dreamy '50s' teen-pop feel to it. When the singer starts asserting his right to do what he likes and stops doing his homework, the guitars crank up a notch and the music takes a more defiant stance. Then, in the middle section, the city and its suburbs, the school, the freeways, disappear entirely. Cuomo leaps through a slightly hilarious Sound of Music soundscape, with choirs of angels echoing over the hills and taped birdsong twittering in the background. Here, the world of custom and convention seems far away — there are no teachers, no grownups, no cops and no record companies. As his voice rings out over the landscape, he starts to wonder if the natural world isn't somehow connected with the source of his own creativity. He feels cramped and constrained by human society, with its rules and regulations. People are always telling him to 'get with the program'. Out here, it quickly becomes obvious that there is no program, and the singer's imagination finally has space to roam. He throws away his schoolbooks and his 'Encyclopedia of Pop', and starts listening to the birds and the bees. Wordsworth: Books! 'Tis dull and endless strife! # Wordsworth THE SWING AWAY from classicism in eighteenth-century England began during Pope's lifetime, as the classical poetry of the day was supplemented by a growing interest in popular ballads of the Middle Ages. The authors of these unruly old poems were mostly unknown, and the verses themselves had changed many times over the centuries as different singers picked them up and adapted them to their purposes. They were rarely written down, mostly because they were considered too rough and bawdy to be proper literature — ballads were not for polite company, and they found no place in the eighteenth-century salons. The ballad's humble birth and lusty swagger landed it on the wrong side of the line dividing the Classical from its uncouth opposite, the Romantic. This made it the perfect vehicle for the poet who would knock the wig-wearers off their perch in the nineteenth century — a man who had no time for classicists or cafes. He announced his arrival in 1798 with a book of Ballads. In William Wordsworth's 'Expostulation and Reply', we find the poet by the side of the road, sitting on a rock, staring into space. A wandering classicist stops to lecture him: shouldn't he be re-reading Horace or refining his couplets? 'Why, William, on that old grey stone, Thus for the length of half a day, Why, William, sit you thus alone, And dream your time away? 'Where are your books? that light bequeath'd To beings else forlorn and blind! Up! Up! and drink the spirit breath'd From dead men to their kind.' The man with the walking-stick wants Wordsworth to stop rambling about the countryside and get back to work. But what he doesn't realise is that Wordsworth is at work — the forest is his office and the lake is his library. He doesn't need the 'spirit breath'd from dead men to their kind', because he's chosen to learn from the living. In the poem's sequel, 'The Tables Turned,' he puts forward his case: Books! 'Tis dull and endless strife, Come, here the woodland linnet, How sweet his music; on my life There's more of wisdom in it. And hark! how blithe the throstle sings! And he is no mean preacher; Come forth into the light of things, Let Nature be your teacher. These two poems first appeared in a book called the Lyrical Ballads , published as a joint venture with Wordsworth's friend Samuel Taylor Coleridge in 1798. This little book would change English poetry forever, and the after-effects of its discoveries would be felt all over the English-speaking world. But the poet's original intentions were humble enough. The previous year, Wordsworth and his sister, Dorothy, had moved to Alfoxden, overlooking the Bristol Channel. Here, Dorothy had begun to keep a journal describing the sights and sounds of the countryside, and Wordsworth began some poems with the intention of doing the same thing in verse. As he worked on these, it became increasingly clear to Wordsworth that the eighteenth-century's rules for good poetry would be of no use to him whatsoever. Imitating Pope imitating Horace might help you make a big splash at the coffee house but out in the countryside, miles away from London and its whirlwind social life, different standards prevailed. It made no sense to describe the lives of tramps and cottage girls in the language of Pope and Dryden — who talks like that? Not the tramps and the little girls, that's for sure. But classicism was equally useless for the task of describing Wordsworth's impressions of nature, the ecstatic sense he had of a great spirit moving through all creation. How could he take a feeling like that and chop it up into pieces to make it fit some prefab idea of classical proportion? So while all the other poets of the day were knocking themselves out trying to nail their poetic diction and get their heroic couplets down pat, Wordsworth was working just as hard to remove every trace of eighteenth-century classicism from his work. That's why he spent all his time sitting on a rock and not at the library. All he would learn from books is how to write like poets who came before him, and Wordsworth had decided that their language, as good as it had been in its day, was of no use to him. He was determined, as he says in 'The Tables Turned', to learn from nature. # Civilisation WORDSWORTH'S REJECTION OF culture in favour of nature doesn't seem that remarkable today. But in the century he was born into, it would have been considered deeply weird. In the eighteenth century, it was taken for granted that modern civilisation had improved and refined nature in every way. The spirit of the age was extraordinarily optimistic. On the 3 July 1750 Louis XVI's minister Jacques Turgot had told his audience at the Sorbonne that the world was getting better, and that if things seemed less than perfect now, it was simply because human civilisation had some growing up to do: ...the whole human species, looked at from its origins, appears to the philosopher as an immense whole, which, like an individual, has its infancy and its progress... The totality of humanity, fluctuating between calm and agitation, between good times and bad, moves steadily though slowly towards a greater perfection. Turgot's theme was a popular one during the eighteenth century, a period of time referred to by historians as The Age of Reason or the Enlightenment. The era was dominated by an enthusiasm for the discoveries of science and a belief that the power of rational thought would transform every area of human life. The thinkers of the Enlightenment didn't claim to know everything. But they maintained, by and large, that everything could be known. Whatever problems mankind faced now, they reasoned, would be solved in the future by fearless rational investigation. At least, that was the idea. In 1749, philosopher and writer Jean-Jacques Rousseau, a native of the city of Geneva, read a notice in the Mercure de France, announcing an essay-writing competition on the topic: 'Has the restoration of the arts and sciences had a purifying effect on morals'. Rousseau pondered this, and soon found that his head was swimming with a thousand ideas. He felt faint. When he'd collected himself, he sat down and wrote twelve thousand words to the effect that, No, the restoration of the arts and sciences had not had a purifying effect on morals. Progress, Rousseau argued, was not improving society — it was making things worse. Rousseau began his essay by pointing out that if the arts and sciences were improving morals, then France, which was supposed to have the best art and the cleverest scientists in the world at that point, ought to be the most moral place on the face of the earth. This, Rousseau insisted, despite appearances, was not the case: There prevails in modern manners a servile and deceptive conformity...politeness requires this thing; decorum that; ceremony has its forms, and fashion its laws, and these we must always follow, never the promptings of our own nature. In the pages that followed, Rousseau offered a scathing critique of his supposedly 'improving' century. Where others saw the peak of civilisation and refinement, Rousseau saw only phoney manners held up to disguise a disturbing lack of real human feeling. But Rousseau also hinted, as in the passage above, at a 'true' human nature which had somehow been left behind or forgotten. He took up this theme in his next crack at the Dijon Academy's essay-writing prize; this time the given topic was 'What is the origin of inequality among men, and is it authorised by natural law?'. In this, his Second Discourse, Rousseau sized up the eighteenth-century man and tried to figure out what, in all his behaviour, is most 'natural'. This, he admitted, was tricky. Human beings had by this point been so altered by the societies they had evolved, that they barely resembled their ancestors. But Rousseau believed that he had discovered, lurking beneath the surface of these modern people, two 'natural' inclinations. These were not arrived at by reason, like our tacked-on modern philosophies, but came as standard with the human soul, part of our original design. One is an interest in our own welfare, the other is the feeling of repugnance at the sight of another's suffering. But on top of this original design, Rousseau says, we have acquired a caked-on crust of false standards, all of which have their basis in our need to acquire privilege and distinction. This is in turn a result of the fact that human beings have, over the centuries, been coming together in greater numbers and living in closer proximity to one another. Now, instead of living naturally, the social man lives 'for others'. This is the case, not only for those on the lower rungs of society — who have to make their way in the world under systems designed for the benefit of the rich and powerful — but also for the privileged few, who judge their worth purely in terms of the power they command over others. From this artificial way of life has come all our law, and the hierarchies of our society. None of it, Rousseau concludes, has any basis in natural law. Here was a resounding 'no' to the essay question posed — and something more. The Second Discourse contains Rousseau's most convincing, and most dangerous idea: that the furniture of eighteenth-century life — royalty, serfdom and myriad class distinctions in between — were not fixed, but moveable. Rousseau died in 1778. Eleven years later, the furniture of France would be thrown out the window. # The French Revolution IN HIS INTOXICATING account of the French Revolution, historian Thomas Carlyle conjures a vivid picture of the forging of France's new constitution, 'amid glitter of illuminated streets and Champs-Elysees, and crackle of fireworks and glad deray'. Carlyle describes: Twelve Hundred human individuals, with the Gospel of Jean-Jacques Rousseau in their pocket, congregating in the name of Twenty-five Millions, with full assurance of faith, to 'make the Constitution': such sight, the acme and main product of the Eighteenth Century, our World can witness once only. For Time is rich in wonders, in monstrosities most rich; and is observed never to repeat himself, or any of his Gospels: — surely least of all, this Gospel according to Jean-Jacques. All sorts of factors were at work in the years leading up to 1789, and in any accurate account of the Revolution's causes, Rousseau's books would have to get in line behind such heavyweights as France's escalating financial crisis, the simmering resentment of the peasantry, the War of Independence in America, and a series of mini-revolutions in other parts of Europe. But there can be no doubt that Rousseau's name was associated with the Revolution from the moment it took place. Whether the actions of the revolutionaries themselves were inspired by his (very popular) books is almost beside the point. The Revolution seemed to put his ideas into practice — right from the start, hereditary privilege and serfdom were abolished, freedom and equality were the slogans. The new constitution's first clause, 'Men are born, and always continue, free and equal in respect of their rights', echoed Rousseau's famous statement in The Social Contract, 'Man is born free; and everywhere he is in chains'. The Revolution, as Victorian critic Matthew Arnold has observed, 'seemed to ask of a thing, "is it rational?"'. In other words it was heralded as the culmination of all the hopes of the Enlightenment. For over a hundred years, philosophers and other thinkers had been talking about a society built on rational principles — now, it seemed, this society was being born. It's impossible to overestimate the optimism with which this was greeted among freedom-loving artists and intellectuals. Bliss was it in that dawn to be alive, But to be young was very heaven... wrote Wordsworth, thinking back on the Revolution's early days, when anything seemed possible. The poet was in Paris for the first anniversary of the storming of the Bastille. He saw, at first hand, the 'fireworks and glad deray' Carlyle described. Then, two years later, in the winter of 1791, he was back. This time, he fell in love — twice — once with a girl named Annette Vallon, and once with the Republican cause. His political enthusiasm, as his biographer, Roger Sharrock, points out, was mingled in his mind with the natural beauty of the landscape he'd seen on his first visit. For Wordsworth the grandeur of nature seemed to point toward the dignity of man in his natural state. The essential rightness of democracy was indicated by the very ground he was standing on and the sky above his head. All of this was brought to a premature halt when Wordsworth's money ran out in 1792, and he was forced to return to England. There, he soon found that his deeply felt republican sympathies had made him a traitor to his own country, as the Pitt government declared war on France in February of the following year. Wordsworth cheered when he heard that English troops had been massacred by the French — and hated himself for it. There were further shocks in store for the lover of freedom and democracy. By this time, France's monarchy had been abolished, and the king himself had been executed. Democracy it seemed was within sight. But the newly reborn nation was in a state of chaos, at war with almost every country in Europe while simultaneously being torn apart by civil strife, hunger and confusion. 'There was no room', as Rupert Christiansen puts it in his book Romantic Affinities, 'for the democracy that allows dissent.' France's future was effectively placed in the hands of Maximilien de Robespierre, the most influential member of The Committee of Public Safety, and a staunch follower of the gospel according to Jean-Jacques. Robespierre had learned from Rousseau's The Social Contract that power came not from kings or governments, but from the will of the people. This, as philosopher and historian Bertrand Russell points out, is a much-misunderstood idea in Rousseau. In practice, it tends to mean that power-hungry individuals, or those with an axe to grind, can claim — by some mystic association — to 'represent' this will of the people, without having to go through all the fuss and bother of ballots and elections. Robespierre was certainly one of these. The people, he maintained, were virtuous, but their virtuous new republic was under threat from aristocrats and royalists — leftovers from the bad old days. Robespierre prescribed a Reign of Terror — a necessary stage in which these counter-revolutionaries, and anyone else who stood in the way of freedom, would be rounded up and disposed of so that France could get on with the business of creating a new society. Robespierre's courts and police were kept very busy. The guillotines worked overtime, and seventeen thousand enemies of freedom were executed in the space of fourteen months. Persecution and mass-killings were nothing new in Europe — but this was something else. Crusades, witch-hunts, pogroms and inquisitions had always been carried out in the name of religion, or authorised by despotic kings; here was slaughter carried out in the name of natural virtue, the will of the people made manifest. Rousseau had always seemed to say (though this is not exactly what he meant) that if you made people free, they would be good and just. But this, it now seemed, was untrue. For many, the brightest hopes of the Enlightenment, the dream of freedom, equality and brotherhood, died sometime in 1793. William Wordsworth, for one, was deeply confused. For him, as for all those who 'had fed their childhood upon dreams', the Revolution had promised nothing less than heaven on earth: ... O times, in which the meagre, stale, forbidding ways of custom, law and statute took at once the attraction of a country in Romance; when reason seemed the most to assert her rights... Now his faith in this vision was being sorely tested. For a while he clung to the idea that the Terror was simply a necessary means to an end, that true freedom and democracy could only be achieved after a difficult, but necessary, clampdown on freedom and democracy. When he realised how untenable this position was, he turned to the political philosophy of William Godwin, who advocated Universal Reason as mankind's brightest hope. In his much-read and discussed 'Enquiry Concerning Political Justice', Godwin argued that reason should be given priority over all other considerations in life, including law, social convention and family ties. What this meant in practice was, as Godwin illustrated in a famous example, that if you could save only your mother or the world's greatest philosopher from a burning building, you really ought to save the philosopher — reason says he will be of more use to the human race in the long run. Despite his initial enthusiasm, Wordsworth soon found he was incapable of being a good Godwinian. He just couldn't quite let go of his emotions — though it wasn't for lack of trying: Thus strangely did I war against myself; A bigot to a new idolatry Did like a monk who hath forsworn the world Zealously labour to cut off my heart From all the sources of her former strength; In his autobiographical poem, The Prelude, Wordsworth explains that the story of his life up to this point had been — like that of the century he was born into — a story about things getting better: This history, my friend, hath chiefly told Of intellectual power, from stage to stage Advancing, hand in hand with love and joy, And of imagination teaching truth. But the Revolution and the Terror had knocked him badly off course. He'd found his youthful idealism diverted toward a cause that made him an apologist for murder. Then, searching for an alternative, he'd embraced a philosophy that required of him that he cut out his heart. This he knew he could not do. Irrational though it might be, the young poet had a feeling his heart would come in useful later on — and he was right. Conor Oberst: 'A special moment governed more by intuition than intellect'. # The Story is in the Soil Lifted or The Story is in the Soil, Keep Your Ear to the Ground is plainly too long for the name of an album — but Bright Eyes' Conor Oberst will always happily sacrifice tidy form to the expression of powerful feelings. In the album's first song, 'The Big Picture', we find the singer riding in the back of the tour van while the driver and the guy in the passenger seat argue jokingly about where this place they're looking for is supposed to be. Something about this conversation gives the singer an idea. He pulls out his guitar and starts picking out some chords, letting the words come as the music builds up a head of steam. It doesn't rhyme properly in places, and some of the words are crammed a little awkwardly into the metre — but the feeling is real, and it's the feeling, not some pre-conceived idea of 'good songwriting' that Oberst follows in bringing this tune to completion. 'The Big Picture' goes on for another six and a half minutes — bringing the whole to a total of eight minutes and forty-seven seconds. Too long, you might say, for a melancholy dirge banged out in the back of the tour bus. But to Oberst in 2004 criticism of this kind meant nothing. 'There's a point where they feel complete, and that's where I stop', he said of his songs. 'Maybe for some listeners they felt complete four minutes ago — they can fade it out.' Oberst is a student of nature, so he's not interested in rules or traditions. He might sound philosophical in 2005's 'I Believe in Symmetry', but really he's expressing a wish to be rid once and for all of philosophy — and all the other stuff they teach you at school. Has any of it, Oberst asks himself, made me happier? An argument for consciousness The instinct of the blind insect Who makes love to a flower bed And dies in the first freeze 'I want to know such simple things,' says the singer, 'no politics, no history.' But ridding yourself of five centuries of tradition is not as easy as all that — politics is everywhere, and history keeps screaming in his ear. In 'Road to Joy', recorded the same year, Oberst decides to scream back. The song is a portrait of a young man with a sensitive heart and a head full of noise trying to get his thoughts down before it's too late: So now I'm drinking, breathing, writing, singing. Every day I'm on the clock. My mind races with all my longings. But I can't keep up with what I got. What he's got is a feeling, not just for himself, but for the whole country, the whole human race. Now, politics has become personal for the singer, and he's turned into a sort of emotional news anchor, reporting on the state of his world as it relates to President Bush's War on Terror. Everything is involved — his parents, his girlfriend, the flowers in the driveway, the dead bodies in the cemetery, everything hums to the tune of his anxious ballad. Oberst has what Wordsworth would call 'a heart that watches and receives', and hearts like this can't help but pick up the world's static. He gives his feeling words, and fits the words to a tune — not one of his own, this time, but one that was written to give voice to a similar mood of turmoil and hope almost two centuries before Conor Oberst was born. In 1785 Friedrich Schiller had just gotten over his last girlfriend, and spring was coming to his village near Leipzig. He was overcome with an incredible surge of happiness and goodwill for the human race, and sat down to write an 'Ode to Joy': Joy, brilliant spark of the gods, daughter of Elysium, heavenly being, we enter, drunk with fire, your holy sanctuary. Your magic reunites what was split by convention, and all men become brothers where your gentle wings are spread. Be embraced, you millions! This kiss for all the world! Brothers, above the starry canopy must surely dwell a loving father. Schiller's lines expressed the highest ideals of his century — the hope that the clearing away of dogma and outmoded institutions would, in time, heal the rifts in modern society and bring an end to war and misery. They also hinted at something new (or something very old, which seemed new); a wish to take leave of one's senses — to dance, to sing, to lose oneself in a happy throng. Five years after he wrote it, young German poets were running through a meadow near the seminary at Tubingen shouting Schiller's poem into the night air, and pausing between stanzas to take swigs from a bottle of wine. Schiller, like Wordsworth, was deeply sympathetic to the Revolution; and like the English poet, he found his convictions impossible to maintain after the Reign of Terror. But if the Revolution shattered his faith in the ideals of the Enlightenment, it convinced him more than ever of the importance of art and poetry: If man is ever to solve the problem of politics in practice he will have to approach it through the aesthetic, because it is only through beauty that man makes his way to freedom. Even after world events had conspired to make Schiller's optimism seem naïve, it was impossible to dismiss out of hand the vision he had presented in 'Ode to Joy'. In fact, as the bright hopes of 1789 receded into the distance, the question of how to make people happy seemed more pressing than ever. Ludwig van Beethoven decided to tackle the problem himself in 1802, announcing his intention to set Schiller's 'Ode' to music. It would be another twenty-two years before he would write to his publisher with good news on this front: Vienna, March 10, 1824. ... These are all I can at present give you for publication. I must, alas! now speak of myself, and say that this, the greatest work I have ever written, is well worth 1000 florins C.M. It is a new grand symphony, with a finale and voice parts introduced, solo and choruses, the words being those of Schiller's immortal 'Ode to Joy', in the style of my pianoforte Choral Fantasia, only of much greater breadth. At the asking price of 600 florins, the publisher had got himself a bargain. The Choral Symphony wedded Schiller's verses to one of Beethoven's most powerful pieces of music. The poem appears in the final movement, which begins with the ugliest blast of discordant noise that had been heard in a concert hall up to that time — which, for Beethoven, symbolised nothing less than all the misery in the world condensed into one gigantic, impossible chord. Then, as the smoke clears and the dust settles, a lone voice pipes up in the darkness, 'Oh friends! No more of these tones! Let us sing something full of gladness!'. A chorus appears out of nowhere and joins the singer as he belts out Schiller's 'Ode' and the whole thing is carried by a magnificent, soaring melody — the same melody, in fact, that Conor Oberst rides in 'Road to Joy'. Beethoven, conducting this final section at the piece's premiere in 1824, got completely carried away — he was still furiously waving his arms in the air long after the orchestra had stopped playing. And Oberst seems to be swept up in the same feeling of wild abandon as his own song comes to its conclusion. 'Let's fuck it up boys!' he tells his band, 'make some noise!' But where Beethoven had his cacophony redeemed by a dream of universal brotherhood, Oberst ends his song with the end of the world. Oberst had always tried to write hope into his sadder songs. But you can hear his optimism fading in the last verses, as he looks around at the world and what we've made of it. The same suspicion with which he regards Western civilisation in 'I Believe in Symmetry' here reaches fever pitch. He sneers bitterly: I hope I don't sound too ungrateful, What history gave modern men. A telephone to talk to strangers, A machine gun and a camera lens None of these are any consolation for the still missing-inaction dream of universal human brotherhood. It's over two centuries since Schiller wrote his poem, one hundred and eighty years since Beethoven set it to music, and three decades years since that music was adopted as the official anthem of the European Community. But the dream it represents seems further away than ever. The cracked howl and burst of noise at the end of 'Road to Joy' signalled a shift in Bright Eyes' music. Oberst had already thrown himself into political activism, performing onstage with his hero, Bruce Springsteen, on the 2004 Vote for Change tour. Later, in May 2005, he released a download-only protest song called 'When the President Talks to God' — a direct critique of the Bush Administration. Then, at the beginning of 2008, he fell back — feeling, as he later described it, 'corrupted and corroded' — and turned his gaze inward again. But the album he recorded — the first to be released since his very early days under his own name — was very different to Lifted or earlier efforts like Letting off the Happiness. Where Oberst used to look inside himself and see a world of trouble, here, on songs like 'Sausalito,' he seemed to have found a measure of self-reliance, even peace. The source of this new strength, it turned out, was nature. In 'Sausalito', the singer describes a camping trip with his girlfriend — they drive out into the desert so as to have the stars all to themselves. Here, Oberst's experience of the landscape becomes almost religious; he has a sense of a spirit moving through creation, a 'sound too soft to hear'. This mysterious 'something' accounts for the new feeling of calm in the songs on his self-titled album, which was recorded in a small cabin in rural Mexico. The music, as Oberst explained to triple j's Zan Rowe, sprang from the landscape itself and the feelings it stirred in him. 'I believe places have an energy to them,' he said. 'I felt at peace, but also inspired.' # A Motion and a Spirit IN JULY 1798 Wordsworth and Dorothy travelled to Bristol to see Lyrical Ballads through the presses. They made their way up the Wye River and stopped at a place called Tintern, not far from the ruins of an old gothic abbey. Wordsworth was overwhelmed by conflicting emotions. On his last visit to Tintern five years previously, his state of mind had been desperate, to say the least. He'd had his heart broken by a woman he had to leave, and by a cause he could no longer believe in. He'd found himself a traitor to his own country, an apologist for tyranny, and an apostate to a faith he'd only recently converted to. He was, in other words, a wreck. Back then, he'd raced around this landscape: Wherever nature led: more like a man Flying from something that he dreads than one Who sought the thing he loved... But since then, much had changed in Wordsworth's life. In 1798 he could look back at the Wordsworth of five years previously with not a little admiration for his hot-headed romantic passion. But he knew he wouldn't trade that for what he'd found since: A presence that disturbs me with the joy Of elevated thoughts; a sense sublime Of something far more deeply interfused, Whose dwelling is the light of setting suns, And the round ocean and the living air, And the blue sky and in the mind of man: A motion and a spirit, that impels All thinking things, all objects of all thought, And rolls through all things. Here, in 'Tintern Abbey', Wordsworth is describing the almost mystical faith in nature that would sustain him for the rest of his life. This feeling runs through even the plainest of the Lyrical Ballads, and it's the key to his admonishment to poets in 'The Tables Turned' to put down the books and go for a walk. Wordsworth's faith, and his ability to write poetry, had been restored by his year in the country. It seemed to him as though the source of life and the source of his creative power were one and the same. But this realisation had not come easily to him — to reach it he had, in a sense, found it necessary to jettison almost four hundred years worth of European history; four centuries in which man's ability to reason was prized above all else, and the kind of simple faith Wordsworth was longing for was thought to be a relic of a (thankfully) long-gone era. Wordsworth's new philosophy turned this attitude on its head. For the second edition of the Lyrical Ballads, Wordsworth wrote a preface explaining his new ideas as they related to poetry. He warned his readers that they were about to enter a poetic universe in which the laws laid down by Pope and the coffee-house classicists did not apply. He had powerful feelings to communicate, feelings which had come to him in the presence of nature; feelings which could not be made to abide by a set of rules any more than nature itself could be made to fit the harsh geometry of an eighteenth-century garden. 'All good Poetry,' Wordsworth insisted in his now-famous preface, 'is the spontaneous overflow of powerful feelings.' But Wordsworth was careful to add a disclaimer to this, saying that poets should make sure they don't get too carried away with their emotions: The end of Poetry is to produce excitement in coexistence with an overbalance of pleasure... But if the words by which this excitement is produced are in themselves powerful, or the images and feelings have an undue proportion of pain connected with them, there is some danger that the excitement may be carried beyond its proper bounds. Here, Wordsworth is insisting on one hand that poetry must come from feeling, while warning on the other that the poet must temper this feeling with a mood of calm repose such as the one in which he wrote 'Tintern Abbey'. Over the coming century, generations of younger poets happily embraced the first part of his formula while completely disregarding the second. This, in a sense, was entirely fair. Wordsworth had revealed that the rules of poetry were a sham, and that the only authority the poet ought to respect was the poet's own feeling for truth. It was a bit late now to start talking about 'proper bounds'. # Romantic WHEN CONOR OBERST said, back in 2002, that people who thought his songs were too long could just fade them down, it's almost as if he was saying, 'I don't care if you listen — these are my feelings'. This is the kind of thing Wordsworth warned about in his preface — the poet's 'excitement carried beyond its proper bounds' perfectly describes Bright Eyes' early music. The singer has rejected formalism, and replaced it with nothing. As a result, Oberst sings too long, confesses too much, cries too easily, and screams too loud. These days Oberst's position is closer to the Wordsworth of 'Tintern Abbey'; his music still comes from feeling, but that feeling is tempered by a sense of spiritual calm. He's even made some concessions to form — although the forms he uses are much more likely to come from the street-level tradition of popular balladry than from any encyclopedia of pop. And yet it's no less personal — everything comes from feeling and the artist's inner life, and he shares it with us because it moved him. No other reason is required. When he starts screaming and hollering in 'I don't wanna die in the hospital' or over-sharing about his sex life in 'Sausalito', the old Conor Oberst is not too far away. Is it a bit much for you? he seems to be asking. Go on, fade it down — see if I care. He can afford to say this because he knows we won't — not all of us anyway. For every hundred souls who fade him down and fade up the new Maroon 5 album, there's at least one or two who stay the distance, and those special few are hooked for life. Music writer Brian Howe has said that being a Bright Eyes' fan is about having 'a sense of being a part of a special moment governed more by intuition than intellect'. Conor Oberst's music is about feelings, not rules; and to love him is to choose the sound of gut-wrenching sadness over polished perfection, to prefer soul-baring excess to cool refinement. To like these things in 2005 made you emo; in 1798 it made you romantic. Romanticism is often seen as a reaction to the Enlightenment — a rejection of the philosophical and literary ideals of the eighteenth century. Its earliest stirrings can be found in the very midst of the Enlightenment itself. Rousseau was, in many ways, a typical Enlightenment philosopher, since he sought to improve life by discrediting assumptions. But because he preferred nature to society and strong passion to rational thought, he was also the first of the romantics. After Rousseau came the Germans, who took things an important step further. Rousseau, as Isiah Berlin has shown in Against the Current, may have rejected the culture of science, but he never abandoned the idea that the world made sense. The Germans of the late eighteenth and early nineteenth century — Haman, Goethe, Schiller, Heine, Hegel, Fichte and Schopenhauer — would not be so squeamish. These writers would replace the Enlightenment's clockwork universe with a world of flux and chaos, and this change was mirrored in the art and literature they produced and championed — classicism was replaced by folklore; refined elegance by untamed nature; good sense by intense emotion. Meanwhile, in England, the achievements of Wordsworth, Coleridge and Southey were followed in the early nineteenth century by stronger stuff from Byron, Shelley and Keats. These poets looked to Wordsworth sometimes as an elder statesman, sometimes as an embarrassing old uncle. They were generally less cautious in their methods and more extreme in temperament than Wordsworth — and they augmented his idea of poetry as a description of the poet's inner life with an interest in darkness, despair, madness and other altered states. This was the legacy of another Counter-Enlightenment tendency — the gothic revival, which had been gaining momentum for almost half a century before the Lyrical Ballads was published. By the time Byron's Childe Harold's Pilgrimage became a surprise smash hit in 1812, romanticism was a craze, and by 1830 any adherents to the school of Pope would be feeling — as literary historian Robert Barnard puts it — 'very lonely indeed'. Romanticism would, in various forms, dominate the artistic and philosophical world of the nineteenth century. By 1900 it had given the world Wordsworth's 'Tintern Abbey', Beethoven's Choral Symphony, Mary Shelley's Frankenstein, Keats's 'La Belle Dame Sans Merci', Goethe's Faust, Schopenhauer's The World as Will and Idea, Victor Hugo's Les Miserables, Eugene Delacroix's Liberty Leading the People, Wagner's Tristan und Isolde, Nietzsche's Also Sprach Zarathustra, Bram Stoker's Dracula, Puccini's La Boheme and Oscar Wilde's The Picture of Dorian Grey. If we could somehow get the authors of all these great works together in a room, we'd have a tough time getting them to agree on anything — and no hope at all of discovering a single artistic principle they all share in common. Romanticism is hard to define, partly for the same reason emo is; it's entirely predicated on the idea that the artist is a unique and special individual, and there's nothing unique and special individuals hate more than the implication that they are somehow one of a 'type'. But even if the artists' objections are ignored, the historian will have a tough time coming up with a definition of 'romantic' that holds true in every case. Romanticism seems to dissolve as it's subjected to scrutiny — a metaphor the romantics, with their suspicion of reason and science, would appreciate: Sweet is the lore which nature brings; Our meddling intellect Misshapes the beauteous forms of things; — We murder to dissect. In fact if one thing could be said to connect the movement's most famous voices — to provide a link between the careers of such singular and unclassifiable personalities as Shelley, Beethoven, Nietzsche, Puccini, Hugo, Friedrich and Keats — it's the idea Wordsworth speaks of here in 'The Tables Turned': Enough of science and of art; Close up these barren leaves; Come forth, and bring with you a heart That watches and receives. Nature is greater than science, emotion more important than reason. The romantic artist favours passion over good sense. He prefers the sound of lusty old ballads to well-observed satires, and he certainly prefers the sight of mountains to neatly trimmed hedges. Wordsworth was by no means the first to express this preference, but in art, timing is everything, and Wordsworth's timing was impeccable. Lyrical Ballads appeared in the midst of one of the greatest upheavals in European history, a period of time in which almost every aspect of life — politics, religion, philosophy and the arts — was fundamentally altered. The crisis of faith Wordsworth had been through in his twenties, when his head led him badly astray and his heart put him back on track, seemed to play out, in microcosm, the crisis of a whole generation. Gerard Way: A world that sends you reeling. # Disenchanted MY CHEMICAL ROMANCE front man, Gerard Way, has only just turned thirty, but we get the feeling he's already seen more of the world and what it can do than he'd care to, as he steps up to the microphone to introduce the next number. 'This is a song about dreams', he tells the audience. 'It's called "Disenchanted".' The occasion is a sold-out concert at Mexico's Palacio de los Deportes on 7 October 2007. My Chemical Romance has been on the road for over a year, playing to hundreds of thousands of fans all over the world. During that time, the band's most recent album, The Black Parade, has never stopped selling — gathering rave reviews and topping readers' polls as it goes. Before the inevitable world tour had even hit the road, Gerard Way was well on the way to the upper echelons of rockstardom. Now, he treads the stage as though he's never been anything less than a glam-rock superhero. He dips a shoulder, and thousands of girls scream. He shares his pain and millions of kids adore him for it. All of which begs the question: what does Gerard Way have to be disenchanted about? All his dreams would appear to have come true — and then some. So what exactly is the problem? A closer listen to The Black Parade uncovers the malaise at the heart of Gerard Way's emotional world, and — more importantly — reveals the means by which he hopes to transcend it. The album is a loosely structured rock opera in the vein of Bowie's Ziggy Stardust, and the star of the show is a little guy called 'the patient' — a shell of a human being, eaten away from the inside by disease, connected by wires and tubes to obscure machinery, counting out his last days staring at the blank walls of a hospital ward. The album's centrepiece is a song called 'Welcome to the Black Parade', in which death finally comes for our hero, in the shape of an undead marching band. Death, Way insists, arrives in the form of our most treasured childhood memory. For the patient, this was the day his father took him into town to see a parade. On that day, he recalls — as he lies in his hospital bed and the machines count out what's left of his life in metrical beeps — his father said something to him that would stay with him forever: He said son when you grow up Would you be The saviour of the broken The beaten and the damned The song starts out reflectively, as the patient describes the day he spent with his father all those years ago, and the promise he made. Then he starts to think about the world as it revealed itself to him in his teens and twenties, those years when, one by one, we are systematically disavowed of the simple dreams of our childhood. The singer's not reflecting anymore — he's snapping and snarling about decimated dreams and bodies in the streets. But this bitter mood is not the one he closes his song with. For the final section of their rock epic, My Chemical Romance shift gears from breakneck punk to anthemic glory. The last sixty seconds of 'Welcome to the Black Parade' are pitched somewhere between the epic grandeur of 'We Are the Champions' and Phil Spector's Wall of Sound as played by the E Street Band on 'Born To Run', and the singer's tone is doomed but defiant. He realises that through it all, no matter how much misery and pain life threw at him, there was one thing that he never let go of — his dream. 'Welcome to the Black Parade' is a story about a vision, glimpsed during the singer's childhood, of a better world. It's a story about how that vision was then betrayed by the failure of the real world to live up to the singer's hopes. And it ends with the singer realising that he couldn't care less what the real world does or says or will or won't let him do. He discovers, at the end of the song, that all he needs is himself: Take a look at me 'cause I could not care at all do or die you'll never make me because the world will never take my heart Gerard Way has found that society, the real world, adult life — whatever you want to call it — cannot provide him with happiness or satisfaction. So he's moved the search for happiness from outside to inside, and has found it, deep within himself, in his own dreams, his own imagination. This is what puts the romance in My Chemical Romance — the rejection of society in favour of the individual. The philosophers call this solipsism — a system of thought that insists that the self is the only possible area of knowledge — and up until the nineteenth century it was regarded as mostly a bad thing. But the romantics, as Oxford professor Alex de Jonge notes in Dostoyevsky and the Age of Intensity, flipped the script: Whereas most philosophies seek to avoid solipsism...the Romantics positively embraced it. They did so because they found themselves in a world in which the self alone seemed to offer a measure of certainty... This was the world Wordsworth found himself living in. In the hundred years before he was born, the Enlightenment had systematically picked apart every mystery of life until it seemed there was nothing left to dismantle but the Enlightenment itself. This was somehow foreseen by Rousseau and achieved by the Revolution — but at a terrible cost. Post-revolutionary Europe now had to live every day with the awful knowledge that nothing — not even such previously rock-solid ideas as king and country, not even God himself, certainly not the widely discredited god of Reason — was a permanent fixture. Wordsworth, having placed his faith in several of these phantoms only to have them melt away into the air, turned his gaze inward. In his rural retreat at Alfoxden, he found his thoughts drifting toward his childhood, which had also been spent in the country. In The Prelude, Wordsworth describes the vivid scenes that were recalled to his mind, a stormy day just before Christmas when he had run up to the top of a hill and sat by an old stone wall: Upon my right hand was a single sheep, A whistling hawthorn on my left, and there, With those companions at my side, I watch'd, Straining my eyes intensely, as the mist Gave intermitting prospect of the wood Visions such as these restored his faith. Exclaims the poet: Oh! Mystery of Man From what a depth proceed thy honours! I am lost, but see In simple childhood something of the base On which thy greatness stands The Prelude, is a rejection of Empiricism, a popular philosophical doctrine of the eighteenth century which maintains that all knowledge is derived from experience, and that the mind is, at birth, a blank slate. Empiricism played a key role in the Enlightenment's belief in the perfectibility of human beings. It also influenced the criticism and teaching of art to an extraordinary degree. The president of London's Royal Academy, Sir Joshua Reynolds, taught that excellence in art could, and must be learned. 'Our minds,' he wrote, 'should be habituated to the contemplation of excellence.' But Wordsworth's contemporary, the poet and engraver William Blake, was having none of it. 'This man,' said Blake of Reynolds, 'was fired by Satan to depress art.' In the margin of his copy of Reynolds' Discourses, next to the sentence just quoted, Blake scribbled furiously: Reynolds thinks that Man Learns all that he knows. I say on the Contrary that Man Brings All that he can have Into the World with him. For Blake the poet or artist is not, and never was, a blank slate — his unique visions come from within, not from without. Wordsworth, too, rejected Empiricism. Like Rousseau, he believed in a sort of original human soul, connected to nature, which has been corrupted and distorted — not improved — by society. That's why his epiphany took place in the countryside, which in turn stirred memories of his childhood — both were a way back to this original state. Having reacquainted himself with it, this original self would become his guide in the wilderness, the one fixed point in a chaotic and unfriendly world. It's this same self-reliance that allows Gerard Way, in 'Welcome to the Black Parade', to look back at the rise and fall, the bodies in the streets, and the world that disappointed him at every turn, and say, as though he really means it, 'I, don't, CARE!' The source of the singer's faith, the one thing he could hang on to in an unstable world, turned out to be hiding somewhere in the depths of his original self. Here, he found dreams and ideals formed long before society, with its books and rules, taught him how to think — and how not to feel. The world can go on being the world — he has his heart — his unique feeling for what is true and right. It's this brave heart that he holds aloft during the final section of the song, as he falls in line with the black parade, and the rat-a-tat sound of their skeletal drum major disappears over the hill. # Paint It Black and Take It Back TO GERARD WAY, the black parade is many things. It's an alter-ego for his band, an image of death, a hope for salvation, and a way to describe his fans. It's also a dream of society in reverse — a place where the broken, the beaten and the damned can be alone together. It's a parallel universe where sorrow is sublime and the good guys wear black. It's fitting then, that the grand tableaux on the album's inner sleeve looks like the cover of The Beatles' Sgt Pepper's Lonely Hearts Club Band printed in negative. In 1967, Peter Blake's iconic pop collage hinted at a bright new world of colour and imagination, with The Beatles playing the national anthem. Mexican fashion designer Manuel fitted the band out in technicolour military jackets which neatly caught the spirit of the times. The Sgt Pepper uniform suggested a historical revolution, but a bloodless one, fought with flowers and buckets of Pop-art colour — nostalgia blended with optimism. For the black parade's uniforms, Gerard Way handed Hollywood costume designer Colleen Atwood a sketch for a Sgt Pepper outfit with all the colour drained out of it. With their fabric a uniform black and the gold details bleached bone white, the ribbing on the jackets had become ribs. The black parade uniform makes a clever visual pun on the cross-braids of a nineteenth-century military jacket by forcing them into a closer resemblance to the stripes of white corpse-paint worn on stage by Gerard Way's heroes The Misfits in the early 1980s. In the sleeve photo, the five band members, now wearing Atwood's creations, embody Gerard's idea of the black parade perfectly — soldiers who are dead before they've even started marching. Atwood — who honed her craft working with director Tim Burton on films like Edward Scissorhands — was clearly the right artist for the job. This image of the band as the black parade was inescapable in 2007 and 2008, as their gigantic tour wound its way around the globe. Then, just as the tour came to a halt, and Gerard, Mikey, Bob, Ray and Frank hung their uniforms up backstage for the last time, their look was stolen by (of all bands) Coldplay. Their costumes were a tad brighter than My Chem's, and a little 'deconstructed' (although that could have been due to the fact that the band made them themselves and aren't very good at sewing) — but the similarity was striking. Chris Martin appeared on the cover of Rolling Stone dressed in his new duds, looking like a doomed young freedom fighter, staring into the distance, hand on his heart. The look he was aiming for was 'Revolution'. With their new album, Viva La Vida or Death and all his Friends, Coldplay had wiped the slate clean — they'd thrown out the hit-making formula of the last two records, turned up their guitars, and were about to make what Martin called 'a slightly angry restart. Or not angry, just passionate.' As part of this mini-revolution, the band selected a very telling image for the cover of Viva La Vida — Eugène Delacroix's Liberty Leading the People. Delacroix's painting is one of the most famous images of France in revolt — it's very likely the first thing that comes into most people's heads when they hear the words 'French Revolution'. The painting shows an armed rabble surging toward the viewer out of a haze of gun smoke; a ragtag mob, students fighting alongside workers, a street kid waving pistols. At their head is Liberty herself — boldly stepping forward with a bayonet in one hand and the tricolour in the other. A popular ode of the nineteenth century described Liberty as 'This strong woman with powerful breasts, rough voice and robust charm'. And this is exactly the figure Delacroix painted — liberty made flesh. This is not the Revolution of 1789, it's the Revolution of 1830. In July of that year, King Charles X had issued an unpopular decree that wound back a number of the hard-won freedoms of 1789 — including the freedom of the press. Several newspapers protested, police were sent in to subdue the rabblerousers, and outraged Parisians banded together to fight them off. 'Paris streets,' writes Tom Prideaux, 'took on the look of the Revolution all over again.' Less than a week later, Charles had abdicated. Delacroix began working up his canvas as the smoke was still clearing and the new 'Citizen-King', Louis Philippe, was being installed. He wrote to his brother: I have undertaken a modern subject, a barricade...and if I have not conquered for my country, at least I will paint for her. Delacroix's painting — like Delacroix himself, was not especially political. The artist was bored by politics, but the uprising of 1830 had just enough sex and violence to appeal to the painter of kinky masterpieces like The Death of Sardanapalus. The finished painting, however, proved to have a little too much of both for his critics. They complained that the rabble was too dirty looking and that Liberty made flesh was a bit too...fleshy. Liberty, they felt, was all very well — but couldn't she put her top back on? Nevertheless, the new government bought Delacroix's painting, with the idea that it would be hung in Louis-Philippe's throne room 'as a reminder to the new king of how he came to be sitting there'. But the king eventually decided that while he approved of the 'people' in principle, he would rather not look at armed workers and revolting peasants all day long. The painting was taken down, and Delacroix sent it to his Aunt Félicité's for safekeeping. Now it hangs in the Louvre — or did, until Coldplay marched in with their spray cans and wrote 'Viva La Vida' all over it. 'From very early on,' says Coldplay's Guy Berryman, 'we had this painting in mind to show a slightly badly organised revolution — with everything a bit homemade and scrappy.' The painting, in turn, matches their homemade scrappy outfits; and the whole package combines to give the feeling of a passionate struggle for (artistic) freedom, and a new world about to be born. In July 2008 the band presented their new music and new jackets for the first time in a live TV performance. MTV's Buzzworthy, while noting that they seemed to have raided Gerard Way's wardrobe, described their new look as 'Napoleon meets American Apparel'. This was most likely the first time that any of the members of Britain's nicest band had been compared to a would-be conqueror of the world. But it would probably have pleased them in a way that a comparison to, say, Hitler or Genghis Khan would not. Napoleon: Always alone among people... # Napoleon THE YEARS BETWEEN France's first two revolutions were dominated, not only in that country, but across the whole of Europe, by the extraordinary figure of Napoleon Bonaparte. He first came to France from the island of Corsica in order to complete his military education. Having been made a general by the Revolution at the age of twenty-four, he then helped to put down the Royalist Uprising of 1795, was promoted to commander of the army of Italy in 1796, and became — in all but name — the dictator of France in 1799. Over the next fourteen years, his armies poured across Europe. By the time he was defeated and exiled in 1814, the map of the continent was completely redrawn, and thousands of Europeans had, as historian Norman Davies puts it, 'a taste for something entirely different'. While the crowned heads of Europe were — justifiably — scared stiff by the seemingly unstoppable Corsican, many of their subjects eagerly anticipated his arrival — for exactly the same reasons. Napoleon came in the name of Liberty, bringing French-style freedom and democracy with him: oppressive monarchies would be toppled, serfdom abolished. He was, in Holland, for instance, exactly the sort of foreign invader you'd want to be conquered by. He was also, as British historian Eric Hobsbawm puts it, Europe's first secular myth. In 1804 Antoine-Jean Gros painted a large canvas recording Napoleon's Egyptian campaign. In 1798, Napoleon had been trying to establish a foothold for France in the Middle East when a large number of his soldiers were infected with plague. Partly as an act of mercy, and partly so as not to be held up, he ordered the stricken soldiers to be poisoned — but the mission proved to be a failure in any case. This less-than-glamorous story is not, however, the one Gros portrays in his 'Bonaparte Visiting the Plague Victims of Jaffa'. Gros shows a resolute but compassionate Napoleon reaching out his hand to touch one of the plague victims in a symbolic gesture of healing. Gros's painting, in which Napoleon has become a Christ-like figure, was dubious as history, but enormously seductive as propaganda. It was this image of Napoleon as part military genius, part supernatural redeemer that captured the popular imagination. Ludwig van Beethoven was, at first, convinced that Napoleon was the real deal, the living embodiment of democracy and freedom. Having heard of the First Consul's expedition to Egypt, he began dreaming up a symphonic tribute to the great man, which he sat down to compose in 1803. His friend Ferdinand Ries visited him around this time and saw, sitting on his work-table, the completed score for a new work with the word 'Buonaparte' written on the title page. But Ries had some bad news for the composer, which he did not take at all well: I was the first person who brought him the news that [Napoleon] had declared himself Emperor. Thereupon, he flew into a rage and cried out, "He too is nothing but an ordinary man! Now he will trample underfoot all the Rights of Man and only indulge his ambition: he will now set himself on high, like all the others, and become a tyrant!" Beethoven went to the table, seized the title-page from the top, tore it up completely and threw it on the floor. Napoleon was, at this time, a highly contentious figure for the romantics. Blake and Wordsworth were opposed to him for much the same reasons as Beethoven, while others — like Delacroix, or the German philosopher Hegel, whose native Prussia had been completely crushed by Napoleon's army at the Battle of Jena in 1806 — admired him. But after his defeat and exile in 1814, Napoleon became one of the quintessential heroes of the second wave of the romantic movement. For artists of Wordsworth's generation, who had lived through the Revolution, he was too problematic. But for those who came of age in the first decades of the new century, the exiled emperor seemed to embody the Revolution itself, with all its yet-to-be-fulfilled promise; 'a semi-mythical phoenix and liberator', as Eric Hobsbawm writes. To Lord Byron, born one year before the storming of the Bastille, Napoleon was a hero. When he heard of the emperor's defeat at Waterloo, he said, 'I'm damned sorry for it'. Napoleon's defeat led to the restoration of France's monarchy, and a slow but inevitable winding back of the Revolution's reforms, which would eventually lead to the uprising of 1830. During this time a stifling conservatism overtook public life, not just in France, but across the whole of Europe. The feeling was that disaster had been only narrowly averted, and that peace and stability could only be maintained by a rigid adherence to the status quo. But for the romantics, this mood of dull conservatism only made the image of Napoleon's reign blaze all the more brightly by contrast. New heroes started to appear in the literature and art of this period. Eric Hobsbawm describes them as: Dashing young men in guards or hussar uniforms leaving operas, soirees, assignments with duchesses or highly ritualised lodge-meetings to make a military coup or place themselves at the head of a struggling nation... The hussars were a cavalry force in the Napoleonic wars — the armies of France, Austria and Prussia all included hussar regiments. They were notorious for their reckless behaviour, and instantly recognisable for their jackets — double-breasted affairs with horizontal stripes of gold braid across the front, inspired by Hungarian fashions of the late eighteenth century. Napoleon himself was known to wear them — some accounts of his last farewell before being exiled have him wearing a hussar guard's jacket as he made his way down the marble staircase and bid his officers adieu. If Napoleon was often intentionally confused with Christ by his mythographers, then this departure scene, as Norman Davies has observed, was his Last Supper. It came to symbolise the end of an era, and did much to popularise the idea of Napoleon as a martyr — an idea the emperor himself had already succumbed to by the time he wrote this letter to his first wife, Josephine, on the weekend before he sailed for Elba: They've betrayed me one and all...adieu, ma bonne Josephine. Learn resignation as I have learned it, and never banish from your memory the one who has never forgotten you, and will never forget you. Napoleon was always seen to be a different kind of military hero. The poets and artists who admired him during his reign 'did not depict him as a victor', as art historian William Vaughn observes: But as a man of emotion, anxious in mid-battle, compassionately visiting the plague-stricken, or expressing horror at the consequences of war. And despite the exaggerated nature of some of these portrayals, Napoleon, as his letters show, was an emotional man — as a young man, especially so. In 1785 the seventeen-year-old army recruit confided thoughts of loneliness to his journal: Always alone among people, I return home to dream by myself, and submit to the liveliness of my own melancholy. In these moments, when the teenaged Napoleon felt most isolated from his fellow human beings, he found solace in a small book called The Sorrows of Young Werther. He wasn't the only one — Werther had, since its publication in 1774, become a runaway bestseller. Its readership was mostly made up of moody young men, and the key to its appeal lay in the fact that the book's protagonist, the young Werther of the title, was, like them, solitary, introspective and over-emotional. The book purports to be a series of letters written by this sensitive young man to a close friend, telling the story of his unhappy love affair, his descent into despair, and his eventual decision to end his life. The preface explains that the author's purpose in presenting these letters is to provide consolation for those similarly afflicted: And thou, good soul, who sufferest the same distress as he endured once, draw comfort from his sorrows; and let this little book be thy friend, if, owing to fortune or through thine own fault, thou canst not find a dearer companion. Napoleon read it seven times. Twenty years later, no longer just a lonely young man, but a lonely young master of the Continent, Napoleon finally got to meet the author, J W von Goethe. In fact, he'd just invaded and conquered Goethe's country, so there was not much chance of the writer refusing the invitation. On 2 October 1808 Goethe and the emperor met over a large round table while the latter was eating his breakfast. Napoleon told the fifty-seven-year-old author that he looked young for his age. Having got the small talk out of the way, the Emperor owned up to how many times he'd read Goethe's novella. Werther being a tragedy, the talk then moved on to tragedy in general, which, in Thomas Carlyle's account, Napoleon told Goethe, 'ought to be the school of kings and peoples'. He declared that there was no greater subject for a tragedy than the death of Caesar, and complained that Voltaire had not really done the story justice. 'A great poet', Napoleon insisted, 'would have given prominence to Caesar's plans for the regeneration of the world, and shown what a loss mankind had suffered by his murder.' That Goethe and Napoleon should have started out discussing the story of an emotional young artist who commits suicide on account of a hopeless love (Werther) and ended up talking about the assassination of one of the most powerful men in history (Caesar) might seem incongruous. But Caesar, Werther and — as he seems to have known himself — Napoleon are all, in the romantic imagination, tragic heroes. They have earned their place in the pantheon — alongside Hamlet, Don Giovanni and Lord Byron — because they are all, as Eric Hobsbawm puts it, 'trespassers beyond the limits of ordinary life'. The romantic hero is solitary. He retreats into himself because the world has failed to satisfy him, to live up to his dreams. A 'weak' romantic figure like Werther dies through inaction, because he can no longer cope with the divide between himself and the world. A 'strong' romantic hero like Caesar goes out into the world and tries to reshape it according to his vision, but he, too, is inevitably crushed by reality. This conflict between the individual and society, as Napoleon correctly guessed, was to be the basis of tragedy in this new century. Because no matter how hard the hero fights, and no matter how brave his heart is, in any contest between the self and the world, the self will come off second best. The romantic hero is always doomed, because his adversary is reality itself. # This Tragic Affair IN THE VIDEO for 'Welcome to the Black Parade', the patient lies dying in a hospital bed. His life has been full of struggle and heartbreak, yet he clings to it as though it might still have something to offer him. Suddenly he looks up and sees the salvation he's been waiting for. If this were a religious painting from the Middle Ages, an angel would be hovering over his deathbed. But this is America in the twenty-first century, so the light beaming down on him from above comes from a TV. On the screen, the patient sees Gerard Way in his black parade uniform, singing the story of his life. He reaches up to touch the vision, the real world disappears, and he finds himself on a frozen road, with the black parade marching toward him. The Black Parade contains a dangerous idea. It suggests that life might be a struggle for which there is no reward, a bad joke at best. In this world, where dreams are made to be broken, and the promise of happiness is an illusion, our only possible salvation lies in death. Which is not the same thing as writing a song that says; kill yourself. The Black Parade is, as Gerard and the band have pointed out many times, a very life-affirming record. It accepts that living is impossible, but insists that we must be brave enough to do it anyway. On the album's final song, 'Famous Last Words', Gerard recoils from 'a life that's so demanding', but refuses to give up the fight — which of course is the old romantic stand-off between the solitary hero and the cruel, cruel world: I am not afraid to keep on living I am not afraid to walk this world alone The Black Parade is a complex work, full of contradictions. Like any great work of art, it refuses to lie still and play the part of an illustration for a single idea. But the mass media has a way of flattening out the subtleties in art so it can be more easily squeezed into the grid of the six o'clock news bulletin. Sometime in 2007, having been put through this process, My Chemical Romance became a band who dressed like zombies, wrote songs about death and played them for a fan base primarily made up of your children. They also became, much to their dismay, an emo band — which in the UK was already tabloid code for 'suicide cult'. When thirteen-year-old My Chem fan Hannah Bond took her own life early in 2008, the band, and their album, had no hope of a fair trial. The Black Parade's complex array of meanings was reduced to a series of wildly inaccurate sound bites — it was, according to one journalist, 'the place where emos believe they go when they die'. Fans organised a day of action, holding up banners displaying the lyrics from 'Famous Last Words' and testifying to the power of the band's music to save lives. But it was too late. In the eyes of the British public, Gerard Way was a cult leader, a glamouriser of death, and a very bad influence on the youth of today. In 1774, Johann Wolfgang von Goethe found himself in a similar position to Gerard Way. Goethe's novella, The Sorrows of Young Werther, was a runaway bestseller — Germany's first, in fact. More than that, it was a novel that seemed to speak for its age, to articulate the feelings of confusion and hopelessness that lurked beneath the surface of eighteenth-century life. It was the kind of book, as Goethe himself had predicted with his short introductory note, that people took to heart — young people especially. It gave rise to new behaviours — the emotional, death-obsessed youths who loved it became more emotional and more obsessed with death. They were easy to spot — Werther fans had Werther faces — they were dreamy, gloomy, cut off from the world, in love with their own misery. But Wertherism wasn't just a lifestyle — if taken to its logical conclusion it became a death-style. Werther, it was said, triggered a kind of suicide epidemic in the late 1770s. Of these, the most disturbing for Goethe was the case of a woman who drowned herself in a river not far from where he lived. When her body was dragged out of the water, she was found to have a copy of The Sorrows of Young Werther in her pocket. Even after this initial panic died down Werther continued to generate trouble for its author. Goethe was pestered on and off for the rest of his life about this little book — partly because of its shocking content, and partly because its story was rumoured to have come directly from Goethe's own life. This was true. In 1772 the author, then still a student, had moved from Frankfurt to a country town named Wetzlar. Here he pursued his legal studies by sitting in on sessions of the Court of Justice. He also became involved in the social life of the students and court administrators, and it was in this company that the young Goethe fell deeply in love for the second time. Goethe met Lotte Buff, the twenty-year-old daughter of a court official and surrogate mother to her many brothers and sisters, at a ball at Wolpertshausen. They danced, she gave him flowers, and the two stared into each others eyes on the carriage ride back to town. Goethe was convinced that Lotte was the love of his life, and his subsequent discovery that she was more or less engaged to a secretary in the Hanoverian Legation did not deter him in the slightest. Even when Lotte's fiancé, Kestner, returned from his posting abroad, the smitten young poet continued to hang around the house — even going so far as to strike up a close friendship with Kestner. This arrangement worked out for a little while but eventually the situation became intolerable for Goethe — and as a result of his increasingly hysterical behaviour, he began to frighten Lotte and alienate himself from Kestner. Seeing no way out of the impasse, Goethe left the house for good on 10 September. Back at home, and in a truly disturbed state of mind, he heard news of a fellow student of his from the University of Leipzig: Of a moody temperament, disheartened by failure in his profession, and soured by a hopeless passion for the wife of another, he had borrowed a pair of pistols under pretense of a journey, and had shot himself on the night of October 29. The news of this young man's lonely suicide had an electrifying effect on Goethe — for obvious reasons. He soon wedded the story of his unhappy affair with Lotte to the grim tale of his acquaintance from Leipzig, and within four weeks, The Sorrows of Young Werther was complete. Goethe had become Werther, (an artist now, not a poet), Lotte had become Charlotte, and Kestner had become Albert, but in most other respects the story of the romantic young man and his impossible love affair was the same. Young Werther, like Rousseau, Wordsworth, and Goethe himself, is a student of nature: She alone is inexhaustible, and capable of forming the greatest masters. Much may be alleged in favour of rules, as much may be likewise advanced in favour of the laws of society: an artist formed upon them will never produce anything absolutely bad or disgusting; as a man who observes the laws, and obeys decorum, can never be an absolutely intolerable neighbour, nor a decided villain: but yet, say what you will of rules, they destroy the genuine feeling of nature, as well as its true expression. At the start of the book, nature is for Werther a source of sublime joy. He longs to disappear into the tranquil scenes before his eyes, and, like Conor Oberst, he finds himself dreaming dreams of insect bliss: Every tree, every bush, is full of flowers; and one might wish himself transformed into a butterfly, to float about in this ocean of perfume, and find his whole existence in it. The next day, he floats further into raptures. Walking through the valley at sunset, he flings himself to the ground and presses his ear to the soil: ...as I lie close to the earth, a thousand unknown plants are noticed by me: when I hear the buzz of the little world among the stalks, and grow familiar with the countless indescribable forms of the insects and flies, then I feel the presence of the Almighty, who formed us in his own image, and the breath of that universal love which bears and sustains us, as it floats around us in an eternity of bliss... Werther has a hard time getting all this down on paper. He has 'a heart that watches and receives' but it has no filter, no way of limiting or controlling the sensations that come his way. He feels overwhelmed, and finds it difficult to draw: Oh, would I could describe these conceptions, could impress upon paper all that is living so full and warm within me, that it might be the mirror of my soul, as my soul is the mirror of the infinite God! O my friend — but it is too much for my strength — I sink under the weight of the splendour of these visions And this is the carefree, happy Werther! Clearly he's an excitable young man. His problem, as he admits to his penfriend, is that he is too sensitive. 'I treat my heart like a sick child,' he writes, 'and indulge its every whim.' Later, staring at another scene of birds, beetles and rolling green hills, Werther is completely overcome, and begins to hallucinate: Stupendous mountains encompassed me, abysses yawned at my feet, and cataracts fell headlong down before me; impetuous rivers rolled through the plain, and rocks and mountains resounded from afar... Everything around is alive with an infinite number of forms; while mankind fly for security to their petty houses, from the shelter of which they rule in their imaginations over the wide-extended universe. Poor fool By this point, as you might have guessed from the apocalyptic tone he's adopted, Werther's heart has begun to break. It's become perfectly clear to him that Charlotte will never leave Albert for him, and that his love is hopeless. And because everything — from the Homer he reads in the garden to the garden itself — is a big deal to Werther, this is a really big deal. As he sinks into despair, his relationship to the natural world undergoes a remarkable change. He's just as alive with sensitivity to the life of nature as he ever was; only now, if he steps on an ant, he immediately spirals into thoughts of cosmic despair, and begins to see nature as 'a monster, devouring her own offspring'. Universal love has become universal chaos. Werther still sees the landscape as a mirror for his soul. It's just that now Werther's soul is clouded over with misery, and he's begun to find that rolling green hills and pretty butterflies just don't do it for him anymore. He doesn't go out early in the morning or at sunset anymore — he waits until it's dark — and if the weather has turned bad, so much the better to suit his foul mood. Here Werther is in luck, as the book moves to its grim conclusion the leaves begin to fall from the trees and autumn gives way to winter. It is even so! As nature puts on her autumn tints it becomes autumn with me and around me. Then in mid-December there comes an unexpected thaw, and the river bursts its banks. The town is plunged into chaos, and Werther, on the stroke of midnight, sets out into the freezing dark to survey the devastation. And when the moon shone forth, and tinged the black clouds with silver, and the impetuous torrent at my feet foamed and resounded with awful and grand impetuosity, I was overcome by a mingled sensation of apprehension and delight. With extended arms I looked down into the yawning abyss, and cried, 'Plunge!' For a moment my senses forsook me, in the intense delight of ending my sorrows and my sufferings by a plunge into that gulf It's around this time that Werther becomes convinced that the only course of action left open to him is to 'quit his prison' once and for all. 'Yes, I feel certain, Wilhelm,' he writes, 'and every day I become more certain, that the existence of any being whatever is of very little consequence.' As he hangs around Charlotte's household moping and pining, he contemplates all sorts of crazy ideas. He wonders what would happen if he just swept Charlotte off her feet and kissed her; he wonders if he might have to kill Albert; he wonders, in his darkest moments, if he might have to kill Charlotte. But somehow he can't bring himself to do any of these things. He just wants to disappear. I am ill; and yet I am well — I wish for nothing — I have no desires — it were better I were gone. Young Werther is not very old, but he's already seen enough of life to know what's in store: more pain, more misery, and above all, more disappointment. What is the destiny of man, but to fill up the measure of his sufferings, and to drink his allotted cup of bitterness? Charlotte implores Werther to be more reasonable — his love is hopeless — why wallow in misery? Werther agrees that his love is hopeless, but rejects Charlotte's conclusions — he refuses to get over it because this implies a walling off from feeling, a denial of his emotions, that he cannot accept. Instead, he has chosen to see his love to its grim conclusion — he has begun to see that he must die. Frustrated at every turn, Charlotte feels there is nothing more she can offer Werther but her pity — but he already knows there is one more thing she can do for him. He wants her, in short, to finish him off. After his final confrontation with Charlotte, he has his servant visit her house with a request to borrow Albert's hunting-pistols 'for a journey': The arrival of Werther's servant occasioned her the greatest embarrassment. He gave Albert a note, which the latter coldly handed to his wife, saying, at the same time, 'Give him the pistols. I wish him a pleasant journey,' he added, turning to the servant. These words fell upon Charlotte like a thunderstroke: she rose from her seat half-fainting, and unconscious of what she did. She walked mechanically toward the wall, took down the pistols with a trembling hand, slowly wiped the dust from them, and would have delayed longer, had not Albert hastened her movements by an impatient look. She then delivered the fatal weapons to the servant, without being able to utter a word. Werther, upon receiving this final gift, falls into raptures: 'They have been in your hands you wiped the dust from them. I kiss them a thousand times — you have touched them. Yes, Heaven favours my design, and you, Charlotte, provide me with the fatal instruments. It was my desire to receive my death from your hands, and my wish is gratified...' The stage is set for Goethe's tragic hero to have his final showdown with the world that broke his heart. Goethe: 'It is impossible to describe one's feelings save in the flash and fire of the moment'. # Passion Incapable of Being Converted into Action EIGHTEENTH-CENTURY EUROPE HAD been dominated, politically and culturally, by the French — the rationalism of the Philosophes was the intellectual fashion. Werther was a book about feelings, told from the point of view of a character who is ruled by his emotions to an unprecedented degree. It was a revolt against French ideas, and the founding work of the German romantic movement. German critics August Wilhelm von Schlegel and Friedrich Schlegel were among the first to reclaim the word 'romantic' as a positive term. To Sir Joshua Reynolds, it would have meant, as art historian William Vaughn puts it, 'those emotive extremes that lay beyond the proper sphere of the artist to depict'. For the Schlegels, emotive extremes would characterise the art and literature of the new century. They seized on the romantic as being closer to the spirit of the age than the watered-down classicism that had been in vogue for so long. In his Vorlesungen über dramatische Kunst und Literatur (1809), August Schlegel praised Goethe for jettisoning the tired rules of neoclassicism in favour of 'organic form'. Thanks to Schlegel's promotion, and the fact that his work was so widely translated, Goethe's name soon became indelibly connected with the romantic movement in England, Italy, Spain and — most surprising of all — France. Goethe's reputation was consolidated by Madame de Staël's appreciation of his work in her De l'Allemagne, which, as Martin Swales points out, virtually inverted the supremacy of France over Germany in the world of letters in one fell swoop. By 1826 Goethe was being praised by French critics for having revived that country's literature by replacing the old classical insistence on learning and imitation with a new approach that drew on personal confession, 'finding the subject matter within oneself'. For the young Goethe, there was simply no other way to write. As Barker Fairley has shown in his study of the author, Goethe was fiercely anti-intellectual as a youth. At the age of eighteen, he had already realised that books had nothing to teach him, and that everything he had to offer the world could be found by plumbing the depths of his own soul. He was extraordinarily creative — but the idea of editing or refining his work, let alone ordering it or subjecting it to intellectual scrutiny, was absolutely abhorrent to him. 'It is impossible to describe one's feelings save in the flash and fire of the moment,' he wrote in 1775. By the time he died, Goethe was seen, in most European countries, as the father of German romanticism. The irony in this is that Goethe enjoyed being called 'romantic' about as much as Gerard Way likes being called 'emo'. He very quickly tried to distance himself from Werther's emotional excess, maintaining that it was a crazy book written at a crazy time in his life. The older Goethe never read from Werther in public, and admitted once or twice that he was almost scared to open the thing, in case the terrible mood that had inspired it was somehow trapped between its pages, and might overtake him again. Thomas Carlyle would have agreed that Werther was better left on the shelf. Not that Carlyle didn't admire Goethe, in fact, he did more for the cause of Goethe in Britain than anyone, including Madame de Staël. De Staël had unintentionally done Goethe a disservice by presenting to her English readers a version of his Faust that played up the work's 'satanic' overtones at the expense of its more important ideas. This merely confirmed Wordsworth and Coleridge's suspicions that there was something offensively immoral in Goethe. Carlyle's translations and essays improved Goethe's reputation in Britain a great deal. But Carlyle was not unbiased in his appreciation. 'Carlyle', writes Swales, 'saw in Goethe's career a reflection of his own spiritual development that led from gloomy despair to the recognition of community service.' In other words, Carlyle saw Werther as a phase that Goethe had grown out of, and that the literary world had — or ought to have — as well. In England Werther had been a smash hit. It had run to fourteen editions and been turned into a popular play. It was, for a while, inescapable: like something in the air, you could catch it just by walking around and breathing — although young men with good educations and nothing to do seemed more susceptible than most. Lord Byron didn't even have to read the book to understand its importance — he couldn't have, in any case, since he'd never learned to do anything other than swear in German. But Byron instinctively picked up on the mood of gloomy introspection in Goethe's novella, and rode the same unhappy bandwagon all the way to the bank, which in turn led to another wave of tortured poetry by young men with lots of feelings — all of them bad. Years later Carlyle, fed up with all the sobbing and moping Werther had inspired, made an example of Byron as an English 'sentimentalist' — hopefully, he said, the last: For what good is it to 'whine, put finger i' the eye, and sob,' in such a case? Still more, to snarl and snap in malignant wise, 'like dog distract, or monkey sick?' Why should we quarrel with our existence, here as it lies before us, our field and inheritance... Carlyle is effectively telling the sentimentalists, the Werther faces and the Byronic brooders to grow up and get over it. And this is almost exactly what Charlotte, when she can no longer take his hysterics, says to Werther. In life, as in art, Werther follows his heart exclusively, and refuses to be bound by manners, good taste or commonsense — all of which he sees as every bit as deadening to life and love as they are to art. But by refusing to see reason and indulging his feelings to the exclusion of all else, Werther drives everybody crazy — in the real world, it seems, being emotional is not okay. Charlotte, who at first finds that her kind heart will not permit her to turn Werther completely away, eventually runs out of patience: 'Oh! why were you born with that excessive, that ungovernable passion for everything that is dear to you?' Then, taking his hand, she said, 'I entreat of you to be more calm: your talents, your understanding, your genius, will furnish you with a thousand resources. Be a man, and conquer an unhappy attachment toward a creature who can do nothing but pity you.' Charlotte, in other words, sees that Werther needs to take control of his life. She wants him to stop standing around emoting and do something — anything. But Werther is paralysed by feeling. This, as Carlyle himself admitted, is why Werther is an important book. Werther, Carlyle insisted, attempted the more accurate delineation of a class of feelings deeply important to modern minds, but for which our elder poetry offered no exponent, and perhaps could offer none, because they are feelings that arise from Passion incapable of being converted into Action. The hero's 'quarrel with existence' is not one that can be resolved by practical means, because his revolt is a revolt against the practical world — he demands the right to be unreasonable in the Age of Reason. # Sentimentalists ROMANTIC LOVE, SAYS Rupert Christiansen, began sometime between 1763 and 1774. The love we know from blockbuster movies, perfume commercials and daytime soaps — all passion beyond reason and waves of feeling bursting the banks of everyday life — was totally unknown in the early eighteenth century. Love, in the literature of Pope and Johnson's time, was a contract in the parties' mutual interest, in which the occasional bawdy romp was part of the give-and-take. But Rousseau's novel Julie and — fast on its heels — Goethe's The Sorrows of Young Werther, changed all this for good. In these books, love is not a part of the normal fabric of social relations, it's something so powerful and irrational that society can barely accommodate it. And where an earlier age might have used these eruptions of emotion to teach the reader a lesson about the wages of sin, here the moral universe of Christianity was reversed. Rousseau taught that feeling was more important than reason, and implied that the sensitive individual who feels more deeply than others is privy to a deeper, truer moral wisdom. Werther took this idea to extremes. Werther's emotionalism made him a hero — a martyr even. 'In Werther,' writes historian Walter Benjamin, 'the bourgeoisie finds the demigod who sacrifices his life for them.' Reasonable people told him to get over it and do something useful with his life — but Werther would rather die of feeling than not feel anything. For this reason, Werther became the founding work of a late eighteenth-century cult of the emotions called sentimentalism. In his book Romantic Affinities, Christiansen presents a fascinating document of the sentimentalist mindset — a collection of love letters written in 1777. Mary Hays and John Eccles fall in love but soon find that their perfect love is threatened by the dull world, which considers them too young and too poor to marry. Ignoring the advice of her friends and family, who urge her to calm down and get over it, Mary goes half-hysterical with frustrated passion. Of course, she knows they're right — that she could, if she chose, set her sights on finding a husband with better prospects, settle down, raise a family, and leave all this sentimental nonsense behind. But Mary would rather live in sorrow for the rest of her life than suppress her true feelings for even a day. Half the world have no souls. I envy them not their dull insipid calmness — rather would I suffer all those heart-rending, exquisite distresses, which too often flow from sensibility. It is, as Christiansen admits, enough to make you gag. From this small piece of evidence, Carlyle's objections to sentimentalism are easy enough to understand. At the heart of the sentimentalist's philosophy is the trendy Rousseauistic idea that human beings are naturally good, but have been corrupted by society. Rousseau's novels, as Norman Davies points out, made unprecedented links in the public imagination between nature, feeling and virtue. Emotions are natural, society is artificial. Therefore, to the sentimentalist, feelings are sacred, and nobody can tell him otherwise. But this is a selfish philosophy which in the end leads to a profound estrangement, not only from society and its problems, but from other people. This is exactly what happens to Werther. I could tear open my bosom with vexation to think how little we are capable of influencing the feelings of each other. No one can communicate to me those sensations of love, joy, rapture, and delight which I do not naturally possess; and, though my heart may glow with the most lively affection, I cannot make the happiness of one in whom the same warmth is not inherent. Werther is full of feeling, but he doesn't treat people very well — because they're not him, and they don't understand how he feels. As Martin Swales notes, this 'dreadful solipsism' of Werther's ruins his relationship with Lotte before it's even begun. Again and again he cries to heaven; Why, how can she not feel how I feel? For all his ability to empathise with her, she might as well not exist. # Across the Sea RIVERS CUOMO'S SENTIMENTALIST phase reached its logical end point with Pinkerton. Here, the exploration of his emotional world that he'd begun in the privacy of his bedroom some years earlier bore its bitter fruit. Cuomo's over-sensitivity to feeling, combined with his estrangement from the world, resulted in songs like 'Across the Sea', where the hero despairs of ever being able to connect with another human being, Why are you so far away from me? I need help and you're way across the sea. But as he admits on the album's last song, a beautiful acoustic lament called 'Butterfly', it's mostly his fault. He spins an allegory about going out into the garden with a mason jar to catch a butterfly. He snares one — a real beauty. But after a couple of days, it's dead. This kind of thing, the singer tells us, happens to him all the time, 'Everytime I pin down what I think I want it slips away.' It always feels like he's doing the right thing. 'I did what my body told me to', he reflects. He acts naturally, according to his emotions, but it doesn't make him happier. 'Butterfly' contains an important insight — that a philosophy based on feelings almost inevitably leads to romantic despair — and with Pinkerton, Cuomo had ridden the snake all the way to the bottom of the board. It's little wonder that, after he'd recovered his strength again a couple of years later, he had no desire to go back down there. Like Goethe, he must have had a slightly superstitious feeling that he'd simply stuffed the nightmare into those songs and shoved a cork in the top — why risk letting it out again? 'The most painful thing in my life these days,' he told Rolling Stone magazine in 2001, 'is the cult around Pinkerton.' It's just a sick album, sick in a diseased sort of way. It's such a source of anxiety because all the fans we have right now have stuck around because of that album. But, honestly, I never want to play those songs again; I never want to hear them again. 'It's so weird being at total loggerheads with your fans,' he said of the Pinkerton obsessives who were so frustrated by his new unemotional music. I don't know how to deal with it. I don't want to say anything that would sound condescending, but those fans are probably younger and they probably just want to hear that extreme emotion from moment to moment. They need to hear that excess. Any definition of emo, as music writer Andy Greenwald has discovered, has to begin with the fans. This is the only thing that could be said to connect the hundreds of bands who have found themselves described as such. What do My Chemical Romance, Weezer, Sunny Day Real Estate and Dashboard Confessional have in common? Almost nothing. But their fans all want the same thing — feelings. In Nothing Feels Good: Punk rock, teenagers, and emo, Greenwald meets four young friends from Plainview, Long Island, united by their love of Dashboard Confessional and their belief in the importance of emotion. While all the other kids are drinking beer and watching football, Ian, Howie, Anthony and Justin sit around playing acoustic guitars, writing poetry and...feeling stuff. The girls at their high school, sick and tired of pretending to be interested in sports, start to look their way more often. Justin is not surprised. 'They'd never seen real people that are emotional,' he explains. Elsewhere, fifteen-year-old Tracy drives the point home — emotional people are real people — everyone else is fake. I don't care what anyone else thinks anymore — I'm not gonna be fake. I'm gonna be real. I made all new friends because I didn't want to have fake friends, and all of them are themselves too. In 1973 theatre critic John Weightman noted that the world is full of people who subscribe to a basically Rousseauistic philosophy without ever having heard of him. He was thinking of hippies, but emo is, in many ways, even closer to the mark. Emo fans share a strikingly similar language to that of the sentimentalists, and an almost identical set of priorities. Greenwald quotes an online review of Sunny Day Real Estate's Diary, which insists that the album: Strikes at the heart of all that makes us human, begging us to profess our deepest sympathies and dearest sensibilities. Two hundred years ago they were saying the same thing about Rousseau's Julie. Emo fans, like their sentimentalist forebears, have evolved a system of values in which powerful feelings automatically have moral superiority, because they're seen as authentically human. This is why emo culture is inherently terrifying to parents, and an irresistible target for news organisations who profit from their secret fears. Emo is a very specific sort of teenage longing, a romantic and ultimately self-centered need to understand the bigness of the world in relation to you. To the kid who feels that her emotions are the most important thing in the world, the singer says: it's true. 'What has AFI taught you?' asks a thread on the goth-punk band's website. 'The main thing for me,' says kXa, 'was I don't need to live up to anyone's standards. I don't have to put on a fake smile and go through my day.' 'They've taught me that being emotional is okay,' says edenforever, 'and to express myself any way possible.' Davey Havok: As nature puts on her autumn tints, it becomes autumn with me and around me. # Love Like Winter LIKE MY CHEMICAL Romance, AFI is not a band anyone would have thought to call emo ten years ago. But then, ten years ago, AFI was a very different band. As music writer Neil Strauss has observed, the AFI of today barely resembles the group that recorded 'I Wanna Get a Mohawk (But Mom Won't Let Me Get One)' in 1997. Since 1999, Havok's steady diet of mostly English bands — The Cure, The Cult, Bowie and Morrissey — has pushed AFI's music in a completely different direction from its scrappy punk roots. These days, AFI is larger than life, mythic. Like Gerard Way, Havok is a proper glam-rock superhero. His look is an ultra-stylised synthesis of kinky goth-wear, punk ink and Misfits-style Halloween corn. His band, likewise, eschews the snot-nosed pop-punk typical of recent emo bands for a much more eclectic sound. 2006's Decemberunderground was their most ambitious effort to date, sounding by turns like The Cure, Depeche Mode, Bowie and Bon Jovi — with even a hint of Timbaland-style stadium R&B in the album's intro. Like his hero Robert Smith, Havok is not really interested — from an artistic point of view — in the outside world or its problems, his subject matter comes entirely from his turbulent inner life. The letters that make up the band's name stand for A Fire Inside, and it's this fire — and never the candidate on TV or the war in Iraq — that their songs attempt to describe. From inside, outside can just fall apart. Here is where AFI finds its common ground with emo. Davey Havok's estrangement from twenty-first century society, from the world as it is, has driven him to the bleakest extremities of nineteenth-century thought. 'It is against a sense of a hostile, alien and valueless reality that romanticism mounts its various strategies of escape,' writes Alex de Jonge. 'It is the reality of the romantic age that inspires some of the richest and most ambitious attempts to deny reality that the west has known.' Two centuries later, reality is still a big problem, and the poet's attempts to escape its grip are no less ambitious. The Black Parade is one such attempt, AFI's Decemberunderground is another. Where, or when, is this 'Decemberunderground'? On the band's website, Havok explains: It is where the cold can huddle together in darkness and isolation. It is a community of those detached and disillusioned who flee to love, like winter, in the recesses below the rest of the world. Who are these detached and disillusioned? They are those, like young Werther, who are cursed with an excess of feeling and an inability to make compromises. Real people, emotional people. Decemberunderground, Havok told Rolling Stone, describes 'a sort of exclusive, unique type of feeling that certain people have'. These lucky few, he says, know a special love: '...it's that strange love, it's that dark love, it's that cold love, it's the outlook that it's completely different than what most people perceive as something maybe even positive.' What most people perceive as positive, if the images on TV are anything to go by, is equilibrium, a sense of stability. They want money in the bank, a good school for their kids, a holiday once a year, white teeth and a tan. To the romantic, such people are a mystery. Mary Hays, in 1777, did not envy their quiet uneventful lives. She preferred misery to calmness — because at least when you're miserable you're feeling something. She would much rather suffer the 'exquisite distresses' that result from her attachment to feeling than lead a normal life. Davey Havok's lyrics are full of these exquisite distresses — intensely pleasurable feelings derived from a surrender to sadness. He's a connoisseur of misery. Like his hero Robert Smith, he's found that not only does he not fight it anymore, he actually looks forward to it. He admits in 'The Interview' that he's always 'waiting for disaster'. Later we find him, Swimming, bathing Drowning in sorrow. Here is that 'cold love' Havok described earlier. The idea gets a further workout in the first single from December-underground, 'Love like Winter'. 'Warn your warmth to turn away,' he sings, 'here it's December everyday.' In the song's big-budget video, Havok is seen dressed in his customary black, his face the usual whiter shade of pale, wandering through a snow-bound forest in the depths of winter. Like Young Werther, he seems quite at home in this frozen landscape — if nothing else, it suits his mood. Life has turned cold for the storm-singer, and his feeling for nature can now only be satisfied by scenes of decay and desolation. The nu-folksingers can have their sandy beaches — he's only at home where it's dark and cold. As the snow whirls around him and the storm clouds brood overhead, our singer starts to hallucinate — he sees, as though in a dream, a beautiful woman walking toward him, dressed in a hooded cloak. They share a moment together, an unspoken understanding. Could it be that he's found true happiness at last? Has he finally found a way out of his dreadful solipsism? Has he realised that impossible romantic dream, an actual connection with another human soul, someone with whom he can be alone — together? Well, yes and no. The beautiful woman pulls back her hood — she has removed the last of her defences, the way has been cleared for a pure union of souls. And in that very instant, when the singer's happiness seems assured, catastrophe strikes. The ice gives way beneath his feet, he sinks into the freezing darkness. And as he struggles, his exquisite distress is made more exquisite by the reappearance of the beautiful woman, holding him down under the water. She smiles, and holds him in a tender, but deadly embrace. In real life, the woman is a model from the Ukraine. In an interview for a making-of special, the video's director refers to her as 'the ice-bitch'. But to Davey Havok, she is nothing less than 'the beautiful physical embodiment of hopelessness'. # Alone and Palely Loitering IN 1816 SEVENTEEN-YEAR-OLD Thomas Keats began receiving letters from a mysterious French girl named Amena Bellafila. She claimed to have met him — though he didn't remember meeting her — through a mutual acquaintance, Thomas's school friend Charles Wells. The letters were written in a curiously old-fashioned idiom, like Elizabethan sonnets — Amena told Tom that he was her knight in armour, that he would rescue her, and that she, in turn, would soothe him. Tom travelled over France trying to find her, but never could. Two years later, Tom was dying. The doctors said he had tuberculosis, but he fancied he knew better. In October 1818 he told his brother John the truth — he was sick because his heart was broken. He was in love with Amena Bellafila, but Amena had disappeared. He was dying of unrequited love. John Keats was twenty-three and a promising poet. He was utterly devoted to his younger brother and never left his side during these final days — but something about this 'Amena Bellafila' made him uneasy, and he took time to make some investigations regarding Tom's disappearing French girl. The results confirmed his worst suspicions: there never was an 'Amena'. The letters were fabrications, written by Charles Wells in exaggeratedly feminine handwriting, and composed in the mock-Elizabethan language of his and Tom's schooldays. After Tom's death, Keats finally got to see the letters for himself. 'It is a wretched business,' he wrote to his other brother, George. 'It was no thoughtless hoax — but a cruel deception.' A few days later Keats wrote to George again. If he was still preoccupied with the cruel trick played on his dead brother, there was little sign of it. Keats described the review he'd just written of a new parody of Wordsworth's Peter Bell, A Tale, and shared his thoughts on a diorama of the North Pole he'd recently seen. Then, in the middle of all this casual chat, he copied the stanzas of a poem he'd just written called 'La Belle Dame sans Merci'. It was a ballad, told in a style that showed the influence of Wordsworth (who was probably on Keats's mind because of the review), as well as another of Keats's heroes, the Elizabethan poet Edmund Spenser. The poem's action is set in the depths of winter — the leaves have fallen off the trees, the birds are silent. In the midst of this desolate scene, the narrator meets a knight-at-arms 'alone and palely loitering', who tells the tale of how he came to haunt this particular patch of frozen ground: IV I met a lady in the meads, Full beautiful — a faery's child, Her hair was long, her foot was light, And her eyes were wild. The knight tells of a forest idyll — he made the lady a garland of flowers and set her on his 'pacing steed'. She said, 'I love thee true.' IX And there she lullèd me asleep And there I dream'd — ah! Woe betide! The latest dream I ever dream'd On the cold hill side X I saw pale kings and princes too, Pale warriors, death-pale were they all; They cried — 'La Belle Dame sans Merci Thee hath in thrall!' XI I saw their starved lips in the gloam, With horrid warning gapèd wide, And I awoke and found me here, On the cold hill's side. XII And this is why I sojourn here, Alone and palely loitering, Though the sedge is withered from the lake, And no birds sing. 'La Belle Dame' condenses a number of Keats's fears and preoccupations. Much like a dream, it seems to give form to anxieties even the poet himself was not consciously aware of at the time — though some of the motifs are more easily recognisable than others. The knight-at-arms is most likely a version of Tom, dressed up to suit the conceit of the 'Amena' letters, 'haggard and so woe-begone' from a long illness exacerbated by a broken heart. It follows, then, that the 'Belle Dame' is Amena Bellafila. She certainly fits the part — a vision of perfect love which turns out, as soon as the knight falls asleep, to be a trick. The knight's terrifying dream expresses the full extent of Keats's horror at his brother's death. Tom had died broken-hearted, feeling that the happiness he'd sought had been denied him. But the worst of it was that this happiness had never really existed. In the end, as Keats's biographer Robert Gittings points out, the lady in the meads is no simple allegorical figure representing Tom's disappearing girl. She is, when all literary hints from other sources, when all the accidental events of Keats's day-to-day life are exhausted, the symbol of the eternal fusion between love and death. # A Forest THE JOURNEY INTO despair that Robert Smith made between 1980–82 began with a walk in a forest. The first single from Seventeen Seconds was already a fairly gloomy proposition, and not the follow up to Boys Don't Cry that most critics were hoping for. Julie Burchill, writing for the NME, accused Smith of 'trying to stretch a sketchy living out of moaning more meaningfully than any man has moaned before'. Burchill was right on two counts — 'A Forest' is full of moaning and full of meaning. The song describes a dream in which the singer hears a woman's voice calling his name. He follows the sound into the trees, and the band takes up the theme, conjuring the feeling of a headlong rush into a dark world. Electronic whooshes zip past like drifts of fog. But as Smith searches for his mysterious lover in the forest, a terrible thought occurs to him: Suddenly I stop But I know it's too late I'm lost in the forest All alone The girl was never there It's always the same I'm running towards nothing Again and again and again and AGAIN 'A Forest' illustrates an important idea in romantic thought. Many of the romantics — Rousseau and Wordsworth being the clearest examples — idealised nature as a symbol of all that is good, pure and true in human nature, and a standard by which our behaviour can be measured. But there is a flip-side to this, in which the romantic individual, having rejected society, runs out into nature and finds that he is not really at home there, either. Since he is now out of options, his philosophy becomes one in which reality itself is hostile. His goals — happiness and love — exist somewhere outside of the world. All of this makes love complicated for the romantics — and makes it more likely that they will end up confusing love with death and despair, as Keats, Robert Smith and Davey Havok have all done. Love, in the romantic imagination, is pure and natural, part of the mysterious world of feeling and the human heart, which society cannot touch. But to think this is to overburden love with a weight of unrealistic expectations. Ideal love, like nature, becomes one of the romantics' escape destinations — existing outside of the cares of the world, or as Davey Havok would have it, in the cold regions below. But this, like all the romantics' attempts to escape reality, is doomed to fail, since there is no real escape from the world other than death. As a consequence, love becomes an illusion, forever out of reach — The girl was never there It's always the same I'm running towards nothing — or fatal, pulling the poet toward oblivion. The singer finds his perfect love, the ice cracks, the poet and his fatal lover sink into darkness. John Keats: Fallen lemons in my path. # Lemonade AMONG THE STRANGE brew of ingredients that went into the writing of 'La Belle Dame sans Merci' was a copy of Robert Burton's Anatomy of Melancholy — one of the most popular and most talked-about books of the seventeenth century. Keats's copy was heavily marked up, 'almost as if with a personal application to himself' as his biographer Robert Gittings puts it. It was Burton's book, according to Gittings, that gave Keats the image of the sorrowful knight 'alone and palely loitering'. It also provided him with the subject matter and imagery of another of his great odes — 'Ode on Melancholy'. As this poem begins, Keats dismisses some of Burton's medicinal cures for sadness. His argument is a little like some of the current debates about anti-depressants — wolf's-bane and the like might leave him in a state of happy forgetfulness, but they would also 'drown the wakeful anguish of the soul'. What does he need that for? Keats's answer is simple: poetry. But when the melancholy fit shall fall Sudden from heaven like a weeping cloud, That fosters the droop-headed flowers all, And hides the green hill in an April shroud; Then glut thy sorrow on a morning rose, Or on the rainbow of the salt sand-wave, Or on the wealth of globed peonies; Or if thy mistress some rich anger shows, Emprison her soft hand and let her rave, And feed deep, deep upon her peerless eyes. In 'Ode on Melancholy', Keats advises the reader not to drown his sorrows with wine, but to put them to work. This is exactly what Weezer's Rivers Cuomo did in the mid '90s — though he got the idea not from Robert Burton or John Keats, but from his mum. When Rivers was having a rough time growing up (which he frequently did), his mother would cheer him up with a little homespun philosophy: 'If life gives you lemons, make lemonade'. One day in 1992 she said it again, and Cuomo turned his mother's saying into a song. Back when he was younger, he reflects in 'Lemonade,' he used to just let his rage and frustration rot where it fell: Till the day I couldn't pass Fallen lemons in my path So with my mom I now agree And use the lemons life gives me Cuomo had begun to understand that 'making lemonade' was exactly what he'd been doing with his recent songs, the songs that would soon appear on Weezer's debut. With 'Buddy Holly', 'In the Garage' and 'Say It Ain't So', Cuomo was turning bitter experience into sweet treats; great pain into little songs (or as he would later call them 'angst muffins'). Weezer's second single, 'Buddy Holly' is lemonade of a fine vintage, drawn partly from Cuomo's high school days (where he and his brother Leaves 'got the crap beaten out of them'), but mostly from an incident that took place during his time at Santa Monica College. Already in Weezer, Cuomo also enrolled in the college choir where he made friends with a girl named Kyung He. The song recalls a day when the Weezer guys were making fun of Kyung He's accent. Your tongue is twisted Your eyes are slit You need a guardian Cuomo, furious at his band mates and feeling protective toward his friend, wrote a lyric which said, in the plainest terms possible, 'it's okay to be different — we can make it together'. It was deceptively simple powerful stuff, set to an irresistible tune. Pain had been turned into poetry — more than that — a huge hit which finally made all of Rivers Cuomo's rock-star dreams come true. The lesson was not lost on the singer. Even after he'd abandoned the confessional mode of Pinkerton for the classicism of the Green Album, Cuomo continued to recognise the importance of melancholy as a songwriting resource. In 2002 he assured the readers of Spin magazine (in an obvious bit of emo-baiting) that he still had feelings 'like everybody else', but had carefully compartmentalised them. 'I like to exploit them and use them for my own purposes,' he explained. A couple of years later, when producer Rick Rubin suggested to Cuomo that he try transcendental meditation as a way of focusing his mind on the recording process, Cuomo was horrified. 'I sent him a very anxious page, saying, "Rick, no. I cannot get into meditation because it will rob me of the angst that's necessary to being an artist."' Cuomo's reasoning was sound — after all, you can't make angst muffins without angst. But the idea, taken too far, can turn ugly. If great songs come from unhappiness, is it now the songwriter's job to remain unhappy? Most Weezer fans, after the Green Album, would have answered in the affirmative. They hated the new classical Rivers, with his well-constructed hits. They wanted more suffering and emotional excess, like on Pinkerton, the album he wrote in the mid '90s when he was incredibly miserable. One fan, lavishing praise on Pinkerton while writing off Weezer's current output as 'horrible pop songs', remarked, 'depressed people sure do write good music'. There is, it has to be said, something ghoulish about this. Cuomo's fans didn't care in the least about his wellbeing, they wanted him to suffer so that their appetite for intense emotion could be satisfied. This Cuomo steadfastly refused to do. Weezer could have cashed in on the cult of Pinkerton many times over — they could have, at the very least, included the occasional Pinkerton song in their shows to keep the fans happy. But as Pete Wentz from Fall Out Boy recently admitted to Rolling Stone, 'it's really a fine line between being an inspiration to your fans, and creating an industry out of misery.' Wentz, being a bass player in an emo band, would know all about it. Billy Corgan: Misery Industry. # Anatomy of Mellon Collie THIS 'MISERY INDUSTRY' is a tough gig. Entertainers have it easy with their pantomime of emotion. All they have to do is turn it on at the start of the show, then turn it off when they get backstage. For them, it's just a job. But for the romantic poets, with their commitment to emotional truth, the pain has got to be real, the tragedy drawn straight up from the well of their unique sensibility, their ability to feel. Smashing Pumpkins' Billy Corgan is one of these fearless explorers of inner space. Like Keats, Corgan learned to make use of his moods — but in a more dramatic fashion that seems closer to German poet Heinrich Heine's secret formula for sturm und drang poetry; Aus meinen groszen Schmerzen Mach' ich die kleinen Lieder. [From my great pain I make little songs.] Corgan would accept no other method for his art — every song came from real emotional experience. But not just any old emotional experience. 'A good song, a smiling face, a true feeling, doesn't do it.' He said in 1994, 'People want to see things smashed to bits. They want to see you rip your heart out'. Corgan had just done more or less exactly that with the Smashing Pumpkins 'Disarm'. The song is an aching ode to childhood innocence. Corgan surveys the wreckage of his teens and twenties, and wonders how everything went so wrong. 'I used to be a little boy,' he screams. Of course, his parents and teachers couldn't let him stay that way, they 'cut that little child' out in order to prepare him for the so-called real world. But something went wrong with Corgan's de-programming. He's like one of those science fiction secret agents whose memory was supposed to have been erased, and who then finds he starts remembering things — things responsible grown-ups aren't supposed to remember. A world of sunlight, ice-cream and daydreams; a world where it was okay to be emotional. He knows too much, and this knowledge has made him lonely; and prolonged exposure to loneliness has made him angry and vengeful. But because he's a poet, Corgan, instead of taking up arms against his oppressors, took up his diary and wrote a song. And as the song took shape, Corgan realised he had to turn bitterness into beauty. Later, he said: The reason I wrote "Disarm" was because I didn't have the guts to kill my parents, so I thought I'd get back at them through song. And rather than have an angry, angry, angry, violent song I thought I'd write something beautiful and make them realise what tender feelings I have inside my heart. This kind of bold personal confession, Corgan felt, was the future of music. Speaking to Richard Kingsmill in 1998, he insisted that extreme personal emotion was 'the only place left to go in rock and roll'. But this had already taken its toll on the singer. Remember, Heine says that from great pain you only get little songs, and Smashing Pumpkins' fans wanted to hear little songs like 'Disarm' over and over again, which, for Corgan, meant several emotional apocalypses a night for a six-month stretch. The singer found that, while his sadness was infinite, his ability to keep exploiting it was not. At 1994's Lollapalooza festival, Corgan found himself confronted with an impossible choice — to keep giving his audience the 'real thing' and risk bleeding himself dry; or to 'fake it' for the sake of his sanity. Unfortunately, this second option was something his romantic commitment to emotional truth would never allow. 'I do feel a responsibility to best articulate what I feel,' he said. So, he couldn't just walk away or do the show on autopilot, he saw it as his job to express suffering, and the suffering had to be real. He was furious at finding himself trapped like this, and decided to channel the rage into his performance. But what good does that do? He can snap and snarl all he wants on stage, but in the end he's just a performing bear — worse, a rat in a cage, running hopelessly on his little wheel so as to satisfy his audience's taste for extreme emotion. Hey, rat in a cage, that's pretty good. He explores this image and the angry, bitter feelings associated with it, and pretty soon he has...another song — that people like! That his fans want him to perform onstage every night of the week for the rest of the year! When, he wonders, will his torment end? On one level, Smashing Pumpkins' 'Bullet with Butterfly Wings' reflects fairly standard rock-star anxieties about Corgan's relationship with his audience and the music industry that pushes him in front of them. 'Secret destroyers,' he sings, 'hold you up to the flames.' But here, already, the imagery has become somewhat biblical, and when Corgan screams that he cannot be saved and starts comparing himself to an Old Testament prophet, it becomes clear there is more at stake than just the perils of show business. 'Bullet with Butterfly Wings' was, as Corgan later admitted, written about his experience at Lollapalooza. Nirvana was meant to headline the festival, but when lead singer Kurt Cobain, similarly trapped between his own anxieties and his audience's expectations, took his own life, Smashing Pumpkins was invited to fill the void. The grisly implications of this were not lost on Billy Corgan. When, in 'Bullet with Butterfly Wings,' he accused the whole world of trying to kill him, and then made a simile between himself and Jesus, it's because he was all too aware of the way Cobain had become a martyr to his audience's expectations, and had a terrible feeling he might be next. In the early '80s Corgan's hero Robert Smith had felt himself being pushed toward the edge of a similar precipice. He too had become a pressure valve for his audience's anxieties by writing with unflinching honesty about the horrors of life. But writing about horror meant living with horror while you recorded it, and then touring the horror for another twelve months. Smith began to wonder just how much horror he could take. And looming over it all was the spectre of Joy Division's Ian Curtis, who'd hanged himself in 1980. Curtis had collapsed under the same pressure Smith was now feeling. In fact, Smith had a feeling he'd only arrived at this point because Curtis was gone, and that he was next in line. Later he said: I hate the idea that you'd die for your audience, but I was rapidly becoming enmeshed in that around the time of Pornography; the idea that Ian Curtis had gone first and I was soon to follow. With the news of Kurt Cobain's death still fresh in his mind, Corgan couldn't help feeling that history was repeating itself. The story of Billy Corgan and the Smashing Pumpkins was starting to take on the characteristics of a modern tragedy, in which he, as the hero, must perish. This might seem absurd — after all, they're just songs, and he's just a singer, right? Wrong! For Corgan, his life was his art and his art was his life. To an eighteenth-century spectator, the idea that a singer would die for his audience would be absurd. But in a post-romantic world, it's entirely plausible. Ziggy Stardust: The rake's reward. # Rock and Roll Suicide IN 1971 DAVID Bowie was dreaming of re-inventing musical theatre with a story about an alien rock star. But time got away from him, and before he got the chance to write the story out properly, his rock musical had become a rock album. 'There was no time to wait,' he later told Mojo's Paul du Noyer. 'I couldn't afford to sit around for six months and write a proper stage piece, I was too impatient.' Still, as Bowie admits, the resulting album, The Rise and Fall of Ziggy Stardust and the Spiders from Mars, does retain a rough dramatic structure. The first act deals with the arrival of a man from the stars who picks up a guitar and decides to become a star of the more worldly variety. As the album plays on we watch as Ziggy ascends to the heights of fame, and enjoys all the rarefied pleasures it has to offer. Everybody loves him, everybody wants him — and all of this is going to Ziggy's head. So by the time the third act rolls around, and we hear that he's been 'making love with his ego', we know our hero is headed for a fall. He dies, spectacularly, at the end of his theme song 'Ziggy Stardust'. 'He took it all too far,' Bowie reminds us. To an eighteenth-century audience the tale of Ziggy Stardust, despite confusing references to twentieth-century phenomena like radios, ray guns and vaseline, would be instantly familiar. Ziggy is clearly a gross libertine, who indulges his every appetite. His exploits are disgraceful, but since the story advances a useful moral theme, these can be overlooked (and not-so-secretly enjoyed). The useful moral theme is this: that a dissolute rake like Ziggy will eventually get his just desserts. On 29 October 1787 Wolfgang Amadeus Mozart conducted at the premiere of a new opera — whose premise was exactly the one described above — at The National Theatre in the old city of Prague. The title on the program that evening was 'Il dissoluto punito', or 'The Rake's Reward', but the opera later came to be known by the name of its protagonist — the rake of the title — Don Giovanni. The opera was a hit, but that was virtually a foregone conclusion, since Don Giovanni was a retelling of a story that had been playing to packed houses in Europe for almost two hundred years — the seventeenth-century legend of the great seducer, Don Juan. Mozart's opera begins with a scene where Don Giovanni sneaks into the apartments of Donna Anna, the daughter of the Commandant of Seville. He means to make her the latest of his many conquests, but she screams, and her father comes to her rescue. A fight breaks out between Don Giovanni and the furious Commendatore, and the older man is killed. Don Giovanni escapes, his thoughts already turning to his next seduction. A little later, we learn the full extent of Don Giovanni's debauchery in a scene where his servant, Leporello, recites the history of his amorous adventures in front of a horrified Donna Elvira — six hundred and forty in Italy, two hundred and thirty-one in Germany, a thousand and three in Spain... Then, in the second act, after yet another narrow escape, Don Giovanni and Leporello rendezvous in a churchyard near a stone statue erected in the memory of the dead Commendatore. Incredibly, the statue comes to life, and informs Don Giovanni that he will be dead before the sun comes up. Leporello sees that under the statue is written the inscription: 'Here I await vengeance upon a vile assassin'. Now, the Commendatore's long wait is over. But Don Giovanni treats this warning the way he treats everything else — as a laugh. His response to the threat of eternal damnation is to invite the statue round for dinner. To Leporello's horror, the statue proves to be as good as his word. Hearing some commotion, he goes outside to investigate, and comes back to his master with terrifying news — he's so frightened he can barely speak. 'The stone...man...all white,' he stammers. The statue pounds on the door, and since Leporello is too scared out of his wits to answer it, Don Giovanni does it himself, while his servant hides under the table. Don Giovanni, still maintaining the light-hearted and cynical attitude that has carried him through life thus far, offers the stone man a seat. But the ghost is not here for dinner: 'A graver purpose than this, another mission has brought me hither.' He tells Don Giovanni that he must come with him, and extends his hand. 'Will you in turn come and sup with me?' asks the stone guest. Don Giovanni, not one to be called chicken, accepts. As his hand touches the statue's, he becomes locked in a vice-like grip. 'Tis colder than the tomb!' he exclaims. The Commendatore demands three times that the wicked rake repent for his sins, and each time Don Giovanni answers with a defiant, 'No!'. 'Your time has come!' roars the ghost. Right on cue, a crack opens up in the floor, and Don Giovanni feels himself being pulled downward. It starts to get hot. 'Whence come these hideous bursts of flame?' he cries. From the unfathomable depths below the earth comes an answer: No doom is too great for your sins Worse torments await you below Don Giovanni is dragged down to hell with a blood-curdling scream. Mozart seals the villain's fate with a couple of conclusive chords — and this really sounds like the end of the opera. But it's not. With Don Giovanni's final terrified yell still ringing in our ears, Leporello, Donna Elvira, Donna Anna, Massetto and Zerlina come come back out on the stage to let us know what they're up to now that Don Giovanni is locked out of harm's way. Donna Elvira says she's off to enter a convent, Zerlina and Massetto are going home for a nice quiet dinner and Leporello tells us he's going to the pub. Why are they telling us all this stuff? Because they have a useful moral theme to deliver. Zerlina, Masetto and Leporello: Let the scoundrel remain below with Proserpine and Pluto; and we, good people, will gaily sing the ancient moral. All: This is the evil doer's end! The death of sinners will always match their life That's the end of Don Giovanni — or at least, it was when Mozart wrote it. But if you'd gone to see the opera in, say, 1816 — you'd find this ending had vanished. For most of the nineteenth century, Don Giovanni concluded with the no-good rake being consumed by the flames of hell — Leporello, Zerlina, Masetto and their useful moral theme were nowhere to be seen. In the twenty years since Don Giovanni made its debut in Prague, the world had changed dramatically, and popular taste had changed with it. Nineteenth-century music-lovers still wanted Don Giovanni — but they didn't want a moral enforcing the importance of moderation in all things. What they wanted was a story about a man who did whatever he liked, in defiance of society's rules, and died heroically as a result. The German writer E T A Hoffman (the 'A' stands for 'Amadeus') defined the role of this new Don Giovanni in his 1813 story Don Juan. Here, a concertgoer watching Mozart's opera is treated to a special director's commentary by the ghost of Donna Anna. She reveals that Don Giovanni was not a bad man who got his just desserts, but a hero. He lived a life of passion and inspiration while everyone else just went to work and paid their taxes. In this version of the tale, Don Giovanni is preferred to the society whose limits he refused to accept. As a result, he becomes the very definition of the tragic romantic hero — the inspired individual who picks a fight with society, a fight which he can't possibly win. The romantic hero is always outnumbered, because he is always alone. Like Don Giovanni, Bowie's Ziggy Stardust comes with an epilogue. The hero takes it all too far and is destroyed — not by demons from hell, but by the mass of grabbing hands just below the stage. In his live shows, Bowie illustrated Ziggy's demise even more dramatically with a cover of the Velvet Underground's 'White Light/White Heat', in which the rock star hero is burned to a crisp by the sheer velocity of his lifestyle. But just when you think he's gone forever — his threat to the status quo safely locked away in the depths of the earth or burned away into space — he's back. He sits on the side of the stage, smoking a cigarette and snapping his fingers to the beat. He sings a song called 'Rock and Roll Suicide'. Ziggy didn't actually take his own life — but then he also knew (or should have known) what he was signing up for when he decided to become 'the special man', to live among humans while reserving the right to ignore all of their moral laws and social conventions As he tells his story, the song builds and builds — and then comes to a halt. He waits a beat, leans in to the audience, and delivers his message to the kids: You're not alone! Just turn on with me, and you're not alone Here is a great paradox, but one that Bowie, and the artists he inspired — Robert Smith, Billy Corgan and Gerard Way — would attempt to resolve. The singer tells us that he was a solitary rebel who died tragically in his one-man war against society. But he didn't do it so he could end up as an illustration for a moral principle — he did it for everybody else who ever felt like they didn't fit in. He died for us, and for the idea that someday, some way, we might find a way to be alone — together. But in the meantime, Ziggy reminds the faithful, though the world is cruel and will rob the young romantic hell raiser of his dreams, one must carry on regardless. This is the one part of Ziggy's story that would have been deeply confusing and disturbing to our imaginary eighteenth-century audience. In 'Ziggy Stardust', it's the libertine — and not the shopkeeper or the nun — who has the last laugh. This, to an eighteenth-century spectator, would seem shockingly immoral. But sometime around the first decade of the nineteenth century, things changed — permanently, it would seem. Because in 1973 David Bowie could virtually guarantee that everyone in his audience would be rooting for the dissolute rake in his story, and not for the society whose rules he refused to accept. He took it all too far — but the moral of 'Ziggy Stardust' is not 'pride comes before a fall' or 'fiery doom is the rake's reward'. After the curtain comes down on Bowie's low-budget rock opera, there's no little party of regular folk to deliver a lesson about the virtues of hard work, abstinence and commonsense. In rock and roll, this rarely happens — in glam rock (which Bowie more or less invented with 'Ziggy') it never does. The rock star is not a moral scarecrow — he's a hero in death, because his non-stop sinning is a protest against the limits of ordinary life. In AFI's 'Miss Murder', the singer contemplates his own rock and roll suicide. The song weds the melody of Depeche Mode's 'Strangelove' to the sound and rhythm of Gary Glitter's glam classic, 'Rock and Roll Part 1'. Havok's lyrics tell the tale of a beautiful, otherworldly rock star. 'How they all adored him,' he sighs. As the chorus kicks in and the band 'whoah-ohs' like they're Bon Jovi, Havok poses the mother of all romantic dilemmas: Hey, Miss Murder can I make beauty stay if I take my life? Two hundred years of the romantic tradition says: yes you can. It's not just about dying young and leaving a good-looking corpse. The existential dilemma of a Don Giovanni or a Ziggy Stardust comes from the fact that society wants him to be useful, whereas he respects only beauty. As both of these romantic ur-myths show, there is only one way out of this dilemma, but by taking it, the hero becomes a martyr to the principle by which he lived his life, which is the pursuit of strange beauty at all costs. In 'Miss Murder', death, once again, takes female form. The full-length version of the song's video incorporates a prelude. We see, in a series of cuts, a beautiful woman seated at an ornate writing desk where she writes a letter, as Davey Havok stalks the marble floors of a cathedral. The singer has been betrayed — by the woman, by love, by the world in general — we don't know, but he's mad as hell, and as he walks, he sings a poisoned lullaby. He's decided to quit this world — but before he does, he has one more thing to ask of his beloved: ...you may forget me I promise to depart just promise one thing Kiss my eyes and lay me to sleep As the last notes of his lullaby die away, and the glam rock goose-step of 'Miss Murder' fills the cathedral, we see that the singer has been granted his wish. The 'Miss Murder' of the title, we now realise, is the woman sitting at the writing desk. She folds a piece of paper, swallows it — and it instantly appears in the singer's mouth. He pulls it out, unfolds it, and sees an image of three black rabbits arranged in a circle — a symbol of death in AFI lore. Havok doesn't break down in tears at the news of his imminent demise, and he doesn't scream like Don Giovanni. He adopts an air of melancholy resignation, as though he knew, all along, that this is where his flaunting of society's rules would lead him (which of course he did). This is the 'tortured poet charisma', which Matt Diehl, in his book My So-Called Punk, insists is key to AFI's appeal. Havok most likely copped the pose from careful study of the posters of Bowie, Morrissey and Robert Smith that line the walls of his vocal booth during recording sessions. But this 'tortured poet charisma', while being fairly familiar in the world of rock and roll, is something relatively new in the world of poetry. It would have been completely alien to the poetry-lover of the eighteenth century — in the days of Pope and Johnson, poets were not tortured. Poets were sharp-eyed observers of society, but they always knew they belonged in society — what could they have to be tortured about? Wordsworth undermined this assumption in 1798, but it was completely overturned in 1812 by a much younger poet — a man much admired by such literary heavyweights as Goethe and Nietzsche — but also by a large reading public, who enshrined him as the archetypal romantic hero of the eighteen-hundreds. Thanks to George Gordon Byron, 'tortured poet charisma' quickly became the only kind the nineteenth century cared about. # Screamin' Lord Byron IN THE DECADE following the success of 'Ziggy Stardust', Bowie killed Ziggy off twice, reinvented himself as a blue-eyed soul singer, then a Teutonic robot, and then a tragic Pierrot. Finally, he transformed himself into the only thing that could really surprise his fans — an ordinary bloke. In Julian Temple's extended video for his 1984 single 'Blue Jean', Bowie plays Vic — an ordinary bloke — who, while working up a ladder one day, falls in love with a girl. Trouble is, she looks straight past him — right over the top of his scruffy head in fact — to the huge poster of exotic pop singer Screamin' Lord Byron across the street. So the resourceful Vic blags his way into a date with the girl by making up a cock-and-bull story about being a relative of Screamin's', and promises to use his influence to get some free tickets to the show. Because he's in no position to do any such thing, he eventually resorts to breaking into the club where Screamin' is doing his show. Vic crashes through the ceiling into the horrified singer's dressing room in a hail of plaster and a shower of dust. Screamin' Lord Byron is the opposite of Vic in every way. The singer is not so much a human being as an assemblage of affectations and complications — heavily made-up, even more heavily medicated, terrified of human contact and absolutely scared stiff of Vic with his cheery cockney brusqueness. But this cringing mess is also a superhuman god. Later, we see him on stage at the Bosphorous Rooms where he holds the audience in thrall with his deep voice and hypnotic gestures. The rapt fans raise their hands up as if to worship him — he clicks his fingers to the beat and they all do the same, never taking their eyes off him for a moment. They worship him, but he barely notices them. He leaves the stage without so much as a wink or a smile. He waits in his dressing room until almost everyone has left before venturing back into the room, where Vic and his date are waiting, hoping for an audience. Vic leaps to his feet on seeing Screamin' — but the singer doesn't see him at all. He goes straight for the girl, sweeps her off her feet, and stalks out of the club — leaving the furious Vic hurling insults at the star's flashy car as it disappears down the street. 'Your record sleeves are better than your songs!' The joke is that Screamin' Lord Byron is also played by David Bowie. This is a great bit of casting, because we all know that before he was a regular bloke, Bowie was a narcissistic, antisocial rock and roll superman. And just as the superman's name was a dream amalgam representing Bowie's rock and roll ideal — in which the glam fantasy of the Original Stardust Cowboy met the Dionysiac excess of Iggy and the Stooges — so 'Screamin' Lord Byron' weds the show-business flash of early British rock 'n' roll ('Screamin' Lord Sutch') to the name of the nineteenth-century poet who most resembles a rock star before the fact — Lord Byron. Byron: 'melancholy and sullen detachment'. # Lord Byron IN THE MONTHS before he wrote 'La Belle Dame sans Merci', John Keats had been in a terrible mood. His friend Benjamin Robert Haydon, the painter, remembers that around this time Keats became 'morbid and silent', though he was also prone to outbursts. When a family friend made a comment on Keats's growing reputation, saying to Mrs Brawne, 'O, he is quite the little poet,' Keats angrily exclaimed, 'You see what it is to be under six foot and not a Lord!' Time has been kinder to Keats's poetry than Byron's. But in 1818 Keats had good reason to be jealous. It wasn't just that Byron was tall (he was not quite six foot in fact) or that he was of noble birth. The real reason why Keats was feeling so sensitive was that he'd just learned that Byron had sold over four thousand copies of the last canto of Childe Harold's Pilgrimage. This poem, the first two parts of which had been published six years earlier, had already made him incredibly famous. Byron had written Childe Harold while travelling in Spain, Malta, Turkey and Greece between 1810 and 1812. While abroad he'd lived the life of a libertine: frolicking with olive-skinned youths on the beach, being courted by Turkish warlords and enjoying 'fooleries with the females of Athens', as Byron put it. As he returned home to England to claim the estate that came with his title, with all the responsibilities that entailed, those two carefree years in the south started to seem more and more like the best years of his life. The future, on the other hand, was almost too grim for the twenty-three-year old Byron to contemplate. He felt done with life. He'd seen the world, and was now looking, he told a friend, for 'the most eligible way out of it'. In this gloomy frame of mind he moved into his dilapidated gothic abbey at Newstead. While travelling, Byron had been working on a poem. He'd recently changed the name of the protagonist from Childe Burun to Childe Harold, but there was no mistaking him for anyone but his creator — he stands to inherit a title and a 'venerable pile', and his travelogue, as described in the poem, is similar to Byron's. This worried Robert Dallas — a family friend of Byron's who was arranging for the poem's publication. Dallas loved the poem, but was concerned about the state of its author's soul. If Byron had done half the things his literary alter ego claimed to have done, he was going straight to hell. Ah me! In sooth he was a shameless wight, Sore given to revel and ungodly glee; Few earthly things found favour in his sight Save concubines and carnal company, And flaunting wassailers of high and low degree. Byron told him he was right to be worried. 'My whole life has been at variance with propriety, not to say decency', he admitted, with melancholy resignation. This last was the real trick up Byron's sleeve. The poet Samuel Rogers, who read the proofs of Childe Harold, predicted that it would be a flop because the hero was both an unrepentant sinner and a misery guts. Who would want to read about the doings of a man like that? But this double whammy of debauchery and despondency, as Colin Wilson insists in his book, The Misfits, is exactly what caught the public's imagination. Childe Harold was not a 'cheerful voluptuary' in the mode of Mozart's Don Giovanni. He sinned, but he did so with an air of sorrow and detachment, as though there were some terrible sadness in his past that he could never quite escape. This drove the ladies wild. When Childe Harold was published on Saturday 7 March 1812 Byron was a nobody. By the following Monday, he was famous. Childe Harold was well on the way to selling out its first print run, and its author was presented in his rooms with a salver full of visiting cards. 'Women,' as Wilson writes, 'begged for introductions.' Dallas had guessed that the close identification of Byron with Harold would increase the poem's appeal, and he was right. He was correct, too, in believing that the poem was something new under the sun. To Dallas, as Byron's biographer Peter Quennell writes, Childe Harold 'seem[ed] to catch and concentrate an unresolved element in the life of the period, something to which no novelist or versifier had yet been able to give a literary shape...' This was probably the same 'unresolved element' that Goethe had isolated thirty-five years earlier — or at least a very similar one. But where the celebrity hunters would later look in vain for Werther in Goethe, they found Childe Harold in Byron. There was, as Goethe himself observed, an unconscious quality to Byron and his work, as though the one simply sprang fully formed from the other, unmediated by any normal artistic process. This makes his poetry a little unsatisfying when compared with Keats's or Wordsworth's. But neither of those two achieved, in their lifetime, anything like the fame and notoriety of Byron. Byron was a star, courted by society, endlessly propositioned by female admirers, and studiously imitated by young men. In the London of 1812, the Werther face had been replaced by the Byronic limper. Byron had been born with a club foot that gave him a curious and distinctive dragging gait. That his young admirers should start imitating this, his least attractive physical feature, might seem strange. But Byron's deformity, his 'mark of Cain' as he called it, was actually the key to his whole 'look' — and much more besides. The club foot had been a source of endless torment for Byron in his childhood. Doctors had prescribed various cures involving braces and harnesses, all of which were physically painful and — much worse — socially crippling. His unlovable mother did nothing to help matters by calling him a 'lame brat'. All of this left him with a desperate need for approval on the one hand, and a deep-seated conviction that he was doomed to be lonely and unhappy on the other. So it made no difference to him how famous he became, how many books he sold, how many times his portrait was painted, or how many girls — or boys — he slept with. He pursued all these things vigorously, but none of them, not fame, money or pleasure, could compensate for the blow he'd been dealt at birth. As Quennell writes, 'The admiration he might arouse while stationary must vanish, he felt sure, when he crossed the room.' His solution to this was to stand still — and here was the origin of the famous Byronic look — the pose people still imagine when they hear the words 'romantic poet'. Peter Quennell describes it vividly in Byron: The Years of Fame: As he leant on one elbow, his small white hand clenched beneath his cheek, meditative, immobile...in the anteroom of some brilliant London party — melancholy and sullen detachment pervaded his attitude... This stance communicated volumes. As Quennell points out, a young, healthy-looking man like Byron must, it would be assumed, have a good reason for standing still. But since there was nothing obviously wrong with him, his audience was forced to assume that he was paralysed by existential boredom — which was not too far from the truth. This pose, combined with his extremely pale complexion — a side-effect of his brutal skin care regime — created the impression that Byron was a creature from another world. In his heart of hearts, Byron longed to be in the world, to relate to others as an equal. But since he knew this would never happen (because of the terrible curse), he further entrenched himself as an outcast by creating mythologised versions of his suffering self in his poems. These reinforced the impression his insecurities had created until it was impossible to tell where the myth ended and the man began. Tortured poet charisma starts here. There was in him a vital scorn of all: As if the worst had fall'n which could befall, He stood a stranger in this breathing world, An erring spirit from another hurl'd; By portraying himself as a solitary, inspired individual, forever cut off from society, Byron was simply acting out the dilemma of all poets and artists since the revolution. But Byron, with his flair for publicity and his gift for self-mythologising, was the first to make this idea popular with a middle-class public. So much he soar'd beyond or sunk beneath, The men with whom he felt condemned to breathe, And long'd by good or ill to separate Himself from all who shared his mortal state; The source of his estrangement, as he explains in 'Lara', lay in his childhood. He was born with a double handicap — a deformed leg and an oversized heart. his early dreams of good outstripped the truth and troubled manhood followed baffled youth The world had already broken Byron's heart when he was only twenty-three. From there, things could only get worse, and sure enough, they did. His insatiable appetite for kicks conspired with his desperate need for attention to produce a series of scandals that culminated in an affair with his half-sister Augusta in 1814. After that, he went from society darling to social pariah in record time. He left England for the continent shortly after, and remained in exile for the rest of his life. This, of course, only confirmed his belief that he was a man apart. Like Werther, Byron ascribed almost cosmic significance to his emotions, and the feelings stirred up by the Augusta affair led him to his most spectacular conclusion. He loved Augusta, and for that society denounced him as a sinner. Since his feelings couldn't be wrong, he must be a sinner, and since, as the philosopher Bertrand Russell says, 'he must be remarkable, he would be remarkable as a sinner, and would dare transgressions beyond the courage of the fashionable libertines whom he wished to despise'. He would become a super-sinner. Byron had seen his way marked out for him even before a furious Madam de Staël told him he was 'un demon' — he had already compared himself in verse to a fallen angel. Like Don Giovanni, Byron was hell-bound. But unlike the light-hearted seventeenth-century rake, Byron knew in advance where he was headed, and would get there on his own terms. For the hero of the modern tragedy, there's no question of survival — he's doomed before the lights go down. But he can decide how he wants to go out. Werther and the emo singers simply take the path of least resistance and let the world roll right over them. They're paralysed by the sheer pointlessness of everything, and by the world's refusal to live up to their expectations. So they wait until life has them boxed into a corner, and slip quietly into oblivion with a heavy sigh. Werther can barely bring himself to commit suicide; he prefers to think that he's allowed Charlotte to kill him. Byron started out this way: returning from his pilgrimage in 1812 he asked nothing more of the world than a way of walking out of it, and wondered if, somewhere in London, he might find someone who'd be willing to save him the trouble. But after the Augusta scandal his position had changed. He'd become a 'strong' romantic, the kind who sees that society cannot accommodate him, and so sets out to oppose everything that society stands for. If he's already doomed, he's going to do whatever he likes and make as much trouble as possible along the way. What's more, he'll have the last laugh. Life might be impossible for the romantic outsider, but he can still go out in a blaze of glory — or hellfire, as the case may be. # Give Them Blood THERE IS A twist in the tragedy of Ziggy Stardust: when he finally goes down, he doesn't overdose on smack or choke on his own vomit. Even the threat of his jealous band mates turns out to be a red herring. In the end it's the kids — Ziggy's own fans — who finish him off. This is what Bowie means when he sings about Ziggy being 'a leper Messiah'. His fans, no longer satisfied with admiring his 'snow-white tan' from afar, took him up on the offer he seemed to be making of his body. They all wanted him, so they each grabbed a piece and ripped him apart. And in 'Rock 'n' Roll Suicide' he explains that he gave himself willingly, gave up his body as a sacrament so that the fans might finally achieve the communion they craved. Ziggy Stardust has endured as a rock and roll myth because it has its basis in something terrifyingly real. Bowie had already lived through the deaths of Jimi Hendrix and Jim Morrison, 'strong' romantic heroes who — quite self-consciously in Morrison's case — pushed at the limits of life and paid the price. He knew better than most that the message of these stories is not a moral one. And Ziggy is prophetic, too. When Bowie first unveiled the album in 1973, the world had yet to witness the likes of Ian Curtis and Kurt Cobain. These were singers who, in a very real sense, died because of their audience's emotional expectations. They expressed suffering on behalf of their fans, and their deaths came to be understood as the logical end point of that suffering. The moment of panic Billy Corgan experienced at Lollapalooza, as recorded in 'Bullet with Butterfly Wings' was a result of his realisation that he'd been cast as the lead in a Ziggy-like tragedy, in which his own fans demanded his head so that the show they'd come to see would have a proper ending. Back in the days of the Ancient Greeks, the first tragedies were performed to accompany the ritual sacrifice of a goat. The name itself comes from the word tragodia, meaning 'goat song'. So when Gerard Way invites us along to witness a tragic affair in the opening moments of The Black Parade, two thousand years of tradition say that blood will have to be spilled. And by insisting that the suffering we are about to witness must be his own, the romantic artist has put himself forward as the most likely candidate. By the end of the album Way has come to understand this. Like Ziggy Stardust, The Black Parade ends with an amoral conclusion. After the smoke has cleared, the Patient is wheeled back onstage on his hospital gurney to perform a jaunty cabaret number called 'Blood': Blood, blood, gallons of the stuff, give them all that they can drink and it will never be enough Earlier in the album, Gerard had promised blood — he just had no idea the kids would want so much of it. Here, the tragic hero resigns himself to his fate; his fans want blood, and he'll give it until his veins run dry. He is, he explains, their favourite dish. Billy Corgan felt the same way, and expressed the feeling in similar terms — though without Gerard's weary resignation. 'Bullet with Butterfly Wings' contains a dizzying mix of metaphors, which is part of its charm. The singer's terror and paranoia will not allow him the peace of mind needed to choose his images carefully — they tumble out of him in a great flood. When he compares himself with Jesus, he's evoking the image of the sacrament, the idea that he gave up his body and his blood so that his fans might mosh. But this is a thought that's grown out of an earlier one — the great statement Corgan delivers, a cappella, at the beginning of the song: 'The world is a vampire'. Corgan evokes the sexualised threat of the vampire to describe the way the music industry, the media, and even his own fans seem to be slowly draining him, body and soul. It's an image Gerard Way has returned to again and again. The horror hospital scenario of 'Blood' and the nocturnal cannibalism described in 'The Sharpest Lives' are the last drops of a great tide of vampire imagery that flowed through My Chemical Romance's early work. They became so well known for it, in fact, that Gerard swore no more for The Black Parade. But somehow, a couple of those pesky bloodsuckers crawled in there. They are, as Way explained in an online interview with X V Scott, almost unavoidable in his line of work: ...there's just something about the bloodsucking walking dead that can say so much to people. There are really so many people trying to get control over you on a daily basis and...take a part of you. Count Dracula: 'He stood a stranger in this breathing world'. # The Vampyre CONSIDERING HOW LITTLE time he had for romanticism of any stripe, it seems odd at first to learn that Goethe admired Lord Byron. This, after all, was the same Goethe who wrote off the flood of gloomy sentimental prose that appeared in the wake of The Sorrows of Young Werther as 'the literature of despair', and who blasted the French romantics of Delacroix's generation for perpetrating 'aesthetics of the grotesque'. For Goethe 'romantic' usually meant 'deranged', if not merely 'badly done'. If he was so down on despair and derangement, what could he possibly find to admire in the author of Childe Harold's Pilgrimage? Goethe, as Gerhart Hoffmeister has shown, did not renounce any of his convictions for Byron's sake — but he did make an exception for him. To Goethe, Byron was a special case, for the simple reason that he was a natural genius, to whom no rule could apply. He belonged, not with other poets, but with Napoleon. Creatures such as these, Goethe believed, could not be judged by society's laws, because their very purpose in life is to break free of those laws — and any others they might find along the way — forever. It's this highly idealised Byron — spontaneous genius and rebel angel — that Goethe paid tribute to by including him as a character in the second book of his epic drama, Faust — completed just before Goethe died in 1832. Here, the poet Goethe claimed to love like a son has become Faust's son, Euphorion. Euphorion: let me be springing, Let me be leaping, Pressing on, mounting, Through the clouds sweeping, Strong these desires In my thoughts run Faust: gently, ah gently, Be not too daring, Lest in disaster All of us sharing... True to life, Euphorion does not do as he's told — he's a force of nature. When he dies, it's as a flaming ember shooting up to the stars. White light, white heat; Euphorion appears in Faust only briefly, but burns very brightly. In 1819 Goethe got hold of a new prose work of Byron's called The Vampyre. The story, published in The New Monthly Magazine, had all the elements Byron's fans had come to love. Brooding, black-clad anti-hero of noble birth? Check. Secret sorrow? Check. But The Vampyre had a new twist, the gloomy protagonist was not just deathly pale in the approved Byronic mould, he was actually dead. Or rather, he was undead, a ghoulish parasite feeding on the blood of the living. Like Byron himself, The Vampyre seemed to present a distorted mirror image of ourselves — a creature loosed from moral restraints for whom the only good is what brings pleasure. And as always with Byron, the protagonist was scandalously identified with the author himself. It was a dead giveaway, really — the vampire's name, Lord Ruthven, was the same as the one given to the Byron character in Lady Caroline Lamb's Glenarvon. This, even for Byron, was daring stuff. Putting the story down, Goethe proclaimed it the poet's best work yet. But here Goethe was wrong on at least one count. The Vampyre wasn't Byron's. Well, not exactly. After leaving England in 1816, Byron had paid a visit to the field of Waterloo before stopping in Switzerland at the Villa Diodati by the shores of Lake Geneva. Here, he'd settled in to write the last two cantos of Childe Harold's Pilgrimage. He later described his circumstances, with typical understatement, to the poet Tom Moore: I was half mad during the time of its composition, between metaphysics, lakes, love unextinguishable, thoughts unutterable, and the nightmare of my own delinquencies. I should, many a good day, have blown my brains out, but for the recollection that it would have given pleasure to my mother-in-law... At Lake Geneva Byron was plagued by a horde of celebrity spotters, clamouring for a glimpse of the scandalous poet. There was also the usual parade of not-so-secret admirers, including the highly resourceful Claire Clairmont, who'd enjoyed a fling with the poet in London and had now, it seemed, journeyed eight hundred miles to 'unphilosophize' him, as Byron put it. The good news in all of this was that Claire had come with her stepsister Mary Godwin (daughter of the philosopher William Godwin and the feminist Mary Wollstonecraft) and Mary's husband-to-be, the poet Percy Bysshe Shelley. Theirs was a whirlwind romance — only two weeks earlier the couple had been courting by Mary's mother's gravestone. Now, following a bizarre honeymoon in war-torn rural France, they were staying at a house at the base of the hill below Byron's villa. Byron, bored out of his skull by the company of his physician, John Polidori, welcomed the more stimulating conversation of the Shelleys. Byron and Shelley took walks in the surrounding countryside visiting places they knew from Rousseau's books, and discussed Wordsworth's latest poem, 'The Excursion'. But the weather soon turned nasty, which not only kept the literary conversation indoors, but seemed to demand literature of a type better suited to thunderstorms. Byron suggested that the holiday-makers should each try their hand at writing a ghost story. But according to Mary, the great poet's enthusiasm for the project quickly ran out. He wrote only a small fragment of a story about a vampire. In his biography of Byron, Frederic Raphael makes the interesting suggestion that, in producing a story in which the victim is 'drained of blood by battening predators', Byron may have been inspired by his own situation — besieged by celebrity spotters and groupies who all wanted a piece of him. Raphael wonders whether it was 'this aggrieved sense of being constantly drained', which led Byron 'to concoct a little fragment which added a fanged twist to the Gothick repertoire...' Polidori's contribution to the contest — concerning a woman with a skull for a head, was by Mary's account, laughably bad. But he made up for it later. The temperamental doctor took up the idea Byron had laid aside and expanded it, over the next year or so, into a novella called The Vampyre — the very same one Goethe enjoyed so thoroughly. Goethe can be excused for his confusion over the story's authorship. The New Monthly, without Polidori's permission, published The Vampyre as a new work by Byron. Although he didn't write it, Byron's hand is all over The Vampyre — and not just because the original idea was his. Polidori's tale effectively updated the old folkloric version of the vampire for nineteenth-century tastes by 'Byronising' him. Polidori's Lord Ruthven is exceedingly pale, is burdened by a secret sorrow, and preys on society ladies and young girls. All of this gives him an irresistible allure — he's always being invited to parties 'in spite of the deadly hue of his face', and once he gets there he 'gaze[s] on the mirth around him as if he could not participate therein'. Ruthven is both a very thinly veiled portrait of Polidori's former master, and an attempt to capitalise on Byron's notoriety and the perceived threat of romanticism to the stability of bourgeois society. The Vampyre invites the reader to imagine the damage that could be done to a well-ordered world by a creature with an insatiable passion and no moral qualms to speak of. Polidori's modifications to the vampire would survive intact in his most famous appearance, in Bram Stoker's 1897 Dracula. And since it's Dracula that's won over in the popular imagination as the definitive version of the monster, the Byronic traits that Polidori introduced to his character are now fixtures. For this, we have Hollywood to thank. F W Murnau's expressionist classic Nosferatu (1922) was an unofficial adaptation of Stoker's novel. But the real breakthrough for the Byronic vampire was Tod Browning's 1931 Dracula, starring Bela Lugosi. Lugosi's exotic, cape-wearing aristocrat defined the look and the manner of movie vampires for decades to come, and Browning's Dracula was to become the foundation stone of an entire vampire film industry, which flourishes to this day In June 1983 a film by Tony Scott called The Hunger opened in cinemas, starring David Bowie as Blaylock — a pale, aloof, aristocratic vampire living in 1980s Manhattan. The Hunger received terrible reviews, ('incoherent and foolish,' said the Observer), but it did contain a few striking set pieces, including the opening sequence set in a nightclub and cut to the rhythm of Bauhaus's 1979 single, 'Bela Lugosi's Dead'. The band appeared on screen, performing the song inside a cage under stark expressionist lighting, while singer Peter Murphy — frequently criticised for being a Bowie copyist — did nothing to shake off the comparison. The song, like the movie it appeared in, was brilliant nonsense. Over skittering rhythms and disorienting dub effects, Murphy paid tribute to the world's most famous movie vampire in a voice as deep as the hollows under his cheekbones. Meanwhile Bowie, as Blaylock, surveyed the club's clientele with a mixture of predatory lust and superior disdain. Casting Bowie as Blaylock was an inspired choice. The vampire is an easy role for a rock singer to play, for the same reason that 'rock singer' makes a good disguise for a vampire. Both are a threat to society because of their voracious sexual appetites and their indifference to conventional morality. It would already have been assumed that Bowie — as a rock and roll star — came out at night, preyed on young women, and shrank in terror from crucifixes and holy water — all that was missing was the teeth and the cape (and Bowie wore a few of those in the '70s too). Peter Murphy, as the singer in a darkly glamorous post-punk band, would have been perceived in much the same way, and his pale skin and skull-like features only added to the 'creature of the night' effect. And of course, it went without saying that neither had any time for bourgeois morality. Bowie had spent the last decade demolishing taboos with the disdain of an alien aristocrat, and Bauhaus were already (much to their dismay) seen as standard bearers for a musical movement that dressed in black, came out only at night, and despised nothing so much as middle-class suburban conformity. Over the next ten years, the adherents of this new post-punk religion would turn The Hunger from a box-office turkey into an object of worship. The scene in the nightclub would be endlessly rewound and replayed in teenage bedrooms across Britain: 'undead, undead, undead...' # Goths WHILE The Hunger was playing in cinemas, an unusual nightclub tour was making its way around Britain. The Batcave promised to bring unsavoury sounding entertainment from the likes of Alien Sex Fiend, Flesh for Lulu and Specimen to Sheffield, Birmingham, Manchester, and any other English town foolish enough not to have stocked up on garlic and crucifixes in advance. This Batcave had begun as a Wednesday night happening in a Soho club called, appropriately, The Gargoyle. The flyers promised something 'thoroughly nasty' and guaranteed 'absolutely no funk'. Not that there wasn't dancing — here, 'Bela Lugosi's Dead' had become a certified floor filler along with other nocturnal odes such as The Birthday Party's 'Release the Bats' and Siouxsie and the Banshees' 'Spellbound'. The regulars were described by David Johnson in The Face magazine as looking like 'Dracula meets the Muppets'. The décor was Victorian romance gone to seed, with posters from classic 1930s horror movies on the walls. Very quickly, the denizens of the Batcave came to resemble their surroundings — here you could see make-up by the living dead, clothes by Count Dracula, and hair by the Bride of Frankenstein. 'You had to take a lift up to the top floor, which used to be a hostess club', recalled Soft Cell singer Marc Almond. 'There was a little theatre where stripteases used to take place and they used it to watch gothic movies, or bands would perform there, and you could see people like Robert Smith hanging out at the bar.' In 1983 the makers of a BBC documentary on The Batcave lamented the lack of any proper celebrities to film — but they may have turned up on a bad night — or maybe it was just too dark. 'The usual tykes were spotted flaunting their disease-wracked bodies at the Batcave last Wednesday night', wrote a Sounds reporter in 1983, 'Including (yawn)...a fat Siouxsie Sioux, a dazed Nick Cave'. No-one will ever know exactly who first used the word 'Gothic' to describe the scene that moved into the Batcave and soon became synonymous with it. But the best story comes from The Cult's Ian Astbury. Long before he started fronting the resurrected Doors, Astbury was playing Jim Morrison to Peter Murphy's Bowie in the early goth scene with his first band, Southern Death Cult. Astbury remembers giving Andi from Sex Gang Children a nickname: I used to call him the Gothic Goblin because he was a little guy...and he lived in a building in Brixton called Visigoth Towers. This is an unusual name for a block of flats. The Visigoths were one of a number of Germanic tribes whose migrations across Europe led to the weakening and finally the collapse of the Roman Empire in the fifth century. Inspired by the success of the Ostrogoths at Hadrianopolis in 378, the Visigoth's king, Alaric, successfully sacked Athens in 396, and then Rome itself in 410. This, in turn, put a strain on the Empire's resources, making it easier for the Burgundians to push back the Romans and establish their own Kingdom in the Rhone valley. From here, occasional victories notwithstanding, the Roman Empire went into decline, and Europe as we know it began to take shape. The ensuing historical period — which lasted from the fall of the Roman Empire to the beginning of the Renaissance around the fourteenth century — was considered by most Enlightenment historians to be a sort of temporary interruption in the course of Western civilisation — a lapse into primitivism from which Europe had, thankfully, begun to recover. By the eighteenth century, it was far enough away to seem like a bad dream. Sanity had been restored, and the arts and sciences could begin to reconnect with the knowledge of the ancients, to pick up where the Romans had left off before their glorious civilisation had been overrun by Goths and the world had been plunged into darkness. That's why in the Age of Reason the worst thing you could say about a building, a painting or a poem was that it was Gothic. This was, unsurprisingly, an Italian put-down, coined during the Renaissance to describe the spooky, mystical and (to the refined fifteenth-century mind) not very well-drawn art of the Middle Ages — as well as the suspiciously pagan-looking architecture of the period. The term has its origins in a misunderstanding, whereby the makers of these artefacts were assumed to be Goths. But the word stuck because, as art critic John Ruskin points out in The Stones of Venice, medieval building styles 'appeared like a perpetual reflection of the contrast between the Goth and the Roman in their first encounter'. In other words, the Gothic style was an affront to civilisation. In the midst of the great triumph of science, reason and the classical ideal, Gothic buildings, by their very presence, made an unwelcome reproach — at best, they were like family members you'd rather forget you had but found yourself obliged to invite to Christmas dinner. Toward the second half of the eighteenth century things began to change. Just as the reign of classicism in poetry was slowly but surely undermined by an interest in medieval ballads; in the world of architecture, it gradually became acceptable to admit that Gothic buildings had a sort of primitive appeal. The first significant stirring of this new feeling can be found in a letter written by Horace Walpole, the fourth Earl of Orford, to the Honorable H S Conway in June 1747. Walpole was writing to tell his friend that he'd just moved house: You perceive by my date that I am got into a new camp, and have left my tub at Windsor. It is a little plaything-house that I got out of Mrs. Chenevix's shop, and is the prettiest bauble you ever saw. It is set in enamelled meadows, with filigree hedges... Walpole's 'bauble' was a country house near Twickenham on the outskirts of London. For several years after he wrote this letter, he continued to commute between Strawberry Hill and London, where he sat as a member of Parliament. But Walpole was never that interested in politics — his appearances in the House grew more and more sporadic and by 1768 he had stopped showing up for good. He had plenty of other, better ways to amuse himself — he entertained at the drop of a hat, played cards till two in the morning, and wrote stories and poems which he published with his own press. But the majority of Walpole's time was taken up with his two greatest passions: renovating and decorating, and it was Strawberry Hill itself that would become his life's work. When he wasn't supervising construction work or poring over sketches and architect's drawings, the future fourth Earl of Orford could be found hanging around the auction houses of the greater London area, looking for bargains. As to what it was exactly that he was looking for, his letters give us a pretty good idea. Writing to Horace Mann, he described the state of Strawberry Hill in 1753: The bow-window below leads into a little parlour hung with a stone-colour Gothic paper and Jackson's Venetian prints... From hence, under two gloomy arches, you come to the hall and staircase, which it is impossible to describe to you, as it is the most particular and chief beauty of the castle. Imagine the walls covered with (I call it paper, but it is really paper painted in perspective to represent) Gothic fretwork: the lightest Gothic balustrade to the staircase, adorned with antelopes... Gothic, gothic, gothic. Walpole's passion for all things medieval led him, over the next thirty years, to transform Strawberry Hill, both inside and out, into a strange hybrid of medieval castle and gothic cathedral. Its growth was entirely improvised — Walpole would return from a walking tour of the countryside or a trip to the Continent with a head full of ideas, and would quickly have his architects and builders incorporate what he'd seen into extensions for his rapidly growing country mansion and its grounds. By the time its owner died, Strawberry Hill had grown from its original five acres to forty-six, and the house itself had become an eccentric, ad-hoc tribute from a wealthy and slightly eccentric English gent to what, in his lifetime had been an almost universally reviled form of architecture. By 1763 it was a much-visited and much-discussed tourist attraction. What no-one could have suspected, was that even as Walpole had been transforming Strawberry Hill, Strawberry Hill had begun to transform its owner. One morning in June 1764 Walpole woke from a nightmare. ... I thought myself in an ancient castle (a very natural dream for a head filled like mine with Gothic story) and that on the uppermost banister of a great staircase I saw a gigantic hand in armour. That night Walpole sat down at his desk to write, 'without knowing in the least what I intended to say', and before he knew it, a novel had appeared: a tale of medieval intrigue laced with family curses and underground passageways. The Castle of Otranto is generally regarded as the very first gothic horror novel, although in Walpole's day, that would not have been considered a compliment. Walpole knew as much when he wrote it. It was, as he explained to a friend, a book out of time. 'It was not written for this age, which wants nothing but cold reason.' Walpole, aware of the difficult proposition he had on his hands, decided to publish Otranto anonymously, and to further cover his tracks by not using his own press. He then concocted an intricate ruse to explain the appearance of this superstitious tale in his age of reason. In the book's preface Walpole introduces himself as the anonymous translator of a sixteenth-century text based on a story written during the crusades. This did the trick. As British literary historian Michael Gamer points out, Walpole's conceit allowed the eighteenth-century reader to swallow the impossible events in his story, because the reader could accept that people in medieval times would believe such things. The Monthly Review praised it as an entirely worthy historical curiosity. Unfortunately, Walpole blew it by revealing himself as the true author in the preface to the book's second edition, after which the reviewers changed their tune entirely: While we considered it as a translation, we could readily excuse its preposterous phenomena, and consider them as sacrifices to a gross and unenlightened age. But when, as in this edition, The Castle of Otranto is revealed to be a modern performance, that indulgence we afforded to the foibles of supposed antiquity we can by no means extend to the singularity of a false taste in a cultivated period of learning. It is, indeed, more than strange that an author of a refined and polished genius, should be an advocate for re-establishing the barbarous superstitions of Gothic devilism Walpole had the last laugh. In 1781 Otranto was adapted for the stage as The Count of Narbonne. Its run would last for the next two decades, during which time Otranto itself became a bestseller. Walpole's story inspired a whole new genre of literature, the gothic novel, whose popularity would last well into the next century. Walpole's The Castle of Otranto exploited a chink in the armour of Enlightenment culture which would eventually bring the whole edifice crashing down. His story gently tricked his readers into admitting that there is beauty in terror and darkness, and that there are things in the universe that can never be explained. These tendencies — our desire to be frightened and our need to believe — are irrational, but undeniably human. The Age of Reason could not, for all its efforts, suppress them forever. As historian Norman Davies has observed it seems deeply strange to us now that an entire culture could have been built around the veneration of a single human quality — reason — to the exclusion of all else. But 'Enlightenment' can only be understood in terms of the 'darkness' it was meant to illuminate. The irrational was deeply troubling to Enlightenment thinkers because it was believed that mankind's surrender to the irrational — to superstition, belief in magic and dogma — had created the horrors of the Middle Ages. The reaction to Walpole's novel from critics was symptomatic of the widespread feeling that if the guard was let down even for a moment, the horror might return. Walpole snuck horror in by the backdoor in a way that allowed the eighteenth-century reader to feel virtuous (following the exploits of the hero) while secretly enjoying terror, mystery, blood lust, and a variety of other unreasonable feelings. # Rocky Horror EVEN AFTER THE popularity of the original craze for gothic fiction died down, its conventions survived in other literary forms. William Beckford's Vathek, Mary Shelley's Frankenstein, and the historical romances of Walter Scott are all directly indebted to Otranto. But so, less directly, are Stoker's Dracula and Oscar Wilde's The Picture of Dorian Gray. And closer to our own time, Alien, Raiders of the Lost Ark, The Shining, The Blair Witch Project and the Harry Potter films still make use of the conventions Walpole established. The most crucial of these, as American academic Michael Gamer points out, is his re-imagining of the medieval castle, 'transformed in Walpole's handling from a locus of safety into a place of sexual transgression and supernatural visitation, of secret passageways and political intrigue... It is a place that harbours guilty secrets and unlawful desires'. If Gamer's description sounds a bit like a plot summary of The Rocky Horror Show, it's with good reason. The props of the gothic story — the dark and stormy night, the horror in the dungeon and, of course, the castle itself — were already clichés by the nineteenth century, which makes them perfect materials for satire. Unfortunately, somebody forgot to tell Brad and Janet. In blissful ignorance of the conventions of over two centuries of gothic fiction the squeaky-clean pair shows up at Frank-N-Furter's castle expecting a locus of safety. What they get is a whole lot of sexual transgression and supernatural visitation. The castle, as Gamer puts it, has become a fortress — not for keeping people out, but for keeping people in. Not that we really want Brad and Janet to escape. In Frank-N-Furter's world, conventional morality has been turned on its head, and that's the way we like it. Outside it's the 1950s — a world of white picket fences, high-school hops and the missionary position. In the castle civilisation has disappeared, and Frank-N-Furter instigates a sort of pan-sexual freakout in which Brad and Janet blissfully surrender to the power of the irrational. The lyric Frank-N-Furter sings at this point, 'Don't dream it, be it', was actually used as an epigram for the NME's very first story on the rising Goth scene. That line, as Little Nell (who played Columbia in the film) later recalled, 'hit a nerve' at the time. Rocky Horror tied up a number of ideas that were floating around in the '70s, linking the sexual transgression of glam rock to the androgynous threat of the vampire and other movie fiends. (Richard O'Brien's make up and costume for Riff Raff was modelled on stills of Max Schreck in Nosferatu.) The transsexual from Transylvania became — along with Alice Cooper and David Bowie — one of the spiritual forefathers of goth. Rocky Horror still changes lives. Davey Havok remembers being fascinated 'at a very young age' by his mother's copy of the soundtrack album. Sixteen-year-old Gerard Way first tried on his mother's lipstick after he got dumped by a girl, but he knew he liked it when he looked in the mirror and realised he looked like Frank-N-Furter. 'It definitely reminded me of Rocky Horror and I was definitely into it,' Gerard later recalled, 'and then, uh...then came the clothes, you know?' For both AFI and My Chemical Romance, horror — rocky or otherwise — is an important ingredient. Their undead aesthetic connects the dots from tortured romantic poet, to blood-sucking fiend, to darkly attractive rock star. This explains the appeal, for both Gerard Way and Davey Havok, of The Misfits — the legendary US punk band formed in New Jersey by Glenn Danzig and Jerry Only in 1977. After seeing The Damned play later that same year, Danzig and Only knew where their future lay — in the unholy union of punk rock and fake blood. Danzig let his love of old horror movies run riot over songs like 'Horror Business' and 'Night of the Living Dead'. By the following year, they looked like a punk band fronted by extras from a zombie movie — Only had hollowed his eyes out into black holes, Danzig had transformed himself into a living skeleton using nothing more than a black shirt and a bucket of white house paint. Videos of Misfits shows from around this time look like dispatches from the more forbidding regions of hell. Monsters lurch around on some sort of primitive altar, sending ear-splitting noises into the darkness, while an army of zombies lift their arms and their voices in worship. 'Muhs-fuhts! Muhs-fuhts!' Watching over it all is the image of a horrible grinning skull. This is The Misfits 'Fiend' logo, which the band found on a poster for an old horror movie and had stencilled onto their gear and printed on T-shirts. These continue to outsell the band's records by a considerable margin. My Chemical Romance has performed The Misfits' 'Astro Zombies', as well as paying the band the more significant tribute of adapting Danzig's 'Corpse Paint' for its Black Parade uniforms. For Gerard Way, The Misfits provided a crucial alternative to the political statements that dominated punk when he was growing up. While other punk bands were speaking up about injustice, oppression and social inequality, The Misfits were creating a world in which these things simply didn't exist. The Misfits inhabited the reversed moral world of the gothic, a world where despair, torment, darkness and even death are sublime. The appeal of the gothic, for romantics, is part of the same impulse to escape the world as it is that sends them running out of the city into the forest. But where Wordsworth's or Rousseau's was largely a flight through space, for the lover of the gothic, it's a flight back in time — to a world that existed long before civilisation, and never fails to creep back in wherever reason lets its guard down. This, for the romantic, is where we find the things that really connect us all to one another — the sight of blood, the eye-sockets of a skull, the elemental power expressed in a dark and stormy night. # Vincent Vincent Malloy is seven years old He's always polite and does what he's told For a boy his age he's considerate and nice But he wants to be just like Vincent Price VINCENT MALLOY, WITH his sunken cheeks and shock of black hair, is the claymation star of director Tim Burton's first short film, made while he was working at Disney in 1979. Young Vincent's imagination is steeped in horror movie imagery. When his aunt comes to visit, he smiles indulgently as she pats him on the head. But in his mind, he's Vincent Price slowly lowering her into a vat of hot wax. His mother sees him playing nicely with his dog, Abercrombie. How is she to know that in his imagination he's Dr Frankenstein hooking Abercrombie up to some infernal electrical machine? He throws the switch, and the dog is zombie-fied. Later: He and his horrible zombie dog Can go searching for victims in the London fog. No tale of gothic horror would be complete without the ghost of a dead lover and a scene in a graveyard at midnight. So Vincent, despite the fact that he is seven years old, convinces himself that he has a beautiful wife who has been buried alive, and promptly rushes out into the gloom with a shovel: He dug up her grave to make sure she was dead Unaware that her grave...was his mother's flower-bed. Vincent's mother is not so much upset about the flower-bed as by her son's insistence that he is cursed and alone, condemned — like some tiny Lord Byron — to 'wander dark hallways alone and tormented'. It's not like she's grounded him or locked him in the attic, in fact, she'd much rather he went out and kicked a ball around with the other kids. But young Vincent feels distant and aloof from other people. For this, he has sound philosophical reasons. Vincent, like Byron, has no time for the idea of human perfectibility. Staying indoors wrapped in morbid thoughts of doom and horror might seem unhealthy, but Vincent refuses to 'get better' because it implies that his inner torment can be affected, even extinguished by reason, whereas he knows it cannot. Little Vincent strongly resembles his creator — and not just in the hair. As a kid, Burton spent a lot of time alone. 'I didn't have a lot of friends,' he recalled in 1994, 'but there's enough weird movies out there so you can go a long time without friends and see something every day that kind of speaks to you.' He grew up physically close enough to Hollywood to be able to see at an early age where these weird movies came from. He took a tour of Universal Studios and saw the streets where they shot Dracula and Frankenstein. 'It was a powerful feeling,' he said. He wasn't much interested in school, in fact, one of his very first efforts behind a camera was a film made in response to an essay question on psychology. Burton handed in a montage of shots of schoolbooks shown to the accompaniment of Alice Cooper's 'Welcome to My Nightmare'. Art school wasn't much better. Here, Burton came to the same conclusions as Vincent on the subject of Empiricism, the same ones William Blake had reached as he'd angrily flipped through Joshua Reynolds's discourses on art education over two hundred years earlier. Man comes into the world with something unique, and society, with its rules and systems, does its best to rationalise that something out of existence. I remember going through art school, and you've got to take life drawing, and it was a real struggle. Instead of encouraging you to express yourself and draw like you did when you were a child, they start going by the rules of society. They say, 'No. No. No. You can't draw like this.' Growing up in the California town of Burbank, Burton made important links between the horror movies he loved watching, and the suburban conformity he saw all around him. As a kid, he dreamed of being the actor who played Godzilla so that he could enjoy the thrill of smashing the grown-up world beneath his scaly feet. He already felt that society needed to be destroyed, and nothing he learned as he grew older changed that feeling to any significant degree. That's why when he watched horror movies Tim Burton always sympathised with the monster. Of these, the one to make the biggest impression on him was James Whale's 1935 classic Bride of Frankenstein. Images from Whale's film have turned up in a number of Burton's, though he insists that the similarities are usually not so much a matter of homage as of his ideas and Whale's coming from the same place, meaning that they tend to be expressed in the same way. Burton instinctively connected the monster's rage with his own, and recognised his need to destroy as the necessary flip-side of his need to be loved. The Bride of Frankenstein: 'More capacity for love then earth/bestows on most of mortal mould and birth...' # Frankenstein Bride of Frankenstein begins, not with the Monster or his creator, but with a prologue set in a Swiss villa. It is, of course, a dark and stormy night. The actor Gavin Gordon stands by the window with his chin set at an impressive angle and tosses his curly locks about. He begins to poetise, in a fairly overripe English accent, about the raging storm outside. 'I should like to think that an irate Jehovah was pointing those arrows of lightning directly at my head — the unbowed head of George Gordon, Lord Byron — England's greatest sinner!' A young woman sits across from him, clearly bemused by Byron's posing. She refuses Byron's invitation to come to the window and watch the storm, and asks her fiancé, Shelley, to light another candle. Byron is tickled by this. 'Frightened of thunder! Fearful of the dark! And yet you have written a tale that has turned my blood to ice!' Mary Shelley smiles a secretive smile, as if to say, There's more where that came from. The tale Byron is referring to is Frankenstein — the dark horse of the story writing contest he'd instigated. While the characters and the setting are based on fact, this little scene in the film is a fiction — Mary never imagined a sequel to her horror story. In fact, when the contest was first suggested, she'd despaired of being able to contribute anything at all. Byron had written his vampire fragment, and Polidori — when not flirting with Mary or challenging her pacifist fiancé to a duel — had written the tale of his skull-headed woman. But Mary's inspiration had deserted her. 'Have you thought of a story?' her friends would ask her when she came down for breakfast. 'Each morning,' she says, 'I was forced to reply with a mortifying negative.' One night, still furiously trying to think of something really scary, Mary half drifted off to sleep. As she floated in that strange zone between consciousness and unconsciousness, she saw, in her mind's eye, a series of terrifying visions — a dark shape bent over a corpse, an unnatural twitch, and a pair of watery yellow eyes in the darkness. Scared out of her wits, she sat up in bed and tried to compose herself. Well, she thought, that was terrifying. If only I could come up with something as scary as that, I'd be able to write the best ghost story ever...hang on! The story Mary Shelley set down over the next few days — and eventually expanded into a novel — tells of a young doctor named Victor Frankenstein who sets out to break the ultimate scientific taboo: the creation of life itself. Frankenstein works in darkness and secrecy for two years to get it done. But as soon as his goal is accomplished and his creature begins to twitch with artificial life, he sees that he shouldn't have done it, never in a million years. Not only has he made a terrible mistake — he's made a mistake that walks, a mistake that creeps up to his bed in the dead of night, pulls back the curtain, and reaches a horrible greyish hand in his direction. Frankenstein flees his apartment at the university and spends a cold night on the street, hoping that the nightmare will simply evaporate in the morning. And at first, it seems as though it has — until his friends and family begin to die... When the monster and his maker finally meet again, we are surprised, along with Frankenstein, at how eloquently he expresses himself. The monster learned the rudiments of conversation, it turns out, by eavesdropping on a rustic family as he hid in their barn. But his education was completed by a package of books he found by the roadside one day — that included Milton's Paradise Lost, Plutarch's Lives and Goethe's The Sorrows of Young Werther — which moved him deeply. 'I did not pretend to enter into the merits of the case,' said the monster of Werther's tale, 'yet I inclined towards the opinions of the hero, whose extinction I wept, without precisely understanding it.' The monster, it seems, sympathises in a very profound way with Werther's feelings of apartness and aloneness. He takes Goethe's advice in the book's preface — The Sorrows of Young Werther becomes a friend to the creature, who has no friends at all. It's fitting that Frankenstein's monster should identify with Goethe's angst-ridden young romantic, since young romantics have always found something to relate to in the monster. He is, as Bertrand Russell has observed, virtually the embodiment of romanticism, and the changes he undergoes in the novel demonstrate the trajectory of romantic philosophy in a startling way. Byron's words in 'Lara': his early dreams of good outstripped the truth and troubled manhood followed baffled youth neatly sum up the monster's life up to the point when Frankenstein confronts him. He is barely human, but just human enough to want to be loved like everybody else. And because he was cursed at birth to be freakish and unlovable, humanity lets him down. The monster tells his creator that, after fleeing Frankenstein's apartment, he made several attempts to befriend his fellow beings, all of which ended disastrously — the only exception being a blind man, who only loved him because he couldn't see how ugly he was — which only goes to show how shallow and judgemental human beings are! The whole human race turned away from him in horror, so the monster turned his back on humanity. He sought solace in nature — even the steely-grey sky above the alps seemed more welcoming to him than the people who lived beneath it. But being an artificial creature, he didn't feel at home in nature any more than he did in society. At this point, the monster's natural goodness began to collapse under the strain of his exile. 'The mildness of my nature had fled', he explains, 'and all within me was turned to gall and bitterness... I am malicious because I am miserable. Am I not shunned and hated by all mankind?' And yet even in his despair, the monster believed he could be redeemed, which is why he set out to find Frankenstein — whom he had grown to hate by this point — who held the keys to his happiness. Now, he implores his creator to recognise him and his needs, to create a female who will complete him. Frankenstein, horrified by the idea, rejects the monster's plea out of hand — and this proves to be the last straw. Now, having nothing to live for, the creature's rampage becomes unstoppable. But Frankenstein's creature never becomes an unthinking killer — and he certainly never becomes an unfeeling one. On the contrary, the inhuman monster is full of human feeling. As Bertrand Russell points out in his History of Western Philosophy, no matter how base his actions become, his sentiments are always noble. After he commits the patricide he has been threatening for the entire book, he stands above Frankenstein's corpse and delivers a moving soliloquy. Oh, Frankenstein! Generous and self-devoted being! What does it avail thee that I now ask thee to pardon me? Walton, the ship's captain who observes this, boldly points out that it doesn't avail Frankenstein much at all that the monster is so full of remorse. If you'd listened to your conscience and not killed all those people, says Walton, none of this would have happened. The monster is outraged at this suggestion. Don't you understand, he asks Walton, how I feel? 'Do you think that I was then dead to agony and remorse? He,' he continued, pointing to the corpse, 'he suffered not in the consummation of the deed. Oh! Not the ten-thousandth portion of the anguish that was mine during the lingering detail of its execution... Think you that the groans of Clerval were music to my ears? My heart was fashioned to be susceptible of love and sympathy, and when wrenched by misery to vice and hatred, it did not endure the violence of the change without torture such as you cannot imagine.' Here, romantic solipsism is taken to its most frightening conclusion. The monster's murders are justified by his feelings. No wonder he found so much to admire in Werther — Frank-enstein's creature is in fact a super-Werther, whose physical strength gives him the power to act out his feelings in ways that Goethe's gloomy protagonist could only dream about. In The Smashing Pumpkins' 'Disarm', Billy Corgan invents a fantasy version of this scenario, in which he hacks off his parents' limbs in order to teach them a lesson about what it's like to live your life as a lonely freak. But like Byron and Frankenstein's monster, he doesn't do harm because he's evil, but because he feels too much, and he can't contain his feelings any longer. The singer in 'Disarm' feels 'the bitterness of one who's left alone' — all those years sitting by himself in the school cafeteria — because his heart is full of 'tender feelings'. The way he sees it, his parents gave him the desire for love, and then made him unlovable, condemned to eternity in a lonely Tower of Doom. 'Ooh, the years burn', sighs Corgan. How can they expect him to play nicely with the other children when he is, as Frankenstein's monster puts it, 'shunned and hated by all mankind'?14 The extreme menace in 'Disarm' comes from the idea, never too far from the surface, that a killing spree has been only narrowly averted by cathartic song writing. # Edward Scissorhands IN THE WINTER of 1989 Johnny Depp was sent a movie script called Edward Scissorhands. At that time the actor was stuck in the depths of TV hell, mouthing god-awful dialogue on the set of 21 Jump Street. After reading this script, he believed he could be saved. It was the story of a boy with scissors for hands — an innocent outcast in suburbia. I was so affected and moved by it that strong waves of images flooded my brain — dogs I'd had as a kid, feeling freakish and obtuse while growing up, the unconditional love that only infants and dogs are evolved enough to have. Soon, the nervous actor was meeting with the director, and after talking to him for an hour or so, he realised, 'this hypersensitive madman is Edward Scissorhands'. Here Depp was right on the money. If the script had brought his awkward teenage years flooding back to him, it was because it was heavily inspired by the director's own painful adolescence — a period of time when, as Burton later described it, he felt like he had a big sign around his neck saying 'Leave me the fuck alone'. The whole idea for the film came from a drawing Burton did when he was a teenager of 'a character who wants to touch but can't, who was both creative and destructive'. Depp also recognised in the script a feeling of profound sympathy for 'those who are not others' — just one of the many important lessons Burton learned from his steady diet of B-movies while growing up. I always loved monsters and monster movies. I was never terrified of them, I just loved them... I felt most monsters were basically misperceived, usually they had much more heartfelt souls than the human characters around them. After a short prologue, Edward Scissorhands begins with Avon lady Peg Boggs having a bad day. She drives around her pastel-coloured suburban town in her pastel-coloured car trying to sell make-up to bored housewives and surly teenagers, but to no avail. So, with nothing left to lose, she decides to try the one house in the neighbourhood she's never been to — the evil-looking gothic mansion at the top of the hill. She drives up the winding path and walks through the massive front doors. 'Avon calling,' she cries hopefully into the gloom, her words echoing through the castle's empty halls. Not quite empty as it turns out. Peg finds a strange creature hiding in the shadows — deathly-pale, encased in black from head to foot, with giant scissors where his hands should be. Her heart breaks: Peg: What happened to you? Edward: I'm not finished. Edward, like Frankenstein's monster, is an experiment abandoned by his creator in a half-formed state. His inventor father (played by Vincent Price), died before he was finished, so instead of real hands, Edward is stuck with scissors. His freakish appearance has kept him confined to his crumbling castle, where he lives in a world of imagination and memory, cutting the hedges in the garden into giant pairs of human hands — topiary as dream-wish fulfilment. Peg takes him home and cleans him up, and for a little while it looks like he might have finally broken back into the lovely world below, redeemed himself by being accepted and loved. But eventually the regular folk turn on the monster in their midst — and who can blame them? Those scissor-hands he uses to cut the townspeople's hedges and barbecue their shish kebabs are terrifying weapons. Not that sensitive Edward would ever intentionally harm anyone. But he's pushed and pushed by these shallow greedy people until he can't help it. And then, in the moments where he's most human, when he reaches out to touch or protect someone he loves, he hurts them. As his story unfolds, it becomes increasingly clear that he just doesn't belong with other people, that he has to be alone. Like little Vincent Malloy, he goes back to his tower of doom, where he spends the rest of his days poring over his painful memories. The sculptures he makes in his garden are suffused with all the longing he feels to be a part of human life — to be loved, to be accepted. Of all the director's films, Edward Scissorhands is probably the purest distillation of Tim Burton's worldview. In a telling moment near the start of the film, just as Peg is about to give up on her door-to-door sales and head home for the day, she adjusts her side-view mirror. We see from Peg's point of view out the car window and into the street, where neat pastel-coloured houses roll out as far as the eye can see. Then, in the mirror, we catch a glimpse, as she does, of Edward's spooky, dilapidated home on the hill. Edward's world is the reverse of Peg's — but only the one in the mirror is real for Burton. 'People ask me when I'm going to make a film with real people,' Burton once remarked. Of course, for him, monsters like Edward are real people, the only people worth knowing. All his heroes are freaks or outcasts of some description: Edward is a half-finished science project, Ed Wood is a toothless cross-dresser who makes terrible movies about grave robbers from outer space, and Burton's Batman is a sociophobe with a fetish for latex. In Beetlejuice, Betelgeuse the bio-exorcist is a disgusting undead ghoul, the young couple he agrees to help are ghosts who distort their faces into monstrous masks or rot away before our eyes, and Lydia, the teenage girl who befriends them, can only see the strange and unusual beings haunting her parents' house because she is, herself, strange and unusual. Lydia is an original '80's goth — her decision to wear black and obsess about death is an intentional affront to her shallow, materialistic, style-obsessed mother. Her dad doesn't really understand her either — but he does at least try to keep with her interests. He makes an attempt to cheer her up by promising to build her a darkroom in the attic so she can develop her photos. But Lydia will not be consoled — 'My whole life is a dark room. One. Big. Dark. Room.' One of the film's funniest moments comes when Lydia sneaks into the attic where the ghosts have taken up residence. The ghosts aren't in — but Betelgeuse has made himself at home — Lydia finds the miniaturised ghoul lounging obscenely on a tiny deckchair. At the sight of Lydia, with her funereal get-up and deathly complexion, Betelgeuse perks up. 'Hey,' he says, 'you look like someone I could relate to!' Like Frankenstein's monster reading Werther, the bio-exorcist senses an important connection between himself — a horrible monster shunned by humanity — and the miserable teenage girl in front of him. Betelgeuse hopes to enlist Lydia in his efforts to get out of the underworld and into the game. Lydia wishes they could trade places. 'I wanna be in there!' she says, pointing to the miniature diorama that represents the spirit world in the film's peculiar mythology. Betelgeuse is mystified by this. 'Why?' he asks. 'Well', he goes on, 'I'm sure you have your reasons.' She does. Like Vincent Malloy, Lydia has been banished to the tower of doom. But not by her parents — they want her to get out, get involved in social life, have some fun. No, Lydia has banished herself to the tower of doom, locked the door from the inside, and swallowed the key. She knows she'll never be like all the other kids, so she rejects the possibility of joining their world and moves permanently into her own. Here, she thinks, in darkness and isolation, is where I belong. Lydia shares the fate of all Tim Burton's oddball heroes. She has been kicked out of the garden — and has come to understand that what everyone else calls happiness is not for her. Lydia: Am I not shunned and hated by all mankind? # The Dark Side of Human Things THE TENOR OF Mary Shelley's Frankenstein is profoundly religious. Victor Frankenstein is clearly shown to be a sinner — plunging himself into death and filth in order to create life. His account of his midnight expeditions in search of corpses for his experiments, is permeated by a deep sense of shame. In fact, his whole tale is a confessional. We hear the story through the device of Frankenstein unburdening himself to the captain of the ship which will take him to his final encounter with the monster (and this will become a convention of horror stories for decades to come — My tale is almost too horrible to relate, yet I must confide in you before it is too late...). And to hear Frankenstein tell it, he knew, even at the time, that what he was doing was deeply unnatural and wrong, but he imagined that in his perversion of nature the scientific ends might justify the means, and that he would in the end be rewarded. But he sees now that he could not have been more wrong — his only reward is death — for himself and for those he loves. Frankenstein's monster is the punishment for his sin — which turns out to be one of the oldest — the sin of wanting to know too much. Of course, all of this stuff about sin and punishment is very medieval, part of a way of thinking that was supposed to have vanished long before 1819. But the Shelleys had grown up in an age that had already stretched the eighteenth-century ideal of reason and enlightenment to breaking point. Prior to arriving at Lake Geneva, the couple had seen up close the havoc and destruction that the Revolutionary wars had visited on the people of rural France. The romantics of Wordsworth's generation had already processed the effects of this, watching as the carefully maintained equilibrium of the eighteenth century was destroyed, giving voice to the crisis in their poetry, and finally finding a third position outside it all, the state of grace Wordsworth achieved with 'Lines Written a Few Miles Above Tintern Abbey'. This poem is actually quoted by Frankenstein himself in Mary Shelley's novel — it had by this point become part of the canon, and Wordsworth was an elder statesman. Mary was only nineteen years old when she wrote Frankenstein — she'd never known the world Wordsworth had been born into and was already living in a very different one. By this point, France's monarchy had been restored, and Europe had settled into a period of extreme political conservatism. The cautious reforms that many governments had been implementing in the pre-Revolutionary period were abandoned, and the general feeling was that if you give the people an inch, they'll take a mile. Frankenstein is typical of the new role of the gothic in this era. The early nineteenth-century horror story represents the threat of chaos, and perhaps even a repressed desire to see that chaos unleashed — to smash society, finish the job and see it all come down. This is why Tim Burton instinctively connected the gothic horror of Bride of Frankenstein with his desire to destroy his suburban surroundings. The monster himself is — like his cousin Dracula — an image of romanticism on the rampage, a terrifying, irrational force turned loose on an over-ordered world. But Frankenstein, with its Old Testament morality, also points to another side-effect of the Revolutionary period on the romantic imagination. The failure of the eighteenth-century dream of a society based on rational principles to materialise had been bad enough; that what had emerged had been something closer to a medieval bloodbath was enough to convince the romantics that the ideal of human perfectibility was dead in the water. Indeed, the arguments for seeing humanity as basically flawed and doomed to repeat its mistakes began to look more and more convincing — especially to those who, as Leigh Hunt had said of Mary Shelley, had 'a tendency to look over-intensely at the dark side of human things'.2 # Mystery IN 1988 NICK CAVE was flying from Australia to London. It's a long trip, and it tends to bring out the worst in people. 'The other passengers were basically gearing up to tear my girlfriend and me to bits if we continued to go the way we were going', he later confessed to Simon Reynolds. On this flight, Cave almost became a born-again Christian: Two days sitting on the plane and fifty bourbons later I had this young born-again advocate holding my hand and praying for me at the top of his voice. To hear Cave tell it, the missionary was quite surprised to find that the evil-looking rock singer he'd just latched on to knew his Bible quite well — better, even, than the missionary. He started quoting things from his modern translation which I find really irritating...to find it so utterly demystified by these modern religions keen to allow people of today to understand...it really appals me. People have been trying to demystify the Bible since Descartes, and during the Enlightenment, it became something of a craze. In the seventeenth and eighteenth centuries, the 'mystery' Cave likes so much in his King James Bible was seen as one of those things the human race was better off without — a point of view championed by the English physicist Isaac Newton. ''Tis the temper of the hot and superstitious part of mankind in matters of religion ever to be fond of mysteries, and therefore to like best what they understand least', he once wrote. Mysteries, for Newton, had kept people in the dark for centuries. Therefore as a scientist and a Christian he considered it his business to rationalise his religion. This was no easy task, Christianity had over the past thousand or so years accumulated a lot of strange dogma and superstition, but Newton fearlessly set about trying to strip all of this back so as to reveal the true, rational religion underneath. Miracles, of course, would have to go, as would the doctrine of the Holy Trinity — which to Newton was a sop to the superstitious pagans. He even went so far as to deny the divinity of Christ. The job of bringing Christianity into line with reason would continue well into the next century. This was the time during which the philosophy of Leibniz became popular. Leibniz argued that since God is all-powerful and infinitely good, he must have created our world as the best of all possible worlds. Evil, in Leibniz's system, is thus explained as a necessary part of the greater good. This was a religious philosophy which appealed to the profoundly optimistic mood of the Enlightenment. In his famous Essay on Man, Alexander Pope wrote: All nature is but art unknown to thee; All chance, direction which thou canst not see; All discord, harmony not understood; All partial evil universal good: And spite of pride, in erring reason's spite One truth is clear, whatever is, is RIGHT. The Essay on Man did much to popularise Leibniz's thought in both France and England. But though Pope's verse is impeccable, his philosophy — and Leibniz's — is full of holes. To look at the world with all its trouble and strife and say 'it's all good' seems unconscionable today — and it wasn't much better in 1732. The glibness of Pope's brand of optimism didn't escape the sharper eyed critics of the Enlightenment. The French philosopher and poet Voltaire mercilessly sent up the Leibnizians in his satire, Candide. Candide's tutor, Professor Pangloss, is a metaphysico-theologo-cosmolonigologist, who insists, in the face of a great deal of evidence to the contrary, that he is living in the best of all possible worlds. His philosophy makes him an idiot — worse, a shit of a human being, who won't lift a finger to help the victims of wars or natural disasters. Candide had been wounded by splinters of flying masonry and lay helpless in the road, covered with rubble. 'For heaven's sake,' he cried to Pangloss, 'fetch me some wine and oil! I am dying!' 'This earthquake is nothing new,' replied Pangloss; 'the town of Lima in America experienced the same shocks last year. The same causes produce the same effects. There is certainly a vein of sulphur running under the earth from Lima to Lisbon.' 'Nothing is more likely,' said Candide; 'but oil and wine, for pity's sake!' Voltaire, as historian Norman Davies has said, was expert at using the techniques of the Enlightenment to expose its flaws — which, in a sense, made him the ultimate embodiment of the age. The German philosopher Immanuel Kant, after all, insisted that the obstacle to enlightenment was not our lack of understanding, but our lack of courage in putting that understanding to work, and Voltaire was fearless in his unmasking of outmoded or useless ideas. He saw it as his life's mission to clear away the accumulated junk of Western thought wherever he found it, and he hated dogma and superstition above all. But Voltaire would not abandon God, and he always believed Christianity was compatible with reason. Two decades after Newton's death, he laid out the principles of a rational religion in his Philosophical Dictionary: Would it not be that which taught much morality and very little dogma? That which tended to make men just without making them absurd? That which did not order one to believe in things that are impossible...? This reasonable Christianity, this religion for the Enlightenment, is not Nick Cave's preferred variety. Since that fateful plane trip twenty years ago, Cave has never stopped looking into his Bible. He's quoted it in his songs and caught its grave rhythms in his prose. But he never went to it looking for glorified commonsense, or with a view to hunting down inconsistencies so as to bring them in line with reason. For Cave, a religion which has been purged of its madness, sadness and bloody-minded violence is not a religion at all; and 'believing in things which are impossible' — as Voltaire put it — is both an essential part of religious experience and a key requirement of his day job as a singer of love songs. Speaking at London's South Bank centre in 1999, Cave insisted that the words of the Old Testament Psalms and the words of a song like Kylie Minogue's 'Better the Devil You Know' are both born from the same profoundly unreasonable impulse. The love song, Cave said, 'is a howl in the void for love and comfort, and it lives on the lips of the child crying for his mother. It is the song of the lover in need of their loved one, the raving of the lunatic supplicant petitioning his god.' For Cave, both the love of God and romantic love are 'manifestations of our need to be torn away from the rational, to take leave of our senses'. To an eighteenth-century ear, this lecture would have exposed Cave as an ignorant superstitious goth. Cave, of course, has often been called a goth — even the 'king of the goths' — and just as often denied it. But if 'gothic' means — as it did in 1750 — irrational, superstitious and unhealthily obsessed with hellfire and damnation, then Cave is gothic to the tips of his well-tailored black suits. # Utopia 'THE TIME WILL come,' wrote Voltaire's colleague the Marquis de Condorcet in 1793, 'when the sun will shine only on free men who know no other master than their reason.' Like Voltaire, de Condorcet was a member of the society of Philosophes and a firm believer in human perfectibility. His Sketch for a Historical Picture of the Progress of the Human Mind demonstrated that science and mathematics would improve every aspect of life in post-Revolutionary France, and eventually, the whole world. Population control, sexual equality, religion, law, language and love would all benefit from the application of mathematics to their specific problems, and given time, poverty, civil strife and war would become things of the past. As a utopian, de Condorcet firmly supported the republic, but as a liberal humanist he could not condone the execution of the king. This automatically made him a Royalist in the eyes of the Committee of Public Safety. So, even as he set down his vision of a mathematically perfectible utopia, Robespierre's police were coming for him. He spent most of 1793 in hiding, and then tried to flee France the following year. He was caught, thrown in prison, and found dead in his cell the next morning, having taken poison. The Terror ensured that there would never be another period of sustained optimism like the Enlightenment. But the Philosophes' vision of a society that works remained a powerfully attractive one for many years to come. Long after the romantic movement parted company with universal reason, works such as de Condorcet's had become the basis for the nineteenth century's belief in progress, which survived virtually unchallenged in the world of industry and science right up until the mid-twentieth century. Even in artistic circles, romantic gloom would occasionally give way to bursts of utopian optimism over the next two centuries. Strong traces of Enlightenment thought can be detected in the arts and crafts movement, the Vienna Secession, at the Dessau Bauhaus, among the Russian constructivists and — curiously — in London's post-punk scene of the late '70s. Like the Enlightenment itself, punk is often understood as a reaction to what came before — in this case, the grandiose mysticism of mid '70s prog-rock. 'We tend to keep away from the present', said Genesis's Steve Hackett in 1974, 'we're very hesitant to make any commitment to how we feel about what's happening now.' Punk, on the other hand, would admit no other subject matter than 'what's happening now'. The lyrical abstractions of prog-rock, like the introverted navel gazing of the West Coast groups, seemed to create music with no social purpose beyond pure escapism — and the punks were adamant that music should be about more than that. In theory, if not always in practice, punk bands wrote songs about what it was really like to live on a council estate or what was in the papers or what their record company did last week. Heroic quests, mystical allegories and song cycles were banished, never to return. Having cleared away the useless clutter and mystical obscurantism of prog, it was now left to the groups who emerged in the wake of punk's first wave to build a new songwriting ideal. Now, all bets were off, everything could be questioned. Gang of Four applied the Philosophes' favourite question: 'is it rational' to that oldest of rock institutions, the love song, and found that it was not. Guitarist Andy Gill muses on 'Love like Anthrax': ...most groups make most of their songs about falling in love or how happy they are to be in love...these groups go along with the belief that love is deep in everyone's personality. I don't think we're saying there's anything wrong with love; we just don't think what goes on between two people should be shrouded in mystery. Like Newton, Gill has no time for mystery; mysteries keep people stupid, and the mysteries of love are no exception. When Gang of Four did write about love, they stripped off the ornament and reduced love to a social agreement or a coupling of bodies; there were no hearts and flowers, no burning fire or pure desire in these songs. Love was presented as difficult but never mysterious. In the post-punk love song, as music critic Simon Reynolds writes in Blissed Out, 'the acknowledgement of the dark side was always grounded in progressive humanism, the belief that what was twisted could be straightened out...shadows could be banished by the spotlight of analysis.' According to Reynolds, punk had established the idea that 'demystification was the road to enlightenment.' Nick Cave: Moody and miserable. # Utopiate IF PUNK RE-ENACTED the Enlightenment, then it was left to a 'moody, miserable' kid from rural Victoria to play the part of the entire romantic movement. As early as 1977, punk's year zero, when everyone else was poring over Theodor Adorno, Herbert Marcuse and Guy Debord, Nick Cave started reading the Bible — the King James, of course. The brutal, bitter tales of the Old Testament confirmed Cave's suspicion that human beings are not infinitely perfectible, but born in sin and bound for hell. Cave knew at an early age he was either destined or damned — or maybe, like Napoleon, a little bit of both. As he grew older, he found that this basic fact of his personality remained unchanged, and nothing he saw after that could convince him that we come into the world as 'lumps of dough that are later moulded by our parents and so forth'. Cave's first band, The Boys Next Door, had a hit in 1978 with a song called 'Shivers' — a song that the producers of Countdown refused to allow the band to perform because the lyrics mentioned suicide. 'Shivers', written by guitarist Roland S Howard, is a confessional in the early Byronic mould. The hero is detached and strangely static. He's been thinking about suicide, but he'll only do it if you're watching, and if you think it's fashionable. In the end, he remains paralysed by ennui. Howard takes up the theme with a long, plaintive guitar solo, which sounds like a lethargic replay of Pete Shelley's famous two-note refrain in the Buzzcocks' 'Boredom'. On the day Bon Scott was buried, The Boys Next Door left Melbourne for London, changing their name on arrival to The Birthday Party. Post-punk was in full swing, and superficially, The Birthday Party fitted right in — the lopsided Magic Band guitar parts, the tribal thump of their rhythm section, their singer's anguished, alienated squawk. At a moment when Captain Beefheart's Trout Mask Replica represented the musical ideal, and PiL's Metal Box the cutting edge, The Birthday Party had every reason to think their success was assured. But even as England learned to love The Birthday Party, The Birthday Party were learning to despise England — English bands especially. Cave quickly realised that he hated all the post-punk/new wave groups that were so heavily feted at the time. His old-fashioned sense of sin and retribution chafed badly against then-fashionable topics such as 'personal politics'. For Cave, love was not, and could never be 'a contract in our mutual interest' as one Gang of Four song put it; love was madness, sorrow, despair, violence, a deeply mysterious and irrational force. 'Zoo Music Girl', the first song on the Birthday Party's debut album is a blood-soaked ballad. 'Oh God,' cries Cave 'let me die beneath her fists!' In 'Wild World' the lovers are crucified, in 'Six Inch Gold Blade' the singer sticks a knife in his girl's head. We are already a long way from the world of personal politics. In 'Hamlet Pow! Pow! Pow!' Cave re-casts Shakespeare's tragic hero as a gun-toting gangster. 'Wherefore art thou baby face?' he sneers (having ended up, not only in the wrong century, but in the wrong play). It makes perfect sense for Cave to turn Hamlet into a killer with a gun, because for the singer, the tragic Dane and the murderer are burdened with the same heavy load — passion that can find no outlet in society. They stand side by side in the Nick Cave pantheon with Saint Sebastian, Iggy Pop, Count Dracula, Beethoven, Dostoyevsky, the Hunchback of Notre Dame, Captain Ahab, Robert Mitchum (in Night of the Hunter) and Jesus Christ. Cave will always side with geniuses, freaks, monsters and outcasts — as opposed to the society that could not accommodate them — because to him society is not only hateful, it's a bad bet; doomed to fail, no matter what the positivists, empiricists and neo-Marxists try to tell you. 'To see yourself as part of some greater humanist scheme,' he said to Reynolds in 1988, 'I can't really abide by that myself. I'm someone who has very little concern with any kind of social problems, someone who's very much concerned with their own plight.' Two years, and two extraordinary albums later, The Birthday Party self-destructed — and you can hear it happen on their swansong, 'Mutiny in Heaven'. The lyrics of 'Mutiny' run on from an earlier song called 'Dumb Europe', written with Die Haut in 1983. 'Dumb Europe' describes a night out in Berlin where 'the cafes and bars still stink'. An early draft of the song features a coda, 'Hey! Dumb Europe! Utopiate! European Utopiate!' Here, Cave stands up in his 'bleak Teutonic hole' and calls time on the Philosophes' dream of a heaven on earth. The perfectibility Jacques Turgot promised his eighteenth-century audience at the Sorbonne, the mathematical utopia de Condorcet was still dreaming about as Robespierre's police hunted him down, the hope of Universal Reason Wordsworth clutched at during his crisis of 1795, where are they now? Utopia, Cave puns, is a Utopiate — a drug which has enslaved the European mind as surely as any of the crackpot dogmas it was supposed to destroy. And the positivist is a junky, on the nod in a corner while the Continent falls apart around his ears. In 'Mutiny in Heaven' Cave invites us to look around at dumb Europe and admit that utopia has long since turned into a slum. The place is overrun with trash and rats — and now even the rats are leaving, crawling up his arm in search of higher ground. This is never a good sign. If this is heaven, he says, 'Ah'm bailing out!' But how do you get out of the modern world? Over The Birthday Party's terrifying rumble, accompanied by a guitar that sounds like the peals of a church bell, Cave talks us through it: Well, ah tied on...percht on mah bed ah was Sticken a needle in mah arm Ah tied off! Fucken wings burst out mah back The positivists would have you choose life. Cave — as Mark Renton puts it in Irvine Welsh's Trainspotting — has chosen something else. Siouxsie Sioux: A hostile and valueless reality. # The Degraded Present BECAUSE HE FEELS that society should be destroyed, Nick Cave is also a lover of horror — note the appearance of Count Dracula in his list of favourite things. For Cave, death and darkness — being closer to the truth of the human condition — are sublime. Luckily, his baby feels the same way: My baby is all right She doesn't mind a bit of dirt She says 'horror vampire bat bite'153She says 'horror vampire How I wish those bats would bite' 'Release the Bats' was a certified Batcave floor-filler — for obvious reasons. It would also become The Birthday Party's most important contribution to the goth aesthetic. As Simon Reynolds shows in Rip it Up and Start Again, Goth first emerged as an alternative to two recent developments in post-punk — the rabble-rousing Oi! movement, and the anarchopunk scene centred around Crass. Goth became a home for kids who liked the energy of these bands, but were bored by the politics. Anything with a whiff of romance, darkness and mystery was bound to appeal to them — and 'Release the Bats' had plenty of all three. When the singer in 'Mutiny in Heaven', sprouts his ungodly wings and flies out of the twentieth century, he's offering his listener something no amount of agit-prop or personal politics can provide — an escape route from the world as it is. Siouxsie and the Banshees embodied this same quality. '[Juju] was released at the height of the Thatcherite years,' remarked music writer Keith Cameron, talking to Siouxsie in 2008, 'yet you seemed to be inhabiting your alternative reality, a horror-show phantasmagoria: Halloween, Voodoo Dolly, Arabian Knights...' 'Right!' the singer replied. 'You're saying "Thatcherite years", and I'm going: "Really?!" I wasn't even aware! We were in our own universe.' The flight to this alternative reality is what links together the motley collection of bands who came to be embraced by the 'white faces' in the early '80s, and has been central to the appeal of goth through the decades. The Cure, for example, are not really goth. But to Geoff Rickly of Thursday, growing up in the '90s, they were of a piece with the other goth bands he liked because they seemed to offer an escape route from the present day. 'The goth and British bands I liked had the same visceral kick as regular punk but it seemed more like a place for me, a space you could inhabit. Something far away from reality.' Similarly, for Gerard Way of My Chemical Romance and Davey Havok of AFI, the appeal of goth is precisely this escape from the present day, an escape that punk with a capital 'p' can never allow. In Bauhaus, Southern Death Cult, The Damned, Alien Sex Fiend, The Virgin Prunes and The Sisters of Mercy, the Britain of 1979–84 is only suggested by its absence. All these bands made the leap out of what Alex de Jonge calls 'the degraded present' and into something out-of-time, something eternal and unchanging. Siouxsie and The Banshees' 'Spellbound' invokes the world of the irrational, dreams, magic and madness. Siouxsie asks us to cast our minds back to childhood and the 'beckoning voice' that seemed to call us through the cradle bars. These deep-seated urges, she insists, cannot be ignored — they define us for all time, and all our efforts to civilise ourselves, from pre-school onwards, are reduced to nothing when we hear this siren-sound again. You hear laughter Cracking through the walls it sends you spinning you have no choice The contrast with the Leeds positivists couldn't be more complete. In 'Love like Anthrax', Andy Gill scoffs at the idea that, deep in the human soul, there are permanent emotions that everyone can relate to because they have not changed in thousands of years. If this is true, what hope do we have of perfecting society? None at all, says the goth. We can't change, because we're not lumps of dough, but unfathomable mysteries, full of primitive urges and recurring nightmares. These timeless and tragic emotions have haunted humanity for thousands of years, and will continue to haunt us for thousands of years to come. This, for the punk activist and pop deconstructivist alike, is almost unforgivably backward. That's why, when they went looking for a name for this unwelcome eruption of romantic gloom in their new pop universe, critics of the day settled on 'gothic'. The word had almost the identical connotations it had for the reviewer of Horace Walpole's The Castle of Otranto — 'gothic' meant superstitious, irrational and unhealthily obsessed with love, religion and death. The implication was — haven't we grown out of all that stuff? But these criticisms ignore the fact that the goth's attachment to the timeless and the tragic is the result of the very same 'Enlightenment' that the pop optimist claims to advocate. Nick Cave, like Wordsworth, would eventually find his God, but for the majority of the romantics — including Shelley, Keats, Byron and Goethe — such simple faith was impossible. Romanticism, as Norman Davies has observed, is characterised by a profoundly religious temperament — a longing to believe. But more often than not, when the romantics, having found no satisfaction in the modern world, went looking for God, they found him gone. The blame, as usual, lay with Newton and his followers, who in the rush to rid Christianity of its mystery, had rationalised God into a corner, and finally out of existence. # Blasphemous Rumours THEY MIGHT HAVE been signed to the same label as Nick Cave, but in the musical world of the mid '80s Depeche Mode were clearly aligned with the pop positivists — those who would analyse the clichés and conventions of our behaviour with a view to creating new relations between people. And like their philosophical forebears, Depeche Mode were fearless in their quest to expose dogma and nonsense to the cold hard light of reason, even if it led them — as it had Newton — perilously close to Blasphemy. On side two of their 1984 album, Some Great Reward, Gahan tries to unravel the greatest mystery of all: why, if God is good, do bad things happen to good people? Girl of 18 Fell in love with everything Found new life in Jesus Christ Hit by a car Ended up On a life support machine. Gahan and Gore wrote 'Blasphemous Rumours' after noticing something odd about the church services they'd attended. At the end, the priest would read out a list of those in the congregation suffering from serious illness, 'and the one at the top always died. But still everyone went right ahead thanking God for carrying out his will. It just seemed so strange.' The conclusion was inescapable. In the song's insanely catchy chorus, Gahan sings: I don't want to start any Blasphemous rumours But I think that God's got a sick sense of humour And when I die I expect to find him laughing. Gahan's accusations were so bitter, and the song struck such a chord, that eventually, God's representatives on earth were moved to speak up in his defence. A priest from Depeche Mode's home town of Basildon spoke to the press, saying, 'If we can say God so loved the world that He sent His only son...if he did that, he cannot have a sick sense of humour.' Which is all very well, but Depeche Mode's questions still nag. Why create us with the capacity for happiness and deny it? Why bring us into the world and then visit us with every kind of horror? Here Gahan, as Nick Cave would later put it, calls upon the author to explain. Edward Scissorhands: Did I solicit thee, from darkness to promote me?' # Paradise Lost In Edward Scissorhands, Edward never gets to confront his inventor, never has the chance to accuse him of leaving him stranded in the world, half-finished, with a burning desire for love and hands that prevent him from touching anybody. But Frankenstein's monster does. What's more, his recent reading material, John Milton's Paradise Lost, gives him a powerful language with which to present his accusations: I ought to be thy Adam, but I am rather the fallen angel, whom thou drivest from joy for no misdeed. Everywhere I see bliss, from which I alone am irrevocably excluded. When God created Adam, the monster asserts, he provided for him, gave him guidance, direction and a companion. But Frankenstein sees his own man of clay not as a son, but as a mistake made at work — a botched job he threw in the bin and hoped no more would be said of it. Now the botched job has learned to talk back, and is demanding justice: You, my creator, detest and spurn me, thy creature, to whom thou art bound by ties only dissoluble by the annihilation of one of us. You purpose to kill me. How dare you thus sport with life? It's all right for you, he seems to be saying to Frankenstein — you have a creator who loves and accepts you. The only thing is, we're not sure the monster is right. By comparing himself to Adam and Frankenstein to God, the monster is inviting a comparison that made Mary Shelley's readers realise how little difference there was between the two cases. This is the sting in Shelley's tale; Frankenstein invites us not so much to imagine Frankenstein as God, but to imagine God as Frankenstein — creating a man on a whim, and then kicking him out of doors on another, leaving him to fend for himself, like a failed experiment. The quotation from Adam in Paradise Lost on the title page of Frankenstein makes the point even clearer: Did I request thee, Maker, from my clay To mould me man? Did I solicit thee From darkness to promote me? Milton was a towering figure for the romantics. From 1658, the exiled poet spent seven years writing Paradise Lost, a Christian epic that tells the story of the creation of the world, of Lucifer's fall from heaven and of Adam and Eve's sin and expulsion from Paradise. Milton was deeply religious, yet as Karen Armstrong points out in her fascinating book A History of God the most likeable character in Milton's epic is not God or Christ but Satan. 'Satan has many of the qualities of the new men of Europe,' she writes, 'he defies authority, pits himself against the unknown.' In fact, it's Milton's Satan, the rebel angel, who would later become enshrined as a romantic hero — the prototype of Napoleon and Byron. When Frankenstein's monster tells his maker that he feels he has been cast out of heaven, he is, once again, being very romantic. But if Satan is the hero of Milton's epic, where does that leave God? There is, as Armstrong points out, something truly horrible about the God of Paradise Lost. It's not that there's anything radically out of the ordinary in Milton's portrayal of the deity. He displays the traits that have been attributed to him since the Old Testament — omniscience, omnipotence and all the rest of it. But it's precisely these traits that make him so unlovable. Milton's admirable desire to explain his religion to himself forces him to reconcile God's all-powerfulness with the suffering the human race has had to endure, and this pushes his God into some awkward postures. God explains to his son, for instance, that he has given Adam and Eve their own free will and the power to resist the temptations of Satan. But he knows they won't, because he's God and he knows everything in advance. What's the point of allowing them free choice if he already knows they're bound to fall? God's answer to this amounts to his saying that this way, he gets to have his cake and eat it too. He can keep being all-powerful, but Adam and Eve can't blame him for the bad things that happen to them because they have free will. Thanks a lot, God. It gets worse too. Later the archangel Michael is sent by God to reassure Adam that his descendants will find their way to redemption by discovering the true religion, Christianity. Michael treats Adam to a sneak preview of the next few thousand years of exile and suffering, culminating in God's sending his only son down to earth to redeem humankind. At this point, Armstrong writes, 'It occurs to the reader that there must have been an easier and more direct way to redeem mankind. The fact that this torturous plan with its constant failures and false starts, is decreed in advance can only cast grave doubts on the intelligence of its author.' This God, who appears to be either hopelessly incompetent (he can't prevent suffering) or monstrously cruel (he can, but he won't) was a particular problem of the Enlightenment. It was, Armstrong insists, the attempt to rationalise God's existence which had made him so unbearable. # The Disappearing God BY SPREADING THEIR blasphemous rumours on Top of the Pops in 1984, Depeche Mode incurred the wrath of The Sun newspaper and Britain's self-appointed moral guardian, Mary Whitehouse. But if Martin Gore had published his verses in 1750, he would have found himself in far worse trouble — atheists were routinely locked up during the eighteenth century. And yet many Enlightenment thinkers found that they were inexorably drawn to deny the existence of God. By looking to science for the answers religion had formerly provided, the Age of Reason had already relegated God to a less conspicuous role. Newton's scientific view of the universe held that nature and the physical world operated by a kind of clockwork. The machine was vast and complex, but essentially logical, meaning that its secrets would, given time, be discovered and understood. But the very existence of this clockwork implied for Newton that there must have been, or still be a clockmaker — and this is where God fits into the scheme. Gravity may put ye planets into motion but without ye divine power it could never put them into such a circulating motion as they have about ye Sun, and therefore for this as well as other reasons, I am compelled to ascribe ye frame of this systeme to an intelligent Agent. Newton's 'Rolls Royce' universe dominated the Western imagination until long after his death, but for Scottish philosopher David Hume, writing in 1750, it wasn't nearly good enough. Hume objected to Newton's argument for God from design. If the universe is the work of a supremely intelligent overseer, he asked, then how does Newton account for the existence of evil? Does God make mistakes, or does he mean to see us suffer? Hume, as Karen Armstrong notes, chose to leave his refutations of Newton — which implied his atheism without ever stating it — unpublished, but Denis Diderot was not so cautious. The French Philosophe was imprisoned in 1749 for publishing 'A Letter to the Blind for the Use of Those Who See' — the strongest dose of atheism yet administered to his century. The letter presents an argument between a Newtonian called Mr Holmes and Nicholas Saunderson, a blind professor. 'Diderot', writes Armstrong, 'makes Saunderson ask Holmes how the argument from design could be reconciled with such "monsters" and accidents as himself, who demonstrated anything but intelligent and benevolent planning.' They could lock Diderot up, but by this point, the horse had well and truly bolted. The rational enquiries of the Enlightenment philosophers had left humanity with a God who resembled the one in Depeche Mode's song to an extraordinary degree — a deity who was incompetent at best, malicious at worst. Diderot, for one, declared that he could do without such a being, and many more would come to the same conclusion. The romantics inherited this unlovable and useless God, which was a shame, because with an alienating industrial future rising in front of them, and the bitter disappointments of the Revolution still lingering behind, they could really have used a 'loving father' of the kind imagined by Schiller in his 'Ode to Joy'. The great sense of crisis in romantic literature comes to a large extent from a feeling of having been shot by both sides — betrayed by the cult of reason on the one hand, and by a disappearing God on the other. Keats, in 1819, found himself in exactly this position, as his biographer Robert Gittings describes: He did not believe...in the perfectibility of earthly life; indeed, perfect happiness in life, he saw, would make death intolerable... Yet the Christian idea that the common hardships of this world were only a miserable interlude before the blessed state of another struck him as 'a little circumscribed, straightened notion'. Keats admired Voltaire. At a dinner with Wordsworth and the painter Benjamin Robert Haydon, he had raised his glass in the direction of Voltaire's likeness and drunk his good health. For Keats, Voltaire's determination to do away with the 'pious frauds of religion' made him a hero. Some time later, back at Haydon's and standing before the same painting, he placed his hand over his heart, lowered his head and said of Voltaire, 'There is the being I will bow to.' But while he admired Voltaire's intellectual bravery, the thorough-going rationalism of the Philosophes did not square with Keats's feeling for mystery — a quality he believed to be essential to poetry. 'He could not be satisfied with a complete and negative scepticism,' writes Robert Gittings in his biography of the poet. 'Somewhere, he must find a faith.' # The Age of Simple Faith WORDSWORTH, HAVING REJECTED Revolution and Reason in quick succession, had found the faith that saved his life in nature. Likewise, after the crisis documented in 'Blasphemous Rumours', Martin Gore found himself advocating a spell in the country: Come with me into the trees we'll lay on the grass and let the air pass These lyrics from 'Stripped', a song on Depeche Mode's Black Celebration album, might seem like unusual sentiments coming from pioneers of industrial dance music. But the song's arrangement casts a grim cloud of irony over Gore's Rousseauish lyrics. It's an industrial symphony of steam hammers and stamping presses, and the effect is of the lovers being chased down the road by a factory even as they drive off into the country. For the romantic, the degraded present often implies an ideal past — a Garden of Eden to which we might return. Accordingly in nineteenth-century England where the rise of industry was faster and more widespread than anywhere else, the romantic escape tended to take the form of a flight into nature. In the same way, the torturous problem of trying to accommodate God into a scientific universe led many romantics to attempt a flight back in time, to the simple faith that characterised Christianity in the Middle Ages. As the nineteenth century wore on, and the 'dark Satanic mills' continued their steady march across the landscape while God remained missing in action, later offshoots of romanticism would be motivated by an attempt to combine these two ideal pasts — the pre-industrial society and the age of simple faith — which were really one and the same. Nostalgia for the Middle Ages was as Eric Hobsbawm has noted in The Age of Revolution, one of the three most popular cures for romantic displacement in the nineteenth century — nostalgia for the French Revolution and nostalgia for the Noble Savage being the other two. Its first stirrings could be detected by observing the crowds of tourists trekking out to see Walpole's Strawberry Hill in the 1760s, or the even bigger crowds turning up for the stage adaptation of The Castle of Otranto ten years later. The success of Walpole's gothic novel paved the way for later phenomena such as the historical novels of Walter Scott, whose swashbuckling heroes were important precursors of Byron's. Scott's stories, while not actually set in the Middle Ages, were jam-packed with medieval paraphernalia. Meanwhile, the gothic revival in architecture began to gather momentum — and a new sense of purpose. By 1837 it was virtually the national style in Britain, a moment signalled by A W Pugin's design for the Houses of Parliament. Some years later, critic John Ruskin went further, advocating not just the gothic style, but the whole medieval ethos as one worth returning to. In his essay 'The Nature of the Gothic' Ruskin argued that industry and progress had cut human beings off from the wellspring of their creativity — nature. Instead of shoddy goods made by unhappy people in ugly factories, Ruskin posited a return to the days of the guild and the artisan; decorations, tools and buildings made by passionate individuals with love and creativity. Designer and social reformer William Morris began to put Ruskin's ideas into practice when he opened the doors of his firm, Morris, Marshall, Faulkner and Co. in 1861. Today, Morris is best known for one of the world's most famous wallpaper designs — but the wallpaper was in fact just a small part of a far-reaching scheme to improve the world through arts and crafts. For Morris, one of the most damaging effects of the industrial revolution was the standardisation and mass production of the applied arts and crafts. He hoped to reverse the alienating processes of capitalism and industrialisation by recreating, within his own firm, the world of the medieval artisan's guild. In between, he found time for pamphleteering, experiments in communal living, learning to paint, and writing fiction. His House of the Wolfings is both fantasy and polemic — clearly influenced on the one hand by the romances of Walter Scott, while implicitly using an imagined Middle Ages as a stick to beat the nineteenth century with on the other. Unfortunately, Morris's enthusiasm for all things medieval led him to write the entire book in some kind of archaic eighth-century dialect — which made it pretty tough going for the average reader. What aileth thee, O Wood-Sun, and is this a new custom of thy kindred and the folk of God-home that their brides array themselves like thralls new-taken, and as women who have lost their kindred and are outcast? Who then hath won the Burg of the Anses, and clomb the rampart of God-home? Morris was also closely associated with a group of painters and poets called the Pre-Raphaelite Brotherhood. The Brotherhood began in 1848, with just two brothers answering the rollcall. Painter and poet Dante Gabriel Rossetti tracked down William Holman Hunt after seeing the latter's The Eve of St Agnes, and the two bonded over Keats — upon whose poem the painting was based. They called themselves Pre-Raphaelites because they believed that after Raphael, European art had begun a slide into irrelevance — empty displays of technical bravado and pointless imitations of Greek or Renaissance art — from which they hoped to rescue it. 'Study nature' was their motto, and in early successes like Holman Hunt's Our English Coasts, you could see that they had. But the best known of the Pre-Raphaelites, Rossetti, seemed to abandon this principle quite early on — and many younger artists followed his example. His painting is beautifully observed, but the degraded present disappears, and is replaced by an idealised fourteenth century. Rossetti's The Annunciation (1849–50) recreates the atmosphere of early Renaissance art, the breakthrough paintings of Fra Angelico and Gozzoli, to the letter — from the carefully planned perspective to the golden disc hovering over the Virgin's head. Rossetti's paintings look back at a time when the divinely inspired artist provided people with objects and images they could believe in. The handicrafts Morris designed and sold through his firm were undoubtedly beautiful and true in many ways to the spirit of the medieval artisan's guilds he so admired. But having insisted on the handmade over the mass produced, Morris was forced to sell his goods at many times the price of the competition — placing his wallpaper and ceramics completely out of the price range of the ordinary folk whose lives he hoped to improve. There was, it seemed, no going back. 'Dreamer of dreams', is how Morris later described himself, 'born out of my due time'. This feeling of having tried to turn back the clock — and having failed — dogged both William Morris and the Pre-Raphaelites to the end of their days. To the Pre-Raphaelite painter Edward Burne-Jones, the chances that art and artists might be able to stop the industrial rot and improve the world looked increasingly slim: 'Rossetti could not set it right and Morris could not set it right — and who the devil am I?...' The age of simple faith was long gone by 1850 — eroded by the achievements of the Enlightenment, and finally blown to smithereens by the shock of the Revolution — and all the golden haloes in the world could not bring it back. Robert Smith: The pious frauds of religion. # Faith THE COVER of the Cure's 1981 album Faith looks like nothing at first — a grey, abstract blur to match the indistinct gloom of the album itself. But just as repeated listening to Faith will cause its clouds of sound to coalesce into songs, so too does the cover eventually resolve itself into an image. It's an out-of focus photograph of Bolton Abbey, a Gothic church in North Yorkshire. Bolton was built in 1151, and has been alternately falling apart and being restored ever since. August Pugin, the Gothic Revival architect, did some work on it during the Victorian era. Earlier in the nineteenth century, the ruined abbey had inspired a poem of Wordsworth's, 'The White Doe of Rylstone'. The story is of a woman named Emily whose brothers and parents were killed in a revolt against Queen Elizabeth. Emily finds solace from her despair — and ultimately faith — in the visits of a white doe. The doe was a childhood pet, raised by her in the days when she was still surrounded by family and the world seemed full of hope and promise. Here, as in 'The Prelude' or 'Tintern Abbey', childhood memories, nature, and the passage of time work their magic to restore faith. Wordsworth, as Karen Armstrong has noted in her History of God, was a religious man, who often spoke of God when he was discussing ethics or morality. But the word never appears in his verse. In 'The White Doe of Rylstone', as in many other poems, Wordsworth evokes the spirit that he felt to be moving through all things, the presence 'whose dwelling is the light of setting suns'. But he never calls this 'God'. Wordsworth's Enlightenment side made him distrust organised religion, his romantic side lead him to resist categorising the ineffable. In 'Tintern Abbey,' he simply calls it a 'something'. It's this same 'something' that Robert Smith searches for in the final moments of Faith. But it seems to have got away from him. The whole album feels elusive and faraway, as though it's being heard from a great distance no matter how close you put your ear to the speaker. As the singer retreats into his loneliness, the world goes out of focus. In early Buzzcocks-inspired songs like 'Jumping Someone Else's Train', Smith's lyrics described characters — now, they're just 'other voices'. It becomes hard for him to make distinctions. 'All cats are grey,' he sings. The word that emerges most distinctly from the fog of Faith is 'nothing' — perhaps because it's repeated so many times. The same 'nothing' Smith found when he went running into the forest on 'Seventeen Seconds' has become the whole world of Faith. And yet, in the album's final song — the title track — the singer dares to hope for something more. Faith is the epitome of The Cure's early '80s sound — guitars like distant church bells, a stripped-bare drum kit ticking away in an empty hall — a song that sounds like a memory of a song. Smith's lyrics rise fitfully out of the gloom, describing the singer's final descent into solitude. Outside, it's 1982, and New Wave pop music is taking over the world. 'The party just gets better and better!' Smith observes — he spits out that last 'better' like Johnny Rotten singing 'pretty'. He's turned his back on all of it. 'I went away alone,' he says, 'with nothing left but faith.' Smith, like some kind of post-punk monk (or, as he puts it, 'an unknown saint') has renounced this world of temptation and illusion — now he has only his belief to sustain him. But belief in what? At the beginning of the album, he'd rejected what Keats called the 'pious frauds' of religion, standing up in the middle of a church service, he'd screamed 'a wordless scream at ancient power'. Like Wordsworth and the Romantics, he's renounced dogma and tradition in favour of a direct experience of the ineffable 'something'. This is the kind of faith people knew in the Middle Ages, the kind that drove the hands of the stonemasons who built Bolton Abbey. This is what Smith is searching for in the final seconds of 'Faith'. Even after the rest of the band have disappeared, and the drum machine has run out of batteries, Smith is still wailing in the empty church hall, 'there's nothing left but faith!' Dave Gahan: Personal Jesus. # World in My Eyes BY THE MID-NINETEENTH century, others were more willing than the Pre-Raphaelites to look the modern world's spiritual crisis in the eye. In 'Dover Beach' Victorian poet Matthew Arnold admitted that the age of simple faith was long gone: The sea of faith Was once, too, at the full, and round earth's shore Lay like the folds of a bright girdle furl'd. But now I only hear Its melancholy, long, withdrawing roar... Arnold, wrote William J Long in his History of English and American Literature, 'reflected the doubt or despair of those whose faith had been shaken by the alleged discoveries of science'. In 'Dover Beach' we see clearly the despair lurking behind the uncannily still fantasies of the Pre-Raphaelites. Arnold wrote the poem on his honeymoon, while staring out at the French coast from Dover. This view gave him the poem's central metaphor — the ocean, which Arnold likens to the faith in God that once seemed so boundless. Now, he writes in 'Dover Beach', this faith has drained away from the world like water through a sieve. There is no sign, even, of the solace Wordsworth found in nature or Keats in the imagination. The world is used up, containing 'neither joy, nor love'. Love, where it does exist in 'Dover Beach' is something that takes place outside of the world, in spite of it, almost. It's the love of two people who have turned their backs on the world, who find themselves unable to place their faith in anything but themselves — and each other. Ah, love, let us be true To one another! for the world which seems To lie before us like a land of dreams, So various, so beautiful, so new, Hath really neither joy, nor love, nor light, Nor certitude, nor peace, nor help for pain; And we are here as on a darkling plain Swept with confused alarms of struggle and flight, Where ignorant armies clash by night. This image of lovers united against a hostile and dangerous world would turn up more and more in Martin Gore's songs after Some Great Reward. In fact, the cover of that album already shows a couple who could have stepped out of Arnold's 'Dover Beach'. The scene is of a grim industrial landscape. It's night, but there are no stars — just the glare of halogen lights throwing the forbidding geometry of the factory into sharp relief. In the midst of this desolate scene, dwarfed by the inhuman scale of their surroundings, is a pair of newly-weds — just married, in fact — he in his tuxedo, she in a bridal gown. It would be wrong to describe them as happy, or even hopeful. But given that their love for each other is all they have, what else can they do but stare into each other's eyes and try to will this nightmare world out of existence? The young couple has been left stranded in the world of blasphemous rumours — a world in which God is either cruel or incompetent, and has subsequently been relieved of his duties. This desolate landscape was the one Martin Gore began exploring on Depeche Mode's next album Black Celebration. In 'Nothing' Gore, like Robert Smith, waits hopefully for a word from God and hears only silence. Now his faith is long gone, and with it any meaning life might have held. He resigns himself to his fate. He will 'learn to expect — nothing'. In the space of two albums, Gore has made the whole journey from the optimistic enquiries of the Enlightenment to the despair of the mid-nineteenth century. Gore has moved into the world of 'Dover Beach' — or the even bleaker one of James Thomson's The City of Dreadful Night, in which the poem's narrator is shown around a desolate city by a mysterious guide. The traveller is baffled by what he sees. Here, humanity seems to have reached the end of its tether: the ties that bind people together — love, family, brotherhood — have all finally snapped. Nobody here believes in anything, nothing has any meaning, no-one seems to have any reason to go on living. And yet life goes on, but why? 'When faith and hope and love are dead indeed', he asks his companion, 'can life still live? By what doth it proceed?' ... He answered coldly, Take a watch, erase the signs and figures of the circling hours detach the hands, remove the dial face The works proceed until run down; although Bereft of purpose, void of use, still go... This is Newton's universe turned ugly — a machine with no-one at the controls. Depeche Mode's Dave Gahan echoes the solitary traveller's confusion in Black Celebration when he wonders — a little enviously — how his friends can carry on living in the face of all this cosmic meaninglessness. Your optimistic eyes Seem like paradise To someone Like me Two albums later, Depeche Mode would do away with this simile. In the world of 1990's Violator, the idea of paradise in another's eyes is not just an idle thought, it's the tenet of a new religion. Violator was a landmark album for the band — in which Gahan and Gore seemed to have discovered an antidote, of sorts, to the despair of Black Celebration. The mood of Violator was religious — but God was nowhere to be found. His replacement was a lover. In the album's first song, the scenario of Black Celebration is reversed — the singer tells his beloved that there's no need to give another thought to the meaningless world outside — he's found a better one. 'Let me show you the world in my eyes,' he croons. The singer has already made this leap of faith himself — he's turned his back on worldly temptation in favour of a new religion based on...worldly temptation! In 'Blue Dress' he insists that the meaning of life is nothing more than the feeling he gets looking at a beautiful girl in a beautiful dress. Here, he seems to say, is the faith that will sustain him, and on the album's first single, Gahan takes this idea to its logical conclusion. If a woman could be God for him, then he can be God for you: 'your own personal Jesus'. In the song's towering chorus, Gahan preaches his new religion to the waiting world. God has deserted you, he says, but I'm right here. 'Reach out and touch faith.' Richard Wagner: So might we die together... # We Can Be Heroes Nature, Medievalism, Satanism; of all the possible escape routes from society at the romantic's disposal, none have quite the pulling power of ideal love. Being a solitary, inspired individual acting in defiance of society's laws is heroic, but it also gets lonely — which is why the romantic goes looking for a soul mate, the perfect, untarnished love of two people united in pure feeling, who live in a zone untouched by the world of dull care. It never really works out, but this is hardly the lovers' fault. The world, with its painful compromises, social conventions and moral laws keeps coming between them. Ideal love has a hard time standing up to the onslaught of reality, and eventually becomes impossible to maintain. So love becomes a recipe for tragedy — the now-familiar stand-off between the romantic individual and society is rewritten for two. The odds, sadly, are not much better than before: Though nothing Will drive them away We can be heroes Just for one day. The lovers in David Bowie's 'Heroes' — kissing by the Berlin Wall as the bullets fly over their heads — are doomed; and they know it. The song, as Bowie admitted after it was released in 1977, is about 'facing reality and standing up to it.' But the lyric is full of wrenching sadness, because the singer knows that in this contest between two people and reality itself, they don't stand a chance. Bowie's extraordinary vocal grows by stages from a croon to a scream as the song moves towards its end — bearing witness to his character's slow realisation that, pure though the lovers' love may be, a stone wall is a stone wall. And yet, paradoxically, even as he faces the fact that he and his beloved can't win, he insists that they can. 'We can beat them,' he sings, 'for ever and ever!'The lovers are heroes — but not the kind to save somebody from a burning building or lead a nation to victory. They're martyrs to love; two people who have chosen to preserve their perfect feeling by, as Werther would say, quitting their prison. Their love is too big, and too bold for the world, they must perish. As with 'Rock and Roll Suicide', the sentiment of Heroes — the very idea that passion cannot and should not be contained by the limits of life — would have been baffling to an eighteenth century audience. But a concert-goer of 1865, hearing 'Heroes', would understand instantly why the lovers in the song are heroic — and might not even be too bothered by the screaming guitars. Thanks to composer Richard Wagner, the nineteenth century music lover had already become quite well acquainted with romantic passion and terrifyingly loud noise. Wagner was born in the last days of the Napoleonic wars, and grew up admiring the great culture heroes of German romanticism — Goethe, Hoffmann, and above all, Beethoven, whose Symphony No. 9 he arranged for piano at the age of seventeen. Twelve years later his opera Rienzi became a huge hit. More successes — artistic if not always commercial — followed, with The Flying Dutchman and Tannhäuser. Then, in 1848, Wagner embarked on the greatest project of his life — the three-part saga, Der Ring des Nibelungen. In the summer of 1857 Wagner, short of funds and mentally exhausted by his monumental and as-yet incomplete trilogy, decided to try something different. This new opera would be simpler to stage, cheaper to produce, and much shorter. Compared to the mind-boggling scale of Nibelungen, Tristan und Isolde would eventually satisfy all these criteria. But if Wagner ever thought it was going to be easy to write, he was in for a shock. Tristan was to be an adaptation of the ancient folk tale of the same name — a Celtic love story with a tragic end, kept alive as a metrical romance in medieval Europe. Versions of the Tristan story turned up in France, Italy, England and Germany. And it was a German version by Gottfried von Strassburg written in 1200 that Wagner used for his libretto. Like all medieval ballads, Tristan had acquired a lot of embellishments over the centuries — things that would have been entertaining to an audience of the thirteenth century, but which didn't translate so well in the nineteenth. In adapting Tristan for the stage, Wagner set about stripping away a lot of this incidental action, to reveal the story's core. As he did so, his Tristan and Isolde began to take on a life of its own. 'Child!' he wrote to the poet Mathilde Wesendonk. 'This Tristan is turning into something terrible! This final Act!!!! — I fear the opera will be banned.' The first act of Tristan is musically revolutionary, but dramatically nothing too out of the ordinary. The action begins on a ship. Tristan, a Cornish knight, is ferrying Isolde to Cornwall and his King Mark, whom Isolde has promised to marry. Isolde is secretly in love with Tristan. She would, she says, rather die than marry 'Cornwall's weary King' — but tragic heroines say things like that. Later in the first act, Tristan snubs her, and the furious Isolde decides that he must die, and that she will die with him. They both drink what they think is poison — but is in fact a love potion. This love potion, as the British academic Michael Tanner points out in his study of Wagner, is not so important as people imagine. 'So far as its long term effects are concerned, they might as well have been drinking water — the potion enables them to release their previously hidden feelings for one another instantly, but they do that only because they believe death is imminent.' The lovers embrace, and become completely absorbed in their love for one another. By this point, the ship is landing, and King Mark's royal train is in sight. But Tristan and Isolde are oblivious. In the second act, the lovers meet in secret while King Mark is out hunting. They begin a duet with the words 'O sink hernieder, nacht der lieber' ('O sink upon us, night of love'). Now, the only thing real in the world of Tristan and Isolde is the dream of Tristan and Isolde — everything else, including King Mark who we know must come back from his hunt at any moment — is an illusion, albeit a persistent one. When Mark returns and confronts Tristan over his betrayal, Tristan dismisses the king and his claims as nothing more than 'Phantoms of the day! Morning dreams!' He has determined to leave this sham world, and asks Isolde to accompany him to the 'wonder-realm of night'. To the king, this is all complete nonsense, but the lovers have long ago replaced sense with sensibility. Feeling, to Tristan and Isolde, is sacred — it's the only law they will accept. And since no-one else can understand their feeling, they retreat more and more from the world and go deeper into their love for one another. People say they're crazy, what do they know? Here, the loneliness that comes of solipsism is in theory redeemed by romantic love. But in a sense, Tristan and Isolde are lonelier than ever. The climax of Tristan and Isolde's love-duet finds them singing, 'So might we die together, eternally one without end, without awakening, without fearing, nameless in love's embrace, giving ourselves wholly, to live only for love!' And this is what they set out to do. Tristan fights a battle with Sir Melot, who had betrayed the lovers to King Mark. He allows himself to be beaten, is fatally wounded, and is carried off to Brittany where we find him, slowly expiring, as the curtain goes up on the third act — the one Wagner worried would be banned, or would drive people insane. Tristan's faithful servant Kurwenal has sent for Isolde to heal his master's wound. The dying Tristan hallucinates Isolde's arrival, and this, as Tanner writes, is his happiest moment — 'essentially, he has found the world he wants to be in'. Tristan dies as Isolde arrives. She, devastated, expires over his body, singing her famous 'Liebestod'. The terms of the agreement they reached in the second act are fulfilled — the world could offer the lovers nothing, they will be united in death. Isolde leaves the world of day behind for ever, and joins Tristan as she sinks into 'unconscious, highest bliss!' Throughout Tristan und Isolde, Wagner uses the other characters in his drama to critique the lovers' behaviour. Mark and Kurwenal's dialogue gives voice to the incomprehension 'normal' people must feel in the face of Tristan and Isolde's monumental self-absorption. While Tristan lies unconscious in the third act, Kurwenal sings, 'Oh deception of love! Oh passion's force! The most beautiful of the world's illusions!' Kurwenal's world, the world most of us live in, is the opposite of Tristan's — here romantic love is an illusion, a phantom which is bound to evaporate. As Tanner points out, many critics have taken Kurwenal's stance to be the true voice of Tristan und Isolde. They see the opera, in other words, as a critique of romanticism — as though Wagner knows better than his doomed lovers, and is subtly exposing their self-deception. In his book, Wagner, Tanner insists that nothing could be further from the truth. 'The trouble with accounts of Tristan which view it as in any way a critique or expose of romantic love is that that is not in the least how it feels.' The promise Bowie's King makes to his Queen is the same impossible vow Wagner's lovers make in the second act of Tristan. Their struggle against the world will end in their deaths, but in death their love will live 'forever and ever'. As Tanner has said of Tristan, it's tempting to interpret this as ironic, because if it's not, then the song is a sincere denial of civilisation itself. 'Heroes' indicts the whole world for letting its lovers down, and then dismisses that world as a sham, insisting that the lovers' dream world — Tristan's 'wonder-realm of night' — is their true home, where they can reign as King and Queen forever. It's still possible that all this is meant to be ironic, but as Tanner says of Tristan, that's not how it feels — the quotation marks around the title appear nowhere in the song. Bowie sings it as though his heart is about to burst. # Wagnerian OF ALL THE romantic composers, Wagner is the one most deserving of a place in the history of rock and roll. Others have had their moment in the sun — Beethoven was briefly in vogue in the late '60s thanks to Wendy Carlos's A Clockwork Orange soundtrack, Strauss had one of his tone poems pressed into service as Elvis's walk-on music in the '70s, and Rivers Cuomo, as we'll see later, has always had a soft spot for Puccini. But Schumann? Mahler? Berlioz? None are likely to find a place in the index of even the most exhaustive rock history, let alone have an entire genre of rock music named after them. The phrase 'Wagnerian rock' is generally credited to songwriter Jim Steinman. Steinman adapted Wagner's Das Rheingold into a stage musical in 1974. Three years later, he had teamed up with ex-Rocky Horror Picture Show star Meat Loaf to record Bat Out Of Hell, a hysterically over-the-top ode to American romance that turned Meat Loaf into an unlikely star, and paved the way for future Steinman mini-operas like Bonnie Tyler's 'Total Eclipse of the Heart'. But by the time Bat Out of Hell appeared, the term 'Wagnerian' already had some currency in the world of heavy metal. When The Stalk Forest Group changed their name to Blue Oyster Cult in 1971, rock critic Richard Meltzer suggested a way to lend their new name a bit of typographical panache. 'I said, "How about an umlaut?"' Meltzer later recalled, 'Metal had a Wagnerian aspect anyway'. The heavy metal umlaut — or 'rock dots' as they came to be called — went on to have a life of their own, subsequently adopted by scores of bands from Motörhead to Mötley Crüe for their vague associations of tragedy, paganism, and above all, loudness. By 'Wagnerian' Meltzer most likely meant 'loud and intense' — which is absolutely fair. 'I like Wagner's music better than anybody's,' says Lady Henry in Oscar Wilde's The Picture of Dorian Gray. 'It is so loud that one can talk the whole time without other people hearing what one says.' Wagner's music was frequently dismissed as 'noise' by nineteenth-century critics. A cartoon published in 1869 showing the composer hammering a crotchet into a concertgoer's ear with a mallet sums up a fairly widespread feeling about him at the time. But Wagner intentionally strove for intensity in his music, and just like the metal bands he unknowingly inspired, if the technology of the day wasn't up to producing what he heard in his head, Wagner simply went 'one louder'. He had a specially designed Festival Theatre built in Bayreuth to accommodate his musical vision — the first stone was laid in 1872, and it would be another four years before the theatre saw its first performance. Meanwhile, The New York Times reported excitedly on Bayreuth's radical new design. Wagner had the orchestra sunk below the floor so that the music would rise up before the audience as if from nowhere. The paper informed its readers that future theatregoers would watch Wagner's dramas 'through an invisible wall of sound'. Almost a century later, pop's most famous Wagnerian, Phil Spector, revived The Times' phrase to describe his new hit making formula, first heard on The Crystals' 'He's a Rebel' in 1962. Spector's Wall of Sound was created by a unique combination of multiple instruments, strings, kettledrums and big reverb-soaked spaces — all squeezed into a mono mix. The result was the most overwhelming noise that had ever been heard on the radio, a deep cataclysmic rumble topped with a sweet sprinkling of bells and shakers and a gut-busting vocal. Spector produced hits for The Crystals, Darlene Love and Bob B Sox, before reaching an extraordinary peak with The Ronettes' 'Be My Baby' in 1963. Spector called these songs 'little symphonies for kids', and spoke elsewhere of taking a 'Wagnerian approach' to rock and roll. While the sound was radically new, Spector's fusion of teen-pop and romantic agony was by no means just a gimmick. The eccentric producer had hit on a fundamental connection between the high school kids who bought his records and the operas of Richard Wagner. Spector had realised that in high school, every time a boy looks at you, let alone asks you on a date or dumps you just before the dance, it feels like Tristan und Isolde. So Spector decided to treat these teen tragedies with the dignity their protagonists instinctively felt they deserved. He would tell the teens of America that their emotions were every bit as important as they imagined. The Wall of Sound is not just a sound — it's a sound married to an idea. Spector quit the business (temporarily, it later turned out) after DJs refused to play his masterpiece, 'River Deep–Mountain High', in 1966. He would have taken his Wall of Sound with him if he could, but by the end of the '60s it was no longer exclusively his. A new generation of artists and producers who'd grown up with Spector's songs ringing in their ears was taking his sound in new directions, and using it to tell new stories. By the mid '70s the kids who'd bought 'Be My Baby' were in their twenties. Their lives had become more complicated, their responsibilities were greater, but they all still retained, somewhere in their hearts, the vision of ideal romantic love presented in that song and the sound that carried it — a vision that came to seem all the more tragic as reality closed them in its net. # Born to Run IN 1974 'BE My Baby' still sounded like the future to Bruce Springsteen. The singer was looking for a way to refine the structure of his music while increasing its emotional impact, and Spector's 'little symphonies for kids' seemed to point the way. Not that Springsteen's new songs were 'for kids' exactly. Music writer Greil Marcus once wrote that Springsteen in the '70s took the carefree, drag-racing, soda-jerking teens of the '50s and early '60s and 'dumped fifteen years on them'. Those kids who busted out of their parents' house and hit the road in search of fun and love; what did they find? And where are they now? This was the territory explored by a new song Springsteen had written early in 1974. He asked producer John Landau if he thought a Wall of Sound-style arrangement would work for 'Born to Run'. 'Born to Run' was one of the shortest songs Springsteen had written up to that point, but recording it took almost six months in the studio — twice as long as it had taken to record his last album. The time spent paid off — 'Born to Run', then as now, explodes out of the radio. As with Spector's '60s' productions, the song's deep spaces and tiny details add up to create the effect of a gigantic symphony compressed into the grooves of a rock and roll 45. But the song doesn't just sound like 'Be My Baby', it works like 'Be My Baby'. The music is impossibly grand, but the song is not about great men doing great deeds, it's about young Americans whose emotions will not be contained by the limits of their small-town lives. It's 'Summertime Blues' meets Tristan und Isolde. It's a tragic romance — and something more. Romanticism replaced the Enlightenment's insistence on knowledge with a philosophy based on action. Goethe said, 'I am not here to know, but to do.' In 'Born to Run', Springsteen's lonely rider agrees. Sensible, rational people tell him to knuckle down and get a good job — but this is secondhand philosophy, which is of no use to a young romantic: ... I gotta know how it feels I wanna know if your love is wild Girl I wanna know if love is real. These lines are followed by a terrifying hallucination of America by night, lost souls drift through the mist, cars scream down the highway, the amusement park looms over the lovers like a mechanical monster. The singer and his girlfriend have to get out of this hostile world — he serenades her with the surprising lyric: I wanna die with you out on the street tonight In an everlasting kiss. This world of illusion is not for the singer and his Wendy; he proposes Isolde's 'unconscious, highest bliss' as an escape route. But he also suggests, just before the song's last chorus, that death is not the end, or that they might be headed somewhere after all: Someday girl I don't know when We're gonna get to the place where we really wanted to go And we'll walk in the sun. 'Born to Run' comes on as a tragedy — only to later reveal itself as something else — a religious drama based on a faith that doesn't exist yet. This, as Michael Tanner insists, is what Tristan und Isolde is really about. Wagner's lovers are determined to live with the consequences of their love, to see it to its conclusion. Their death is not a defeat, because they believe in something beyond the physical world. This is clearly a religious idea — but whereas Bach's St Matthew Passion deals with a religion everybody knows, Wagner was creating a brand-new one — a religion of romantic love. 'Born to Run' and 'Heroes' both preach this religion. The singer and his soul mate place their faith in each other, knowing that this will offer them salvation in death. This is how Springsteen and Wendy can die in each others arms and 'get to the place'; Johnny and Tina can 'make it' whether they make it or not; and Bowie and his queen can be beaten and still 'beat them forever'. This 'Passion of Passion' as Tanner calls it shares a few things in common with Christianity: the players in the Passion are forced to make enormous sacrifices and have their faith tested every step of the way. The key difference, according to Tanner, is that the Christian God makes you wait, whereas romanticism demands immediate action, with paradise as the direct result. In 'Thunder Road', the first song on the Born To Run album, Springsteen is standing outside his girl's house, holding out his hand and offering her a ride. She's been praying for a saviour, he tells her to get up off her knees and get in the car. Well now I'm no hero that's understood, All the redemption I can offer girl is beneath this dirty hood... Springsteen is not just talking dirty when he suggests that salvation is right there under the hood of his car, and that heaven's in the back seat. He's saying: you don't have to wait, and you don't need God. He's going out tonight, he tells her, 'to case the promised land'. He's more or less made up a new religion on the spot — and armed with this new faith, he assures her, the threat of death will become no threat at all. Their faith in each other, he insists, will allow them to transcend the material world. Freddie Mercury: Compagnon de miseres. # Pressure FOR DAVID BOWIE, 'Heroes' marked the beginning of a new, bravura-style of singing which record producer Tony Visconti dubbed 'The Bowie Histrionics'. Though there wasn't much room for it on his next album, the surprisingly low-key Lodger, the new style did make a few appearances on 1980's Scary Monsters — especially on the first track, 'It's No Game'. But here, Bowie sounded more demented than heroic, as though the world he thought he could face on 'Heroes' had beaten him down again. The Bowie Histrionics proper didn't really come out of its case again until 1981, and the occasion was not a David Bowie recording session per se, but a visit to fellow rock royalty that unexpectedly turned into a collaboration. In July of that year, the members of Queen were recording at Mountain Studios in Montreux. Bowie dropped by to say hello, and finding themselves with some time to spare, the five musicians started messing around with an idea. Before he knew it, Bowie found they were writing a song together. The music started to cook, the atmosphere in the studio grew heated, egos clashed. 'It was, er...peculiar,' said Bowie later. 'Peculiar' is one way to describe the result of this unlikely collaboration. At first listen, 'Under Pressure' sounds like what it is — the sound of the two greatest hams in rock trying to out-ham each other. But it's not all empty posturing — the lyric is a little vague, but that's just because the scope of the song is so enormous that it's hard for the singers to stay focused. 'Under Pressure' is about all the trouble in the world, and what we, as mere human beings, can hope to do about it. It struck a chord — the single went to number 1 in the UK at the end of 1981. It also topped the charts in Argentina — and stayed there for the entire duration of the Falklands War. This had the leader of Argentina's military junta worried — he attacked 'Under Pressure' as a piece of British propaganda. But while it's full of rage and hope, 'Under Pressure' is not a protest song — or if it is, it's more in the vein of My Chemical Romance's 'Welcome to the Black Parade' than Bright Eyes' 'When the President Talks to God'. The singers' adversary is not a demagogue or a dictator or a corrupt bureaucracy — it's the world itself. The world, Bowie and Mercury tell us: Breaks a building down Splits a family in two Puts people on streets. As the song goes on, the pressure builds and builds. The guitars brood like thunderclouds as Bowie, in his best Hammer Horror voice, sings about facing 'the terror of knowing what this world is about', as though he's finally come to understand the mysterious source of all this global chaos. Then the pressure drops, the song quietens down, and the singers ask themselves how they can live with the horror. They can't turn away and pretend it's not there, and they can't go on insisting that this is the best of all possible worlds when so many people are miserable. Then the music starts to build again, Freddie Mercury makes a vocal noise that approximates the sun breaking through clouds, and a solution comes rocketing out of the heavens — compassionate love! 'Love! Love! Love!', cries Bowie, heralding our salvation. When people understand that they need to change their way of life and start caring about one another, we will finally experience some relief from this terrible pressure. Freddie goes scatting off into the distance and the band leave the white-boy funk riff that started this whole thing lying on the floor — where Vanilla Ice would find it ten years later. Bowie and Queen didn't stick around long enough to explain how this doctrine of compassionate love would work out in practice. But the German philosopher Arthur Schopenhauer had some ideas, which he set out in his essay, 'On the Suffering of the World', published in 1851. He was sixty-one years old, and had spent at least forty of these living with 'the terror of knowing what this world is about'. He did not believe things were about to get any better. If you imagine, in so far as it is approximately possible, the sum total of distress, pain and suffering of every kind which the sun shines upon in its course, you will have to admit it would have been much better if the sun had been able to call up the phenomenon of life as little on the earth as on the moon... Life, Schopenhauer insisted, is so bad that it can only be a mistake, and given that this is the case, and that we are all in the same unhappy boat, we owe it to one another to show a little kindness. Instead of going about calling one another 'sir' or 'monsieur' (or 'dude'), we ought to address strangers as 'fellow sufferer' or 'compagnon de miseres'. Schopenhauer: The terror of knowing what this world is about. # Schopenhauer SCHOPENHAUER HADN'T LIVED very long in 1805, but he'd already seen enough to convince him that Newton's clockwork universe proceeding according to some grand design was a crock. He had a clerical job in a commercial house in Hamburg, and a few weeks of this provided all the proof he needed that life was not an elegant machine, but a constant lurch between pain and boredom. His father, whom he loved, had been found dead in a canal earlier that year, having taken his own life. His relationship with his mother was uneasy at best, bitterly competitive at worst. He'd taken a tour of the Continent, but with the Revolutionary wars still underway, had seen nothing but cruelty and unhappiness wherever he went. He hated his job, he hated his life. Schopenhauer's Hamburg days formed his mature philosophy, a thorough-going pessimism which — as R J Hollingdale has observed in his introduction to Schopenhauer's essays — remained virtually unchanged until his death fifty-five years later. He acquired more knowledge, but nothing altered his basic feeling about our existence. 'Life,' he wrote later, 'is a disagreeable thing — and I have determined to spend it in reflecting on it.' This is exactly what he did. Back in 1776, the young Goethe, sitting on top of Strasbourg Cathedral, had a vision of the universe as 'convulsed with desires knotted like snakes, from which it tries to escape only to entangle itself again'. This bleak view of life is perfectly understandable as a natural outcome of Goethe's youthful angst, but, as Barker Fairley points out, it would be 'hard to sustain as a piece of philosophy'. Amazingly, Schopenhauer would not only sustain it, but perfect it. He started writing his first book in 1814. Four years later it was published — to no reviews and no sales — and this, considering the book's content, was entirely appropriate. In the two volumes of The World as Will and Representation Schopenhauer laid out his vision of life as a constant struggle for which there is no reward. The source of this struggle and the reason why our desires can never be satisfied is for Schopenhauer something very similar to the thing David Bowie and Freddie Mercury sang about in 1981. They call it 'pressure', but Schopenhauer called it 'Will' — the blind, striving, unstoppable force behind all perceptible phenomena. All things in our world, including ourselves, are manifestations of this Will, which means that no matter how hard we try, we can never become masters of our own destiny. Schopenhauer replaced the Philosophes' infinite perfectibility with infinite struggle, the promise of utopia with an insistence that life is hell. German philosophy in the late eighteenth and early nineteenth century was, as the Australian author Robert Spillane has pointed out in his book, An Eye for an I, a form of revenge on the French, who had dominated philosophy as they had all other walks of cultural life in the 1700s. So the Germans decided that if the French were going to be rational, they would be irrational. Immanuel Kant (who died the same year Schopenhauer went to work in his father's office) began to dismantle the apparatus of the Enlightenment by insisting, like his hero Rousseau, that the discoveries of science could have no positive effect on the lives of human beings. Johann Gottlieb Fichte, a little younger than Kant and a lot more hot-headed, proposed a philosophy of action, not knowledge, and replaced the Enlightenment's mechanical universe with a chaotic, meaningless one in which we must freely exercise our will. Schopenhauer, who admired Kant but despised Fichte, inherited the idea of a world without meaning. But where Fichte saw Will as positive — a way of imposing one's forms on reality — for Schopenhauer Will was entirely negative and destructive. The thing behind all appearances, the force that animates all of nature is, in his view, evil. That's why he was a pessimist. Optimists, like Fichte (or Nietzsche or Napoleon), believed they could impose their will on the world, that they could make a difference. But Schopenhauer looked at the actions of the optimists and concluded that their actions were usually harmful in the short term, and didn't amount to a hill of beans in the long term. For Schopenhauer, action was always subject to Will and could therefore lead to no good. So in place of action, he advocated renunciation and compassion: hence his preferred form of address. In 1854 Schopenhauer received a letter from Richard Wagner — enclosed within was a copy of the composer's Tristan poem. Wagner never got a reply — but this, as Robert Gutman observes in his biography of the composer, is not too surprising: Not only must its diction have offended the great stylist, but, when proudly sending off this paean to love, Wagner was obviously unaware that his idol was a confirmed misogynist whose soul had found its mate in a white poodle. Schopenhauer was an enormous influence on Wagner at the time he was writing Tristan — and in many ways, Wagner's hero is much more Schopenhauerish than he is medieval. Tristan has seen behind the veil of illusion that disguises the true nature of the world. He knows life is a sham and cannot provide him with happiness, so he renounces the world and all its willing by allowing himself to die. His beloved soon follows suit, and they achieve redemption in death. No more willing, no more problem. But the truth is, in welding Schopenhauer's philosophy to his tale of tragic love, Wagner took some liberties with the great pessimist's ideas. For starters, suicide for Schopenhauer 'substitutes for a true redemption from this world of misery a merely apparent one'. He didn't think suicide was wrong or immoral — he just didn't think it worked. Secondly, Schopenhauer would have absolutely no time for Wagner's Passion of Passion. Tristan and Isolde reject the world's illusory values, but replace these with something Schopenhauer would see as even less helpful — a transcendent ideal based on sexual love. The music video for Queen and David Bowie's 'Under Pressure' makes the same philosophical blunder — which is a shame, since the song itself is far more faithful to Schopenhauer's ideas than anything Wagner ever wrote. Since Bowie and Mercury barely managed to work together in the studio for the six hours it took to do the song, getting them to commit to a day of shooting together was out of the question. So the director went for the Ed Wood-style solution of assembling a clip out of stock footage, classic films and TV news images. The video shows people rushing through cities and crowding onto trains. We see riots, a burning car, a woman screaming and a building collapsing — the perceptible phenomena of the world as will. (Vampires, for some reason, also make an appearance.) But for the last section, where the singers herald compassionate love as the means to redemption, the video shows a montage of great screen kisses. As the lovers lose themselves in their ecstatic union, the world fixes itself back up again — buildings un-explode on cue. But here, compassion has been replaced with passion. And passion, as far as Schopenhauer was concerned, is the problem — not the solution. # Pinkerton IN 1900, SEVENTEEN years after Wagner's death, Tristan und Isolde was scheduled to be performed for the first time at Milan's famous opera theatre, La Scala. Unfortunately, the great Wagnerian tenor, Giuseppe Borgatti fell ill, so Tristan was postponed, and Giacomo Puccini's La Boheme was substituted in its place. This was fitting, since Puccini was an admirer of Wagner, and strove to match the emotional intensity of Tristan in his own operas. This placed him on one side of a very firm line in turn-of-the-century Italy, where many saw Wagner's influence on opera as a bad one. La Boheme did not go well at La Scala — the cast was in a bad mood to start with, and the 'fatal silence' of the audience didn't make them feel any better. La Boheme limped through nine more performances, after which Puccini tried to put the whole miserable experience behind him. Little did he know there was more misery in store. Shortly before the fiasco at La Scala, Puccini had been struck with a new idea for an opera based on a play he'd seen in London called Madame Butterfly. The subject — an unhappy love affair between an American naval lieutenant named Pinkerton and his Japanese bride, Cio-Cio-San — was well timed to ride the wave of interest in all things Japanese that was sweeping Europe at the time. Puccini had a good feeling about Madama Butterfly. 'I am completely taken with it!' he wrote in March 1901. But Madama Butterfly was plagued with problems, and two years later Puccini was still struggling with it. Then things got worse. On 21 February 1903 the composer was on his way to dinner at a friend's house when his car drove off the road and plunged down an embankment. Puccini was trapped underneath the car with a fractured leg — a nearby doctor patched him up, but it later turned out that the leg had not been properly set and had to be re-broken. He was immobilised for almost three months, and the inertia made his bad mood worse. He despaired of Butterfly ever seeing the light of day, and wondered if anyone would care if he and his unfinished opera just disappeared off the face of the earth. Later that year, he wrote to his colleague Luigi Illica: I am here alone and sad! If you could know my sufferings! I have much need of a friend, and I don't have any, or if there is someone who loves me, he doesn't understand me. I am of a temperament very different from most! Only I understand myself and I grieve; but my sorrow is continuous, it does not give me peace... My life is a sea of sadness and I am stuck in it Somehow the shattered composer managed to finish his opera, and Madama Butterfly premiered on the night of 17 February 1904. It was an even worse disaster than the performance of La Boheme four years earlier. Puccini, leaning on a cane, could hardly hear the music for the laughter, catcalls and boos. The singers could barely hear themselves. The reviews, when they appeared the next morning, were terrible. 'Butterfly,' they wrote, 'the diabetic opera, the result of an automobile accident.' In years to come, Madama Butterfly would come to be seen, along with La Boheme, as one of Puccini's masterpieces. While the version performed at the premiere was marred by structural problems which would later be ironed out, even in this raw state Madama Butterfly was already full of daring formal innovations and sincere personal emotion. But in 1904 it was regarded as a bitter disappointment from the man who had been hailed, only five years earlier, as the successor to Verdi. The critics roasted him for the opera's sentimentality, the smallness of its themes, and for Puccini's failure to grow as an artist. Ninety years later, history repeated itself. Rivers Cuomo, like Puccini, was an admirer of the German romantic composers — though he preferred Mahler to Wagner. He and his band, Weezer, were hailed as the saviours of bubblegum rock in 1994. But Cuomo broke his leg at the height of their success — not in an accident like Puccini, but on purpose, to have it lengthened. He wore a painful brace for almost a year, and entered a period of deep doubt and depression. Under these circumstances, he began writing the songs that would become Weezer's second album — a rock opera called The Black Hole. But the opera idea was scrapped in favour of a concept album loosely based on Puccini's Madama Butterfly. In years to come Pinkerton would come to be seen as Weezer's masterpiece — full of daring formal innovations and sincere personal emotion — but in 1997 it was regarded as a bitter disappointment from the band who had been hailed, only three years earlier, as the successor to The Cars. The critics roasted Cuomo for Pinkerton's sentimentality, and for his failure to grow as an artist. Rivers Cuomo: My life is a sea of sadness. # Butterfly IN Madama Butterfly, Pinkerton travels across the sea and finds himself a beautiful creature. He captures her, and immediately loses interest — he's drinking a toast to finding 'a real American wife' before he's even left Japan. He sails back to America and forgets Cio-Cio-San, and she is left staring out to sea, clinging to the promise he made that he would return 'when the robin makes his nest again'. The opera ends when Cio-Cio-San, having remained faithful to Pinkerton throughout, learns that he has an American wife. Having no hope left in the world, she takes her own life. One of the greatest challenges Puccini had faced in bringing Madama Butterfly to the stage was the percieved lop-sidedness of the story's plot. Puccini's collaborator Illica complained that Pinkerton virtually disappeared for most of the story, 'and his is the drama!' he wrote to Puccini, exasperated.Weezer's second album would, in a curious way, make amends for this. Here, the drama is all Pinkerton — tellingly, his name has replaced hers on the marquee. There is still the unfathomable distance of the Atlantic Ocean between the young American and his Japanese love. But we never see her — while his emotional crises form the entire plot of the album. Butterfly writes adoring letters on cute stationery to her young man, he sits alone in his room and rationalises their relationship out of existence before it's begun — all the while wishing for it to come true. But it's on the album's last song, 'Butterfly', that Cuomo's feeling for Puccini's opera becomes clearest. The singer tells us he keeps going out to catch butterflies — and they keep dying on him. 'Every time I pin down what I think I want it slips away,' he sings. It seems as though there's something wrong with him, as though his wants and his needs are fundamentally opposed. He wants something, he chases it and often catches it. But the object of his desire melts away as he grasps it. Understandably, he's starting to wonder if there's any point doing anything at all. The prevailing mood of Pinkerton is not so much of despair as of resignation. 'Tired of Sex' is a nightmare reversal of the usual rock star brag about all the girls the singer slept with in all the different towns he played. This is not the Don Juan of the Spanish legend or the Don Giovanni of Mozart's opera, but the Don Juan of Byron, who looks back over his many conquests with an air of melancholy detachment. The joke in Byron's Don Juan is that the great seducer is always the seduced — he doesn't really have to try. This is the scenario of 'Tired of Sex', where the singer sounds mostly perplexed about his many one-night stands, and finally, disgusted. He feels he has been taken advantage of but, like Byron, his pose of static detachment prevents him from running away. 'Thursday night I'm naked again,' sings the tired-sounding rock star. What's the point? In 'Why Bother?' the singer reveals that he's just as tired of love as he is of sex. He could ask that girl out, he could pick up the phone, he could lean in for a kiss. But he has already been disappointed so many times that he can't quite bring himself to try, he has already spent enough time chasing happiness to know that it will always elude him. So he gives up. He ignores his urges — which in any case are no match for his inertia — and resigns himself to solitude. Schopenhauer would say that the singer is right — love is not worth the trouble. The problem with love, according to Schopenhauer, is that it's inseparable from the sexual drive, and the sexual drive is part of the great, destructive tide of birth, struggle and death that pushes life along its purposeless course. The sex drive is will manifested in the individual, and since will for Schopenhauer is always negative, allowing oneself to be driven by instinct can only lead to no good. In the world of rock and roll, this amounts to heresy. Rock ballads — with their we gotta and I wanna — place a lot of faith in instinct. 'We gotta get out while we're young', 'we gotta hold on to what we got', 'I wanna rock and roll all night and party every day' — belief in the power of instinct is the legacy of fifty years of rock music. No wonder Cuomo sounds so confused when he sings in 'Butterfly', 'I did what my body told me to'. In chasing his Butterfly, he merely followed the advice of a thousand radio hits — do what you feel, listen to your body, go for it — and the result was tragedy. 'Tired of Sex' and 'Why Bother?' present variations on the same theme. He acted naturally, according to instinct, and all he got was...more unhappiness. Schopenhauer would say that a moment like this is a step on the path to true wisdom. We must accept that the butterfly will always get away, that our goals will always melt into the air when we reach them — like Robert Smith's girl in 'A Forest'. In Pinkerton the singer has come to realise that happiness can only be found in a state of non-willing, which his new attitude of resignation and renunciation ('Why Bother?') will allow. This, as Schopenhauer himself knew, is easier said than done. The so-called 'Buddha of Frankfurt' had a surprising number of affairs and one-night stands, presumably because — like Cuomo — his inertia made him an easy target. Wagner gave Schopenhauer a number of reasons to dismiss his Tristan poem. Even if Schopenhauer had managed to overlook the problems of style and the fact that Wagner wrote grand operas (which Schopenhauer mostly hated), there would still be the hurdle of the composer's insistence on love as a form of salvation from suffering. For Schopenhauer, love is the reason we suffer, and it ensures that we continue to suffer. In this, Schopenhauer is not, like Andy Gill from The Gang of Four, trying to say that love is a superstition which is perpetuated by love songs and grand operas, and can be unlearned. Schopenhauer accepts that our desire for love, like all our other urges and emotions, is real enough. But the fact that we are created with the desire for happiness and dumped into a world of flux and chaos in which that happiness must always remain out of reach is, for him, yet more proof that life is hell, and that it would have been better if the human race had never been born. Human life, Schopenhauer writes in his essay, 'On The Emptiness of Existence', 'is basically a mistake' — and the proof of this lies in the simple fact that no matter how hard you try, you can't get no satisfaction. Mick Jagger: Man is a compound of needs which are hard to satisfy. # Satisfaction ON 9 MAY 1965 Keith Richards woke up in the middle of the night with a riff in his head. He grabbed his acoustic guitar, hit record on a cassette player, and got down sixty seconds of the guitar part he'd caught on the tail of his dream before going back to sleep. 'The next morning I played it back,' Richards later recalled. 'Amongst all the snoring I rediscovered and found the lick and the lyrical hook I'd come up with to accompany it.' The words that had formed in his mind as he'd bashed out his dream riff in that Florida motel room were 'I can't get no satisfaction'. Sitting by the pool the next day — with a cocktail in one hand and a cocktail waitress's phone number in the other — Mick Jagger took the riff and the chorus Richards had written in his sleep, and turned it into a song which expresses better than any other the suffering of the world and the impossibility of happiness. But 'Satisfaction' was not just a song. As the Rolling Stones' former manager Andrew Loog Oldham has pointed out in his memoir, 2 Stoned, 'Satisfaction' contains, in embryo, the entire culture of the late '60s and everything that came after — the profound refusal of the modern world that would characterise most of the important statements of rock and roll from this point on. Keith Richards, according to Oldham, 'changed life as we know it' in his sleep. In his essays, Schopenhauer accounts for the prophetic nature of dreams by explaining that the dreamer, in his unconscious state, is offered a glimpse of the world as will. The baffles and blinds that our conscious mind put around to convince us that life has some structure and meaning are removed, and the dreaming philosopher glimpses the truth of things. Wagner would bestow this ability on the musician as well. He claimed, in his famous essay on Beethoven, that the dreaming artist hears sounds which provide him with staggering insights into the nature of reality. Later, in his conscious state, the composer must act as the mediator between these terrifying cosmic truths and his unsuspecting audience. This is the difficult role the Rolling Stones take on in '(I Can't Get No) Satisfaction', a song which pulls back the curtain on a world without meaning for just as long as we can stand it. The verses describe Mick Jagger trying to impose his considerable will on the world, the chorus informs us of the result: nothing. There's a lot of pushing, a lot of shoving, a lot of trouble; and no satisfaction. 'It was my view of the world,' he later explained, 'my frustration with everything.' In three verses, Jagger dispenses, one by one, with humankind's traditional consolations. Knowledge, material comfort and sex — none, he decides, are worth the trouble. Even fame, which the singer had worked so hard to get up until this point, turns out to be one big hassle. Doin' this and signin' that — who needs it? Here, 'Satisfaction' differs from earlier teenage anthems like 'Summertime Blues' in one very important respect. In the older songs the singer couldn't get what he wanted because he didn't make enough money or because his parents were a drag. But the singer in 'Satisfaction' has already broken free of all those limitations; he's a rock star with money, fame and power — and none of this has made him happy. And since being a rock star is about as good as it gets in 1965, the singer is forced to conclude that all the other goals he might move himself to pursue will turn out to be just as unsatisfying. 'Satisfaction' is not about how the singer can't get his shirts white or make out with the girl — the disgust in his voice tells us he knows, before he's even tried, that neither will make him happy for longer than thirty seconds. The sheer grinding monotony of Keith Richards' riff drives the point home — as soon as we reach our goal, a new one will appear, and the struggle will resume. This kind of thing, according to Schopenhauer, is all the proof we need that human life is pointless. Our achievements leave us feeling unsatisfied precisely because they are, in a cosmic sense, unsatisfactory, our lives have no meaning or significance because the world is in a constant state of flux, and our efforts to impose our will on the world are doomed to fail. The world has its own way of telling us this through the feeling we call boredom. That human life must be some kind of mistake is sufficiently proved by the simple observation that man is a compound of needs which are hard to satisfy; that their satisfaction achieves nothing but a painless condition in which he is given over to boredom; and that boredom is a direct proof that existence itself is valueless, for boredom is nothing more than the sensation of the emptiness of existence. For if life, in the desire for which our essence and existence consists, possessed in itself a positive value and real content, there would be no such thing as boredom: mere existence would fulfil and satisfy us. As things are, we take no pleasure in existence except when we are striving after something — in which case distance and difficulties make our goal look as if it would satisfy us (an illusion which fades when we reach it)... Howard Devoto: Life is hell. # Boredom IGGY POP KNEW all about boredom. On the first two Stooges albums — the self titled debut of 1969 and the towering Fun House released the following year — the singer sounds like a worn out Mick Jagger, whose frustrated attempts to impose his will on the world have reduced him to a near-vegetable state of inertia. Iggy can't even be bothered to advance an argument on the level of 'Satisfaction' for why life sucks. All he can do is blurt out two monosyllabic words. These two words, as it would later turn out, said more about the human condition in the late twentieth century than anyone had managed before or since: 'No fun'. In 'No Fun' Iggy weighs up his options: 'Maybe go out, maybe stay at home, maybe call mom on the telephone'. Who gives a fuck? Looking into the future, he sees no hope for improvement. In '1969' he reflects that he had no fun in 1968 and that he will most likely have no fun in 1969. He gloomily rings in the New Year; 'another year with nothin' to do'.What about girls? What about 'em? For all his snarling and yowling, when Iggy sings about sex on The Stooges' early albums, he's mostly passive. Again, he sounds a bit like Mick Jagger, but he's nothing like the strutting Don Juan of 'Little Red Rooster' or the eager lover of 'I Wanna Be Your Man'. In 'TV Eye' the singer is a victim — it's as though he's frozen on the couch while the girl fixes her predatory stare on him. When he does get horny, his come-on is framed as a submission. Jagger wants to be your man, Iggy wants to be your dog. In the first phase of their existence, The Stooges were a joke in the music industry and virtually invisible to the public. After their demise, they became a legend. By the end of the '70s they were the musical and philosophical godfathers of punk, and Iggy's concerns became the concerns of the entire movement. This put boredom high on punk's agenda from day one, and punk singers came to insist on boredom as the most basic condition of life. The Slits sang about 'A Boring Life', The Clash about being 'Bored with the USA'. Punks were bored with TV, bored with sex ('Here we go,' said a thoroughly ravaged Johnny Rotten, 'another squelch session'), even rock and roll itself had become boring. By 1977 boredom was already enough of a cliché for Howard Devoto to send it up in the Buzzcocks' 'Boredom'. 'Da dum-de-dum,' sang Devoto, thoroughly bored by his own boredom, and even more bored by the boredom of punk — which he ejected himself from in timely fashion only one year later. Devoto formed a new band called Magazine, and broke two of punk's sacred commandments before they'd even played a show. He hired a keyboardist (a jazz keyboardist, to make matters worse), and told the band to play slowly. 'I don't like most of this new wave music,' Devoto told Jon Savage around the time of his great escape, 'I don't like music.' But he kept making it anyway, because he had a feeling that punk had started as a way of 'diagnosing modern forms of unhappiness' — and that there was still work to be done in this area. By saying that everything is boring and that there's nothing to do, punk had opened up a void which Devoto made it his business to explore. His time spent doing nothing (because there's nothing to do) eventually led him to a great insight, which he expressed in Magazine's 'Song from Under the Floorboards' in 1980. Here, the author starts out with a brief self-portrait — 'I am angry I am ill and I'm as ugly as sin' — before going on to explain why: 'I know the meaning of life it doesn't help me a bit.' In 1977 Devoto had told Savage that he was 'trying to find something to get excited about.' By 1980 he had given up. In every case — love, politics, social life — Devoto tallied the reasons for doing anything at all and found them wanting. In this, the meaning of life would prove to be no use to him at all — in fact, it made things worse. Because the meaning of life, as Devoto had realised by this point, is that we are born to suffer. 'Life', he later told cultural commentator Michael Bracewell, 'is hell. I don't think I ever strayed very far from that idea since I was about twenty.' So, in 'Song from Under the Floorboards' Devoto announces that he has given up looking for satisfaction, which he knows will always elude him. He can no longer allow himself to believe in the bright dreams presented in movies, pop songs and grand operas — those chimeras of romantic love and ideal happiness that have tormented humankind through the centuries. Not for him, not anymore. 'Do you remember dreams?' Savage asked Devoto in 1977. 'I take pills to stop myself,' the singer replied. 'In Song from Under the Floorboards,' he explains why. 'I used to make phantoms I could later chase,' sings Devoto. 'And then I just got tired.' The singer renounces the search for satisfaction, and downsizes his expectations to fit the small dark space under the floorboards. # Notes from Underground IT'S 11 APRIL 2006, Morrissey is onstage at the Olympia Theatre in Paris. 'This song,' he says, 'was written before I was born.' He and the band launch into a tough-sounding version of Magazine's 'Song from Under the Floorboards'. He's exaggerating a bit — the song was written a long time ago, but Morrissey is old enough to have seen Howard Devoto playing with the Buzzcocks' original line-up at Manchester's Lesser Free Trade Hall in 1976. That night changed Morrissey's life, and Devoto, with his keen intelligence, literary style and permanent air of dissatisfaction, quickly became a hero for the singer. At the Olympia, 'Song from Under the Floorboards' grinds to a halt, and the audience cheers. 'Do any of you remember that song?' asks Morrissey, a handful of people answer in the affirmative. 'How?' chuckles the singer, mock astonished.Morrissey is making a joke about his age relative to that of his audience — which seems to get younger every year. But there's a sense in which 'Song from Under the Floorboards' really was written before he or any of his audience were born — before Devoto himself, even. Magazine's outsider anthem is a cover — not of a song, but of a book written in 1864 by Russian novelist Fyodor Dostoyevsky. The first paragraph of Notes from Underground reads: I am a sick man... I am an angry man. I am an unattractive man. I think there is something wrong with my liver. Dostoyevsky's book describes, in the first person, the thoughts and exploits of a mean and miserable man — an 'underground man', as he calls himself. He lives in St Petersburg though everyone tells him the climate is bad for his health. He used to have a good job in the civil service, but after he came into some money he decided, not to sell up and move to sunnier climes, but to retire into a small corner of his small house where he now sits, all alone and sick. What's wrong with his liver? He doesn't know — and even if he did, he wouldn't want to get well — the only thing he really enjoys is complaining about how sick he is. Dostoyevsky's narrator is a liar — he makes maudlin confessions to elicit our sympathy, only to admit three paragraphs later that he was only kidding. Did he tell the truth in the first place and then try to cover it up as an afterthought? We'll never really know, so it's hard to get his story straight. But it quickly becomes clear that he was lying about his liver — his illness resides not in his guts, but in his head. He suffers from a serious case of above average intelligence. 'I swear to you that to think too much is a disease, a real, actual disease,' he says. Intelligence is a terrible affliction for the underground man because he would like nothing better than to be as stupid as an insect. He is, as you've probably guessed, a romantic — although Dostoyevsky's is a romanticism stripped of a lot of the romance. The underground man is not so sentimental as to imagine himself as a butterfly — a lowly mosquito with a simple libido would suffice. But while his prose might be less flowery, the wish behind it is the same as the one expressed elsewhere by Emerson, Keats and Conor Oberst. He accepts that he must live with desire — he'd just prefer not to have to think about it. But the simple happiness of the insect is denied him. Even the simple half-happiness of stupid men is more than he can hope for. He is overburdened with intelligence and further handicapped by a good education. Thanks to this deadly combination, there is no hope of his ever being able to 'act naturally'. In fact, most of the time he finds action of any kind virtually impossible. For most people, action is the easiest thing in the world. If you walk up to a normal, healthy, natural man and insult him, he responds in a 'natural' way — he gives you a swift kicking, and the matter is settled. 'I am green with envy of such men,' says the underground man. How do they do it? Because they're stupid, of course. Dostoyevsky explains: I repeat, and repeat emphatically: all spontaneous people, men of action, are active because they are stupid and limited. How is this to be explained? Like this: in consequence of their limitations, they take immediate, but secondary causes for primary ones, and thus they are more quickly and easily convinced that they have found indisputable grounds for their action. Where is the underground man supposed to find the grounds for action? Let's say he did run after the man who insulted him and managed to kick him in the pants. Would it teach him a lesson? No, quite the opposite; and even if it did, so what? There's always going to be violence in the world, always someone big picking on someone small. Surely it's just foolishness to imagine that this act is of some great importance, simply because it has meaning for you, when there's so much aggression in the world that goes completely unchecked. And okay, so maybe every little bit does count. But in the end, what does it matter? We all die eventually, after leading long, painful lives punctuated by occasional moments of joy, and as far as we know, there's no point to any of it. And in any case, in five hundred million years time, this whole planet will crash into the sun, which means there won't even be anyone around to ask all these stupid questions — and won't that be a relief. This is the great legacy of the Age of Reason, the underground man's birthright. He has the extraordinary ability to reason his way out of every natural impulse that comes his way. In the end, he can find no convincing argument for any kind of action at all. So he does nothing. He sits underground and stares enviously up at the men of action as they strive and achieve, turning his loathing and resentment of them over in his mind. Morrissey: Still ill. # How Soon Is Now? ROCK AND ROLL is full of natural men. Elvis Presley — whose genius resided not in his mind, but in his voice and his body — was the first of these. Part of the importance of Elvis as a rock myth is his almost divine naïvety, the way he seemed to act without thinking, to change the world without knowing what he was doing. Elvis, the man of action, makes an appropriate figurehead for early rock and roll, because for the first ten years of its life, rock was all about action. The songs were invitations to dance, incitements to riot, or none-too-subtle propositions for sex. And the singer's desire was always backed up by the music — the most intensely physical music a mainstream white audience had heard up to that point. Sweat, exertion, desire and spontaneous action created the foundation on which rock and roll was built, and over this the music's architects constructed their machines for dancing and doing. By 1965 these included the Chuck Berry duckwalk, the Sun Studio slap, the Phil Spector Wall of Sound and the Bo Diddley beat. Bo Diddley's 'shave-and-a-haircut, two-bits' rhythm is, as Toby Creswell says in his book, 1001 Songs, 'one of the essential parts of the vocabulary of rock and roll'. The famous beat first came to light on Bo's 1957 hit 'Bo Diddley'. His producers, the Chess brothers, made him change some of the song's lyrics so it would get played on the radio: Bow-legged rooster told a cross-legged duck Say you ain't good lookin' but you sure can...crow But even if he'd scrapped the lyrics entirely, no-one who heard the music would be left in any doubt as to what the song was about. The Bo Diddley beat is pure desire. In 1964, Andrew Loog Oldham overheard Keith Richards singing snatches of Buddy Holly's 'Not Fade Away' over a Bo Diddley beat played on his acoustic guitar, and knew he was listening to the next Rolling Stones' single — it was recorded two days later. 'The Bo Diddley feel is a suggestion in Buddy's version,' said Tony Calder in Andrew Loog Oldham's 2 Stoned, 'and a call to arms in the Stones''. 'Not Fade Away' heralded a tough and threatening new sex drive in the Stones' music which would become a hallmark of their sound from this point on. 'I'm gonna tell you how it's gonna be,' sang Mick Jagger, not messing around any more, 'you're gonna give your love to me.' The Stooges' '1969' is also built on the Bo Diddley shuffle. But while the desire in the rhythm is still strong, the simple sense of purpose it had in the Stones' hands is gone: the song still thrusts and kicks, but in a flailing, hopeless fashion. '1969' seems to go on forever, locked in its two-chord drive to nowhere, and the wah-wah guitar solo sounds more like a tantrum than a come-on. In 1984 the Bo Diddley beat was back — though in barely recognisable form. On The Smiths 'How Soon Is Now?', the sound that had framed a litany of desire in 1957 and a call to arms in 1964 seemed finally to have worn itself out — the song sounds like 'Not Fade Away' played on a Walkman with a dying battery. This was a dance record for those who find dancing — along with any other form of spontaneous action — impossible. The singer introduces himself in the first verse: I am the son And the heir Of a shyness that is criminally vulgar I am the son and heir Of nothing in particular. Self-loathing, self-pity, bad jokes; we're a long way from 'you're gonna give your love to me'. Where would an underground man find the grounds for a statement like that? As the song unfolds, the gloomy young man gets some unsolicited advice. Don't sit there tormenting yourself, say the men of action, go out there and have some fun. Dance! Enjoy yourself! 'You could meet somebody who really loves you.'But the singer knows even before he gets in the car and drives to the club that things will end badly. In fact, he's so smart that he's seen into the future, and knows that everything, everywhere will end badly. As with Dostoyevsky's basement-dweller and the singer's own subterranean hero Howard Devoto, Morrissey's world-weariness is a result of his intelligence — which he would gladly trade for the ability to act. 'I'm obsessed by the physical,' he told Simon Reynolds in Blissed Out, by way of explaining his ongoing fascination with criminals and toughs, 'it always works — instead of creeping around and relying on your thesaurus.' But Morrissey has not succeeded in making an insect of himself, he is decidedly not what Dostoyevsky refers to as 'l'homme de la nature'. 'I don't feel natural even when I'm fast asleep,' he sings in 'Sweet and Tender Hooligan'. Time and again, in Morrissey's songs, the hero is about to take action and finds, for one reason or another, that it's impossible. And this would be fine if he'd somehow managed to transcend his earthly desires — to make himself into the Buddha of Manchester. But as he reminds us in 'How Soon Is Now?,' he is still human, and he still needs to be loved. # Why Bother? IN 1997, RIVERS Cuomo went into retreat. He moved into a small apartment under a Los Angeles freeway, disconnected the phone, sealed up the windows and painted the walls black. The singer's decision to isolate himself has always been seen as a reaction to the embarrassing failure of Pinkerton. But Cuomo's new monkish lifestyle was, in a sense, the inevitable result of that album — the philosophy of Pinkerton put into practice. In 'Why Bother?' the singer thinks about finding a girlfriend, but finds insufficient grounds for action. It's like a super-pessimistic version of Wham!'s 'Last Christmas'. Before he's even picked up the phone he's reasoned his way to the following summer, when she'll no doubt dump him and break his heart. So he remains alone. The singer has proved his intelligence while ensuring that he remains miserable. This line of thinking leads to greater and greater inertia — taken to its logical conclusion, the singer must renounce the search for happiness entirely and derive whatever kicks he can from monkish self-denial. Music journalist Chuck Klosterman described Pinkerton in Spin magazine as emo's Sgt Pepper. 'Philosophically, it defined what emo was supposed to feel like.' Emo songs had always rated the ability to feel much higher than the ability to act, but Pinkerton sealed the deal, by suggesting that there was something truly noble in being broken and beaten. The intelligent but highly emotional singer has seen the horror up far too close — how can we expect him to act, let alone fight? We can't, of course, but we can applaud his inner resolve as he shuts himself away from the rest of the world while everyone else carries on with the meaningless comedy of existence. The Get Up Kids are one of the scores of bands who followed the example of Pinkerton, exploring the lonely landscape Weezer had discovered long after Cuomo himself had moved on. In The Get Up Kids' 'I'm a Loner, Dottie, a Rebel', the hero tells us that last night he was in love, and that the possibility is still there. But sitting by the girl's bedside in the morning, he reasons his way out of whatever future they might have together. 'I'm afraid to try,' he admits, 'I'll keep my hands by my side.' A real man, a natural man (a jock, a Limp Bizkit fan) would do something. But for The Get Up Kids and their fans, this kind of 'action' is deeply suspect. As Trevor Kelley and Leslie Simon have observed in their book Everybody Hurts: An essential guide to emo culture, non-athleticism is one of the sacred commandments of emo. It's reflected, on a very simple level, by the fans' fashion accessories. Emo replaces nu-metal's trainers and baseball caps with black-rimmed glasses and Penguin classics. And it means that the heroes of the scene tend to be of the static, intellectual type — Rivers Cuomo, not Zack de la Rocha; Morrissey, not Metallica. In Nothing Feels Good, Greenwald argues that emo's roots can be traced back to an unholy coupling of Washington DC hardcore with The Smiths that took place in the late '80s.This partnership is not as odd as it might first seem. Hardcore, essentially, is about resistance. But as Greenwald points out, there are different types of resistance. Hardcore in its pure strain made a spectacle of political protest. But The Smiths' music, in its fey, unassuming way, mounts a much more challenging refusal. Morrissey might not be a 'natural man', but his stance is, in its way, as tough and intractable as any of the thugs and gangsters he admires so much; and more than a match, as Greenwald insists, for the hardcore bands he found himself sharing shelf-space with toward the end of the decade. Morrissey, in his songs, demands the right to be miserable and alone. This doesn't sound like too much to ask — but the world keeps telling him he has to cheer up and get over it ('There's a club, if you'd like to go'). And since he steadfastly refuses to do this, his position has, over the years, become more and more entrenched. What started out as a polite request has turned into a war of attrition, with Morrissey as the unlikely heroic general, who refuses to give up one inch of his territory no matter what the enemy throws at him. You might say he's taking it too personally, but what other way is there to take it? The world has refused to accept his personality. No wonder he's determined not to give up the fight. In this war, what's at stake is nothing less than the human soul. Dostoyevsky: Smashing things can sometimes be very pleasant. # The Crystal Palace THERE'S A GREAT deal of masochism in the underground man's miserable stance — he enjoys his suffering, and enjoys complaining about it even more. And since his pain is now his only real source of pleasure, he refuses to get over it. Reasonable people could reasonably accuse the underground man of being horribly self-indulgent. But this accusation raises a question — why are reasonable people so offended by an individual who demands nothing more than the right to be broken, bitter and dissatisfied? Whether he set out to do this or not, the man under the floorboards has exposed a terrible flaw in modern society. By the time Dostoyevsky sat down to write Notes from Underground in 1864, reason, empiricism and mathematical perfectibility should all have been well and truly trashed by almost a century of romantic philosophy — from the counter-Enlightenment of Rousseau and Kant to the view of life as chaos and flux expounded by Schopenhauer (who had, by this time, become one of Europe's best known and most quoted thinkers). But while romanticism did much to form popular taste and opinion in the nineteenth century, for all the difference it had made in the world of industry and commerce, it might just as well have never happened. While the romantic individualist was developing his philosophy of feelings, dreams and the irrational; the businessman — who could have no use for this sort of stuff — simply continued the project of the Enlightenment, but to quite different ends. This time the goal was not the perfectibility of human life, but the perfectibility of human life in the pursuit of profit. Business, with its eternal worship of the bottom line, will always look to rationalise and systematise human behaviour. And since the fate of nations was increasingly tied up in the fortunes of business, the shopkeeper mentality gradually spread to influence every aspect of modern society during Dostoyevsky's lifetime. Goethe had remarked, a few years before his death, that 'wealth and speed are what the world admires and strives for'. By 1864 belief in technical progress had become a sort of secular religion in Western Europe, the idea being that the perfection of industry would lead to greater profit, and greater profit would increase the amount of human happiness in the world accordingly. The triumphant symbol of this ideal — the St Paul's cathedral of progress — was the Crystal Palace, a shimmering glass and steel building created to house London's Great Exhibition of 1851. This, according to the marketing puff, would 'unite the whole world in the quest to apply the latest advances in science and industrial production for the benefit of all'. But who was really benefiting? William Morris walked out of the place disgusted, seeing nothing but mass-produced junk, entirely lacking in that particular, personal sense of beauty that had guided the hand of the medieval craftsman in days gone by. The individual's feeling for expressive form had, it seemed, been ruthlessly snipped out of the manufacturing process for the sake of a better looking profit margin — a more efficient machine. For Dostoyevsky, the Crystal Palace was equally hateful, but for slightly different reasons. In 1864, Dostoyevsky had had forty years to think about man's ability to reason and where it had got the human race so far. For him, the Crystal Palace was a monument not to progress, but to stupidity, the extraordinary stupidity of a culture that had convinced itself that the application of reason and commonsense would improve the lives of human beings. What does reason know? Reason only knows what it has succeeded in finding out...but man's nature acts as one whole, with everything that is in it, conscious or unconscious, and although it is nonsensical, yet it lives. The application of reason could guarantee that a railway will run on time or that a factory will produce more ceramic plates per year. But human happiness, he insisted, cannot be calculated mathematically. Our desires are, and always will be, irrational; and a society that tries to systematise every aspect of life will not be able to accommodate them. In Notes from Underground, the underground man — in one of his many imaginary confrontations with the surface dwellers — puts forward the case against the positivists and their cathedral of commonsense: You believe in an eternal and indestructible crystal building, in which you won't be able to stick out your tongue in secret, or even make a rude sign in your pocket. But perhaps I fear that building precisely because it's indestructible and made of crystal, and you won't be able to stick your tongue out, even in secret. Dostoyevsky saw that the attempt to rationalise all of human life 'for the greater good' would eventually lead to a situation in which bad behaviour is no longer tolerated. Spite, malice, shoplifting and the irrational desire to smash things are all bad for business and bad for the state. But all of these things make us happy — as the underground man, who loves nothing better than to complain, knows all too well. And why are you so firmly and triumphantly certain that only what is normal and positive — in short, only well-being — is good for man? Is reason mistaken about what is good? After all, perhaps prosperity isn't the only thing that pleases mankind, perhaps he is just as attracted to suffering...whether it's a good thing or a bad thing, smashing things is also sometimes very pleasant. I am not here standing up for suffering, or for well-being either. I am standing out for my own caprices and for having them guaranteed when necessary. How much of our happiness are we prepared to give up for the greater good? And what is the greater good if not the sum total of human happiness? We'll never know, because you can't calculate happiness. Scientific analysis might prove that a henhouse is enough to keep a man dry when it rains. But that won't stop him from wanting a mansion. You can only stop human beings from wanting what they want, says the underground man, by altering their nature. This, Dostoyevsky believed, was impossible. We are not perfectible, we are ridiculous, and a society that pretends this is not the case — that tries to tell a man that he is better off with a henhouse when he wants a palace or that he should be cheerful when nothing makes him happier than being miserable — does so at its great peril. Eventually, he warns, desire will escape whatever restraints the positivists place on it. I am certain that underground people like me must be kept in check. Though we may be capable of sitting underground for forty years without saying a word, if we do come out into the world and burst out, we will talk and talk and talk... Morrissey has always suspected as much. The Smiths' 1987 single, 'Shoplifters of the World Unite', is a paean to human happiness in one of its most irrational manifestations, and a call to arms for the malcontents of modern society. The shoplifter just has this thing he likes doing — it hurts no-one, but because he stands in the way of money being made, the government says he must be stopped. This is the same government, he notes, that is busy figuring out how to put more missiles in space so as to be able to kill a few more million people in some future war. And they call him a criminal! His self-righteous proclamations are interrupted, he feels a 'heartless hand' on his shoulder, his happy dream shatters at the same moment as the alabaster vase he was busy lifting. The state tries to rehabilitate him, to teach him to accept a reasonable amount of happiness. 'I was bored before I even began,' sighs the singer. He's not satisfied with what he's been offered — and he knows he's not the only one. # The Broken, the Beaten and the Damned THE SMITHS' albums were to the solitary young romantics of the 1980s what The Sorrows of Young Werther was to those of the 1780s — a friend to the friendless. In America they sustained many a lonely soul through the materialist wasteland of the late '80s. And long after the band had broken up, The Smiths continued to speak for those who weren't being spoken for elsewhere. By the '90s, 'alternative' music was everywhere, and misery, alienation and disaffected rage flooded the radio and the mall in ways that would have seemed unthinkable five years earlier. Paradoxically the orgy of self-congratulation that surrounded Lollapalooza and the mainstream success of Nirvana alienated the very people alternative music was supposed to represent. The lonely and disaffected fled the suddenly exposed world of alterna-rock like vampires caught in a searchlight. They sought out the more rarefied pleasures of Pinkerton or Diary, hit 'play' on their copy of Siamese Dream for the hundredth time or turned — as Gerard Way did — to The Smiths. For a kid who wears black and feels different to everyone else, a song like The Smiths' 'Unloveable' is a way of explaining yourself to yourself and maybe, one day, to the world that doesn't understand you. 'I wear black on the outside because black is how I feel on the inside,' sings Morrissey, 'and if I seem a little strange — well, that's because I am.' For Way, Morrissey was more than just a human voice in an inhuman world, he was an inspiration — one of a few guiding lights he would later follow when he started a band of his own. I've always seen My Chemical Romance as the band that would have represented who me and my friends were in high school, and the band that we didn't have to represent us — the kids that wore black — back then. My Chemical Romance would take up Morrissey's plea for acceptance and turn it into a battle cry. In 'Welcome to the Black Parade', the singer sees that, like himself, the kids who wear black are threatened on all sides by behaviourists and rationalists who are determined to rid them of their irrational desires for the sake of the greater good. The singer, still haunted by his father's words — 'will you be the saviour of the broken, the beaten and the damned' — and determined to make good on his promise, leaps into the fray. 'Let's paint it black and take it back!' he shouts. Gerard Way's heroes — Morrissey, Robert Smith and Billy Corgan — all expressed dissatisfaction with modern life. But their protests mostly took on the form of a hunger strike — the singer would suffer publicly until the world recognised his needs. Unfortunately, outside the proscribed limits of indie rock, the world takes no notice of this kind of thing. So the singer in 'Welcome to the Black Parade' has decided to try something new. He has decided to do the one thing the world will not expect from a gloomy young man with a sensitive temperament — to form an army and start marching. This, he has realised, is the only language the world understands. Pretty soon, he has a ragtag mob of fellow revolutionaries, an army of loners which he leads, Napoleon-like, to the gates of civilisation. The word 'cult' was often used by music journalists to describe the bands Gerard loved in high school. Singers that speak for those who don't have a voice tend to attract committed followers, and sometimes slavish imitators. By standing up and saying 'I don't feel like everyone else', Morrissey and Robert Smith became role models for anybody who ever felt like they weren't like everybody else. And since this was the task Gerard Way set himself — to speak for those who are not like others — it was inevitable that My Chemical Romance would attract its own legion of devotees. But Gerard was never comfortable with the word cult — not because of its religious or pagan overtones, but because it seemed offensively small-minded. Cults meet in secret, communicate in code and die tragically in mass suicides. Gerard Way had a world to change, and he refused to see his fans' energy and ambition curtailed by a word. 'You should all know,' he told his audience from the stage in 2006, 'if you support us...you are not a cult; you are a fuckin' ARMY!' Alex: A sprig in a barrel-organ. # Teenagers 'WELCOME TO THE Black Parade' is something new in the world of rock and roll. There have been songs that angrily demand that the kids be granted the right to party, and there have been songs where the singer says he won't go to the party with all the other kids because he's too full of despair. But there's never been a song that angrily demands that the kids be granted the right to be full of despair. This is what the army of the black parade wants as it rattles the gates of the crystal palace. Here is the underground uprising Dostoyevsky imagined, thousands of human individuals who insist on being useless — broken, beaten and damned — in a utilitarian world. Their slogans are carefully calculated to annoy positivists and empiricists: 'We're all gonna die', 'I think I'm gonna burn in hell', 'What's in is despair'. But the right to be sad is one that the modern world can't allow, and as the black parade began its march around the world in 2007, the media began a severe crackdown on sadness. The word 'cult' began to be thrown around. Old folk devils were revived: The Black Parade contained suicidal messages; the singer was using his shows and web forums to encourage his impressionable young fans to dive, lemming-like, into oblivion with him; links were implied between emo (the band members gritted their teeth) and recent high school shootings. The quiet, lonely kid with the overactive imagination, the notebook full of visions of impending doom, the black clothes and the long fringe. 'If you think your child might be at risk, go to our website...' Way, understandably, was spooked by this media panic, which had in fact been building since the release of My Chemical Romance's second album. The huge spike in the band's sales and concert attendances after Three Cheers for Sweet Revenge was largely due to an influx of very young fans, and Gerard felt an enormous sense of responsibility to them. But the singer could see through the fear-mongering of the news networks and the tabloids to a much more serious malaise lying beneath. On tour, he poured his frustration into a song — a Bon Jovi-ish anthem recorded for The Black Parade, in which middle-America, picking up the tune laid down for them by Fox News, bawls out the refrain, 'Teenagers scare the livin' shit outta me!' This song, 'Teenagers', warns us of the lengths the state may go to in order to pursue its war on sadness. They're gonna clean up your looks with all the lies in the books, to make a citizen out of you 'They've got methods of keeping you clean,' sings Gerard Way, hinting at more sinister procedures to come — drugs, surveillance and mind control. Why would they go to all this trouble? Because they're scared of you! If those underground types keep talking, word will get around that the limits imposed on human desire by the state are arbitrary and false, and people will start demanding all kinds of things that modern society is in no position to offer them. Dostoyevsky's underground man warned that happiness is not synonymous with wellbeing. A complete list of the ridiculous activities that make human beings feel good would have to include sulking, stealing and 'smashing things' which, the underground man insists, can sometimes be 'very pleasant'. Alex, the protagonist of Anthony Burgess's A Clockwork Orange, would agree with this. Burgess's 1962 novel accelerates teenage delinquency into a nightmare future, where Alex and his gang of beautiful young men in eye make-up and bowler hats terrorise the city's streets with 'ultraviolence'. Alex is not interested in the greater good — but he knows what he likes: rape, ultraviolence and Beethoven's Ninth Symphony. These things make him happy — isn't that what life is all about? Of course, that's not how the state sees it. After a night of ultraviolence gone horribly wrong, Alex is thrown in prison, and soon becomes a candidate for a very promising new rehabilitation technique. When the underground man argued in 1864 that the state could only stop him from wanting the things he wants by altering his nature, he didn't believe for a second that this might be a possibility. But perhaps he should have — de Condorcet, in his sketch for a mathematically perfectible utopia, had already suggested that careful breeding might, given time, eventually iron out some of the kinks in the human organism. Now, in Alex's time, science has progressed to the point where unreasonable individuals can be 'perfected' more or less on the spot. But making a citizen out of Alex comes at a terrible cost. Dr Ludovico's brutal aversion therapy and high-powered drug injections rip up Alex's head and rob him of his free will. He can't be 'bad' anymore, and while the government might herald this as a great leap forward, the true meaning is not lost on Alex. He finally realises that, in his society's crystal palace, flipping the bird is not allowed. 'They of the government and the judges cannot allow the bad, because they cannot allow the self,' Alex muses. 'And is not our modern history, my brothers, the story of brave malenky selves fighting these big machines?' Alex is a nasty piece of work, but he's no dummy — and he knows his history. The big machines have been a problem for romantic individuals since William Blake wrote his preface to Milton: A Poem in 1804. Blake gave the nineteenth century one of its most indelible images when he described 'dark Satanic mills' rising over 'England's green and pleasant land'.In 1811 one of these bleak-looking intrusions on the landscape erupted in violence: textile workers in Nottingham, angry about the introduction of a new stocking weaving frame that would, it was said, speed up production and reduce the number of workers needed in the factories, took up arms against the new frames. Many of them found themselves sentenced to death (or worse, sent to Australia) as a result. But the frame-breakers — or Luddites as they became known — found themselves with an unexpected champion in Lord Byron, who argued passionately in the House of Lords against the introduction of the new laws, and later took the case to the streets with an article in The Morning Chronicle. This, at first, seems a little out of character for the poet, who had little love for the common man. But a letter to his mother, written around the time of the dispute, reveals the source of his sympathy for the Luddites: If I could by my own efforts inculcate the truth, that a man is not intended for a despot or a machine, but as an individual of a community... I might attempt to found a new Utopia. Here Byron is making a case for the dignity of the solitary citizen over the interests of states or systems. People want more and cheaper stockings, so it makes good rational sense to install machines that will make more stockings more quickly — more people will get what they want. But here we have already lost sight of the individual human being, and individuality is everything to Byron. This helps unravel the paradox behind Byron's support of the Luddites — how he could despise the mob, and yet stick his neck out to help a mob. The former is an individualised mass, the latter is a mass of individuals. England embraced industry more quickly and effectively than any other nation in the nineteenth century, which is why the image of nature opposed to the rise of the machine, and the individual man opposed to totalitarian systems, became such a hallmark of romantic poetry in that country. It's a vein of imagery that can be traced all the way from Blake to Morris and the Pre-Raphaelites to Tolkien — whom they inspired — through to the strain of medievalism that runs through the hippy movement and right up to Pink Floyd. Animals, Pink Floyd's eleventh studio album, released in 1977, painted a bleak portrait of English life after two centuries of progress and industry — from the dark Satanic mills on the album's cover, to the dog-eat-dog world described within. The album begins by wondering what would happen 'if you didn't care what happened to me and I didn't care for you', and quickly gets worse. Later that year, while touring the album, songwriter Roger Waters discovered that he was far from immune to the social collapse he'd just described when he spat on a fan at a show in Montreal. Much soul-searching followed, which eventually lead to the band's next project, 1979's The Wall. Waters' epic study in alienation traces the roots of his character's soul sickness back to the public school system. In the classroom children are treated as though they are empty vessels, ready to be filled up with correct ideas which will equip them for the workforce. Of course, human children, Waters insists, are not empty vessels. They're unique individuals with strange dreams and irrational urges. But since the behaviourist state cannot admit this even for a moment, they have to beat those dreams out of you. Producer Bob Ezrin, fresh from recording the kids choir on Alice Cooper's 'School's Out', assembled a gang of English school children in the studio to sing Waters' immortal lines: 'Hey! Teacher! Leave them kids alone!' Though Waters, like Byron, is hardly a man of the people, he is a staunch individualist — so he will never accept the idea that human beings are merely part of a system. But this is exactly what modern government wants: another brick in the wall, a human being reduced to what Dostoyevsky calls a 'sprig in a barrel-organ' — or, as Gerard Way puts it in 'Teenagers', 'another cog in the murder machine'. With this line, Gerard exposes the real irony in America's War on Sadness. Society considers it dangerous for a band like My Chemical Romance to promote despair, because despair is a drag on productivity — it sends a bad message to the kids who are the workforce of the future. You can't go around telling people life is pointless. It happens to be true, but how will we get anything done if people find that out? The valuelessness at the heart of modern society will be revealed for all to see, the jig will be up, the machine will be prevented from working. But Gerard suspects, as Dostoyevsky did in his day, that the machine itself might be the real reason the kids are unhappy in the first place. By enshrining progress over real values, to the point where nobody knows what values are anymore, science and industry have created horrors that Rousseau, Wordsworth, Morris and Dostoyevsky could barely have imagined. How can we expect the workforce of the future to put on a happy face while contributing to a society that has produced the atomic bomb, missiles in space, the greenhouse effect and the War on Terror? The world produced by reason and commonsense is a nightmare. So, because utilitarianism has proved incompatible with real human happiness, the romantic artist, as Bertrand Russell has observed in his History of Western Philosophy, tends to replace utilitarian standards with aesthetic ones. The earth-worm is useful, but not beautiful; the tiger is beautiful, but not useful. Darwin (who was not a Romantic) praised the earth-worm; Blake praised the tiger. The morals of the Romantics have primarily aesthetic motives. Byron will support the Luddites over the government; Morris the solitary artist over the big factory; Tolkein the hobbits over Saruman's industry; Nick Cave the murderer over the state that wants to reform him; Morrissey the shoplifter over the cops; Jon Bon Jovi the outlaw over the sheriff; Tim Burton the monster over the suburban world that won't accept him. The factory and the police force are useful, but not beautiful. The monster and the sulky teenager are beautiful, but not useful. In any contest between the big machine and Alex's 'brave malenky selves', the romantic has to side with the 'brave malenky selves'. Gerard Way: Making a difference. # I've Gotta Get Out of the Basement! IN AN INTERVIEW conducted shortly after the release of Three Cheers for Sweet Revenge the members of My Chemical Romance were asked about their goals as a band and whether they felt they had met them. Frank Iero had a detailed answer ready to go. When we started this band we set mini goals and then we had our ultimate goal... We met all our smaller goals...we've been able to reach an exorbitant amount of kids that we never thought we would reach. Our major goal was to make a difference, and I think we are on our way to that goal. In the mythical universe of The Black Parade, this goal was entrusted to My Chemical Romance on that fateful day when the singer's father took him to see the marching band. In the real world, the story gets a little more complicated. Gerard Way spent his formative years locked away in his bedroom, living in a make-believe world of comic book superheroes and Dungeons & Dragons (D&D). Later, he discovered music — his first concert was Springsteen in New Jersey, followed by those other local heroes, Bon Jovi. As high school wore on, he moved on to darker, heavier stuff: The Smiths, The Cure, and other bands who spoke for the loners and losers. Way remembers making the hour-long round trip from his parents' house in Belleville, New Jersey, to the nearest mall to buy The Smashing Pumpkins' Siamese Dream on the day it came out. All of these — the comics, the Springsteen and Bon Jovi concerts, the cult bands, the punk 7-inch singles he was starting to collect — would provide important cues for Way when he started making his own music in his late twenties. By that time, Way was living in his parents' basement, trying to make it as a commercial artist. He interned at DC Comics and pitched an idea to the Cartoon Network, which they very nearly picked up, about a flying monkey who talked like Björk and could make breakfast foods appear out of nowhere. But he was unhappy. He drank too much, he popped too many pills, and occasionally thought about suicide. Then, driving in to Manhattan on 11 September 2001, Gerard had an epiphany. 'I've gotta get out of the basement,' he said to himself, 'I've gotta make a difference!' At this point, Gerard Way's story really does begin to resemble the superhero comics he devoured so eagerly as a young man. Way's response to global catastrophe and personal meltdown was...to form his own superteam. He, his brother Mikey and neighbourhood friend Frank Iero banded together as My Chemical Romance. Their mission? 'To deal with the post-traumatic stress disorder of 9/11.' In the beginning, the band's role was therapeutic. Gerard thrashed and howled and let off steam, and felt much better for it. But he also realised that he had created art. Art had healed his bruised psyche and given him a reason to live. Might it not be able to accomplish the same task on a larger scale? The world was in crisis, society was falling apart (again), everybody he met seemed so loaded up with stress that they might explode at any moment. Could art make a difference? The answer was almost irrelevant, since Gerard and the band felt as though they had no choice but to do what they were doing anyway. But after the adrenaline rush of those early gigs had worn out, Gerard began to think more carefully about how to do it. The clues turned out to be in his own childhood and adolescence — in the music, comics and movies that had shaped his imagination. The gothic gloom of The Cure, the horror business of the Misfits, the Old Testament morality of Nick Cave and the Bad Seeds, the high romance of Queen, Bowie and Springsteen, the cosmic allegories of power and responsibility in the superhero comics, and the medieval escapism of D&D. These were the myths that had sustained him, that had given his life meaning and purpose in his darkest moments — and these became the raw materials for the great multimedia art project that would be My Chemical Romance. Role-playing games and comics have never really been cool, exactly. But they got a boost in the early '90s from the patronage of Rivers Cuomo. Weezer's song 'In the Garage' describes the place the singer goes when normal life drives him round the twist. Here no-one can tell him what to do — he's got his Kiss posters, his X-Men comics, and his twelve-sided die. But we wouldn't know that he's into any of that stuff, the singer tells us, because we're not allowed in here. 'In the garage where I feel safe,' he croons, 'no-one knows about my ways.' With 'In the Garage', Cuomo made himself a martyr to geekiness. Thanks to the Weezer singer's groundwork in establishing such nerdy pursuits as part of the aesthetic of twenty-first-century pop-punk-whatever, the modern rock star no longer has to hide his Dungeon Master's guide in the garage. The members of My Chemical Romance are not so shy about their obsessions. While all the other musicians on the 2006 Warped tour were getting loaded and chatting up groupies, Rolling Stone's contributing editor Jenny Eliscu followed the members of My Chemical Romance around the local Wal-Mart as they looked for Spiderman pyjama bottoms and plastic racks to organise their D&D books. 'They prefer to think of themselves as superheroes rather than rock stars,' Eliscu noted, as the band climbed beneath their Teen Titans bedspreads and bid each other goodnight. And, like any respectable superheroes, the members of My Chemical Romance get their own action figures later this year. 'I don't think that having a My Chemical Romance action figure will make a kid start his own band,' Gerard says. 'I like to think it will make him save children from a burning building.' This is about much more than merchandising for Gerard Way — more than music, even. It's about finding a way out of the shallow materialism of his age. It's about giving fans something to believe in. # Myths of the Near Future THE PROBLEM WITH reason — as Rousseau realised — is that it's essentially amoral. Logic, science and mathematics can help us figure out how to do things better, but they can never tell us whether the thing was a good idea in the first place. So a society that enshrines science and reason above all else is, in the end, guided only by the spirit of competition — the race to see who can build a more efficient mill, a faster steam engine. Progress can never tell us how to live or why we should carry on doing so. For that, Dostoyevsky believed, you need religion. But the new ideology of commonsense had rationalised religious faith out of existence. Here again Dostoyevsky saw the dubious legacy of the Enlightenment at work. A society based on rational principles can have no use for things that don't make sense — and Christianity, like all major religions, is full of nonsense. A virgin birth? A man who dies and comes back to life? Three persons who are the same person? Enlightenment philosophers jumped through hoops to reconcile all this mumbo jumbo with reason — and what they couldn't explain, they did away with. This, for Dostoyevsky, was the greatest mistake of the age. The empiricists and positivists had created a world in which behaviour was only tolerated if it was rational and useful. This left no room for tradition or faith, which are irrational and therefore useless. But again, Dostoyevsky asks, how did reason become the measure of all things? Myths, as Isiah Berlin points out in Against the Current, are not 'false statements about reality corrected by later rational criticism'.What's important about mythology will never show up on the rationalists' radar — so of course they'll assume that it's useless. But what if there was something wrong with the system? What if the most important element in our lives, the one thing that can give meaning to what otherwise seems like a brute struggle for existence, cannot be weighed on a scale or calculated by an adding machine? Dostoyevsky believed that the erosion of faith by science had created a world without meaning in which the spirit of competition and one-upmanship was the only rule. Now society, which had once been bound together by real values, was falling apart at the seams. Against his degraded present, Dostoyevsky opposed the image of a specifically Russian Christian tradition, rooted in the soil of the nation and flowing through the veins of its people. This, as Alex de Jonge points out in his book Dostyevsky and the Age of Intensity is closely bound up with his idea of the 'living life' — a Rousseau-ish vision of human beings connected to their natural impulses. The underground man is envious of the natural men and the men of action precisely because they seem to be in touch with this 'living life'. He, on the other hand, is unnatural — a test-tube man — the inevitable result of the fragmented and meaningless world created by reason and enlightenment. Dostoyevsky was not alone in voicing his discontent with modern society — nor was he the only one to posit national folk traditions as the remedy for its ills. Richard Wagner, like William Morris, advocated a return to the artisan's communities of the Middle Ages, and found the materials for his dramas in medieval folklore. He believed these indigenous traditions could provide German people with the kind of spiritual satisfaction that the modern world, with its flimsy material consolations, could not. And Wagnerians — following the composer's lead — loved Wagner's music in a completely different way to that in which, say, the French concert-goer of the eighteenth century would have enjoyed Mozart. Wagner was not entertainment, his operas were, as Eric Hobsbawm puts it in The Age of Empire, 'all-purpose providers of spiritual content'. And Wagner was not just a composer, he was — as he was the first to insist — a maker of myths. This new role for the artist came to seem more and more important as the world created by money and business proved to be not only immoral, but far less stable than the positivists would have you believe. The Great Depression of the late 1800s, as Hobsbawm points out, seems like a barely perceptible blip compared to the financial crises of the twentieth century. But it was enough to shake people's faith in economic progress, and it certainly lead to a great deal of 'I-told-you-so'-ing from the nineteenth-century's discontents. In his study of turn-of-the-century culture in Vienna, Carl Schorske explains how this new instability gave extra momentum to the Wagnerians' cause. 'The crash of 1873', he writes in Fin-Du-Siecle Vienna, 'made particularly attractive [Wagner's] glorification of the Medieval Artisan community against modern capitalist society.' Addressing a meeting of Vienna's Wagner society in 1875, August Sitte told his audience that '[t]he essence of the modern condition being the fragmentation of life, we stand in need of an integrating myth'. Showing how these unifying myths could be created by artists was, Sitte argued, Wagner's great achievement. Just as Wagner's Siegfried forges a new weapon from the fragments of his father's sword: So too must the modern artist generate, by the example of his art, the strength to overcome fragmentation and provide a 'community life-outlook' for people as a whole. In a meaningless and chaotic world, a world in which social relations have broken down and the old faiths have disappeared through neglect, it was the artists' job, the Wagnerians believed, to heal society's wounds. # Gustav Klimt TOWARD THE END of 1895 the Austrian painter Gustav Klimt was commissioned to paint three pictures for the great hall of a Viennese university — one for each of the university's three main faculties: medicine, philosophy and law (jurisprudence). The board of trustees wanted something inspirational, something that communicated in every possible way that the combined forces of knowledge, reason, and intellect would, in the fullness of time, lead humankind out of the wilderness and into the light. Something modern as well — that went without saying — but not too modern. What they got was a shock — to say the least. Klimt's first effort, 'Philosophy', was an overwhelming avalanche of human joy and tragedy — birth, death, agony and ecstasy cascaded past the viewer, springing from nowhere and, it seemed, going straight back there. As Schorske says, 'The ideal of mastery of nature through scientific work was simply violated by Klimt's image of a problematic, mysterious struggle in nature.' The university politely asked for its money back. A painting that represented the thing-in-itself as imagined by an Enlightenment philosopher would be harmonious and elegant. But Klimt had decided to give them Schopenhauer — so it's little wonder the thing turned out looking nasty. For Schopenhauer, the thing-in-itself, the world as will, is senseless, destructive and evil. The university's trustees worried that, faced with such a heavy dose of romantic despair in the great hall, students would simply throw their books in the air, turn around and go back home — there to spend the rest of their days in contemplation of the suffering of the world. This is actually not too far from Klimt's intention, and very close to Schopenhauer's idea of redemption — the one piece of good news in his otherwise gloomy philosophy. Schopenhauer insists that as long as we're pursuing our interests — food, shelter, sex, material possessions or power over others — we're being driven by will, which can never be satisfied. For this reason, he dismisses the Rousseauian idea of a 'natural' state to which modern people can aspire. For Schopenhauer, our natural state is the problem — we've complicated matters by becoming as self-aware as we have, but the source of our unhappiness is the sheer pointlessness of life itself. We must pursue our interests, knowing that they must leave us unsatisfied. But if we can somehow become disinterested, we are no longer willing — and the result is a feeling of bliss. This, Schopenhauer believed, is what art does for us. When an aesthetic perception occurs the will completely vanishes from consciousness...this is the origin of the feeling of pleasure which accompanies the perception of the beautiful... As we contemplate art, we are able to see life — with all its striving and willing — in a detached, aesthetic way. We are freed, briefly, from the desiring that takes up so much of our time, and leaves us so unsatisfied, as we look at life from the artist's point of view. In this way, the suffering of the world becomes bearable, and art, according to Schopenhauer, becomes our most important consolation for the pain of life. It's little wonder that, of all philosophers, he's the artist's favourite. Schopenhauer's formula for redemption through aesthetics explains, among other things, how it is that a song about how life sucks can make us feel good. 'How Soon is Now?', 'Blasphemous Rumours', 'Butterfly', '(I Can't Get No) Satisfaction' — these songs are full of bad news about the human condition, and all have the power to make us feel fantastic. Even as we recognise the sincerity of the artist's view of life, and the honesty with which he's portrayed it, the feeling we get as we listen to his song about how life is hell is not the same as the feeling of living in hell — quite the opposite. It's as though the singer, by giving us such an unflinching portrayal of the world as will, has shifted our position in relation to it. If life could be compared to a giant traffic jam, the song has the effect of lifting us high above the traffic in a helicopter. We can still see the chaos on the roads, but we're no longer directly involved in the struggle — where previously we were interested (because we have to get to work on time), now we are disinterested — and from this new aesthetically detached point of view, the traffic jam becomes beautiful, a glittering mosaic winding its way around the city. We no longer experience the pain of the world as sufferers but as spectators. The university trustees needn't have been so worried about Klimt's Philosophy mural after all. Far from spreading despair, a painting like that — in which the suffering of the world is presented as an aesthetic spectacle — would, if Schopenhauer was correct, become a means of redemption. This idea proved to be enormously popular and durable in the late nineteenth century. It formed the backbone of Wagner's conception of music and opera as a substitute for religion in a fragmented modern world. And it gave a young philologist from the University of Basel — a man much admired by Klimt — the necessary foundation on which to build a career that would take romantic philosophy in undreamt-of new directions. Nietzsche: 'I discovered all these abysses in myself...' # Nietzsche IN 1865 TWENTY-ONE-YEAR-OLD Friedrich Nietzsche walked into a second-hand book shop in Leipzig and purchased a copy of Schopenhauer's The World as Will and Representation. This was good news for Schopenhauer (who unfortunately was dead by this point) because it's safe to say that even if only two people had ever read his books, as long as those two people were Richard Wagner and Friedrich Nietzsche, his place in history would be assured. Schopenhauer's philosophy of disgust spoke to Nietzsche with a voice he was ready to hear, a voice that seemed to confirm what he already suspected — that the optimism of his age was a thin veneer over a meaningless abyss. He stayed up all night sitting on the family sofa, reading The World as Will and Representation over and over again. 'Here where every line cried renunciation, denial, resignation,' he later wrote, 'here I saw a mirror in which I observed the world, life and my own soul in frightful grandeur.' Nietzsche was, by this point, extraordinarily well acquainted with his own soul, having written the autobiography of his emotional life at least six times (he would make it nine by the time he was twenty-three). He had already seen enough to convince him that the world was not an elegant system into which the individual could be inserted like a sprig in a barrel organ, but a dark, mysterious, violent struggle in which he must either fight or perish. Now, in the book he held in his hands, he had finally found a writer who was willing to admit this, and who seemed to offer a solution. Nietzsche came to realise through Schopenhauer that he had sound reasons for being dissatisfied with life — only a stupid man could find life satisfying, and Nietzsche knew he was not stupid. But, Schopenhauer insisted, the man of intelligence could redeem himself through music, philosophy and renunciation. Here was a man who could explain to the young Nietzsche why music had such a tremendous effect on him. Life is unbearable but music, incredibly, allows us to see it as beautiful. Thus, music becomes a consolation for suffering. The man of intelligence must take an aesthetic attitude to life. Finally, Nietzsche thought to himself, someone who understands me! For a period of two weeks, Nietzsche took this business of renunciation more seriously than Schopenhauer himself ever had. He horrified his mother by adopting a monkish lifestyle — keeping a very strict diet and depriving himself of sleep, human company and material comfort. He became obsessed with a certain atmosphere he'd detected in Schopenhauer, 'the ethical air', as he described it, 'cross, death and grave'. Nietzsche loved Schopenhauer for the same reason that he loved Wagner — as a teenager, he'd spent hours at the family piano pounding out the chords of Tristan und Isolde. Both confirmed his instinctive belief that the optimism of the nineteenth century was a sham, and that tragedy and violent struggle constituted the true essence of life. And since Wagner himself was so heavily influenced by Schopenhauer it seemed only natural to Nietzsche to develop a system of aesthetics that incorporated the two. In any case, as Colin Wilson points out in his classic study of the artistic personality, The Outsider, Nietzsche had by this point consigned every other major intellectual figure of his century to the scrap heap — there was no-one else left. 'Nietzsche stood alone,' writes Wilson, 'except for the two men for whom he still felt respect: Schopenhauer and Wagner. Three men against the world...but what men!' In 1868 Nietzsche, now a professor at Basel University, began writing what would become his first book, The Birth of Tragedy — a meditation on the origins of Greek tragedy out of what Nietzsche called 'The Spirit of Music'. Nietzsche had by this point become quite a close personal friend of Wagner, and a firm believer in the composer's propaganda. Wagner, in turn, was enormously impressed by the young professor. After reading the manuscript of The Birth of Tragedy, Wagner declared it to be the finest thing he'd ever read. But then, he would say that — Nietzsche had devoted the last quarter of his treatise on Greek drama and music to building a case for Wagnerian opera as the true revival of tragedy in the modern world and the future of music for Germany. This cost him his career — and in the long run, his sanity. After The Birth of Tragedy was published in 1872, Nietzsche was more or less laughed out of the academic world for good. Despite (or perhaps because of) his enormous popularity, Wagner was not considered cool in the Philology Department of the University of Basel, and Nietzsche misjudged the mood of his colleagues entirely by spending the last four chapters of his book mounting a vigorous argument in favour of a composer who catered to the tastes of emotional young girls and girlish, emotional young men. Nietzsche himself later disowned The Birth of Tragedy completely. He came to regard it as a ridiculous book, in which he'd tried to do something impossible — to reconcile the three things he happened to be into at the time — Schopenhauer, Wagner and Greek Tragedy — into a single system. The young professor jumps through hoops to make Schopenhauer more Greek and the Greeks more Wagnerian, and ends up falling on his face. But while The Birth of Tragedy is a deeply strange book, it's crucial to understanding Nietzsche's mature philosophy, (in which, as Colin Wilson has noted, he eventually came back around to the position he'd staked out in The Birth of Tragedy), and contains many striking insights in its own right, particularly as regards its central subject — tragedy. How are we redeemed by tragedy? asks Nietzsche, and what is the role of the tragic hero in relation to music? His answers have as much to teach us about Wagnerian rock as they do about Wagner himself. Freddie Mercury: Not to be born, not to be, to be nothing. # A Night at the Opera 'COME ONE, COME all to this tragic affair.' Gerard Way's opening words on The Black Parade let us know what we're in for immediately. A man will be wheeled out on stage, and we will be told the story of his life. He'll grow up, fall in love, fall out of love, face terrible obstacles and painful decisions. He'll come to understand, at the end of the show, the meaning of life — which is that life is a joke with a terrible punch line. And this knowledge won't help him a bit, because this is a tragedy, and the rules of tragedy say the hero must die. How can we stand it? The hero's theme song, 'Welcome to the Black Parade', is a multipart epic, which draws on a number of different musical styles. The first part is a stately ballad in the vein of 'My Way', the second act sounds a bit like Green Day, and the final section is pure Wall of Sound — the noise of teen angst inflated to epic proportions. But the single biggest influence on the song — and the album as a whole — is, as many fans and critics have noted, the operatic bombast of mid '70s Queen. In fact, the whole pocket epic form of 'Welcome to the Black Parade' is virtually unthinkable without the precedent of Queen's own rock opera classic 'Bohemian Rhapsody'. In 1977 Queen found themselves recording in the same studio as the Sex Pistols, and Sid Vicious decided to drop in and meet the neighbours. 'Hullo Fred,' said Sid Vicious. 'So you've really brought ballet to the masses then?' 'Ah, Mr Ferocious!' replied the flamboyant frontman. 'Well, we're trying our best, dear!'Welcome to the Black Parade's Queen-meets-punk arrangement imagines a parallel universe where Mercury had invited Mr Ferocious in for a cup of tea, and the two bands had ended up writing a song together. Two years earlier, in 1975, Queen's producer Roy Thomas Baker had dropped by Freddie Mercury's apartment in Kensington. The singer sat down at his piano and told Baker that he'd like to play him something new he'd been working on: 'So he played the first part and said "this is the chord sequence"... He played a bit further through the song and then stopped suddenly, saying, "This is where the opera section comes in". We both just burst out laughing.' It took weeks of painstaking work in the studio for Baker and the members of Queen to get Mercury's ambitious new song into shape. In the process, the 'opera section' grew and grew — 'just one more Galileo!' Mercury would insist, while Baker watched the master tape wear away to nothing. When it was finished, the band liked it so much they decided it would be their next single, an idea which was met with hoots of derision from their record company. A six-minute single with an opera in the middle of it? Are you mad? In the end, however, EMI's hand was forced by Capitol Radio presenter Kenny Everett, who broke the song on his show. 'Bohemian Rhapsody' was rushed into stores, and spent eight weeks at the top of the British charts. 'Bohemian Rhapsody' begins with the singer pondering the eternal romantic dilemma. His life is awful, but his dreams are beautiful — is it possible he has been deceived? Are his dreams real, and 'reality' simply a sham? 'Do I wake or sleep?' He tries, like Tristan, to wave these morgentraume away only to find that he cannot quit his prison so easily. The only escape route lies in death. Our hero is not quite ready to commit suicide, but now that the world has been exposed as a cruel deception, he can't really be bothered doing anything with his life either. What's the point? 'Anyway the wind blows', he sings, 'doesn't really matter'. The tragic hero has been afforded a glimpse behind the screen of bourgeois life and has seen the eternal chaos and flux of Schopenhauer's world as will — how can he be expected to show up to his classes or clean his room now? Here, the opera section kicks in, and 'Bohemian Rhapsody' turns from a lyrical poem into a mythic drama, in which Beelzebub and a chorus of angels battle it out for the hero's soul. This metaphysical argy-bargy recalls many similar scenes in Goethe's Faust. Faust, like the hero of Queen's epic, believes that life can show him nothing, and thus becomes the subject of a wager between God and Mephisto. God believes Faust's disillusionment with earthly pursuits will eventually lead him to religion. The Devil is certain he can get Faust interested in something. It was these alarmingly casual chats between God and Mephisto that led early British critics to condemn Goethe's drama for its 'blasphemous levity'. The light-hearted tone of Faust suggests that human life might be no more than a joke — and 'Bohemian Rhapsody' (which was funny long before it appeared in Wayne's World) leaves us with much the same feeling. In any case, the hero eventually storms out of his own scene of heavenly judgement, insisting that he can beat anything the Devil throws his way. He's gone from being a hopelessly static whinger, like the Byron of Childe Harold, to a proper romantic hero, the later Byron so admired by Goethe for his determination to push against all natural laws. 'So you think you can stop me and spit in my eye!,' he snarls. But he quickly runs out of steam. Reality, it seems, is a stone wall. Our hero falls back, exhausted, and makes ready to die — his tragedy has run its course. In The Birth of Tragedy Nietzsche repeats Schopenhauer's assertion that life can only be made bearable by philosophy and art. Nietzsche insists that this is what art is for; to create a space from which we can begin to see suffering on an enormous scale as an aesthetic phenomenon. He then goes on to fuse Schopenhauer's idea of redemption through non-willing with his understanding of Tragedy, formed during those marathon sessions at his parents' piano, and further honed by his long walks and talks with Wagner at Tribschen. For Nietzsche, the role of the tragic hero is to confront the world as will head on. He sees behind the veil of illusion, stares into the horror and is crushed by it. By doing this, he effectively takes the whole weight of existence on his back, and relieves us of its burden momentarily. By contemplating the nature of this burden, Nietzsche finds the link between Schopenhauer and the world of the Greeks that forms the foundation of his thoroughly mad, but strangely convincing book. Schopenhauer, bored out of his skull by his dumb clerical job in Frankfurt, had looked around him and realised that the meaning of life was precisely nothing. And given that life is painful, boring and pointless, he concluded that it would be far better in the grand scheme of things if the human race had never existed. The Greeks, Nietzsche insists, were well acquainted with this truth. Not the Greeks as the eighteenth-century classicists liked to imagine them, the white-marble world of clarity and Apolline perfection so admired by the likes of Joshua Reynolds and Gottfried Lessing; but the real Greeks, who knew Apollo as just one of many deities, and not the wisest among them. In The Birth of Tragedy, Nietzsche writes, According to the old story, King Midas had long hunted wise Silenus, Dionysus' companion, without catching him. When Silenus had finally fallen into his clutches, the king asked him what was the best and most desirable thing of all for mankind. The daemon stood silent, stiff and motionless, until at last, forced by the king, he gave a shrill laugh and spoke these words: 'Miserable, ephemeral race, children of hazard and hardship, why do you force me to say what it would be much more fruitful for you never to hear? The best of all things is something entirely outside your grasp: not to be born, not to be, to be nothing. But the second-best thing for you — is to die soon'. This, for Nietzsche, is the burden the tragic hero takes up on our behalf. And this is why he must die — because no-one can go on living knowing what he knows. The wisdom of Silenus condemns the hero to death, because understanding — as Nietzsche says of Hamlet — kills action. That's why we know, sometime between the moment when the young Bohemian realises that nothing really matters and the bit where he wishes he'd never been born at all, that he will not survive this drama. And if Gerard Way's spruiking his rock opera as a 'tragic affair' wasn't enough of a giveaway, we should know by the last line of the first verse where things are headed. As the song's acoustic strum gives way to an avalanche of orchestral noise and Brian May-style multitracked guitars, Gerard screams: 'When I grow up I want to be NOTHING AT ALL!'. This is the wisdom of the woods — or, if you like, the wisdom of the Frankfurt clerical office — which no-one can survive. The tragic hero knows it so that you don't have to. # The Wisdom of the Woods IN SEEKING THE roots of tragedy in Greek art and music, Nietzsche discovered that the Greeks knew 'two worlds of art, utterly different'. On the one hand is the Apolline world of the representational arts, of painting and prose; on the other is the wild abandonment of the Dionysiac, which produces music. Nietzsche, as Colin Wilson points out in The Outsider, understood the Dionysiac instinctively. Listening to Wagner, he felt the pull of the dance, of the passions, of the half-crazy impulses that lurk beneath our civilised exterior. It was the spirit of Dionysus, Nietzsche believed, that had inspired the folk dances that swept through Germany in medieval times. Nietzsche writes admiringly of the way the dancers allowed themselves to be pulled along by instinct, leaping, singing, shouting at the top of their lungs; they indulged their senses and lost their minds. It was this spirit, too, that Schiller caught in his 'Ode To Joy' — which is why the young German philosophers of the early nineteenth century liked to recite it while getting drunk and dancing around in the fields. But Nietzsche also observed that the spirit of Dionysus is not for everyone: Some people turn away with pity or contempt from phenomena such as these 'folk diseases', bolstered by a sense of their own sanity. These poor creatures have no idea how blighted and ghostly this 'sanity' of theirs sounds when the glowing life of the dionysiac revellers thunders past them. This is still a problem in nightclubs today, as The Chemical Brothers' 'The Salmon Dance' shows. Typically when poets address small animals in verse no-one really expects the animal to answer back. All this changes in 'The Salmon Dance' where the poet, MC Fatlip, actually invites the fish into the studio to trade a few lines. Unfortunately, the fish has not a scrap of romantic sensibility, he talks like a nature documentary produced for school children in the '70s. 'My peeps spend part of their lives in fresh water, and part of their lives in salt water,' he drones. 'Wow, very interesting,' says Fatlip unconvincingly. Fatlip never promised us poetry; he told us we were going to learn fun facts about salmon, and a brand new dance. But the facts about salmon are less fun than we had been led to expect, and the brand new dance is, at first, a disaster. Fatlip finds that he is the only one in the club doing the salmon, hands pressed to his sides, swaying like a fish swimming upstream. Everyone else just stands there looking at him (bolstered by a sense of their own sanity). 'What the fuck is that?' they say to each other. But by the end of the song, everyone is dancing like a salmon. What changed? Simple, Nietzsche would say. The people in the club simply surrendered to the Dionysiac urge. They gave up the struggle to maintain a rational attitude to an irrational world, and immersed themselves in the ceaseless flow of life, the same flow that sends a salmon swimming upstream, the very will that pushes the world and everything in it along its purposeless course. Singing and dancing, man expresses himself as a member of a higher community: he has forgotten how to walk and talk, and is about to fly dancing into the heavens...he gives voice to supernatural sounds: he feels like a god. The romantics tended to see all art as an attempt to say things that could not be said any other way. Paintings, poems and symphonies will all, in the end, resist our attempts to analyse and explain them — as though turning art back into ordinary language were like translating a newspaper article from French into English. And of all the arts, music is the hardest to explain because music, unlike painting for example, doesn't represent the world. Music does not present facts — it is a fact. (Musicians still use this romantic article of faith as a way of not answering interview questions when they say 'the song speaks for itself' or 'it's all there man...') This lead Schopenhauer to the curious (but poetic) idea that music must be made of the same stuff that life is made from. It is, in other words, a pure expression of will. This explains music's effect on us: in surrendering to the power of music, we feel ourselves transported back to a primitive state, outside custom and convention, pulled along by the same forces that cause the grass to grow and the fish to swim upstream. Nietzsche, in The Birth of Tragedy, agrees with this theory, and uses it to explain how it is that the lyric poet comes into contact with the world as will, in order to bring back the terrible insights that he later shares with us. Nietzsche observes that Schiller, when asked how he composed his poems, replied that he never started with a preconceived idea, but rather with a certain 'musical mood' that came over him. This, Nietzsche notes, squares with the origins of lyric poetry itself, which the Greeks always recited to the accompaniment of music. The conclusion he draws is that the poet, when this musical mood comes over him, is absorbed in the spirit of music — which is a manifestation of will. Thus he confronts the great metaphysical truths hidden from the rest of us, and somehow lives to tell the tale. This all sounds a bit far-fetched, to be sure — but it would explain how it was that Keith Richards accessed the wisdom of the woods in a hotel room in Florida, and how the lyrics that came to him from that musical mood could subsequently go on to change the world. In The Birth of Tragedy Nietzsche quotes Wagner as saying: 'Civilisation is annulled by music as lamplight is annulled by the light of day.'Lamplight is a symbol of our mastery of nature; scientific man on the move, shining a light into the dark spaces of the world. Music is nature's revenge. It sneaks up on us, attacking via — what Nietzsche called 'the organ of fear' — the ear. Music cracks our civilised veneer, one blast, and we turn back into cave people, standing dumbstruck before a thunderstorm. Nietzsche's view of the world as will is, as you've no doubt noticed, slightly different from that of Schopenhauer. Right from the beginning, his descriptions of the flux and chaos of life are shot through with a feeling of excitement that would be entirely abhorrent to the older philosopher. Nietzsche still sees will as irrational — and in some ways he still sees it as evil. But when he contemplates this evil, he finds that it makes him feel good. In 1865, the same year he discovered Schopenhauer, Nietzsche wrote to his friend von Gersdorff: Yesterday an oppressive storm hung over the sky, and I hurried to a neighbouring hill called Leutch... At the top I found a hut, where a man was killing two kids while his son watched him. The storm broke with a tremendous crash, discharging thunder and hail, and I had an indescribable feeling of well-being and zest... Lightning and tempest are different worlds, free powers, without morality. Pure Will, without the confusions of intellect — how happy, how free. Even as he was being seduced by Schopenhauer's view of reality, Nietzsche was turning it on its head by treating it as positive, rather than negative. Instead of renouncing life, he would embrace it — all of it. That's why Nietzsche approves of music and dancing, but also of electricity and bloody violence — all represent will, and will for Nietzsche is sublime. Dave Gahan: a new sense of power. # Personal Jesus FOR TEN YEARS after the fiasco of The Birth of Tragedy, Nietzsche embraced philosophy — truth became his goal, and thought was exalted above emotion. But he found he couldn't keep this up for long. In 1882's The Gay Science he declared himself disgusted with the 'will-to-truth' of the philosophers. From this moment on, he said, he would embrace life, not thought. 'I wish to be at all times hereafter only a yea-sayer,' he wrote. By being determined to say 'yes' to life, Nietzsche became the natural enemy of Christianity with its seven deadly sins and its thou-shalt-nots. For him, what Christians call 'good' behaviour is perverse. In 1883's Also Sprach Zarathustra, Nietzsche's wild-eyed prophet drops this reversal of morals on the unsuspecting townspeople. 'It is not your sin, but your moderation that cries to heaven,' Zarathustra tells his baffled audience. This news has turned their world upside down, but it's all in a day's work for the prophet. It's a sin! the people say. So what? says Zarathustra. Sin more! 'Your very meanness in sinning cries to heaven!' Zarathustra sets himself as a Christ-in-reverse: 'It may have been good for that preacher of the petty people to bear and suffer the sin of man. I, however, rejoice in sin as my great consolation.' Zarathustra's reversal of morals is almost incomprehensible to his audience — but quite familiar to us after fifty years of rock and roll. The rock star knows instinctively that one must say 'yes' to life, and his career serves to demonstrate Nietzsche's philosophy in practice. In 1989 Depeche Mode singer Dave Gahan — the lapsed church-goer whose faith had been shattered five years earlier by the experiences described in Blasphemous Rumours — decided that Christian morality was something he could do without. He decided, as he put it, 'to become a monster... I wanted to live that very selfish life without being judged'. Depeche Mode's 1990 album, Violator, had cemented his anti-religious stance with searing indictments of the confessional ('Policy of Truth') and of Catholic guilt ('Halo'). And on the previous year's 'Strangelove', Gahan had hinted that he might try being a sinner — if only to stave off the boredom of a meaningless existence: I give in to sin Because you have to make this life liveable. After the extraordinary success of Violator and the Music for the Masses tour, Gahan suddenly found himself with the means to find out just how much sinning he could do. Accordingly, on the band's next tour, Gahan drank, snorted and shagged his way around the world, alienated everyone who ever cared for him, became addicted to heroin, destroyed hotel rooms, broke up his marriage and nearly got kicked out of his own band. And it all felt...fantastic. 'I'd be lying if I said it didn't make me feel...like I'd never felt before. Like I belonged. To what, I've no idea.' Thanks to Nietzsche, we are in a position to fill in the blanks in Gahan's account. The singer felt like he belonged because he was living authentically, according to his desires. He had said yes to life, and this, as Nietzsche discovered on that day when he watched the storm break, feels incredible. This is romantic optimism in a nutshell. In his History of Western Philosophy, Bertrand Russell says that the individual who throws off social bonds and indulges his instincts gets 'a new sense of power from the resolution of inner conflict'. He already sees himself as actively inspired, rather than passively 'getting along'. Now he wonders if he hasn't become some new kind of human being — outside convention, social bonds and even morality. Nietzsche would say that he has; he called these exceptional individuals 'artist-tyrants', and insisted that we could not expect them to operate according to society's laws. 'Morality,' Nietzsche declared, 'is the herd instinct in the individual.' To live in society while demanding the right to ignore its rules makes no sense — and the romantic individualist knows it. That's why he tends to justify his irrational philosophy by claiming the authority of a mystic, or a prophet. The 'fire inside' from which romantic poets and philosophers draw their inspiration is not unlike the voice that speaks to the prophet or the saint, which is why romantics — from Goethe to Nietzsche, Springsteen to Gerard Way — slip so easily into those roles. To do what they do, these individuals believe they must obey the true voice of feeling in their hearts, which inevitably means they must, to a large extent, renounce the practical, material world. When the strain of keeping this up becomes too much, they quickly make the transition from saint to martyr — which is why Bowie describes Ziggy as a 'leper messiah', and Billy Corgan imagines 'secret destroyers' roasting him over flames in 'Bullet with Butterfly Wings'. Gahan, who ascended to megastardom on the back of a spooky glam-rock stomp called 'Personal Jesus' saw his way clearly marked out for him. In 'Personal Jesus' he'd proposed himself as a secular messiah: first in the intimate context of the song, as one to another; and then, inevitably, in the stadium, where he'd invited his fans to reach out and touch faith. Now, Gahan began to grow his hair long and to cultivate a beard. He took to appearing onstage shirtless, with his arms spread in a crucifixion pose. The follow-up to Violator was full of gospel choirs and lyrics about repentance and salvation, and the band named it Songs of Faith and Devotion. In 'Walking in My Shoes' the singer's stance is Byronic — he's done bad, bad things, he tells us. But don't think for a second that he was just having a good time. He also suffered terribly for his reversal of traditional values, and his belief that moral laws should be destroyed: I'm not looking for a clearer conscience Forgiveness for the things I do but before you come to any conclusions try walking in my shoes. Gahan's story — like all rock and roll tragedies, had a spiritual rather than a moral purpose. The singer was not a moral example; he was a martyr to a new, anti-Christian faith. And like the Nazarene preacher he not-so-subtly evoked on stage, he inspired followers. Both Marilyn Manson and Nine Inch Nails' Trent Reznor were as much inspired by Gahan's gloomy nihilism and Dionysiac excess as they were by his band's industrial-strength synth-pop. Both deliberately set out to explore the limits of morality by identifying with anti-social monsters like Charles Manson and flirting with images of evil — Nazism and Satanism. Marilyn Manson's reversal of good and evil inevitably lead him to Nietzsche, whose aphorisms he paraphrased in interviews, and who could have written the lyrics to 'Beautiful People' himself: It's not your fault you're always wrong The weak ones are there to justify the strong. Like Gahan, Manson practised what he preached, embracing hedonism with a vengeance, and going so far as to style himself as 'The God of Fuck'. But by 2004 he was feeling like a martyr to his own revolt. In an interview for Kerrang! entitled 'Twilight of the Gods' (a reference to Wagner's opera of the same name), he claimed that his new greatest hits collection represented 'ten years of fighting to get where I am', and that he'd decided to cover 'Personal Jesus' because, '"Personal Jesus" says more than anything I could say myself right now'. Nietzsche — who spent many years in exile, driven by neglect into a state of acute paranoia, reached the end of his life in a similar state. In 1889, just before he went completely insane, he began signing his letters 'The Crucified One'. It was no great leap for Nietzsche to imagine himself as a martyr, because he had always seen himself more as a prophet than a philosopher. Philosophers sit down with the works of other great philosophers and subject their methods to empirical tests to see if they hold true — if they don't, they reason their way to new truths. For Nietzsche, such people were 'blockheads'. His insights came from his direct experience of the world, visions that descended upon him as he contemplated nature and his own soul. 'I have seen thoughts rising on my horizon the like of which I have never seen before,' he wrote in a letter, around the time he was working on The Gay Science. He went on: The intensity of my emotion makes me tremble and burst out laughing. Several times I have been unable to leave my room for the ridiculous reason that my eyes were swollen — and why? Each time I have wept too much on my walks of the day before — not sentimental tears, but actual tears of joy. I sang and cried out foolish things. I was full of a new vision in which I forestalled all other men. Nietzsche's temperament was religious, but it was also artistic. His insights came from intuition rather than intellect, and this put him much closer to the poets, painters and composers than to most philosophers. Because of this, he understood instinctively that the artist is always amoral. This, as you can imagine, was a quality he admired a great deal. In The Gay Science he asks: 'Do you suppose that Tristan and Isolde are preaching against adultery when they both perish by it?' Nietzsche saw that for the great artist, the pursuit of beauty outweighed all other considerations, including moral ones. Furthermore, he knew that this approach was fundamentally right — that is, that it was desirable to take an aesthetic attitude to life. That's why, to him, the artist was a 'higher man'. Artists have no use for morality, because their only allegiance is to beauty. And since beauty redeems us from suffering, no-one could say that their attitude is wrong. The only problem for the 'higher man' is that of the masses — the little party of regular folk from Don Giovanni who seek to impose their standards of normalcy on the fearless artist-hero. This is what finally defeated Nietzsche — his books were full of earth-shattering revelations, but the critics and academics saw only the ravings of a lunatic. This, too, was the snag Dave Gahan ran into as he tried to live his life artistically. In 'Condemnation', the artist–martyr shakes his fist at the heavens and demands to know why he is made to suffer. Of course, he already knows the answer: 'My duty was always to beauty,' he confesses. 'That was my crime.' Kanye West: Power increases, resistance is overcome. # Stronger WHILE HE ADMIRED Napoleon a great deal, Nietzsche had nothing but contempt for the French Revolution, with its Liberty, Equality and Fraternity. Democracy to Nietzsche is a travesty. In a democracy, or any other variation of 'rule by the people', the vision of a great artist-tyrant can be compromised and undone by the petty wants of the bungled and the botched — the masses. Nietzsche says the suffering of one great man is more important than the suffering of millions of ordinary people. 'What do the rest matter?' he asks. 'The rest are merely mankind — one must be superior to mankind.' Nietzsche always sides with the individual genius against the world that doesn't understand his vision. This is exactly the point of view expressed by Kanye West in his 2007 hit, 'Stronger'. 'There's a thousand yous there's only one of me,' raps the artist-tyrant. Accordingly, when it came time to give his song a chorus, Kanye fused the musical mood of Daft Punk's 'Harder, Better, Faster, Stronger' with one of Nietzsche's best-known aphorisms: 'What does not kill me makes me stronger.' The song's action takes place at a nightclub. We fade in on Kanye West tuning the 'black Kate Moss'. Like all romantics, Kanye takes a dim view of convention and the artificial refinements of modern life — looking around, all he sees is fake shit. 'Does anybody make real shit anymore?' he asks, rhetorically. For Kanye — as for Nietzsche and Dostoyevsky — authenticity is a big deal. But while he knows he's more authentic than everybody else, he's also become lonely as a result. He believes he can still be redeemed by love, and yet it seems that the perfect union of souls he imagines is constantly under threat from the world, with all its worn-out morality. Before he and the black Kate Moss have even swapped digits, he's already lost in narcissistic fantasies in which he and his soul mate fly away to another world: Let's get lost tonight You can be my black Kate Moss tonight... Y'all don't give a fuck what they all say, right? Awesome, the Christian and Christian Dior. Kanye, as is fairly well known, is a Christian. But at this same party, barely thirty seconds into the same song, we find him standing on a table with a few Cristals under his belt, espousing some very un-Christian sentiments: Bow in the presence of greatness, 'Cause right now thou hast forsaken us You should be honoured by my lateness Kanye's philosophy as presented in 'Stronger' is clearly much closer to Nietzsche's than to any flavour of Christianity. As a student of philology, Nietzsche was always fascinated by the Titans — the monstrous race of super-beings who spawned the Greek deities. He was, as Rüdiger Safranski observes in Nietzsche: A Philosophical Biography, far more impressed by those who make gods than by the gods themselves. In The Birth of Tragedy he relates how the Titans became the first tragic heroes of the Greek stage. Oedipus and Prometheus pushed against all natural laws, and when they died, they died heroically, for the sake of their ambition. Nietzsche warns his readers at the start of the book that if they approach the world of the Greeks looking for the type of morality found in the New Testament — or in the sickly productions of the nineteenth-century stage — they will be sorely disappointed. In stark opposition to the myths of Christianity — in which humankind is always punished for its sins — Nietzsche places the myth of Prometheus, who is heroic in his determination to push against the limitations placed on him by Zeus, who sees him as a threat. Hence Nietzsche's admiration for strength, and the importance he places on the testing of will: 'What does not kill me, makes me stronger.' In this, the tragedy of Kanye West is exactly the type Nietzsche would admire. In 'Stronger', Kanye's will is constantly being tested by haters. But the rapper insists that he will prevail: he will continue, in the face of ridicule and indifference, to preach 'the new gospel', and he will impose his forms on the world. This applies to his public life as much as his art. At 2007's Video Music Awards, Kanye — having learned that he'd lost the best video award to Justice vs Simian's 'We Are Your Friends' — crashed the stage. 'Oh, hell no!' he exclaimed, interrupting Justice's acceptance speech and angrily protesting that his video for 'Touch the Sky' should have got the gong. He started out listing its merits, 'This video cost a million dollars! I got Pam Anderson! I got 'em jumpin' across canyons and shit!' But this was not really the point. Kanye knew his video should have won because he knew it was the best video. Even as he was apologising to the bemused members of Justice whose acceptance speech he'd hijacked, he was insisting that his video was better than theirs, even though he'd never seen it. 'It's nothin' against you man, I've never seen your video.' Kanye then went on to suggest that the show's judges were effectively sabotaging the credibility of their own show by not giving him the award, after which he suggested that the whole show might as well go fuck itself. Couldn't he be more polite and gracious? Wouldn't that be good? What do you think Nietzsche would say to that? What is good? All that heightens the feeling of power, the will to power, power itself in man What is bad? All that proceeds from weakness What is happiness? The feeling that power increases — that resistance is overcome. By his determination to impose his forms on the world, regardless of the petty complaints of the bungled and the botched, Kanye has become one of Nietzsche's 'Higher Men'. Nietzsche first encountered the higher man in the person of Lord Byron. In his youth Nietzsche had an intense admiration (as all gloomy young men of the nineteenth century tended to do) for the author of Childe Harold and Manfred. Nietzsche admired Byron's energy and drive, and the way he seemed to live the exploits described in his poetry and to embody the characters he described — as though his life and his art were one. It could be argued that Byron was playing a role. But it was a role he could only play because he knew it. 'One cannot guess at these things.' Nietzsche wrote, 'One simply is it or is not.' To Nietzsche, Byron seemed superior to other men because he was active, immoral and free from restraint. Later in life, after shifting from his initial ultimate yes, to a brief dalliance with ultimate no, and then back to yes again, Nietzsche started to dream into existence his ultimate yea-sayer — a man who could say yes to all of life, for whom there were no limitations, no restraints. Would such a man, he wondered, be something a little more than human? In coming up with a name for this new creature, Nietzsche revived a word he'd first used to describe Byron in his student days — Ubermensch — Superman. Richard Strauss: The dance that everybody forgot. # Also Sprach Zarathustra IN THE SUMMER of 1972 Elvis Presley played a record-breaking run of dates at the Sahara Tahoe hotel in Nevada. Elvis was in high spirits, the band was on fire, and the set-list — including 'You've Lost that Loving Feeling' and 'The Impossible Dream' — was generally regarded as topnotch. But what really set the tone for the show was a number Elvis didn't sing on at all. It was his walk-on music, a majestic theme beginning with a simple but tremendously powerful sequence of three notes — C, G, C. This little fanfare was written not by bandleader Joe Guercio or any one of the dozens of songsmiths-for-hire in Freddy Bienstock's little black book but by a Munich-born romantic composer named Richard Strauss. Like every other German-speaking composer of the late nineteenth century, Strauss grew up in the enormous shadow of Richard Wagner, and his operas and symphonies sometimes sound like they're trying to out-Wagner Wagner for sheer emotional drama and intensity of sound. Sometimes, he comes close — as in the piece Elvis included in his show at the Sahara Tahoe. The King's walk-on music began life as the opening theme for a symphonic tone poem Strauss wrote in 1896 called Also Sprach Zarathustra — inspired by Nietzsche's book of the same name. Strauss had first read Nietzsche four years earlier while on a holiday in Greece, and it had transformed him — and his music — immediately. After reading Nietzsche's denunciations of Christianity, Strauss tore up the last act of his (very Wagnerian) opera, Guntram, and re-wrote it so that it ended with the hero turning his back on society and organised religion and going it alone. Friends were horrified, and advised him to go back to his Bible — but Strauss, ever the romantic individualist and now a confirmed Nietzschean, refused to budge. Three years later he started work on his homage to Also Sprach Zarathustra. Strauss later explained his aims in composing the piece: I did not intend to write philosophical music, or to portray in music Nietzsche's great work. I meant to convey by means of music an idea of the development of the human race from its origin, through the various stages of its development, religious and scientific, up to Nietzsche's idea of the superman. The human race, Nietzsche explained in Also Sprach Zarathustra, is in a transitional phase. Or rather, the human race is a transitional phase. We are, as Nietzsche puts it, standing on a makeshift rope over a great abyss. As we sway uncertainly over the deep, we look behind us and see our past — the ape — and when we do this we feel pretty proud of ourselves. Zarathustra points us toward the other end of the rope, ahead of us in the distance. That, he says, is where your destiny lies; the new form humanity must take; the means by which man will be superseded — just as man superseded the ape. 'Behold,' says Zarathustra. 'I teach you...the superman!' Humankind, in Zarathustra's opinion, has wasted a lot of time and effort trying to safeguard the future of humankind. What we should really be asking ourselves is: 'how shall humankind be overcome?' The superman is this next stage in human evolution. And it's this great event that Strauss's magnificent fanfare is meant to announce. Joe Guercio, who would arrange Strauss's piece for Elvis's band and lead them through it hundreds of times in the '70s, first heard Also Sprach Zarathustra the same way most of us did — watching Stanley Kubrick's 2001: A Space Odyssey. The film begins on prehistoric earth, at the moment when man first learns to impose his will on the world (by picking up a bone and using it to smash things), and eventually progresses to impose his will on others (by smashing them in the head with the very same bone). Kubrick marks this as the moment when man takes the first step toward his destiny by scoring the scene with Strauss's Zarathustra — and then cheekily suggests that we haven't got much further in the last few hundred thousand years by cutting from a shot of the bone flying through the air to a bone-shaped spaceship tumbling through the void. Kubrick's vision of humanity in the twenty-first century is a little like Nietzsche's estimation of the nineteenth. 'Even now,' he wrote, 'man is more of an ape than any ape.' Sitting in the movie theatre watching 2001, Joe Guercio heard Strauss's massive chords heralding the arrival of humanity's successor and thought of Elvis. Elvis, watching the film a few days later, thought of himself. This is not too surprising. The idea that Elvis might in fact be some completely new kind of human being — or not a human being at all — had been implicit from the moment he first appeared on TV in 1956. (Charles Laughton once introduced him to the audience as 'that man, Elvis Presley' as though he didn't know what he was.) Ten seconds later, he'd awakened teenage America's suppressed longing for Dionysiac revelry, lifted the Judeo-Christian God's veto on the passions, and signalled a complete reversal of morals that would last, happily, until the present day. 'Yeah, that was the dance that everybody forgot,' said country singer Butch Hancock — echoing Wagner. 'It was the dance so strong it took an entire civilisation to forget it, and ten seconds to remember.' An American evangelist famously grumbled that Elvis was 'morally insane'. Zarathustra had warned in 1882 that the superman would be 'a destroyer of morality', and that his arrival would be heralded by madness and lightning. Ziggy Stardust: Free power, without morality. # Homo Superior IN 1969 RCA records released a single called 'Space Oddity' that was perfectly timed to cash in on the popularity of Kubrick's film. The song was a strange, psychedelic folk number — a meditation on cosmic alienation, sung by a man who lives outside everything in a tin can in space. 'Space Oddity' was a hit and David Bowie, who'd been mounting a series of increasingly desperate-looking attempts on the charts since the mid '60s, finally breathed a sigh of relief. But Bowie was not quite home and dry. Two years later, 'Space Oddity' was starting to feel more like a millstone around his neck than a foot in the door to superstardom. He was in danger of disappearing into the 'where-are-they-now?' file: David Bowie? Oh yeah, the guy with the stylophone! What happened to him? In 1971 Bowie released The Man Who Sold the World, and its cover appeared to be yet another attention-grabbing Bowie stunt — the singer wearing a flowing blue dress and reclining on a divan, playing with his hair. The photo was based on a painting by onetime Pre-Raphaelite Dante Gabriel Rossetti, and the mood of medievalism persisted on listening to the album. But this was no exercise in Tolkienesque whimsy — Bowie's new songs had taken on an apocalyptic tone — he sang about madness, death, and in the last song, a race of long-dead supermen. The following year, Bowie finally caught a break: a song he'd written and sold to ex Herman's Hermits singer Peter No-one became an unlikely hit. 'Oh You Pretty Things' is one of Bowie's catchiest songs, but it's also one of his most frightening. What starts out as a normal day, getting dressed, making breakfast, takes an extraordinary turn in the very first verse. The singer, stirring his coffee, looks out the window and sees a great hand coming out of a crack in the sky, reaching toward him. The singer has been chosen as a prophet, his task is to announce the end of the human race as we know it. 'Homo sapiens', says the singer, 'have outgrown their use. You'd better make way for the homo superior'. The singer's prophecy soon came true, as the ubermensch arrived on earth only six months later. He did not appear, as Nietzsche might have imagined, on a dramatic mountain peak silhouetted by a flash of lightning — but in the slightly less impressive surrounds of the Toby Jug pub in Tolworth. The lightning was provided by a hand-painted banner hung from the back of the stage — a red flash zigzagging across a white disc. In front of this stood a rock and roll band with 'The Spiders from Mars' painted on their drum kit. And towering over all (thanks to a pair of shiny stack-heeled boots) was the homo superior: David Bowie, now reborn as the alien rock singer Ziggy Stardust. Bowie as he later admitted, 'always had a repulsive need to be something more than human'. Bowie had read his Nietzsche, along with his Brecht and Burroughs. He'd also spent a lot of time watching Stanley Kubrick films like 2001 and A Clockwork Orange, from which he'd picked up the idea, as he later put it, that 'nothing was true'. Over the next two years, the singer would set out to prove it. What really pushed Bowie over the top was a 1972 interview in Melody Maker wherein the singer declared: 'I'm gay', and a nation choked on its tea. This was the first open admission of homosexuality by a British pop star. It was also, it later transpired, a flat-out lie — but Bowie had a point to make, and the fuss his confession caused served his purpose well. His new plastic pop star had to be seen to be a destroyer of morality because morality, as Nietzsche said, is the herd instinct in the individual, and Bowie was never going to be one of a herd. The Melody Maker writer picked up on this immediately. Noting that Bowie, while claiming to be gay, refused to identify with or make himself available to the cause of gay lib in Britain, Michael Watts concluded that the only cause David Bowie was really interested in was David Bowie. 'It's individuality he's really trying to preserve.' 'Starman', the first single from 1973's The Rise and Fall..., was the sound of Ziggy beaming in a message through the static and the space junk. The kids huddled around their radio in the middle of the night can just make out Ziggy's hazy cosmic jive. 'Let the children lose it...fzzzzzt...let all the children boogie...' It's been suggested that Ziggy's message, and his mission on earth, was supposed to bring peace and love to humanity, but this is not at all what he meant to do. Ziggy's arrival (heralded by a new star in the sky), his martyrdom and his resurrection undoubtedly make him a Christ-like figure, but his message is not one of tolerance, forgiveness and brotherly love. He wants us to realise our potential ('use it'), discard our moral standards ('lose it') and — in a final directive which combines the previous two in their highest form of expression — dance ('boogie'). But he is not a man of the people — he is aloof, superior and aristocratic. British critic Herbert Read has suggested that Christ's sermon on the mount — 'love thy neighbour' — contains the essence of the democratic ideal. Ziggy, like Nietzsche, wants nothing to do with this ideal. He is the 'special man' — the strong individual who acts in defiance of his community, the one who realises his visions at the expense of others. Ziggy is able to do this partly because he's an alien — but mostly because he's an artist. In 'Star', Ziggy contemplates the suffering of the world, and realises that it can only be redeemed by art. 'I could make it all worthwhile as a rock and roll star,' he muses. His ability to view suffering as an aesthetic phenomenon means he is unlikely to be troubled by morality. Bowie chose the lightning bolt as Ziggy's insignia because lightning is the perfect symbol for such an individual. Lightning, for Nietzsche, represented 'free power, without morality'. His superman was a result of his attempt to imagine a man who could accept, and conduct such vast energies. Nietzsche is not asking us to imagine some God-like being — a blonde giant with lightning coursing through his veins. The Superman, for Nietzsche, is nothing more than a man who can accept everything — beauty, sadness, joy, madness, the awful, destructive force of nature itself, and still say 'yes' to life. That's why Zarathustra asks the people: Where is the madness that will cleanse you? Where is the lightning to lick you with its tongue? behold: I teach you The Superman he is this madness, he is this lightning For his follow up to Ziggy Stardust, Bowie fused these two supremely anti-social motifs — madness and lightning — into one image. The cover of Aladdin Sane shows Ziggy with a lightning-flash painted across his face and a mercury tear pooling on his snow-white collarbone. Bowie's magpie eye had first spied this lightning logo on the equipment cases used to carry the band's gear to his shows — 'Danger: High Voltage'. Contemplating the vast energies that surge through the mains power supply and into the Spiders' amplifiers, Bowie had an epiphany — a modern-day variation on Nietzsche's lightning-storm. Bowie's rock star ideal would be a man who could accept these vast energies and dispense them freely, joyfully, immorally. It worked — as Mojo's Ben Fisher must have realised when, in 1997, he described Bowie's guitarist Mick Ronson as 'a Nietzschean ubermensch, hatched straight from Bowie's consciousness'. And it may have worked too well — within a couple of months the lighting flash seemed to have been appropriated by a group of rival super-beings. 'I was not a little peeved when Kiss purloined it,' Bowie later recalled. 'Purloining, after all, was my job.' # Destroyer LIKE WAGNER, PAUL Stanley understood the need for new myths in the wasteland of modern life. His first band, Wicked Lester, was going nowhere precisely because it had failed to grasp this principle. One night in 1972, Stanley's bandmate Ace Frehely symbolically killed Wicked Lester when he wrote the band's new name — which Stanley had just recently come up with — over the old one on a poster outside a club. Frehely took out a texta and wrote the word 'KISS', stylising the two 's's to make them look like twin lightning-bolts. In re-inventing themselves as a readymade rock and roll myth, the members of Kiss combined two ideas — both of which are Nietzschean. One is the lightning motif, the other is the Superhero. Inspired by the Marvel and DC comics they loved, the members of Kiss transformed themselves from a mere rock band into a league of supermen — the demon, the cat-man, the space-man, the star-child. Kiss was the first band to spell out the connection between power-chords and super-powers, first on their album covers and in their live shows; and later when they became stars of their own Marvel Comics series in 1977. The rock and roll superman now shared shelf-space in the newsagents with the real Superman. In Men of Tomorrow, Gerard Jones traces the development of Jerry Siegel and Joe Schuster's original man of steel, Superman, to Nietzsche's ideas, which by the 1930s were popular not just in Germany — where he had been ignored for so long — but in the United States as well. In Siegel's 1932 short story, The Reign of the Superman, the teenage author re-imagined Nietzsche's ubermensch for the readers of his self-published magazine, appropriately titled, Science Fiction — The Advance Guard of Future Civilization. Like many Nietzscheans of the early twentieth century, Siegel misunderstood the ubermensch — in The Reign of The Superman, he becomes a fantasy of personal power instead of a religious idea of transcendence. But Nietzsche might still have enjoyed Siegel's story, in which the protagonist is not the upholder of 'truth, justice, and the American way' he would later become, but a moral monster, who uses his super-powers to trample and destroy. Superman, in his final form, would vow to constrain his elemental power in the interests of good-old-fashioned morality. But the danger implied by such power would be constantly invoked over the next seventy years of his life — not just in the Superman comics, but throughout the Superhero universe. This barely concealed threat is what draws the rock singer to the super-being. In the storylines of Superhero comics, the questions of power and responsibility that nag at the thoughtful young rock star are played out on a cosmic scale. The rock singer, as a romantic outsider, senses the tremendous power that awaits the individual who throws off social bonds forever and enjoys free energy without morality. But the singer is also burdened with a feeling of social responsibility — a burden which becomes heavier as the band's audiences start to fill stadiums. In the comic books, these two conflicting ideas are usually split between the Supervillain and the Superhero — the first is a threat because he is powerful and selfish; the second is still powerful but feels he must help others. Glam rock stars like Bowie or Kiss tend to be more like Supervillains than superheroes — they're a-moral destroyers and corruptors, whose existence poses a threat to the status quo. But in the mythical universe of My Chemical Romance, it's not so simple. By embracing symbols of death and evil, and by portraying himself as a Byronic super-sinner, Gerard Way at first seems to be of the same type as Ziggy — who he admires intensely. But Gerard is too moral (and, as we'll see later, too democratic) to embrace this idea completely. He knows that a Kiss comic book will very likely inspire its young reader to want to play guitar, score with groupies and flirt with Satanism. But he hopes that the kids who buy My Chemical Romance action figures will grow up to be super-heroes, not a super-villains. They might accidentally destroy civilisation with the awesome power of their shredding; but they'll use that same power to rescue babies from the rubble after they've done it. Rilke: Love your loneliness. # Such a Special Guy IN 1992, RIVERS Cuomo was still hiding in the garage with his X-Men comics and Kiss posters. Back then, he was worried we might call him a nerd or a dork. Now, he couldn't care less what we think of him. He has become the thing he used to dream about — an axe-guitar-wielding superman, a not-so-teen Titan. He can do what he likes — in or out of the garage. In 'Pork and Beans' he sings: I'm-a do the things that I wanna do I ain't got a thing to prove to you With 2008's The Red Album, Rivers Cuomo seemed finally to have busted out of the underground. The self-loathing and self-denial of the late '90s was long gone. Part of Cuomo's disgust with the cult of Pinkerton stemmed from his belief that emo's insistence on misery and the inability to act was profoundly unhealthy. Andy Greenwald has observed that post-Pinkerton emo — the period that produced lyrics like 'I'm afraid to try, I'll keep my hands by my side' — was defined by 'an arrogance derived from superior humility'. Greenwald's description of emo ethics here echoes Nietzsche's thoughts on Christianity in his On the Genealogy of Morals. In Nietzsche's view, Judeo-Christian morality is a fairy story invented by the weak to justify their weakness. 'They are stronger,' say the oppressed, 'but we are more virtuous.' For Nietzsche, nothing could be further from the truth: All truly noble morality grows out of triumphant self-affirmation. Slave ethics, on the other hand, begins by saying "no" to an 'outside' an 'other' and that "no" is its creative act. A rock-and-roll superman could have no use for such perverse ethics. With The Red Album, Cuomo traded slave morality for triumphant self-affirmation — emo self-denial for the Nietzschean philosophy of Queen's 'We Are the Champions'. The new Cuomo had no time for losers. 'One look in the mirror and I'm tickled pink,' he sang in 'Pork and Beans'. His mood was proud and defiant. No wonder he never fit in — he's not one of a herd, but a lone, inspired individual. In 'Troublemaker', Cuomo boils the romantic philosophy down to one sentence: 'There isn't anybody else exactly quite like me.' While the self-asserting superman of The Red Album might seem worlds away from the human wreckage at the centre of Pinkerton, one very important trait connects the old Rivers to the new Rivers. In 'Troublemaker' he reminds us he's a big star, and that everyone wants a piece of him. But the grabbing hands will never touch him. You won't see Rivers out having fun like everybody else — and even if you do, you might as well not be there. Here's why: When it's party time Like 1999 I'll party by myself because I'm such a special guy In this one important respect, the new Rivers is not that different to the old Rivers. Because whether he's living in a black box, hiding in the garage, meditating, or just partying by himself, Cuomo needs solitude. But it's not just because he enjoys it, and it's not even because he needs the angst. The success of The Red Album shows that meditation wasn't so bad for his art after all. Turns out it wasn't the angst he needed so much as the isolation that produced it. It took him fifteen years to figure this out — Cuomo might have saved himself some trouble by reading Rainer Maria Rilke's Letters to a Young Poet. Rilke was a Czech-born poet whose first mature work was produced at a time when Nietzsche's influence was virtually inescapable. Rilke's poetry is steeped in Nietzsche's proto-existentialism — his insistence on life over thought, his search for redemption in this world rather than the next. Rilke also inherited Nietzsche's supreme subjectivity — his belief that the artist creates truth rather than merely recording or revealing it. In 1903 Rilke received an unsolicited book of poems from a young soldier named Franz Kappus, with a note asking whether the poet would mind reading them and sharing his thoughts with the author in the form of a critique. Rilke politely refused, but he and Kappus struck up a correspondence in which Rilke, while never dealing specifically with Kappus's poems, offered the young man a lifetime's worth of advice on the subject of being a poet. First of all, Rilke advised, you should stop asking for advice. You are looking outside, and that is what you should most avoid right now. No one can advise or help you — no one. There is only one thing you should do. Go into yourself. Rilke returns to the theme in his sixth letter, sent shortly before Christmas. Knowing that Kappus would be alone for the holiday season, Rilke urged him not to be frightened of loneliness, but to embrace it. What is necessary, after all, is only this: solitude, vast inner solitude. To walk inside yourself and meet no one for hours — that is what you must be able to attain. Over and over in the course of the ten letters, Rilke returns to this theme. 'Love your loneliness,' he says. This is not an easy thing to do. But Rilke suggests to his protégé that if he's really a poet, he'll find that solitude suits him, that he'd rather be there than anywhere else. We already have an idea why. Being weird and lonely at school was a blessing in disguise for Rivers Cuomo (as it was for Billy Corgan and Gerard Way) because it gave him the experience and the insight to write his earliest songs. He made use of his melancholy by writing profoundly affecting music about his condition. But just when it seemed like he'd broken through his loneliness and connected to the world, he went out of his way to make sure that he stayed lonely — moved out of the big world, and back into the garage. Now he's triumphantly solitary — indeed, he insists on aloneness as a condition of his existence. That's because Cuomo feels that being lonely is an important part of his job, and Rilke would agree. Rilke insists that for a good poet there is no poor subject matter since all his experience is filtered through the unique prism of his own sensibility. That's why he advises the young man to look inside himself to find out whether or not he is a poet. Cuomo insists in 'Troublemaker' that he doesn't need books because he learns by studying the lessons of his dreams, and Rilke confirms this. The outside world is overrated, he says: learn to do without it. ...even if you found yourself in some prison, whose walls let in none of the world's sounds — wouldn't you still have your childhood, that jewel beyond all price, that treasure house of memories? Turn your attentions to it. Try to raise up the sunken feelings of this enormous past; your personality will grow stronger, your solitude will expand and become a place where you can live in the twilight, where the noise of other people passes by, far in the distance. And if out of this turning-within, out of this immersion in your own world, poems come, then you will not think of asking anyone whether they are good or not. It's exactly this self-reliance that Cuomo developed during his long and painful apprenticeship in the garage. In 'Troublemaker' he realises it was this, and not any of the dumb stuff they tried to teach him at school, that made him an artist. At school, they'd tried to teach him arts and crafts; in his garage, he taught himself to shred on his axe guitar while gazing up at his posters of Kiss and reflecting on his emotions. Who looks stupid now? 'You wanted arts and crafts?' sings Cuomo. 'How's this for arts and crafts!' He unleashes a face-melting guitar break, and the world is put in its place. This, he explains in 'The Greatest Man...' is how it's going to be from now on. If you don't like it — you can shove it, But you don't like it — you love it Cuomo knows he is a great artist because...he knows he is a great artist. Society's pronouncements on his worth have proven to be consistently unreliable — why should he care what we have to say about him or what he does? He is expressing himself authentically. What could be more important than that? Conrad Veidt: The notions of sick brains. # Expressionism THE AUSTRIAN PAINTER Oskar Kokoschka had an intense admiration for children's art. When children draw people, they don't worry overly much about details — unless the details are emotionally important to them. An aunt's curly hair will be obsessively laboured over if that's what the artist loves about the aunt — who cares how many fingers she's got or whether or not she has a nose? Kokoschka would maintain this attitude to portraiture all his life. 'When I paint a portrait,' he declared in his autobiography, 'I am not conerned with the externals of a person.' For Kokoschka, such things were the business of the photographer, or the lawyer drawing up a will. In his autobiography, he admits he may have given one of his portrait subjects only four fingers on one hand: 'Did I forget to paint the fifth? In any case, I don't miss it. To me it was more important to cast light on my sitter's psyche than to enumerate details like five fingers, two ears, one nose.' Kokoschka was a truth-teller — as Viennese painters were expected to be. But his truth was internal, not external, and was celebrated as such. 'I am proud of a Kokoschka's testimony,' wrote the poet Karl Kraus, 'because the truth of a genius that distorts is higher than the truth of anatomy, and because in the presence of art reality is only an optical illusion.' Kokoschka first made a name for himself as an artist in 1907 with a book of poems and woodcuts called The Dreaming Boys. This highly symbolic little book, which the artist dedicated to his mentor Gustav Klimt, was inspired by Kokoschka's life as a teenage monster. To his horror, at the age of thirteen, the artist had grown freakishly tall, sprouted hair in obscene places, given voice to guttural growls and bat-like squeaks, and felt overcome by unspeakable longings. In his verses Kokoschka described: A hesitant desire/the unfounded feeling of shame before what is growing/and the stripling state/the over- flowing and solitude Just as he dispensed with traditional rhyme schemes in his poems, in the wood-cut illustrations that accompanied The Dreaming Boys Kokoschka took gross liberties with the proportions of the human body to give his readers a sense of how isolated he felt, and how freakish and unwieldy his new body seemed to him. Inspired by Van Gogh's portraits, and by his own training as a children's art teacher, he simplified his drawing into brutal, expressive shapes and thick black lines. In 'The dreaming boys', Kokoschka warned polite Viennese society that a monster had grown up in its midst, and was even now staring hungrily at those well-fed children through the gap in the hedge: When the evening bell dies away I steal into your garden into your pastures I break into your peaceful corral The teenage Tim Burton also expressed his alienation from society by imagining himself as a monster — first in his sketch for the character who would become Edward Scissorhands, and later in the character of Vincent. Vincent, as Burton's poem reminds us, only looks like a normal seven-year old boy — inside, he's a diabolical fiend, obsessed with visions of madness, death and despair. Actually, Vincent doesn't really look that normal either, with his sharp, angular cheekbones, deathly pallor and sunken eyes. But it's hard to tell, because there is no objective reality in Vincent — everything — including our image of Vincent himself, is distorted through the prism of Vincent's sensibility. And whatever there was of external 'reality' at the start has disintegrated by the end, as Vincent's torments overtake him. In the 'nightmare' sequence in his Tower of Doom, Vincent's psychic stress expands out from his head to warp the architecture of his little room. As he struggles to get to the door, the walls and ceiling become horribly distended, and the door itself looms up like a crooked tombstone. Here, as always, Burton insists on showing us how things feel rather than how they look. In Edward Scissorhands, the castle's lurching architecture is impractical — if not impossible. In his previous film, Batman, the city belches steam and the buildings loom threateningly over the populace. Elsewhere, trees curl up their branches like Art-Nouveau latticework, and hallways disappear into infinity — in Tim Burton's world, reality does not conform to the evidence of photographic records. But this approach is not without precendence. Burton rejected the idea that art could be taught or learned. But he learned that he didn't need to be taught — that he should trust the peculiar visions in his own mind — by watching old horror movies. In the classic horror films of the 1920s and 30s, extreme emotions — fear, paranoia, madness — tend to be expressed in 'stressed out' visual forms. In Bride of Frankenstein, Dr Praetorius's laboratory is architecturally insane, but perfectly expressive of the mood in which he works. And as his efforts to create a mate for the monster reach fever pitch, the walls themselves seem to pull back in horror at his perversion of nature. The roots of this approach can be traced back to one of the very first horror films, Robert Wiene's The Cabinet of Dr Caligari. Almost sixty years separate Wiene's silent classic and Burton's first films — but the family resemblance is so strong that, looking at stills from Caligari, with its lurching cityscapes and gothic curlicues, we half expect to see Winona Ryder peering out of the shot. Conrad Veidt as Cesare, dressed from head to foot in black, his face a white mask slashed with black marks, looks uncannily like Johnny Depp in Edward Scissorhands, and the pasty, shabby-looking Dr Caligari seems to be the not-too distant ancestor of Danny DeVito's Penguin in Batman Returns. Burton, with his intensely subjective approach to movie making and his artist's flair for visual interest, was always going to be susceptible to the idea contained within Caligari, that form can — and should — be altered by feeling. Of course, they never showed old German silent films at the drive-in in Burbank where Burton grew up. But they did show Tod Browning's Dracula and James Whale's Frankenstein — both of which owe a considerable debt to Caligari. And considering that these two films are the foundation upon which the whole monster/horror film genre is based, it's no wonder Burton should end up showing the influence of Caligari in his films — even if he didn't actually see it until much later. The story of Caligari is presented to us in the film by the mentally deranged Francis — and Francis's world, acccordingly, looks completely deranged on the screen. When the mysterious Caligari makes his way through the city, the city itself seems to close in on him — the houses and shopfronts behave more like trees in fairy-tale forest than buildings. In the background, staircases disappear at impossible angles, and windows curl upwards into crooked smiles. Later, we catch Caligari's sleepwalking servant Cesare in the act of committing a horrible crime. He steals into a woman's bedroom and stabs her with a knife. But even before he does, it looks as though the whole room is conspiring to murder the hapless victim, shadows point threateningly to the corner of the room where she sleeps, even the crazy angle at which the window frame bisects the glass suggests violence. In Caligari, as in Kokoschka's portraits, 'correct' appearances are pushed out of shape by intense feelings. This is no coincidence — the artists who created Caligari's striking backdrops moved in the same circles as Kokoschka. Weine commissioned designs from three artists: Hermann Warm, Walter Rohrig and Walter Reimann, all of whom were associated with a Berlin art magazine called Der Sturm. Der Sturm was the brainchild of Herwath Walden, a tireless promoter of modern art in Germany, whose Cologne Sonderbund exhibition of 1912 had introduced many German painters for the first time to the works of Gaugin and Van Gogh. He also organised and promoted Oskar Kokoschka's first exhibition in Germany. The artists associated with Der Sturm could be said to have one important idea in common — all were trying to find a way of expressing intense psychic or emotional states using paint on canvas. And by 1911, people had started calling this kind of art Expressionism. It was never a movement as such, but the term now serves to cover a number of 'mini-movements' active in Germany before, during and after the First World War. These included the group centred around Walden's gallery and publishing house — including Kokoschka, Wassily Kandinsky, Franz Marc and Arnold Schoenberg, as well as another group from Dresden known as Die Brücke ('The Bridge'). For the founding members of Die Brücke, extreme emotion was the only place left to go in art. Objective reality, which had already been undermined by the Impressionists, had been completely discredited by Munch and Van Gogh. Now, the artist must turn inward in search of truth. Art teachers could teach them nothing — as Fritz Schumacher, who instructed many of the early Die Brücke artists in drawing, would discover. Schumacher recalls a particularly heated exchange with a young Erich Heckel: When I criticized the drawing for its carelessness he invoked his right to stylise. I put it that a person must be able to draw correctly before going on to stylisation...but I did not convince him. He said that the only important thing so far as he was concerned was the seizure of a total expression. Heckel's colleague, Emil Nolde agreed. 'The art of an artist,' wrote Nolde, who joined Die Brücke in 1906, 'must be his own art.' In their paintings, Kirchner and Nolde took the lessons they'd learned from Klimt, Munch, Gaugin, African and Islander art and Gothic prints, and synthesised them into terrifying visions of the human soul under stress. Early Expressionist painting takes a crucial extra step away from so-called 'objective' reality. Where in Van Gogh and Munch reality is strained, but remains recognisable, in Kirchner and Nolde the world of appearances seems fatally cracked. Perspective collapses, shadows abruptly reverse direction, human faces are sawn off into primitive masks or stripped of their flesh to reveal grinning skulls. This decisive move from outer appearance to inner truth opened up exciting new vistas for artists, but it lead to enormous problems when it came time for these painters to meet their public. Expressionism precipitated what the art historians like to call 'a crisis of subjectivity' — which in layman's terms means that viewers, by and large, thought Expressionist art looked horrible, and that the artists who made it were incompetent — if not actually insane. Who looks at a beautiful woman's face and sees a flat mask with a stripe for a nose? What kind of degenerate is this painter if he can't even put a wall at a right angle to the floor? That Expressionist art was generally thought to represent the visions of madmen made it the perfect visual language for Caligari, which, after all, is a story told to us by a madman. This made its radical designs acceptable to the public, in the same way that atonal noise, while deemed inappropriate for the dinner table or the nightclub, is regularly used to indicate warped mental states in thrillers and horror movies to this day, where it goes by virtually unnoticed — at least on a conscious level. One reviewer of Caligari noted that: The idea of rendering the notions of sick brains...through expressionist pictures is not only well conceived but also well realised. Here this style has a right to exist... But Wiene let his expressionist scene painters have the last laugh. As Siegfried Kracauer points out, the final episode of Caligari is not told from Francis's point of view. So, since we're no longer seeing the world through the eyes of a mental patient, the wonky chimneys of Caligari, should — in theory — straighten themselves out in accordance with the laws of 'correct' visual perception. But this is not what happens. Expressionism, Weine seems to be saying, has a right to exist in any case, because this is what life in the early twentieth century feels like — and if it looks horrible or insane, that's because modern life is horrible and insane. # The Pain Threshold It is hard to live in the age of psychoanalysis and feel oneself detached from the dominant public savagery. In this way, at least, the makers of horror films are more in tune with contemporary anxiety than most poets. A ALVAREZ WROTE those words in 1962, the same year A Clockwork Orange was published. In Burgess's novel, when Alex likes something a lot, he says it's 'real horrorshow', and Alvarez, too, had the feeling that horror might be closer to modern truth — and therefore beauty — than what we usually think of as beautiful. This is an idea that was unthinkable in 1750, already in sight by 1850 and artist's gospel by 1900. When Gustav Klimt was accused of flinging filth in the faces of Vienna's youth with his philosophy mural, his defenders argued that the vision of horror in Klimt's painting was simply the truth, and that a society can only ignore the strong and bitter realities presented by artists at its own risk. This kind of argument was guaranteed to hit the late nineteenth century bourgeois where it hurt. Thanks to Wagner, the Viennese middle classes had come to accept as gospel the idea that art is a form of spiritual instruction, and the artist a kind of prophet, whose dire warnings the populace ignore at their peril. The artist's stance as misunderstood prophet only grew more entrenched as the twentieth century got under way. 'Our age seeks much,' wrote composer Arnold Schoenberg in 1910. 'What it has found above all is: comfort. That permeates full-scale into the realm of ideas and makes it too comfortable for our own good.' When Schoenberg talked about lighting a fire under the Viennese bourgeois's comfortable behind, he wasn't just striking a pose. Schoenberg was many things — a rebel, an outsider, a pretty good expressionist painter, a great musical mind and a hugely influential writer and teacher, but no-one could make a case that either he or his music were in any way pleasant. Nor was he popular; in fact, he often seemed to go out of his way not to be. Aside from some early efforts in commercial dance music, Schoenberg was swayed very little by the currents of popular taste, critical opinion or his own economic situation. He was influenced, as far as possible, by only one thing: himself. He set out his case for self-expression at all costs in a letter to the painter Wassily Kandinsky in 1911: ...art belongs to the subconscious! One must express oneself! Express oneself directly! Not one's taste, or one's upbringing, or one's intelligence, knowledge or skill... In this, Schoenberg was very much a product of the Viennese milieu that produced Kokoschka and Gustav Klimt. Like them, he believed that the purpose of art was to 'show modern man his true face', and that the artist had no hope of doing this if he was not, first and foremost, honest with himself. He felt he could no more ignore the dictates of his heart and his emotions than a biblical prophet could ignore the voice of God. In his music, Schoenberg faithfully and unflinchingly presented his appalling truths — the things he saw in the darkest recesses of his heart. But he found that, by and large, the people who heard them turned away — closed the door on his truth and settled back into their easychairs. Now he knew he was right! Of course his art was terrifying to them — life is terrifying. The more uncompromising and truthful Schoenberg was in his art, the greater his isolation from society became. But he kept going, because he believed in only one kind of truth — subjective truth. And even as his own psyche began to fall apart under the strain, he found himself more and more determined to confront the fact of his own crack-up in his art, because here, he felt was the greatest thing he had to offer the world. In 1912, Schoenberg found the perfect vehicle to express these anxieties. He was commissioned by a Viennese actress named Albertine Zehme to compose some piano music for a recitation she was planning. Her songs were to be adaptations of a series of poems by the Belgian Albert Giraud. In Schoenberg's hands, Pierrot Lunaire became, as the expressionist painter Paul Klee put it, 'a mad melodrama'. Pierrot, being a clown, is in the entertainment business. But he chafes against this — he stops doing comedy and starts expressing his emotions because he feels he has important truths to share with humanity. The audience, predictably, hates this. 'Do something funny!' they demand. Funny? Pierrot climbs up on an altar and rips open his clown-suit: The hand, consecrated to God, tears the priest's habit to celebrate the gruesome eucharist by the dazzling glare of gold. With a gesture of benediction he shows to the fearful souls the dripping red Host with bloody fingers: his heart —, to celebrate the gruesome eucharist. The spectators would much prefer that the sad clown put his horrible guts away and got on with the business of making them laugh. But Pierrot will not be deterred. Of course they would rather be pleased than appalled, of course they hate him for showing them what they would prefer to see hidden. But he has a heart that is bigger and braver than anything they can throw at him. 'Look,' he seems to be saying, holding this still-beating heart out to the people. 'This is what I am prepared to do for you.' Gerard Way strikes a similar pose in 'Welcome to the Black Parade'. The singer looks around at the chaos of the world, and tries to make sense of what seems like madness. He sees 'the rise and fall, the bodies in the streets', and a great tide of misery and hate. But these horrors only make him more determined than ever to carry on showing the world its true face. This, after all, is his job. Gerard — who once confessed he was 'addicted to truth and honesty' — fulfils the role society has demanded of artists since the late nineteenth century — to be the bearer of bad news. In a conversation with Liza Minnelli in Interview magazine in 2007, Gerard made his position clear: I think we just went into it with the attitude that we're going to be different to everybody else because we're simply going to be ourselves. We're going to sing about things that other people wouldn't sing about...that is to say, we're going to sometimes put extremely difficult subjects in pop music... Gerard means to make us uncomfortable. He knows we'd like it if he just sang about 'driving a truck, smoking weed and objectifying women' — as he put it on another occasion. But he knows that what we need is the truth, and that this truth is, by nature, unpleasant. # Sprechstimme IN PIERROT LUNAIRE, Schoenberg portrayed a performer who was out of phase with his audience's expectations by giving his singer music that was out of tune. This was no accident — Schoenberg knew what he was doing, and had already been doing it for a couple of years. Back in 1864, Wagner's Tristan had suggested, with its famous opening chord, that the rules of music could be broken if they could no longer contain the force of the composer's emotion. Strauss, inspired by Wagner's example, would bend musical relationships further out of shape in Salome. But it was left to Schoenberg to break the chords that bind forever with his first atonal works in 1908. These pioneering works of Schoenberg's were sometimes referred to as 'expressionist'; and like the painters with whom he associated, Schoenberg found that by allowing his emotions to dictate the form of his music in defiance of all rules, he had completely alienated himself from his public. But as with Wiene's canny use of Expressionist décor in Caligari, Schoenberg discovered he could get away with his mad music within the context of Pierrot, since he was using it to express madness. Here, this style had a right to exist, and Pierrot was — unusually for Schoenberg — a hit. But atonality wasn't the only trick Schoenberg had up his sleeve in bringing the eerie emotional world of Pierrot to life. Instead of arranging Giraud's poems as songs, Schoenberg gave his vocalist what he called 'Sprechstimme'. Here, the singer talks in rhythm over the music, suggesting — rather than singing — the pitches, and only occasionally holding sustained notes. Again, within the context of Pierrot, this unusual technique worked a treat. As Allen Shawn puts it in Arnold Schoenberg's Journey: What pushes [ Pierrot Lunaire] over the edge into the world of the sublimely bizarre is how the music combines with the singer who isn't quite singing. Here, a kind of universal madness has been fixed on paper with clarity and art. Appropriately for a meditation on the divide between art and entertainment, Sprechstimme was a technique Schoenberg had become familiar with when he was employed in show business. Between 1901 and 1902, the young composer had worked for a cabaret company — Baron von Wolzogen's Überbrettl — writing popular songs for drinking and dancing. Later, after he'd made the most gut-wrenchingly confessional music of his early career, and then forced himself to confront the fact that people just wanted him to be an entertainer, he would refer to his cabaret music again, pressing it into service to describe his painful alienation from his audience. The Sprechstimme technique he'd learned writing for the cabaret was used to suggest a singer who was somehow dangerously detached from his material, and by extension, his life. Schoenberg's music for Pierrot Lunaire is cabaret gone horribly wrong — the singer was supposed to pull out his hits — but what's this? He's pulled out his heart! Today, musicians routinely espouse Schoenbergian philosophy. Billy Corgan echoes his insistence that 'one must express oneself' ('I do feel a responsibility to articulate what I feel') and draws the same conclusions from this as Schoenberg — that is, that if the artist's emotions lead him into territory that is alienating or confusing to the listener, then the listener had better suck it up. Corgan insisted that the future of rock and roll lay in music 'so emotionally explosive it's hard to listen to'. But, as we've seen, not one of the thousands of kids in Zero shirts at Lollapalooza had any trouble listening to Corgan's emotionally explosive music. In fact, they wanted him to keep exploding. The situation of Pierrot Lunaire was reversed; here, the audience insisted that the singer rip out his heart. This demand for emotional intensity nearly killed Robert Smith in the early '80s, and it scared Rivers Cuomo away from emotional music for almost a decade. In the meantime, the modern misery industry grew to spectacular proportions. In Saves The Day's 'Jukebox Breakdown', Chris Conley — who was starting to feel like some kind of automated human unhappiness dispenser — accused his audience of conspiring to kill him. All you want from me Is a broken heart and a mouthful of blood. Yes, we do! said the kids in the crowd. Schoenberg didn't know how easy he had it. He railed against his audience for wanting the happy clown when he was dying inside. He had to rip out his heart to show them just how bad he was feeling. But in the twenty-first century this situation is reversed. The modern emo audience demands nothing less than the artist's still-beating heart, served up fresh, every night of the week for the length of a twelve-month tour. Needless to say, ripping out your heart on a nightly basis is hard to sustain. By the end of My Chemical Romance's The Black Parade, Gerard Way is starting to feel the long-term effects of all this soul-baring and blood-letting. On the album's final number, he sings: Give them blood, blood, gallons of the stuff give them all that they can drink and it will never be enough. The situation is terrifying, but Gerard sings this in a light, jaunty ironic way — you can almost see him tipping his hat and twirling his cane. The arrangement has just a touch of dissonance to indicate the artist's impending crack-up, and Gerard deploys a little Sprechstimme to give the song that weightless 'I've lost too much blood and I'm getting dizzy' feel. The music is pure cabaret — a small group wheezing out a steady oom-pah. Gerard's voice is even put through an effect that makes it sound like he's singing through a megaphone — a popular means of getting the audience's attention in the pre-amplification days of the literary cabarets. But the effect the band are aiming for is not so much the Überbrettl of 1901 as the Troika circa 1930. Siouxsie Sioux: Brand new people... # Everything Collapses MY CHEMICAL ROMANCE weren't the only post-emo outfit to revive the spirit of the literary cabarets in the early twenty-first century. The Black Parade appeared hot on the heels of Panic at the Disco's debut album, A Fever You Can't Sweat Out — a record shot through with hot jazz rhythms and dressed to the nines in high literary style. Songwriter Ryan Ross poured ironies on his agonies and crammed so many sub-clauses into his parentheses that singer Brendan Urie just barely managed to fit the words between the beats. And yet the band always kept their cool — Ross's tales of bad sex, cheap laughs and existential boredom were presented with a lip-sticked pout and a mascaraed wink. Emo kids the world over fell head-over-heels in love. Caberet, as music writer Norman Lebrecht has observed, thrives in societies in decline. This might seem odd at first. Civilisation is falling down around your ears — is this really the best time to be drinking absinthe until two in the morning and experimenting with make-up? But for the cabaret singer, there is no better response to social collapse. Panic at the Disco know this instinctively. 'Looks like the end of history', sings Urie on Panic's 2008 single, 'Nine in the Afternoon'. 'Oh, no — it's just the end of the world.' Urie delivers these lines like he's seen it all before, and will see it again. In a sense, he has, and he will. In Berlin, following the end of the First World War, the world was also about to end. Faced with a pile of war debts — debts which the shattered nation was in no position to repay — the German chancellory came up with the novel solution of simply printing more money. The value of the Deutschmark plummeted, and the moral standards of the capital fell quickly in its wake. Dostoyevsky, it seemed, had been right. The modern world's shopkeeper philosophy had effectively replaced real values with monetary ones. Now that money was worth nothing, life had become meaningless. 'Standards and values disappeared,' writes musicologist Douglas Jarman. 'Berlin was transformed into the Babylon of the world.' But the collapse of morals turned out to be good news for the owners of nightclubs, where — as Jarman observes — business continued as usual. In the Berlin cabarets of the early 1920s, jaunty, jazz-inflected pop tunes told tales of political subversion and sexual perversion. Even after the mark stabilised in 1923, and Germany regained some sense of order, the cabarets continued to flourish as hotbeds of satire and sleaze. But after the Wall Street crash of 1929, Germany's economy spiralled out of control again. Predictably, in the cabarets, business boomed. Once again, all bets were off, and everything was permitted, with the sole exception of bourgeois conformity — the cabaret's arch-enemy. In 1931 — with unemployment creeping toward the six million mark and the capital edging toward civil war — an English expat named Christopher Isherwood described a typical night on the town, in his novel, Goodbye to Berlin. The couples were dancing with hands on each other's hips, yelling into each other's faces, streaming with sweat. An orchestra in Bavarian costume whooped and drank and perspired beer. The place stank like a zoo. Christopher's closest friend in Goodbye to Berlin is a nightclub singer called Sally Bowles. Sally seems to survive on a diet of Prairie Oysters and cigarettes. She refuses Christopher's offers of more substantial fare by saying: I just don't want to eat anything at all. I feel all marvellous and ethereal, as if I was a medieval saint or something. You've no idea how glorious it feels... Have a chocolate, darling? Like many of Christopher's friends in Goodbye to Berlin, Sally has lost the trick of acting naturally, if she ever had it. But to compensate, she has become very good at acting — and not just on the stage. Sally is unconvinced by life — reality seems unreal to her. Nazi Putsch or Communist Revolution? Eat and live or starve and die? Have another chocolate, darling? But despite Sally's inability to take it seriously, reality won't go away either. So Sally — being a consummate professional — has resolved to put in a convincing performance. Isherwood had no way of knowing it, but his snapshot of Berlin on the brink would go on to become one of the founding texts of glam rock. Isherwood's portrait of a decadent society in decline in Goodbye To Berlin, his characters' ironic, detached attitude to life — even the book's title — would provide David Bowie with the atmosphere of his sequel to Ziggy Stardust, Aladdin Sane. In Goodbye to Berlin, the news is all bad; one newspaper headline reads EVERYTHING COLLAPSES. And this is precisely the reason why no-one seems to care very much about anything besides having a good time. In the face of the apocalypse, what else can you do? Bowie's Ziggy Stardust had begun with the news that the world would end in five years. Now, in Aladdin Sane, Bowie announced that there was no reason why you shouldn't do whatever you liked. Life was simply a tragedy, which would soon be over. Might as well have some fun. 'Panic in Detroit' describes a society in its death throes — the kids turn up at school and find that their teachers have simply stopped work. Sure, children are the future — but what if there's no future? The kids scream, run out into the street and start smashing things. Bowie reports on this with his usual combination of alien detachment and high melodrama. On the album's best song, 'Time', Ziggy puts the fate of humanity into perspective. It's a cabaret number — the singer sits on a bar stool smoking a cigarette while pianist Mike Garson throws Schoenbergian shapes over his keyboard, complicating 'Time''s Weimar-jazz arrangement with expressionist dissonances. 'Time!' sings Ziggy, He speaks of senseless things His script is you and me 'Time', like many of Bowie's songs, betrays the influence of composer Kurt Weill — a songwriter who virtually epitomised the cultural world of the Weimar Republic. In his memoir, A Little Yes and a Big No, the painter George Grosz recalled that 'you could hear [Weill's] songs everywhere you went in those days'. Weill inherited the Wagnerian ideal of music-theatre as a means to repair a fragmented society. But he had no time for Wagner's emotional excesses. Weill replaced the Ring of the Niebelungen with the Threepenny Opera, the most famous of his collaborations with Bertolt Brecht. The opera's best-known song, 'The Ballad of Mack the Knife', is typical Brecht–Weill. It observes the worst aspects of human behaviour in a society on the brink, but does so over a tune that once heard, never leaves your head. Later in the 70s, Bowie would record Weill's 'Alabama Song', perform many of his songs live, and come very close to starring in a film adaptation of The Threepenny Opera. Bowie's interest in Weill's music was part of a larger fascination with Weimar culture. He was (and still is) a keen admirer of expressionist painting and film — the set designs for his Diamond Dogs tour were strongly influenced by The Cabinet of Dr Caligari and Metropolis. Later, when glam rock went mainstream, he said he felt very upset that 'people who'd obviously never seen Metropolis and had never heard of Christopher Isherwood were becoming Glam Rockers'. In the summer of 1976 he and Iggy Pop moved into an apartment in Isherwood's old neighbourhood, the Schoneburg district of Berlin. On the cover of Heroes, he made an explicit homage to the expressionists, posing in imitation of an Erich Heckel painting he'd seen at Die Brücke museum. In the photo, Bowie's hands are raised at an awkward, theatrical angle, bent out of shape by the stress of modern life. His face has been reduced to a black and white mask, his cheeks are hollow, his eyes have a haunted look. This portrait of Bowie is also a very close cousin to the one of Iggy Pop on the cover of The Idiot — an album Bowie produced and played on while the two were living in Berlin. On 'Nightclubbing', the pair takes us with them on one of their nocturnal escapades through the divided city: We're seeing people Brand new people They're something to see. The brand-new people do brand-new dances, he tells us, 'like the nuclear bomb'. This, as you've probably guessed is the dance of the damned. Why not twist your body out of shape and wreck yourself with pills and liquor? With everything collapsing, anything goes. Bowie's timing was, as usual, impeccable. The same year, the Sex Pistols declared that, in the face of the apocalypse, morality was bunk. 'When there's no future, how can there be sin?' asked Johnny Rotten. 'We're the future,' he insisted, pointing to the nocturnal freaks crawling out of the city's garbage, 'your future!' Sure enough, Iggy's new people had appeared, growing out of the city the way Cesare seemed to grow out of the expressionist décor in Caligari. Drawn by the aura of sin and subversion around the Sex Pistols and their manager, Malcolm McLaren, a strange crowd of teenagers from the London suburb of Bromley joined the group's entourage early in 1976. Siouxsie Sioux was eighteen at the time, and was determined to propel herself as far as possible from the stifling conformity of Bromley. She'd already tried to find work as a model, but the agencies had rejected her because she was too skinny and wore too much make-up. Anyone else might have been disheartened by this, but Siouxsie was already cultivating a personal style that had nothing whatever to do with the blonde, healthy, suntanned look the agencies were after. She'd first seen the light three years earlier while lying in a hospital bed, recovering from a serious illness. Switching on Top of the Pops, she saw David Bowie singing 'Starman'. It was, as Siouxsie's future collaborator Robert Smith once told Richard Kingsmill, the sort of thing that changes lives. Bowie looked deathly pale, painfully thin, generally unwell...and fabulous! Siouxsie cheered up instantly. 'I'd lost so much weight and got so skinny that Bowie actually made me look cool.' By the time she'd fallen in with the Sex Pistols, Siouxsie had turned ill-health into a fashion statement — and something more, an existential protest against bourgeoise comfort. Pale skin and dark eyes said to the world: 'I am a creature of the night — I stay up too late and punish my body in unthinkable ways. But I do it because I will never settle for the half-life of the suburbs, the stifling comfort of work, dinner, TV, sleep, work...' Siouxsie and her friends — John, Blanche, Tracey and Berlin — knew they were better than this, and set out to prove it in every way. 'The only thing that was looked down upon,' she told Jon Savage, 'was suburbia. I hated Bromley: I thought it was small and narrow-minded. There was this trendy wine bar called Pips, and I got Berlin to wear this dog-collar, and I walked in with Berlin following me, and people's jaws just hit the tables... People were scared!' Berlin was only fifteen years old, but had already reinvented herself as a Weimar-era nightclub singer in bowler hat and fishnets. Cabaret, as Norman Lebrecht has observed, thrives in societies in decline — which is why Bowie used it as one of the harbingers of the apocalypse on Aladdin Sane. Berlin from Bromley sensed this connection between England circa 1976 and Germany 1929 intuitively. 'I can't tell you the parallels between those days and Goodbye to Berlin,' she told Jon Savage. 'We were living it out, the whole bit.' Berlin knew what a society in decline felt like from Isherwood's novel. But she knew what it looked like because she'd seen the movie. Goodbye to Berlin had first been adapted as a popular play called I Am a Camera, then a musical with songs by Fred Ebb and John Kander in 1966 (which Bowie saw as a teenager and loved), and finally, as a film directed by Bob Fosse and released in 1972 as Cabaret. Joel Grey, who plays the MC at the Kit Kat Klub, was one of the few actors to be retained for the movie from the stage version. In the film, Grey wears a thick mask of black and white make-up that pushes his face into an exaggerated smirk, like an expressionist painting come to life. 'Wilkommen, Bienvenue, Welcome!' he sings, by way of greeting the assortment of local scene-makers and gawking tourists who've come to see the show. He hams, winks and mugs his way through his duties, a model of ironic detachment and nocturnal sleaze. For the Bromley contingent, Grey's show stealing performance embodied a whole philosophy of life. Liza Minnelli: Where are your troubles now? # Life Is a Cabaret IN Cabaret, THE world is about to end. But in the Kit Kat Klub, it might as well not exist. 'Where are your troubles now?' Grey asks the audience after another bawdy, gut-busting tune: Forgotten! In here, life is beautiful... The song the audience have just heard is 'Life is a Cabaret', the singer is Sally Bowles — played in the film by Liza Minnelli. In the song, Sally tells a story about a friend she knew in Chelsea who did whatever she liked and never thought about tomorrow. Elsie lived in a tiny rented room and died alone. At her funeral all the decent respectable folk from the neighborhood snickered self-righteously; 'Well, that's what comes of too much pills and liquor'. But this is not the moral of the story. The moral of the story is: Elsie from Chelsea is a hero. Why? Because life is pointless, work is futile, love is fleeting, and the world is steadily marching toward the brink of war. Again. So why not kill yourself with booze and pills? Why not sin, and sin proudly? At least you'll enjoy yourself while you're here. All you respectable folk can do what you like, says the singer, her voice starting to tremble, her eyes starting to pop. As for me I made my mind up Back in Chelsea When I go I'm goin' like Elsie At this point, Liza Minnelli's performance takes on an almost religious intensity, as she sings us through Sally's epiphany, the means by which she has learned to transcend her meaningless life and the world's meaningless collapse. Life is a cabaret old chum It's only a cabaret old chum And I love a cabaret Now, as the MC says, life is beautiful. Cabaret is not just about distracting yourself from the threat of social collapse with a bit of harmless fun. It's about learning to see life aesthetically. It's about seeing the suffering of the world turned into a song, and then beginning to understand that the song is not only a consolation for suffering — it justifies the suffering. If this sounds a little Nietzschean, it ought to. Cabaret — the style, and by extension, the movie — owes a great deal of its character to Nietzsche. German cabaret began with Baron von Wolzogen's Überbrettl in 1900. Brettl is the German word for the 'little boards' upon which the cabaret performers plied their trade — Wolzogen attached the Nietzschean prefix, über, to his company name, to show that, while the boards might be little, the ambitions of his new cabaret were anything but. The Nietzschean overtones, as Peter Conrad explains in Modern Times Modern Places, were absolutely intentional. 'The Berlin cabarets, explicitly invoking Nietzsche as their founder, encouraged the uprising of rude, savage nature against anaemic society.' The German cabaret was intended as a place where one could sin boldly, and reflect on suffering with the mocking laughter of the 'higher man'. The high romantic irony of Bowie's 'Time', Kander and Ebb's 'Life Is a Cabaret' and Lou Reed's 'Satellite of Love' (which Bowie produced), all retain this Olympian perspective — a residue of the cabaret's original manifesto — which is crucial to their appeal. In the brief period of optimism preceding the crash of 1929, Brecht and Weill re-imagined the cabaret as having a constructive social function. Cabaret during this time took on a strong left-wing flavour. But the presence of Nietzsche in the cabaret's history explains why the genre tends to appeal to those with an aristocratic, rather than a democratic attitude to life and art. In Stardust, Tony Zanetta remarks on Bowie's superior attitude to his audience at his first Ziggy shows — so different from the 'I'm just a regular guy like you' image cultivated by the Californian bands of the day. Bowie, says Zanetta, 'projects an upper-middle class patrician quality, and seems impressively elegant.' This aristocratic quality in Bowie made it very easy for him to slip into the role required of him in a song like 'Time', that of the aloof observer to whom suffering is merely a form of play. It also made him an important reference point for those who were caught up in the energy of punk, but unable to relate to its democratic ideals. The Clash used to invite their fans, en masse, to sleep on the floor of their hotel rooms after shows. Siouxsie and her Banshees were having none of it. 'I mean, no!' laughed a horrified Steve Severin in Simon Reynolds' 'Rip it Up'..., 'we'd let them stay out in the rain!' When Siouxsie says that the only thing she and he friends looked down upon was suburbia, she echoes Nietzsche's horror at 'the mob hotchpotch'. 'Oh disgust! Disgust! Disgust!' wrote the most un-democratic philosopher in 1884. In 1972 Bowie's Ziggy Stardust and Bob Fosse's Cabaret were ideal companions. Both offered a view of a society on the brink of collapse, and both suggested an aesthetic view of suffering as the solution. Both end heroically — the world is shown to be cruel, dangerous and corrupt, but the hero takes the stage for one more song with a great 'nevertheless'. Liza Minnelli's 'Life is a Cabaret' and Bowie's 'Rock and Roll Suicide' both insist that tragedy will redeem us and justify our suffering. But where Ziggy Stardust ends with a grand flourish of strings and a decisive 'home' chord, Cabaret strikes a more ambiguous note. 'Where are your troubles now?' asks the MC. 'Forgotten!' We have to admit he's right. The singer has stood up on stage and said 'yes' to life — to all of life. In doing so, she has become that thing that Nietzsche could never have imagined (because he was scared of women) — a superwoman. Here is a woman, we think, who cannot only cope with madness, death and societal collapse, but actually enjoy it — even laugh at it. We imagine, as we watch her, that we might be that brave and that bold. And here, in the cabaret, where everything is permitted, it seems possible. To be always a yea-sayer, to take the worst that life can throw at us and laugh at it. But as Nietzsche knew, being a yea-sayer is not the same as saying 'all is for the best in the best of all possible worlds' or 'it's all good'. To know whether you have what it takes to be a superman or superwoman, you have to be able to grasp suffering on an endless loop, you have to be able to swallow the notion that the war you have just lived through will be followed by another, and another and another. Now, as the MC goes backstage to wipe off his make-up, Fosse's camera pans around the bar, and our new aesthetic attitude to life is put to its most gruesome test. Refracted through the prism of a whisky glass, we can see a group of men in brown shirts with red armbands sitting in the corner of the bar. Life, Fosse seems to be saying, will not be beautiful much longer. Can you say yes to World War Two? # Mother War GERARD WAY FIRST saw Cabaret when he was just 'a little kid'. It didn't just happen to be on TV one night, it was shown to him — as part of a broader education in art, theatre and music — by his grandmother, Elena Lee Rush. The film had a tremendous impact on him — the sound and style of Kander and Ebb's songs, the atmosphere of the Kit Kat Klub, and the strange combination of power and fragility in its leading lady. All of these things would stay with him forever, and would go on to shape his own music in important ways. Right from their humble beginnings, My Chemical Romance stood out from its punk and hardcore peers because of Gerard's exaggerated sense of melodrama, and his understanding of the importance of wardrobe and make-up. In fact, Gerard was still avidly studying Cabaret as My Chemical Romance wrote and recorded The Black Parade — and it shows. There are plenty of stylistic references — the hot jazz rhythm and Sprechstimme of 'Dead' and 'House of Wolves', the demented Bavarian oom-pah of 'Mama' and 'Blood', the jaunty phrasing of 'Teenagers'. But the influence goes deeper — beyond the surface of the music and into the realm of ideas. The Black Parade is full of suffering, but in almost every case Gerard presents his pain as a show. He invites his audience to watch the spectacle of his demise with the knowing wink and insinuating leer of Joel Grey's MC. He knows they'll get their money's worth. 'Gather 'round piggies and kiss this goodbye,' he sings in 'The End', 'I'd encourage your smiles I'd expect you won't cry!' His tone is mocking, superior and ironic...most of the time. Gerard didn't learn all of this by sitting on the couch watching Cabaret on video — a lot of it he learned the hard way, by treading the brettl. Here, again, he had his grandmother to thank. Elena didn't just introduce him to musicals — she gave him the confidence to star in them. Elena encouraged Gerard to try out for the leading role in his school's production of Peter Pan — she made him a costume and everything. This was another milestone in Gerard's life. 'I discovered I could sing,' he later recalled, 'which was pretty interesting.' What's even more interesting is that Gerard should find himself, in 2007, telling this story to Liza Minnelli, the star of Cabaret. Liza with a Z has a few Peter Pan anecdotes of her own. 'When I see you,' she says to Gerard, 'remind me to tell you about the dress rehearsal for Peter Pan with Sandy Duncan.' Liza and Gerard are pals now ('you are my new baby, who I adore' she gushes) since the legendary actress made a guest appearance on The Black Parade. Minnelli's cameo appears in 'Mama' — one of the strangest, fiercest and most emotionally raw songs on the album. 'Mama' begins with the sound of a city under siege, the muffled crack of distant explosions. The band picks up the rhythm implied by the steady fall of the bombs, and pretty soon they're playing a song. Gerard steps into the spotlight and — in the time-honored tradition of the avant-garde artist — tells it like it is. 'Mama,' he sings, 'we're all gonna die.' It's not just a figure of speech. The song is addressed to Mother War, a Shiva-like goddess of destruction who forms part of the pantheon of The Black Parade. On the album's inner sleeve photo, she stands alongside the band and the other characters from the album wearing a Victorian gown and a gas mask. Like the Belle Dame of Keats's poem, or the embodiment of hopelessness in AFI's Love Like Winter video, Mother War is a variation on that great romantic obsession, the femme fatale — the eternal fusion of love and death. In 'Mama', Mother War is played by Liza Minnelli. She tells the singer, her voice muffled through the gas mask, that she wishes he would call her his 'sweetheart'. But the singer resists, because he's seen through Mother War and the things she promises. She's the personification of the will to war in both society and the individual, the collective insanity that drives people to commit murder in the name of their country or their religion. The singer has had enough of it. He's seen the truth — that there's no glory in war, no victory, no eternal reward for the soldiers or anyone else, just a pile of corpses as far as the eye can see. 'Mama', he sings, pouring on the scorn, 'we're all full of lies, Mama we're meant for the flies.' Here, something goes badly wrong with Gerard Way's ironic detachment. He wishes he could be Ziggy, observing human suffering with his Martian cool, or Nietzsche, admiring the strength and power of the 'will to war' in a troop of soldiers marching off to battle. But he can't stand it. His disgust cracks the form of the song — Sprechstimme goes out the window, he starts howling like a dog or a baby, 'Ma-Maaa!, Ma-Maaaa!' The world is going to hell, and everyone just keeps drinking and dancing. Why won't anyone listen to him? More importantly, why won't they do something? This is not the aloof stance of the artist tyrant, but the rage and disgust of a man who, having set out to change the world, finds himself trapped in the music hall. This is where we leave Gerard as The Black Parade draws to a close — stuck in the endless limbo of entertainment. He ripped out his heart to show the audience the truth of modern life, and they just sat there and clapped, and yelled for an encore. He sounds crushed, because he's realised — as Nietzsche always knew — that tragedy doesn't improve the world. The audience in the club or the opera theatre are redeemed, but outside the world will continue to suffer. It has to, so that the artist's tragic attitude to life can be maintained. Gerard Way will not accept this. He loves art, but he's too democratically minded to be able to stomach the idea that it justifies the suffering of the world. The engine that drives My Chemical Romance is powered by the perpetual push-and-pull of Gerard's contradictory impulses — his desire to create tragic spectacles on the one hand, and his need to help the world on the other, his love of the emotional and the irrational versus his determination to make a difference in the real world. The Black Parade is, in a sense, an album length rewrite of 'Life is a Cabaret', a record in which the hero takes everything the world can throw at him, and does his best to celebrate suffering, to always be a yea-sayer. But in 'Mama', looking around the bar and listening to the bombs fall outside, he realises he can't say yes to a third World War. Is there some way, he wonders, of making art and making a difference? This after all, is why Gerard started the band in the first place. On the morning of 11 September, 2001, Way had officially renounced art for art's sake, and resolved to help the world. Something just clicked in my head that morning... I literally said to myself, 'Fuck art. I've gotta get out of the basement. I've gotta see the world. I've gotta make a difference!' George Grosz: Fuck art! # Artists are Cleaners THE GERMAN PAINTER George Grosz decided, at the end of the First World War, that he could not simply stand around sketching the collapse of his society. It gradually became clear to Grosz — as it did to many Euopeans during the '20s — that the unfinished business of the First World War was about to result in a second. Surely, he thought, as an artist, there was something he could do to prevent this catastrophe. His prewar art training had not prepared him for this at all. He'd learned how to describe, how to decorate and — thanks to Van Gogh and Gaugin — how to express himself. But what the world now needed was art to inflame, art to shock, art that would slap the people on the street in the face, make them realise the danger lurking on the horizon and show them how to do something about it. Grosz's work took on a cold-blooded objectivity. 'I considered myself a natural scientist,' he explained. 'I spared no one.' Grosz's new art wasn't pretty — but what good was pretty art? What good was art at all, unless it helped the world? 'I considered all art senseless,' Grosz later recalled, 'unless it served as a weapon in the political arena.' By this point, Grosz had become one of the central figures in the Berlin Dada movement. Dada had begun as a night of noise and nonsense in a Zurich nightclub in 1916. The club's name, 'Cabaret Voltaire', signified the movements aims — combining the anarchic spirit of the cabaret with the sharp-eyed skepticism of the Enlightenment. Dada quickly opened branches in Paris, New York and Berlin, adapting to its surroundings wherever it went. In Paris, for example Dada was an absurd, existential protest — a revolt against sanity in a world gone insane. In Berlin, it took a more constructive approach — pointing fingers and naming names. The Berlin Dadaists could have no use for a movement that simply hoped to smash the world up, because it seemed to them that their world was already pretty well smashed. Berlin Dada became a program of action, designed to shake people out of their stupor. Following the form of other Dada invasions, the Berlin contingent put on Dada 'revues' — nights of artistic entertainment — or so the punters thought. 'They came expecting to see a show,' Grosz later recalled in his memoir, 'we simply told them the truth.' Raoul Hausman would walk out on to the stage, point to the audience, and say, 'Would you just look at this big crock of shit before us!' The Berlin Dadaists threw themselves into the life of the city — chewing it up, vomiting it back out, and presenting it on a plate for the populace. They sliced up the newspapers and magazines and stuck them back together in frightening new combinations, to show people what they were really looking at over their morning coffee. Not all of Dada's targets were political or social. Among its countless announcements, broadsides and manifestos came a curious document called Dadaistisches Manifest, authored by Richard Huelsenbeck, demanding an end, not to war-mongering, fascism, racism or any of the other countless problems of the day, but to expressionism. What, Huelsenbeck asks of the German people, have these expressionists done for you lately? Have the expressionists satisfied our expectations for an art of the present day? No No No The problem with expressionism, the Dadaists felt, was that it was too emotional. The Dada painter Francis Picabia insisted that painters who use emotional themes in their work (love, heartbreak, tragedy) are guilty of manipulating their audience, and that you, the art-lover, are an idiot if you'll allow yourself to be fooled by such a cheap trick. Feelings, Picabia wrote in 1920, are a dime a dozen: You are always looking for an emotion that has already been felt, just as you like to get an old pair of trousers back from the cleaners, which seem new, as long as you don't look too close. Artists are cleaners, don't be taken in by them. If emotional art is a con, then the whole German romantic tradition becomes deeply suspect. The dadaists mercilessly sent it up in their Pan Germanic Poetry Contest. Eleven poets appeared on stage, reciting poetry at the top of their lungs. 'They made gestures, brushed tears from their eyes, held hands over their hearts.' Who will win? the audience wondered. Who has the greatest sorrow, the heaviest heart, the most serious case of einsamkeit? In the end, the dada judges declared the contest a draw. Romanticism was bad enough, but expressionism — a kind of uber-romanticism in which emotion becomes the artist's only consideration — was much worse. Kokoschka painted a whole world viewed entirely through the prism of the artist's feelings. His portraits of other people were, essentially, self-portraits — an attempt to represent the emotions stirred in him by other people, rather than the people themselves. 'Therefore,' concluded Dada theoretician Johannes Baargeld, 'Kokoschka can now with certainty be considered the inventor of the automechanical leech "self-help".' For the dadaists, expressionism, by peddling art as emotional therapy, was counterproductive. Expressionist art turned pain into an aesthetic spectacle, thereby making it beautiful. The spectator settles into his easychair, confident that he understands suffering. Meanwhile, out on the street, the world gets worse, and suffering continues. Expressionists, Huelsenbeck declared, are people 'who prefer their armchair to the noise of the street.' Expressionist art inspires contemplation, where for Grosz and the dadaists, art — if it was to be of any use at all in the world — must inspire action. # Distress Cries Aloud EMO HAS A fondness for expressionism — which makes sense, since both represent romanticism run riot, the expression of feelings elevated above every other concern. The very first emo band, Rites of Spring, adorned the cover of their 1991 album End on End with an expressionist woodcut — a perfect visual equivalent for the music inside. The sleeve of Saves the Day's In Reverie also featured expressionist prints. The booklet illustrations show Kokoschka-like femme fatales drawn with violent gestures and a gloomy young man with haunted eyes above sunken cheeks, his mind being slowly strangled by his own heart. Expressionism has also proved useful to what Neil Strauss calls 'the eyeliner punk pack'. The Alkaline Trio promoted their 2008 album The Agony and the Irony by posing for photos in front of what looks like the set from The Cabinet of Dr Caligari. AFI signalled the beginning of its transition from cartoon punk to gothic revival with an expressionist-style illustration of a graveyard at midnight on the cover of The Art of Drowning. Inside is more emotional art — one drawing shows a haunted-looking young man standing in a bleak cityscape wearing a Misfits-style 'Devilock' — Munch as a punk. My Chemical Romance prefers a more meticulous, comic-book style of art for their sleeves. But their band logo is expressionist to the core — the letters are formed from violent brushstrokes, as though painted in a great burst of inspiration, or as a cathartic release from some enormous psychic pain. The expressionists talked of 'the charging of every action with significance and soul' — everything, from a great triptych to the title page of a hastily written pamphlet — was to be done with passion and emotional force. The message of My Chemical Romance's logo is exactly this: we are sincere, emotional people, everything we do is intense. My Chemical Romance's videos and performances also bear a strong resemblance to expressionist theatre — which flourished during the same period as the art movement and began with a play written by a painter — Kokoschka's 'Murderer, Hope of Women'. Photos from a production of Ernst Toller's expressionist play Die Wandlung , staged in Berlin in 1919, look uncannily like stills from My Chemical Romance's tour film, The Black Parade Is Dead. One image from Toller's play shows four soldiers in corpse-paint making violent, agonised gestures against the backdrop of a flaming wreck. The figure in the middle, with his shock of white hair, black make-up under the eyes, and panicked expression, could be Gerard Way. The story of Die Wandlung also has a familiar ring about it. The play tells the story of Friedrich, a young man with a heart full of dreams. His early optimistic view of life is completely shattered by the war. His dreams are full of armies of skeletons marching through the darkness, troop trains full of undead soldiers. Friedrich returns home, and tries to make art. He is busy working on a statue representing victory when he is interrupted by two war cripples begging for alms. This pathetic display shocks our hero into a new conception of life. 'Fuck art!' he says, smashing the statue to bits, 'I've gotta make a difference!' Much soul-searching ensues, until finally Friedrich realises what he must do. He grabs some kind of megaphone and rushes out into the city square, shouting: You are all of you no longer men and women; you are distorted images of your real selves. And yet you could still be men and women, still be human, if only you had faith in yourselves and in humanity Inspired by Friedrich's call to arms, the broken, beaten and damned join together and sing a revolutionary anthem. But Friedrich knows that singing alone will not change the world. 'Go to your rulers and proclaim to them with the organ tone of a million voices that their power is but an illusion,' he urges. 'Now march! March forward into the light of day!' Toller's play puts the lie to the dadaists' assertion that all expressionists prefer their armchairs to the noise of the street. Die Wandlung is just one example of a politically committed variety of expressionism that flourished in the wake of the war. Just like the dadaists, most expressionist painters had their outlook and conception of art profoundly altered by the First World War. Some, it's true, retreated into themselves — but others believed that the emotional impact of their art could be used to bring about real change in the world. In 1919 Max Pechstein designed the cover for a collection of expressionist statements and manifestos entitled An alle Kunstler! ('To All Artists!') A romantic young man, his face gaunt, his limbs straining, crawls out of the wreckage of a smouldering, oppressive city-scape. One hand reaches for the sky, the other clutches his own burning heart, which lights up the picture with dazzling red flames — the hero's sole guide in the wilderness. The message is: the world must be saved, and the artist, having brought the community together with his appeals to the spirit, will lead the way. Three years earlier, Hermann Bahr had written: Never yet has any period been so shaken by horror, by such a fear of death. Never has the world been so silent, silent as the grave. Never has man been more insignificant. Never has he felt so nervous. Never was happiness so unattainable and freedom so dead. Distress cries aloud; man cries out for his soul; this whole pregnant time is one great cry of anguish. Art too joins in, into the great darkness she too calls for help, she cries to the spirit; this is expressionism. This was undoubtedly stirring stuff in 1916. But as the interwar period dragged on, and the prospect of another conflict began to loom on the horizon, the romantic rhetoric of expressionism began to sound suspiciously vague. It was all very well to talk about crying out into the great darkness, or to speak — as the Blauer Reiter group did — of bringing about 'spiritual renewal' through art — but what were artists actually doing to help the situation? To Georg Lukacs, writing in 1934, such expressionist hogwash reflected 'a general estrangement from the concrete problems of the economy, a concealment of the connections between society, economy and ideology, with the result that these questions are increasingly mystified.' Marxist critics like Lukacs believed — as punk-era critics later would — that it was the task of the modern artist to identify actual problems in society, reveal the hidden causes of these problems, and propose solutions. Ernst Toller realised this. Art and theatre, he felt, could not simply go on saying 'everything is collapsing, and this is how I feel about it' Toller felt that art must follow up its emotional appeals by providing concrete ideas for a new society. But Toller was unusual. Most expressionist art in the interwar period seemed merely to give voice to feelings of crisis and chaos in a time of crisis and chaos. The world is falling apart, and the expressionist says to his public, 'it feels like the world is falling apart'. 'Yes!' says the sensitive aesthete, 'I feel the same way.' The result is an army of people full of feeling, but with no real idea how to help the world. This, expressionism's critics believed, could only lead to trouble. # Rock Stars Are Fascists, Too IN 1924 A lonely, angry dreamer — a failure as an artist, a failure as a revolutionary — sat in his prison cell and considered what it would take to make his people rise up and build a better world. He came to the conclusion that, while intellectuals and politicians might attach great importance to ideas, what really moved the masses was emotion. ...all great movements are popular movements, volcanic eruptions of human passions and emotional sentiments, stirred either by the cruel goddess of Distress or by the firebrand of the world hurled among the masses. This, in turn, led him to understand the importance of propaganda. Keeping the people informed would cause them to think too much, and nothing would be done. But propaganda, if used correctly, could stir those 'human passions' necessary to bring about radical change: Its effect for the most part must be aimed at the emotions, and only to a very limited degree at the so-called intellect. In place of rational solutions to the problems of an increasingly complex world, he would offer people feelings, images, myths and symbols. These he felt confident they would accept, because the real choices before them were so difficult. The author, Adolf Hitler — writing in his memoir Mein Kampf — was using language that we have become quite well acquainted with in this story so far. His preference for 'emotional sentiments' over the 'so-called intellect' and his belief in the power of 'human passions' all have a familiar ring about them, as does his insistence elsewhere on the restoration of myth and symbols as a cure for modern soul-sickness. Nazism, as Bertrand Russell points out, is really just the logical outcome of romantic philosophy applied on a mass scale. Nazism sees the romantic worship of passion and intensity, the elevation of the individual genius above the herd, and the feeling for nature and landscape, turned into politics. This explains why Joseph Goebbels, Hitler's Propaganda minister, made a serious attempt to cultivate expressionism as a national style. 'The fascists, with some justification, see expressionism as a heritage they can use,' wrote Lukacs in 1934. Expressionism, after all, was just the latest permutation of the romantic movement — and what could be more authentically German than romanticism? Goebbels was something of a romantic himself; he wrote his doctoral thesis on Romantic Drama, and in 1929 he wrote a sentimental novel set in the Alps called Michael. Goebbels recognised in the expressionists' landscape paintings a similarly sentimental feeling for the fatherland which, he believed, would make them ideal for propaganda purposes. He cannot have failed to see, also, that the rhetoric of expressionist protest art: 'the world is in chaos; we must unite!' — was exactly the kind of emotional bait a fascist leader could use to drum up support. The expressionist call to arms bypassed the 'so-called intellect' and went straight for the heart. Although Hitler's innate conservatism would quickly cause Nazism and expressionism to part ways, Goebbels' patronage left the style tainted by association for decades to come. David Bowie must have known this at the time he became interested in expressionist art, but had thus far done nothing to distance himself from the work's overtones. In fact, Bowie, much to fans and critics' horror, had already embraced fascism. In 1975 he was living as a virtual recluse in Los Angeles, studying the occult and Nazi theology. The following year he returned to Britain after touring America, and made what looked to many like a Nazi salute to his fans waiting outside Victoria Station. Later, while working on The Idiot in Paris, Bowie told Rolling Stone's Cameron Crowe: I'd adore to be prime minister...and yes, I believe very strongly in fascism. The only way we can rid ourselves from the sort of liberalism that's hanging foul in the air at the moment is to speed up the progress of a right-wing, totally dictatorial government and get it over with as soon as possible. People have always responded better under a regimental leadership. Bowie had a typical Nietzschean artist's sense of morality in that he didn't really have one. His duty, to paraphrase Dave Gahan, was always to beauty — it would never occur to him to alter a line in a song or an image in his show because it could be seen as 'immoral'. If it sounded good, if it looked good, if it meant something to him, it was in. This is no less than we expect of a romantic artist. 'The artist's feeling,' declared the German painter Caspar David Friedrich in 1818, 'is his law.' But Friedrich never played the Hammersmith Palais. Bowie did, and here his insistence on subjectivity took on a new significance. He'd created art in defiance of all moral laws. Now it seemed that there were thousands of people who recognised the truth of that art, who agreed with the things he was saying. Society is going to the dogs, morality is bunk, nothing is true, everything is permitted. 'Yes, yes!' said the thousands of Ziggy clones in the audience. What conclusions could he draw from this? Obviously, he was even more special than he thought he was. He was blessed with a unique ability to express the unconscious desires of this community he'd created. It wasn't as though they'd elected him, and he never had to ask them what they wanted — he seemed to be able to 'express' them directly. There was no need for voting or debate or any other boring democratic process. Ideas like his would never survive in a democracy, because they are too bold, too unique. But here he was, with thousands of kids who all seemed to want the same thing in front of him. Imagine what they could do... As long as it was confined to the realm of music and art, Bowie's philosophy was sound — nobody ever wrote a song like 'Starman' by ballot. When he started imagining his art turned into action, the rock star was on shakier moral ground, but Bowie had already decided morality was something he could do without. Now, there was nothing to stop him from making a flat-out admission that he admired fascism. With his highly aesthetic view of the world, the amoral rock star had found in Hitler a political figure he could relate to. 'Rock stars are fascists too,' he told Crowe. 'Adolf Hitler was the first rock star.' Twenty years later, AFI showed us what Bowie's rock-star-as-dictator might look like in their video for 'Miss Murder'. While recording the song, AFI invited members of their fan community, the Despair Faction, to take part in the sessions. The Despair Faction's 'whoa-oh's and 'hey's can be heard in the song's chorus, suggesting a political rally, and the video expands on this idea. We see Davey Havok, his black fringe combed down over his face, singing in front of what looks like a Nazi rally at Nuremberg. Huge banners hang down in the background, and torches light up the faces of his fans, who raise their arms in salute to their leader. 'Miss Murder' is disturbing, and arguably in poor taste. But Havok at least deserves points for honesty, for being bold enough to follow the implications of his art to their conclusion. Thousands of people, attuned to his thoughts, singing his words. If this weren't just a recording studio or a rock concert, if this were the real world, imagine what we could do... # The Black Parade Is Dead FOR MY CHEMICAL Romance, The Black Parade was a conscious attempt to embrace the dynamics of large-scale stadium rock. Gerard Way, as a confirmed Freddie fan, had doubtless watched Queen's performance at Live Aid many times and wondered — as any performer would — what it would be like to do that. To stand on stage in front of thousands of people and hold every single one of them in the palm of your hand, to have them hanging on every word you sing, every gesture you make. At the famous benefit concert in 1985, Freddie Mercury lead the audience in a vocal warm-up. 'Ay-o!' he sang. Seventy-two thousand people sang 'Ay-o!' back. Mercury went on to sustain this for about ten times as long as anyone would have thought possible, his confidence was supreme, his manipulation of the crowd was breathtaking. Later, he had seventy-two thousand people clapping in unison to Radio Ga Ga — just like the worker drones in the song's Metropolis-inspired video. Watching it on DVD is a thrill, being there would have been incredible. But to be the singer, to be Freddie Mercury on that day in front of that crowd, would be to feel superhuman, to feel that anything was possible. At the Hammersmith Palais, at Mexico's Palacio de los Deportes, and at the final show of the tour on 9 May 2008 at Madison Square Garden, Gerard Way must have known something of what it was like to be Freddie at Live Aid — to wield that power, to stand in front of tens of thousands of people and have them say, 'you express us better than we can express ourselves, we surrender to you.' But Way is not exactly like Freddie. He won't be satisfied by the knowledge that he's united his audience in a collective feeling. He wants to take that feeling and do something with it — and not just vocal warm-ups. He wants to make a difference. And on the Black Parade tour it must have felt to him as though this might really happen. He'd created an army — an army that he would have to lead. It was, after all, his unique insight, his extraordinary ability to identify what was wrong with their world and how to make it right, that had brought them together in the first place. Thousands of kids, a secret army of the broken, beaten and damned ready to follow them anywhere, do anything. Fascism, as Herbert Read has observed, turns on a subtle combination of sadism and masochism. After any revolution, there is an opportunity for freedom — old institutions have been trashed, the field is open, anything is possible. Ideally, Read says that the outcome of this should be a form of communism — people co-operating with one another to build the kind of world they want to live in. But since this is so difficult to do, in most cases people will settle for the far easier option of being lead. They start looking around for a strong leader. Lo and behold, the people's masochistic desire to be bossed around finds its ideal companion in the leader, whose sadistic desire to impose his will on others makes him the logical choice. The problem, in the case of The Black Parade, is that Gerard Way is not a sadist — he just doesn't have the stomach for it. The line 'teenagers scare the living shit out of me' is only partly delivered in character. It's also a sincere statement on Gerard's part, an admission that he realises his audience virtually demands of him that he tell them what to do, and that this terrifies him to his soul. The band aren't cut out to be sadists either — they can play at being 'a little shittier to the audience' as guitarist Ray Toro puts it; but like Roger Waters after the famous spitting incident, they're too sensitive to keep it up. But as Toro later admitted, the Black Parade got out of control. Having set the thing in motion, the band members felt compelled to act out the roles required of them. They'd created a monster, dreamed a comic-book army of the undead into existence. Now, like Frankenstein, they found this thing knocking on their door every night, demanding that its parents recognise their demon offspring. The pressure was intense, the sense of responsibility enormous. Somewhere in South America, Ray Toro said to himself, 'Fuck, this shit is trying to kill us!' At this point, rock history began to repeat itself. The Black Parade is in many ways a twenty-first century retelling of Bowie's The Rise and Fall of Ziggy Stardust and the Spiders from Mars. Both are loosely structured concept albums which end with the tragic and symbolic death of the hero. And both required a change of identity for the performers themselves in bringing it to life. The Ziggy Stardust concerts were Ziggy concerts, not Bowie concerts — he played a role on stage, a role that allowed him to be 'a little shittier to the audience'. For My Chemical Romance it was the same. They didn't just perform The Black Parade, they became The Black Parade. If there was any problem with Ziggy Stardust it was that it worked too well. By presenting the Passion of Ziggy onstage, Bowie created a cult, a religion with devotees who proclaimed the resurrected Ziggy their leader. 'I began to think he was a new kind of messiah,' recalled one teenage Bowie fan in 1976. 'I really thought he had some kind of infinite wisdom.' 'Julie', speaking to Fred and Judy Vermorel for their book of fan fantasies, Starlust, went on. 'Bowie was magic and he was supreme. He had the qualities of a type of ruler.' Bowie had always perceived these qualities in himself, and would admit to them again in the future. But sometime in 1973 his megalomania had been fatally undermined by his drug use and his low self-esteem. He was too much of a nervous wreck to lead anybody anywhere, and the thought of all those grabbing hands reaching out to him filled him with horror. Life seemed to be imitating art. At this point Bowie realised he'd written an escape clause into his new religion. His fans wanted Ziggy, and he was Ziggy. But since Ziggy was just a character, it would be easy enough for Bowie to kill Ziggy (for real this time) and escape through the back door to live another day. This is exactly what he did at the Hammersmith Odeon on 3 July 1973. 'Not only is it the last show of the tour,' he said, looking down at his devotees from the stage, 'but it's the last show we'll ever do.' 'Noooooo!' screamed the kids. Sometime in 2008, the members of My Chemical Romance realised that, just as Bowie had done thirty-five years earlier, they had to kill the monster they'd created — their army of the undead had to die. Onstage at Mexico's Palacio de los Deportes, Gerard Way roared, 'the Black Parade...is dead!' 'Noooooo!' screamed the kids. Then, after playing the last show of the tour, the band packed up their floats and banners and threw Colleen Atwood's uniforms in the laundry basket; perhaps hoping — like Spider Man ridding himself of the black suit — that the evil fascist impulses that had come over them on tour were somehow contained in its fabric and would simply come out in the wash. (Ray Toro's is still in the garage — 'I don't want to bring it into the house!' he said.) My Chemical Romance looked forward to being an ordinary rock band again, and playing ordinary rock shows in front of relatively small groups of fans. The army of the broken, the beaten and the damned that Gerard had called into existence, which now seemed to be massing its ranks in every city in the world, would have to lead itself. To imagine Gerard Way in this moment is to imagine two great romantic heroes in one person. Gerard was both Napoleon Bonaparte declaring himself emperor, and Ludwig van Beethoven, tearing up his manuscript in disgust at his own violation of 'the rights of man'. The truth was that Gerard — as much as he liked the idea of having his own army — could not be the leader of a mass movement. The cult of leadership is nice for politicians and leaders of military coups, but not much good for people, and no good at all for artists. Leadership implies a mass who must be lead, and a mass denies individuals their right to be solitary. An artist like Gerard can't accept this, since solitude is what got him the gig in the first place. Rivers Cuomo: Emotion recollected in tranquillity. # Insulation and Disaffection IN HIS 1964 book, To Hell With Culture, the English art critic Herbert Read tackled a question which is central to any understanding of romantic poetry — or any other kind of emotional art. That is, Read asked himself: what good is self-expression? By the late twentieth century, the importance of self-expression was almost taken for granted by the artistic avant-garde. Read himself, in one of his many long-running arguments with his friend, the sculptor and typographer Eric Gill, had been bold enough to suggest that it might be the sole purpose of art. But searching himself, Read realised that it didn't make sense. Artists would not be tolerated in society if they didn't contribute something useful to it — and what use could society have for self-expression? By simply expressing themselves while everybody else was working, artists would effectively be saying: 'I am more important than everybody else'. Society should, in theory, have no reason to put up with such behaviour. And yet we can't quite tell these selfish individuals to get lost. On The Red Album, Rivers Cuomo's stance is exactly the one described above. In fact, he's gone further. He insisted that he be allowed to express himself and told us in no uncertain terms that this makes him better than you and me — 'the greatest man that ever lived'. Then he dared us to look him in the eye and tell him we don't like it. We have to admit we cannot. How did he get away with this blatantly antisocial self-expression? The story of Pinkerton provides a clue. In 1995, Rivers Cuomo had gone deep into his own emotional world. He came back with an album of pure self-expression, an album so completely driven by the need for self-revelation that, to many listeners, it barely seemed to have songs. It should, in theory, have been as useful to society as a brick thrown out of a window into a crowd. Who would want a thing like that? Surprisingly, quite a lot of people wanted it. Weezer's difficult second album became, over the next decade, a sort of secret road map with which the lonely and alienated could find one another. It identified something many people had in common — something which was emotional, irrational and deeply disturbing — but instantly familiar to those concerned. Listening to Pinkerton, these individuals began to understand that the strange, unnameable things that nagged at them as they did their jobs or went to school were not delusions, but secret truths. There were others, they realised, who felt alienated from society, who felt like life might be a struggle for which there is no reward, who felt like screaming for no reason at all. Realising this, they no longer felt alone. Read would say that what Cuomo did with Pinkerton was to reveal 'the collective instincts which underlie the brittle surface of convention and normality.' This kind of thing, he argued, is very, very useful to society. 'It is the artist's business,' he wrote in To Hell With Culture, 'to make the group aware of its unity, its community.' Artists, he reasoned, only think they're expressing themselves. What they're in fact doing is expressing life — the secret life of their society. With the artist's map of this invisible country in their hands, individuals become connected to one another in ways that a society based on money and production can neither predict nor replicate. The alienating effects of modern life are reversed, a fragmented world is put back together again for the length of a song. The artist's real job, Read insisted, is not self-expression, but life-expression. Having figured out what society could expect from modern artists, Herbert Read next asked himself what modern artists could expect from society, what conditions they required in order to be able to do what they do. In Rilke's Letters to a Young Poet, and in Wordsworth's Preface to the Lyrical Ballads — two documents which neatly bookend the romantic movement — he discovered the answer. 'Poetry,' wrote Wordsworth in 1800, 'takes its origin from emotion recollected in tranquillity.' Here, Wordsworth, like Rilke, seemed to be saying that the poet's insights about the nature of life could only be attained by his remaining at one remove from that life. Therefore the artist's job description, Read concluded, could be said to be: To communicate something as essential as bread, yet to be able to do so only from a position of insulation and disaffection. Rivers Cuomo knows instinctively that this 'position of insulation and disaffection' is the first requirement of his work. He created a cure for loneliness by identifying and isolating certain emotional tendencies in certain people between the ages of sixteen and twenty-one. But he didn't find out what those things were by studying demographics or doing surveys or talking to teenagers in focus groups. He didn't ask his fans what they wanted either (if he had, you can bet they wouldn't have asked for anything like Pinkerton). In fact, he didn't really talk to anyone in the period when he was making Weezer's second album. He discovered the things that would later bind Pinkerton's community together by studying the lessons of his dreams. And to do that, he had to be alone. Cuomo's art — like Wordsworth's — comes from emotion recollected in tranquillity. 'Probably most anyone doesn't go through a week without getting upset about something,' he explained to Jenny Eliscu in 2002, 'and that's what I do. I wait for those moments, and then I pounce.' But while emotions might be a dime a dozen, tranquillity — the contemplative calm required to identify those emotions and catch them before they melt away into the air — is not easy to come by in the modern world. In Los Angeles, it's almost impossible. But Cuomo has always found a way. Locking himself in the garage, breaking his own leg, taking a vow of celibacy, 'partying by himself', blacking out the windows in his apartment — all of these, for the Weezer singer, were simply ways of maintaining the position of insulation and disaffection his job requires. The irony in all this for the rock and roll poet is that the isolation he needs to do his job is constantly threatened by the community his poetry has created. As the cult of Pinkerton grew during the late 1990s, the group sought to elect a leader — and naturally they turned to the genius who had made the group aware of its unity. Cuomo 'expressed the group', therefore he ought to be at its head. He ought to, at the very least, acknowledge his constituents by tossing them a bone every now and then. But Cuomo did not accept this role. He hadn't accepted it the first time around, when the success of The Blue Album had required of him that he be the spokesperson for 'geek-rock' (whatever that was); and he wouldn't accept it this time around either. He went to extraordinary lengths to alienate his new fans, to make it clear he would not, under any circumstances, be the godfather of emo. Cuomo was not just being perverse. He refused to be a part of the group he'd created because he saw that if he did, his alone-ness would be compromised. But by refusing to give up his solitary status, Cuomo struck a blow for solitude, not just for himself, but for everyone. Because in those moments when a group forms and the group wants to be lead, it's not just the artist's independence that's at risk, but the independence of his fans as well. Pinkerton created a community of people who did not feel at home in society. It would make no sense to then turn those people into a 'mass' — since it was everything 'mass' that they revolted against in the first place. They had rejected the world of the average, the reasonable, the world of 'what's best for everyone'; and attached themselves to this strange, particular sense of truth that the artist had revealed. Now they were banding together, organising, massing. The world of 'what's best for everyone' would be the logical outcome of this process, the Pinkerton fans would become more like each other. Cuomo, who had already seen this three years earlier — when he'd looked out into the crowd and seen thousands of kids wearing his glasses — could see where things were headed. Gerard Way: The saviour of the broken, the beaten and the damned. # Leave Them Kids Alone. BRUCE SPRINGSTEEN HAD the best possible training for a rock and roll poet. Because he was painfully shy and socially awkward as a young man, he went into himself. Here, he found words — floods of words — endless poems about youth, freedom, landscape, faith and love. He thought he was expressing himself, if he thought about it at all. It was only later, when people started to hear those songs, that he understood what he was really doing. Talking to Springsteen in 1999, Mojo's Mark Hagen remarked, 'I've never been to one of your shows or listened to one of your LPs, without it connecting with something in me I didn't know I felt.' Springsteen replied: That's the writer's job. The writer collects and creates those moments from out of his own experience and the world he sees around him....and you present those things to your audience, who then experience their own inner vitality, their own centre, their own questions about their own life, and their moral life.... That's what you're paid for — somebody says, "Hey, I'm not alone." Sometime in 1985, sitting in his seat (or maybe standing, punching the air) at New Jersey's Bradenburg Arena, Gerard Way was one of those 'somebody's. Fifteen years later, following his own apprenticeship of loneliness, introspection and overlong poetry, Way would start getting paid to do the same thing. He would discover, like Springsteen, that it's not as easy as it looks. It's hard enough in the rehearsal room, or in some tiny club in front of an audience of your peers. It gets harder when your records start to sell and your audience goes from hundreds, to thousands to hundreds of thousands. Sometime before the recording of The Black Parade, Gerard realised, as Springsteen had realised around the time of Born to Run, that his actions had implications. 'I'd meet these kids that were outsiders,' said the singer. 'And I realised they're looking to us for the answer. It started to scare me.' In the stadium, ideas are amplified along with sounds. And in the same way that you can't turn the volume up on a record without cracks and pops getting louder too, My Chemical Romance found that as their ideas were broadcast on a larger scale, the flaws and contradictions that had always been there were amplified in proportion. The romantic ideals that drive the band and its music, the things that put the romance in My Chemical Romance, were forced to account for themselves. How can we look death in the face and still say yes to life? How can we reject society without dying of loneliness? And how can we insist on our right to self-expression when the world is falling apart? Incredibly, the band chose not to pull back from these questions, but to go further into them — to embrace the contradictions at the heart of their songs and the movement those songs had created, and to watch as the consequences played themselves out in the real world. They created a teenage revolution — not a revolution in the name of civil rights, an end to violence, more drugs, less bombs, more fun or free love. Their revolutionary war was waged in the name of loneliness. The members of The Black Parade were standing up — to paraphrase Dostoyevsky — for their own caprices, and for having them guaranteed where necessary. The world was treated to the extraordinary spectacle of thousands of kids demanding the right to be sad. British fans organised a day of action to protest the unfair tactics of the Daily Mail's counter-offensive strike on sadness. They held up banners saying: 'We're not a cult, we're a fucking army!' What were they all doing there? asked one reporter. A fifteen-year-old My Chem fan explained, 'We're all alone together.' At this point, it became Way's job to show how this mind-boggling idea might work out in practice. He couldn't be a leader — he couldn't say to say to his fans 'it's ok to be weird and lonely and different to everyone else' and then deny them their alone-ness. But he couldn't pretend that what he'd created was of no importance either. He had to honour the movement he'd set in motion, and demonstrate, through his art, how its aims might be accomplished. It might seem like a lot to ask of a rock star — but nobody is better qualified. After all, it was his idea. It's tempting to say that the idea of a world where everybody is 'alone together' is one that could only have been dreamed up by a teenager. Teenagers, who are selfish, amoral, sentimental, obsessed with childhood, dreams and the significance of their own emotions. Only a teenager could elevate these principles above the sensible, utilitarian ones on which our modern society is based. But all of these things are equally important to the artist. A poet like Springsteen or Gerard Way needs them in order to do his job: 'to communicate something as essential as bread, yet to be able to do so only from a position of insulation and disaffection.' When institutions, politicians, society and rational thought itself let the poet down, his faith and creative power can only be restored by returning to the world of childhood, imagination, memory, and emotion — to the small 'h' humanity of a lyric like 'when I was a young boy'. And since the world will not recognise these things, the poet is forced to become a revolutionary. The poet, as Herbert Read writes, 'is compelled to demand, for poetic reasons that the world be changed. We cannot say it is an unreasonable request: it is the first condition of his existence as a poet.' Now, in the twenty-first century, the romantic poet has found new recruits to his cause. Emotional teenagers are the natural allies of the poet. They want the same things — dreams, nostalgia, intense emotions, solitude. They already understand, like the poet, that there is no political party on earth that can guarantee their happiness because they demand a kind of happiness that a society based on production and money doesn't understand. This is Thomas Carlyle's 'passion incapable of being translated into action'. This is passion that can find no useful outlet in society since it refuses to recognise or participate in a society which has proven itself consistently incapable of providing for real human needs. For such individuals, 'Welcome to the Black Parade' — a song which denounces modern life, asserts the importance of dreams, solitude and emotions, and demands the creation of a new world in which those things are recognised — is a call to arms which cannot be ignored. 'Welcome to the Black Parade' is thrilling and irresistible because it dares to imagine that art might change everyday life. Gerard Way could not, in good faith, sing that song unless he believed poetry was that important, that life could be fundamentally altered by a set of words. But the singer believes this because he has to. He believes it because he knows that if it it's not true, then art is useless. Gerard Way is not just engaging in some scene-politics pissing contest when he says that 'Emo is a pile of shit'. For Way, emotion is important. The poetry of 'Welcome to the Black Parade' takes its origin from emotion recollected in tranquillity, and the poet wants you to feel what he felt when he wrote it. But where, for an emo band, the mission would be accomplished at this point, for Gerard Way it's just the beginning. He knows that if the process stalls here, the promise of a better world glimpsed in the last thirty seconds of the song will never be more than thirty seconds of noise on the radio. He can't commit himself to politics, because he's demanding the kind of freedom no political party on earth can allow. He can't get bogged down in the kind of Marxist analysis advocated by the post-punk positivists; if he does, the poetry that drives his music will disappear. But he can't accept the notion that music begins and ends with the sharing of feelings, as though rock and roll were a giant talk show or group therapy session. If art has no power to improve life, then art — as Herbert Read says — will never be anything more than 'self-expression'. Which is another way of saying that until we start taking poets and their irrational demands seriously, everything will be emo. # Notes Emotional People 1. Jason Pettigrew, 'Dead to see another day', Alternative Press, July 2008. 2. Gerard Way, 'My Chemical Romance brand emo "shit".' NME News, nme.com. September 2007. 3. Ronen Kaufmann, 'Blood Runs Deep', Alternative Press, July 2008. 4. Tim Karan, 'Blood Runs Deep', Alternative Press, July 2008. 5. Andy Greenwald, 'Emo: We Feel Your Pain', in Michael Sia (ed.), Spin: 20 Years of Alternative Music, Three Rivers Press, New York. 6. Gwyn Tyme, 'My Chemical Romance', Musicpix.net, May 2005. Buzzcocks 1. Jon Savage, England's Dreaming, Faber and Faber, London, 1991. 2. Buzzcocks: 'What do I Get?', Manchester — So Much to Answer For: The Peel Sessions, Strange Fruit LP, 1990. 3. Annie Zaleski, 'Blood Runs Deep', Alternative Press, July 2008. The Cure 1. Dave Thompson, In Between Days: An Armchair Guide to the Cure, Helter Skelter, London, 2004. 2. Ibid. 3. Ibid. 4. Ibid. 5. Ibid. Weezer 1. Rivers Cuomo, Sleevenotes for Alone: The Home Recordings of Rivers Cuomo, Geffen CD, 2007. 2. Ed Masley, '10 Essential "Disappointing Albums"', Alternative Press, July 2008. 3. Rivers Cuomo, op. cit. 4. Andy Greenwald, Nothing Feels Good: Punk Rock, Teenagers and Emo, St Martin's Press, New York, 2003. 5. Harry Thomas, 'Not So Serious Rivers Cuomo', Rolling Stone, June 2001. 6. Chris Mundy, 'Weezer's Cracked Genius', Rolling Stone, September 2001. 7. Songmeanings.com, 2002. 8. Ibid. 9. Rob Mitchum, 'Weezer: Make Believe', album review, Pitchfork.com, May 2005. The Classics 1. William J. Long, Outlines of English and American Literature, Gutenberg.org 2. Robert Barnard, A Short History of English Literature, B. Blackwell, New York, 1984. 3. Walter Horatio Pater, Essay on Style, Gutenberg.org Troublemaker 1. Steve Kandell, 'Heck on Wheels', Spin, June 2008. 2. Weezer, 'Troublemaker', The Red Album, Geffen CD, 2008. 3. Rivers Cuomo, Sleevenotes for Alone: The Home Recordings of Rivers Cuomo, Geffen CD, 2007. 4. Weezer, 'Dreamin'', The Red Album, Geffen CD, 2008. Wordsworth 1. William Wordsworth, 'Expostulation and Reply' in Wordsworth, William, Lyrical Ballads, with other poems, (1800 edition) Gutenberg.org 2. William Wordsworth, 'The Tables Turned' in Wordsworth, William, Lyrical Ballads, with other poems, (1800 edition) Gutenberg.org Civilisation 1. Norman Davies, Europe: A History, Pimlico, London, 1997. 2. Nicholas Dent, Rousseau, Routledge, New York, 2005. The French Revolution 1. Thomas Carlyle, The French Revolution: A History, Gutenberg. org, 2006. 2. Graeme Fife, The Terror, Portrait, London, 2004. 3. Mathew Arnold, Selections from the Prose Works of Matthew Arnold, ed. Johnson, William Savage, Gutenberg.org, 2004. 4. Roger Sharrock, Selected Poems of William Wordsworth, Heinemann, London, 1968. 5. Rupert Christiansen, Romantic Affinities, Pimlico, London, 2004. 6. Eric Hobsbawm, The Age of Revolution: Europe 1789–1848, Weidenfeld and Nicolson, London, 1962. 7. William Wordsworth, 'The French Revolution' in Sharrock, Roger (ed.), Selected Poems of William Wordsworth, Heinemann, London,1968. 8. William Wordsworth, 'The Prelude, or, Growth of a Poet's Mind' in Ernest de Selincourt (ed.), Oxford University Press, London,1960. 9. Ibid. The Story Is in the Soil 1. Gavin Edwards, 'Rock's Boy Genius', Rolling Stone, October 2002. 2. Bright Eyes, 'I Believe in Symmetry', Digital Ash in a Digital Urn, Saddle Creek CD, 2005. 3. Bright Eyes, 'Road to Joy', I'm Wide Awake, It's Morning, Saddle Creek CD, 2005 4. Norman Davies, Europe: A History, Pimlico, London, 1997. 5. Rupert Christiansen, Romantic Affinities, Pimlico, London, 2004. 6. William Vaughn, Romanticism and Art, Thames and Hudson, London, 1994. 7. Ludwig Van Beethoven, Beethoven's Letters, 1790–1826, Gutenberg.org, 2004. 8. Bright Eyes. 'Road to Joy', op. cit. 9. Triple j interview, Zan Rowe 10. Ibid. A Motion and a Spirit 1. William Wordsworth, 'Tintern Abbey', in Sharrock, Roger (ed.), Selected Poems of William Wordsworth, Heinemann, London, 1968. 2. William Wordsworth, 'Tintern Abbey' in William Wordsworth, Lyrical Ballads, with other poems, (1800 edition) Gutenberg.org 3. William Wordsworth, 'Preface to the Lyrical Ballads' in William Wordsworth, Lyrical Ballads, with other poems, op. cit. Romantic 1. Brian Howe, 'Bright Eyes: Noise Floor' (review), Pitchfork.com, October 2006. 2. Isiah Berlin, Against the Current: Essays in the history of ideas, Hogarth Press, London, 1979. 3. Robert Barnard, A Short History of English Literature, B. Blackwell, New York, 1984. 4. William Wordsworth, 'The Tables Turned' in Roger Sharrock (ed.), Selected Poems of William Wordsworth, Heinemann, London, 1968. 5. Ibid. Disenchanted 1. My Chemical Romance, 'Disenchanted', The Black Parade is Dead, Warner/Reprise DVD, 2008. 2. My Chemical Romance, 'Welcome to the Black Parade', The Black Para de, Warner/Reprise CD, 2006. 3. Ibid. 4. Alex De Jonge, Dostoyevsky and the Age of Intensity, Secker and Warburg, London, 1975. 5. Wordsworth, 'The Prelude', Oxford University Press, London, 1964. 6. Ibid. 7. Herbert Read, To Hell With Culture, Routledge, London, 2002. 8. William Vaughn, Romanticism and Art, Thames and Hudson, London, 1994. Paint It Black and Take It Back 1. Dan Stapleton, 'My Chemical Romance', Rolling Stone, February 2007. 2. Shirley Halperin, 'Coldplay talk "Viva la Vida"'. Ew.com., June 2007. 3. Tom Prideaux, The World of Delacroix 1798–1863, Time Incorporated, New York, 1966. 4. Ibid. 5. William Vaughn, Romanticism and Art, Thames and Hudson, London, 1994. 6. Tom Prideaux, op. cit. 7. Shirley Halperin, 'Coldplay talk "Viva la Vida"'. Ew.com., June 2007. 8. Mtv.com 'Buzzworthy', June 2007. Napoleon 1. Norman Davies, Europe: A History, Pimlico, London, 1997. 2. Eric Hobsbawm, The Age of Revolution: Europe 1789–1848, Weidenfeld and Nicolson, London, 1962. 3. H C Robbins Landon, Beethoven: A Documentary Study, Thames and Hudson, London,1974. 4. Bertrand Russell, History of Western Philosophy, Routledge, London, 2006. 5. Hobsbawm, op. cit. 6. Russell, op. cit. 7. Hobsbawm, op. cit. 8. Davies, op. cit. 9. William Vaughn, Romanticism and Art, Thames and Hudson, London, 1994. 10. Davies, op.cit. 11. Johann Wolfgang von Goethe, The Sorrows of Young Werther, Gutenberg.org, 2004. 12. Thomas Carlyle, Introduction to Johann Wolfgang Von Goethe, Truth and Fiction relating to my life. Gutenberg.org, 2004. 13. Ibid. 14. Hobsbawm, op. cit. This Tragic Affair 1. My Chemical Romance, 'Famous Last Words', The Black Parade, Warner/Reprise CD, 2006. 2. Tom Rawstorne, 'Why no child is safe from the sinister cult of emo', Dailymail.co.uk, May 2008. 3. Rupert Christiansen, Romantic Affinities, Pimlico, London, 2004. 4. Thomas Carlyle, Introduction to Johann Wolfgang Von Goethe, Truth and Fiction relating to my life. Gutenberg.org, 2004. 5. Johann Wolfgang Von Goethe, The Sorrows of Young Werther, Gutenberg.org, 2004. 6. Ibid. 7. Ibid. 8. Ibid. 9. Ibid. 10. Ibid. 11. Ibid. 12. Ibid. 13. Ibid. 14. Ibid. 15. Ibid. 16. Ibid. Passion Incapable of Being Converted into Action 1. William Vaughn, Romanticism and Art, Thames and Hudson, London, 1994. 2. Gerhart Hoffmeister, 'Reception in Germany and Abroad' in Lesley Sharpe, The Cambridge Companion to Goethe, Cambridge University Press, Cambridge, 2002. 3. Ibid. 4. Ibid. 5. Barker Fairley, A Study of Goethe, Oxford University Press, London, 1950. 6. Hoffmeister, op. cit. 7. Thomas Carlyle, Introduction to Johann Wolfgang Von Goethe, Truth and Fiction relating to my life. Gutenberg.org, 2004. 8. Johann Wolfgang Von Goethe, The Sorrows of Young Werther, Gutenberg.org, 2004. 9. Carlyle, op. cit. Sentimentalists 1. Rupert Christiansen, Romantic Affinities, Pimlico, London, 2004 2. Walter Benjamin, 'Goethe' in Michael Jennings (ed.), Walter Benjamin: Selected Writings Volume 2, Part 1, Harvard University Press, Cambridge, Mass., 2005. 3. Christiansen, op. cit. 4. Johann Wolfgang Von Goethe, The Sorrows of Young Werther, Gutenberg.org, 2004. 5. Martin Swales, 'Goethe's Prose Fiction' in Lesley Sharpe, The Cambridge Companion to Goethe, Cambridge University Press, Cambridge, 2002. Across the Sea 1. Weezer, 'Across the Sea', Pinkerton, Geffen CD, 1996. 2. Weezer, 'Butterfly', Pinkerton, Geffen CD, 1996. 3. Chris Mundy, 'Weezer's Cracked Genius', Rolling Stone, September 2001. 4. Ibid. 5. Andy Greenwald, Nothing Feels Good: Punk Rock, Teenagers, and Emo, St Martins Press, New York, 2003. 6. Ibid. 7. John Weightman, The Concept of the Avant-Garde: Explorations in Modernism, Alcove Press, 1973. 8. Greenwald, op. cit. 9. Greenwald, op cit. 10. Posted in 'The Despair Faction', despairfaction.com, November 2007 Love Like Winter 1. Neil Strauss, 'AFI: Decemberunderground' (album review), Rolling Stone, June 2006. 2. Alex De Jonge, Dostoyevsky and the Age of Intensity, Secker and Warburg, London, 1975. 3. Davey Havok, 'Decemberunderground', AFI official website, afireinside.net 4. Austin Scaggs, 'Davey Havok: Q&A', Rolling Stone, June 2006. 5. AFI, 'Summer Shudder', Decemberunderground, Interscope/Universal CD, 2006. 6. AFI, 'Love Like Winter', Decemberunderground, Interscope/Universal CD, 2006. 7. The Making of 'Love Like Winter' Part 1. YouTube.com Alone and Palely Loitering 1. Robert Gittings, John Keats, Penguin, London, 1968. 2. Ibid. 3. John Keats, 'La Belle Dame sans Merci' in Robert Gittings (ed.), Selected Poems and Letters of Keats, Heinemann, London, 1966. 4. Ibid. 5. Gittings, op. cit. 6. Ibid. A Forest 1. Dave Thompson, In Between Days: An Armchair Guide to the Cure, Helter Skelter, London, 2004. 2. The Cure, 'A Forest', Standing on a Beach, Warner CD, 1986. 3. The Cure, ibid. Lemonade 1. Robert Gittings, John Keats, Penguin, London, 1968. 2. John Keats, 'Ode on Melancholy' in Robert Gittings (ed.), Selected Poems and Letters of Keats, Heinemann, London, 1966. 3. Rivers Cuomo, Sleevenotes for Alone: The Home Recordings of Rivers Cuomo, Geffen CD, 2008. 4. Rivers Cuomo, 'Lemonade', Alone: The Home Recordings of Rivers Cuomo, op. cit. 5. Rivers Cuomo, Sleevenotes for Alone: The Home Recordings of Rivers Cuomo, op. cit. 6. Ibid. 7. Rivers Cuomo, 'Buddy Holly', Alone: The Home Recordings of Rivers Cuomo, Geffen CD, 2008. 8. Andy Greenwald, Nothing Feels Good: Punk Rock, Teenagers, and Emo, St Martins Press, New York, 2003. 9. Jenny Eliscu, 'Rivers Cuomo's Encyclopedia of Pop', Rolling Stone, June 2002. 10. Alpha, Centri, posted on Songmeanings.com, August 2002. 11. Brian Hiatt, 'The Boys with the Car Crash Hearts', Rolling Stone, March 2007. Anatomy of Mellon Collie 1. William J Long, Outlines of English and American Literature, gutenberg.org 2. Amy Hanson, Smashing Pumpkins: Tales of a Scorched Earth, Helter Skelter, London, 2004. 3. Smashing Pumpkins, 'Disarm', Siamese Dream, Hut/Virgin CD, 1993. 4. Billy Corgan, from transcript of 1993 Rage interview, quoted in Starla.org. 5. Richard Kingsmill, The J-Files Compendium, ABC Books, Sydney, 2002. 6. Hanson, op. cit. 7. Ibid. 8. Dave Thompson, In Between Days: An Armchair Guide to the Cure, Helter Skelter, London, 2004. Rock and Roll Suicide 1. Paul Du Noyer, 'Contact', Mojo, July 2002. 2. David Bowie, 'Ziggy Stardust', The Rise and Fall of Ziggy Stardust and the Spiders from Mars, RCA LP, 1972. 3. W A Mozart, Don Giovanni, CBS Masterworks, 1979. 4. Ibid. 5. David Bowie, 'Rock and Roll Suicide', The Rise and Fall of Ziggy Stardust and the Spiders from Mars, RCA LP, 1972. 6. AFI, 'Miss Murder', Decemberunderground, Interscope CD, 2006. 7. AFI, 'Prelude', Decemberunderground, Interscope CD, 2006. 8. Matt Diehl, My So-Called Punk, St Martin's Griffin Press, New York, 2007. Screamin' Lord Byron 1. David Bowie, 'Blue Jean' (music video, dir. Julian Temple), Best of Bowie, EMI DVD, 2002. Lord Byron 1. Robert Gittings, John Keats, Penguin, London, 1968. 2. Peter Quennell, Byron: The Years of Fame, Penguin, London, 2001. 3. Ibid. 4. George Gordon Byron, 'Childe Harold's Pilgrimage' in The Poetical Works of Lord Byron, Murray, London, 1948. 5. Ibid. 6. Quennell, op. cit. 7. Colin Wilson, The Misfits: A study of sexual outsiders, Grafton Books, London, 1989. 8. Quennell, op. cit. 9. Wilson, op. cit. 10. Quennell, op. cit 11. Quennell, op. cit. 12. Ibid. 13. Ibid. 14. George Gordon Byron, Selected Poetry and Prose of Byron, W H Auden (ed.), Signet Classics, New York, 1966. 15. Ibid. 16. Selected Poetry and Prose of Byron W H Auden (ed.), op. cit. 17. Bertrand Russell, History of Western Philosophy, Routledge, London, 2006. 18. Quennell, op. cit. 19. Ibid. Give Them Blood 1. David Bowie, 'Ziggy Stardust', The Rise and Fall of Ziggy Stardust and the Spiders from Mars, RCA LP, 1972. 2. Norman Davies, Europe: A History, Pimlico, London, 1997. 3. My Chemical Romance, 'Blood', The Black Parade, Warner/Reprise CD, 2006. 4. Smashing Pumpkins, 'Bullet With Butterfly Wings', Mellon Collie and the Infinite Sadness, Hut/EMI CD, 1995. 5. X V Scott, 'My Chemical Romance' posted on mychemicalromanceweb.yaia.com The Vampyre 1. Martin Swales, 'Goethe's Prose Fiction' in Lesley Sharpe, The Cambridge Companion to Goethe, Cambridge University Press, Cambridge, 2002. 2. Gerhart Hoffmeister, Gerhart, 'Reception in Germany and Abroad' in Lesley Sharpe, The Cambridge Companion to Goethe, Cambridge University Press, Cambridge, 2002. 3. Johann Wolfgang von Goethe, Faust (Part Two), Penguin, London, 1959. 4. George Gordon Byron, Selected Poetry and Prose of Byron, W H Auden (ed.), Signet Classics, New York, 1966. 5. Frederic Raphael, Byron, Thames and Hudson, London, 1982. 6. Mary Shelley, Frankenstein in Peter Fairclough (ed.), Three Gothic Novels, Penguin, London,1986. 7. Rapahel, op. cit. 8. Christopher Frayling, Vampyres: Lord Byron to Count Dracula, Faber and Faber, London, 1991. 9. Christopher Sandford, Bowie: Loving the Alien, Warner Books, London, 1996. Goths 1. David Johnson, 'Dean Street' in Hanif Kureshi and Jon Savage (eds), The Faber Book of Pop, Faber and Faber, London, 1995. 2. George Gimarc, Post-Punk Diary 1980–1982, St Martin's Press, New York, 1997. 3. Johnson, op. cit. 4. Susan Colon, 'The Gloom Generation', Details magazine, July 1997. 5. Sounds magazine, August 1983. 6. Dave Thompson and Jo-Anne Green, Interview with Ian Astbury, Alternative Press, November 1994. 7. Norman Davies, Europe: A History, Pimlico, London, 1997 8. John Ruskin, Stones of Venice, Gutenberg.org, 2003. 9. Horace Walpole, Letters of Horace Walpole, Gutenberg.org, 2003. 10. Ibid. 11. Michael Gamer, Introduction to Horace Walpole, The Castle of Otranto, Penguin, London, 2001. 12. Ibid. 13. Ibid. 14. Ibid. 15. Davies, op. cit. Rocky Horror 1. Michael Gamer, Introduction to Horace Walpole, The Castle of Otranto, Penguin, London, 2001. 2. Ibid. 3. Simon Reynolds, Rip it Up and Start Again: Post-punk 1978–1984, Faber and Faber, London, 2005. 4. Austin Scaggs, Davey Havok Q&A, Rolling Stone, June 2006. 5. Gerard Way, MTV 'VMA Virgins' interview, 2005. 6. Misfits video [YouTube] Vincent 1. 'Vincent', Cinema 16, American Short Films, Warp DVD, 2006. 2. Ibid. 3. Ibid. 4. Mark Salisbury, Burton on Burton, Faber and Faber, London, 1997. 5. Ibid. 6. Ibid. Frankenstein 1. Bride of Frankenstein, Universal DVD, 2001. 2. Ibid 3. Mary Shelley, Frankenstein in Three Gothic Novels, Penguin, London, 1986. 4. Ibid 5. Bertrand Russell, History of Western Philosophy, Routledge, London, 2006. 6. George Gordon Byron, Selected Poetry and Prose of Byron, W H Auden (ed.), Signet Classics, New York, 1966. 7. Mary Shelley, op. cit. 8. Russell, op. cit. 9. Mary Shelley, op. cit. 10. Ibid. 11. Smashing Pumpkins, 'Disarm', Siamese Dream, Virgin CD, 1993. 12. Billy Corgan, from transcript of 1993 Rage interview, quoted in Starla.org 13. Mary Shelley, op. cit. Edward Scissorhands 1. Mark Salisbury, Burton on Burton, Faber and Faber, London, 1997. 2. Ibid. 3. Ibid. 4. Ibid. 5. Ibid. 6. Edward Scissorhands, Twentieth Century Fox DVD, 2007. 7. Salisbury, op. cit. 8. Beetlejuice, Warner Brothers DVD, 2000. 9. Ibid. 10. Ibid. 'The Dark Side of Human Things 1. Rupert Christiansen, Romantic Affinities, Pimlico, London, 2004. Mystery 1. Simon Reynolds, Blissed Out, Serpents Tail, London, 1990. 2. Ibid. 3. Ibid. 4. Karen Armstrong, A History of God, Vintage, London, 2000. 5. Yasmine Gooneratne, Alexander Pope, Cambridge University Press, Cambridge, 1976. 6. Voltaire, Candide: or, Optimism, Penguin, London, 1947. 7. Norman Davies, Europe: A History, Pimlico, London, 1997 8. Nicholas Dent, Rousseau, Routledge, New York, 2005. 9. Armstrong, op. cit. 10. Nick Cave, 'The Secret Life of the Love Song' in Nick Cave: The Complete Lyrics, Penguin, London, 2007. Utopia 1. Antoine Nicolas de Condorcet, 'Sketch' in John Carey (ed.), The Faber Book of Utopias, Faber and Faber London, 1999. 2. Toby Creswell, 1001 Songs: The Great Songs of All Time, Hardie Grant Books, Melbourne, 2005. 3. Gang of Four, 'Love Like Anthrax', Entertainment!, EMI LP, 1980. 4. Simon Reynolds, Blissed Out, Serpents Tail, London, 1990. 5. Ibid. Utopiate 1. Nick Cave, 'The Secret Life of the Love Song' in Nick Cave: The Complete Lyrics, Penguin, London, 2007. 2. Simon Reynolds, Blissed Out, Serpents Tail, London, 1990. 3. Clinton Walker, Stranded: The Secret History of Australian Independent Music, Pan McMillan, Sydney, 1996. 4. Nick Cave, 'Zoo Music Girl' in Nick Cave: The Complete Lyrics, Penguin, London, 2007. 5. Nick Cave, 'Hamlet Pow Pow Pow!' in Nick Cave: The Complete Lyrics, Penguin, London, 2007. 6. Janine Barrand, Nick Cave Stories, Victorian Arts Centre Trust, Melbourne, 2007. 7. Reynolds, op. cit. 8. Barrand, op. cit. 9. Nick Cave, 'Mutiny in Heaven' in Nick Cave: The Complete Lyrics, Penguin, London, 2007. 10. Ibid. The Degraded Present 1. Nick Cave, 'Release the Bats' in Nick Cave: The Complete Lyrics, Penguin, London, 2007. 2. Simon Reynolds, Rip it Up and Start Again: Post-punk 1978–1984, Faber and Faber, London, 2005. 3. Keith Cameron, 'Siouxsie Sioux — The Mojo Interview', Mojo, October 2007. 4. Andy Greenwald, Nothing Feels Good: Punk Rock, Teenagers, and Emo, St Martins Press, New York, 2003. 5. Alex De Jonge, Dostoyevsky and the Age of Intensity, Secker and Warburg, London, 1975. 6. Siouxsie and the Banshees, 'Spellbound', Juju, Polydor CD, 1989. 7. Norman Davies, Europe: A History, Pimlico, London, 1997 Blasphemous Rumours 1. Depeche Mode, 'Blasphemous Rumours', Some Great Reward, Mute LP, 1984. 2. Dave Thompson, Depeche Mode: Some Great Reward, Sidgwick and Jackson, London, 1995. 3. Depeche Mode, 'Blasphemous Rumours', op. cit. 4. Thompson, op. cit. Paradise Lost 1. Mary Shelley, Frankenstein in Three Gothic Novels, Penguin, London, 1986. 2. Ibid. 3. Ibid. 4. Ibid. 5. Karen Armstrong, A History of God, Vintage, London, 2000. 6. Ibid. The Disappearing God 1. Karen Armstrong, A History of God, Vintage, London, 2000. 2. Bertrand Russell, History of Western Philosophy, Routledge, London, 2006. 3. Armstrong, op. cit. 4. Ibid. 5. Robert Gittings, John Keats, Penguin, London, 1968. 6. Ibid. 7. Ibid. The Age of Simple Faith 1. Depeche Mode, 'Stripped', 101, Mute LP, 1988. 2. Eric Hobsbawm, The Age of Revolution: Europe 1789–1848, Weidenfeld and Nicolson, London, 1962. 3. John Ruskin in Elizabeth Gilmore Holt, From the Classicists to the Impressionists: Volume III of A Documentary History of Art, Doubleday Anchor, New York. 4. William Morris, The House of the Wolfings, Gutenberg.org, 2005. 5. The Pre-Raphaelites, Tate Gallery, London, 1973. 6. Ibid. 7. Ibid. 8. Ibid. Faith 1. 24 Hour Museum Staff, 'Rare Wordsworth Manuscript Secured by Wordsworth Trust', 24hourmuseum.org.uk, April 2007. 2. Karen Armstrong, A History of God, Vintage, London, 2000. 3. William Wordsworth, 'Tintern Abbey' in William Wordsworth, Lyrical Ballads, with other poems, (1800 edition) Gutenberg.org 4. The Cure, 'All Cats Are Grey', Faith, 7 Records LP, 1981. 5. The Cure, 'Faith', Faith 7 Records LP, 1981. 6. Ibid. 7. The Cure, 'The Holy Hour', Faith, 7 Records LP, 1981. 8. The Cure, 'Faith', Faith, 7 Records LP, 1981. World in My Eyes 1. Matthew Arnold, 'Dover Beach' in Matthew Arnold: Selected Poems, Crofts Classics, New York, 1951. 2. William J. Long, Outlines of English and American Literature, Gutenberg.org 3. Arnold, op. cit. 4. Ibid. 5. Depeche Mode, 'Nothing', 101, Mute LP, 1988. 6. James Thomson, 'City of Dreadful Night' in John Hayward (ed.), The Penguin Book of English Verse, Penguin, London, 1958. 7. Depeche Mode, 'Black Celebration', 101, Mute LP, 1988. 8. Depeche Mode, 'World in My Eyes', Violator, Mute LP, 1990. 9. Depeche Mode, 'Personal Jesus', Violator, Mute LP, 1990. We Can Be Heroes 1. David Bowie, 'Heroes', Heroes, RCA LP, 1977. 2. Peter and Leni Gillman, Alias David Bowie, Henry Holt and Co., London, 1987. 3. David Bowie, 'Heroes', op. cit. 4. Michael Tanner, Wagner, Harper Collins, London,1996. 5. Ibid. 6. Ibid. 7. Ibid. 8. Ibid. 9. Ibid. 10. Ibid. 11. Ibid. 12. Ibid. 13. Ibid. Wagnerian 1. www.theatreworldawards.org 2. Don Root, 1980s Rock 'n' Roll knowledge cards, Pomegranate Communications, Petaluma CA, 2006. 3. OscarWilde, The Picture of Dorian Grey, Gutenberg.org 4. New York Times, 22 June 1874. 5. Phil Hardy, The Faber Companion to Twentieth Century Popular Music, Faber and Faber, London, 1995. 6. Philip Dodd, The Book of Rock, Pavilion Books, London, 2001. Born to Run 1. Greil Marcus, In the Fascist Bathroom, Penguin, London, 1993. 2. Mark Hagen, 'The Midnight Cowboy', Mojo, January 1999. 3. Robert Spillane, An Eye for an I: Living Philosophically, Michelle Anderson Publishing, Melbourne, 2007. 4. Bruce Springsteen, 'Born to Run', Born to Run, CBS LP, 1975. 5. Ibid. 6. Ibid. 7. Michael Tanner, Wagner, Harper Collins, London,1996. 8. Bruce Springsteen, 'Thunder Road', Born to Run, CBS LP, 1975. 9. Ibid. Pressure 1. Peter and Leni Gillman, Alias David Bowie, Henry Holt and Co., London, 1987. 2. Daryl Easley, 'Under Pressure' in Mojo Classic: Queen, The Inside Story, 2005. 3. Ibid. 4. Queen and David Bowie, 'Under Pressure', Greatest Hits, Elektra LP, 1981. 5. Arthur Schopenhauer, Essays and Aphorisms, Penguin, London, 2004. 6. Ibid. Schopenhauer 1. R J Hollingdale, Introduction to Arthur Schopenhauer, Essays and Aphorisms, Penguin, London, 2004. 2. Ibid. 3. Barker Fairley, A Study of Goethe, Oxford University Press, London, 1950. 4. Robert Spillane, An Eye for an I: Living Philosophically, Michelle Anderson Publishing, Melbourne, 2007. 5. Ibid. 6. Robert Gutman, Richard Wagner: The Man, His Mind and His Music, Penguin, London, 1971. 7. Arthur Schopenhauer, Essays and Aphorisms, Penguin, London, 2004. Pinkerton 1. William Ashbrook, The Operas of Puccini, Oxford University Press, Oxford, 1985. 2. Ibid. 3. Ibid. 4. Ibid. 5. Ibid. Butterfly 1. William Ashbrook, The Operas of Puccini, Oxford University Press, Oxford, 1985. 2. Ibid. 3. Weezer, 'Butterfly', Pinkerton, Geffen CD, 1996. 4. Weezer, 'Tired of Sex', Pinkerton, Geffen CD, 1996. 5. Arthur Schopenhauer, Essays and Aphorisms, Penguin, London, 2004. Satisfaction 1. Andrew Loog Oldham, 2Stoned, Vintage, London, 2002. 2. Ibid. 3. Arthur Schopenhauer, Essays and Aphorisms, Penguin, London, 2004. 4. Robert Gutman, Richard Wagner: The Man, His Mind and His Music, Penguin, London, 1971. 5. Toby Creswell, 1001 Songs: The Great Songs of All Time, Hardie Grant Books, Melbourne, 2005. 6. Schopenhauer, op. cit. Boredom 1. The Stooges, 'No Fun', No Fun, Elektra LP, 1980. 2. The Stooges, '1969', No Fun, Elektra LP, 1980. 3. Jon Savage, England's Dreaming, Faber and Faber, London, 1991. 4. Jon Savage, Time Travel, Vintage, London, 1996 5. Ibid. 6. Magazine, 'Song From Under the Floorboards', Virgin LP, 1980. 7. Savage, Time Travel, op. cit. 8. Michael Bracewell, The Nineties: When Surface was Depth, Flamingo, London, 2003. 9. Savage, Time Travel, op. cit. 10. Magazine, op. cit. Notes from Underground 1. Morrissey, 'Live at the Olympia Theatre, Paris, 11 April 2006', YouTube.com 2. Ibid. 3. Fyodor Dostoyevsky, Notes from Underground, London: Penguin, London, 1972. 4. Ibid. 5. Ibid. 6. Ibid. How Soon Is Now? 1. Toby Creswell, 1001 Songs: The Great Songs of All Time, Hardie Grant Books, Melbourne, 2005. 2. Creswell, op. cit. 3. Andrew Loog Oldham, 2Stoned, Vintage, London, 2002. 4. Ibid. 5. The Rolling Stones, 'Not Fade Away', Rolled Gold, Decca LP, 1979. 6. The Smiths, 'How Soon is Now', Meat is Murder, Rough Trade LP, 1985. 7. Ibid. 8. Simon Reynolds, Blissed Out, Serpents Tail, London, 1990. 9. Ibid. Why Bother? 1. Andy Greenwald, Nothing Feels Good: Punk Rock, Teenagers, and Emo, St Martins Press, New York, 2003. 2. Get Up Kids, Something to Write Home About, Zomba CD, 1999. 3. Trevor Kelley and Leslie Simon, Everybody Hurts: An essential guide to emo culture, HarperCollins, New York, 2007. 4. Greenwald, op. cit. 5. Ibid. 6. The Smiths, 'How Soon Is Now', Meat is Murder, Roughtrade LP, 1985. The Crystal Palace 1. Walter Benjamin, 'Goethe' in Michael Jennings (ed.), Walter Bengamin: Selected Writings Volume 2, Part 1, Harvard University Press, Cambridge, Mass., 2005. 2. 'World's Fairs' exhibition, National Gallery of Victoria, December 2007. 3. Fyodor Dostoyevsky, Notes from Underground, London: Penguin, London, 1972. 4. Ibid. 5. Ibid. 6. Ibid. 7. Ibid. 8. The Smiths, 'Shoplifters of the World Unite', World Won't Listen, Rough Trade LP, 1986. The Broken, the Beaten and the Damned 1. The Smith, 'Unloveable', World Won't Listen, Rough Trade LP, 1986. 2. Gerard Way, 'Future of Music' Q&A, Rolling Stone, November 2007. 3. Gerard Way, Wikiquote, wikipedia.com Teenagers 1. My Chemical Romance, 'Teenagers', The Black Parade, Warner/Reprise CD, 2006 2. Antoine Nicolas de Condorcet, 'Sketch' in John Carey (ed.), The Faber Book of Utopias, Faber and Faber, London, 1999. 3. Anthony Burgess, A Clockwork Orange, Penguin, London, 2002. 4. William Blake, Poems of William Blake, gutenberg.org 5. George Gordon Byron, Life of Lord Byron with his Letters and Journals, Thomas Moore (ed.), gutenberg.com 6. Pink Floyd, 'Pigs on the Wing', Animals, CBS LP, 1977. 7. Sylvie Simmons, 'Goodbye Blue Sky' in Mojo Special Edition: Pink Floyd, 2004. 8. Pink Floyd, 'Another Brick in the Wall: Part 2', The Wall, CBS LP, 1979. 9. My Chemical Romance, 'Teenagers', op. cit. 10. Bertrand Russell, History of Western Philosophy, Routledge, London, 2006. I've Gotto Get Out of the Basement 1. George Koroneos, 'My Chemical Romance Interview', February 2006, lifeinabungal.com 2. Gerard Way, 'Future of Music' Q&A, Rolling Stone, November 2007. 3. Andy Greenwald, Spin magazine, May 2005. 4. Jenny Eliscu, 'Teen Titans', Rolling Stone, July 2005. 5. Weezer, 'In the Garage', The Blue Album, Geffen CD, 1994. 6. Eliscu, op. cit. Myths of the Near Future 1. Karen Armstrong, A History of God, Vintage, London, 2000. 2. Isiah Berlin, Against the Current, Hogarth Press, London, 1979. 3. Alex De Jonge, Dostoyevsky and the Age of Intensity, Secker and Warburg, London, 1975. 4. Eric Hobsbawm, The Age of Revolution: Europe 1789–1848, Weidenfeld and Nicolson, London, 1962. 5. Ibid. 6. Carl E. Schorske, Fin-Du-Siecle Vienna, Vintage Books, New York, 1981. 7. Ibid. 8. Ibid. Gustav Klint 1. Carl E. Schorske, Fin-Du-Siecle Vienna, Vintage Books, New York, 1981. 2. Ibid. Nietzsche 1. Colin Wilson, The Outsider, Pan, London, 1970. 2. Rudiger Safranski, Nietzsche: A Philosophical Biography, Granta, London, 2002. 3. Wilson, op. cit. 4. Ibid. 5. Ibid. 6. Robert Gutman, Richard Wagner: The Man, His Mind and His Music, Penguin, London, 1971. 7. Friedrich Nietzsche, 'Attempt at a Self-Criticism' in Friedrich Nietzsche, The Birth of Tragedy, Penguin, London, 2003. 8. Wilson, op. cit. A Night at the Opera 1. My Chemical Romance, 'The End', The Black Parade, Warner/Reprise CD, 2006. 2. Martin Aston, 'In the Lap of the Gods' in Mojo Special Edition: Queen, 2005. 3. Mark Cunningham, Good Vibrations: A History of Record Production, Castle, London, 1996 4. Ibid. 5. Mojo Special Edition: Queen, op. cit. 6. Queen, 'Bohemian Rhapsody', Greatest Hits, Elektra LP, 1981. 7. Gerhart Hoffmeister, 'Reception in Germany and Abroad' in Lesley Sharpe, The Cambridge Companion to Goethe, Cambridge University Press, Cambridge, 2002. 8. Queen, 'Bohemian Rhapsody', op. cit. 9. Friedrich Nietzsche, The Birth of Tragedy, Penguin, London, 2003. 10. Ibid. 11. My Chemical Romance, 'The End', The Black Parade, Warner/Reprise CD, 2006. The Wisdom of the Woods 1. Colin Wilson, The Outsider, Pan, London, 1970. 2. Friedrich Nietzsche, The Birth of Tragedy, Penguin, London, 2003. 3. The Chemical Brothers, 'Salmon Dance', We Are The Night, Virgin CD, 2007. 4. Nietzsche, op. cit. 5. Ibid. 6. Ibid. 7. Ibid. 8. Wilson, op. cit. 9. Ibid. Personal Jesus 1. Friedrich Nietzsche, The Gay Science, Penguin, London, 2003. 2. Friedrich Nietzsche, Thus Spoke Zarathustra Penguin, London, 2003. 3. Ibid. 4. Ibid. 5. Phil Sutcliffe, 'Never Let Me Down Again', Mojo Special Edition: Depeche Mode + The Story of Electro-Pop, 2005. 6. Depeche Mode, 'Strangelove', Music For the Masses, Mute LP, 1987. 7. Sutcliffe, op. cit. 8. Bertrand Russell, History of Western Philosophy, Routledge, London, 2006. 9. Nietzsche,, Thus Spoke Zarathustra, op. cit. 10. Depeche Mode, 'Walking in My Shoes', Songs of Faith and Devotion, Mute CD, 1993. 11. Marilyn Manson, 'Beautiful People', Interscope CD, 1996. 12. Tom Bryant, 'Twilight of the Gods' in Kerrang!, December 2004. 13. Colin Wilson, The Outsider, Pan, London, 1970. 14. Nietzsche, Thus Spoke Zarathustra, op. cit. 15. Depeche Mode, 'Condemnation', Songs of Faith and Devotion, Mute CD, 1993. Stronger 1. Friedrich Nietzsche in R J Hollingdale (ed.), A Nietzsche Reader, Penguin, London, 2003 2. Kanye West, 'Stronger', Def Jam CD, 2007. 3. Hollingdale, op. cit. 4. West, op. cit.. 5. Ibid. 6. Ibid. 7. Ibid. 8. 'Kanye West Throws Diva-Like Tantrum in Europe', Rolling Stone, March 2006. 9. Ibid. 10. Hollingdale, op. cit. 11. Mario Praz, The Romantic Agony, Oxford University Press, London, 1978. 12. Rudiger Safranski, Nietzsche: A Philosophical Biography, Granta, London, 2002. 13. Ibid. Also Sprach Zarathustra 1. Peter Guralnick, Careless Love, Abacus, London, 2001. 2. Michael Kennedy, Richard Strauss, J M Dent and Sons, London, 1988. 3. E C Snow, Sleevenotes for Richard Strauss, Also Sprach Zarathustra, Decca LP. 4. Friedrich Nietzsche, Thus Spoke Zarathustra, Penguin, London, 2003. 5. Ibid. 6. Guralnick, op. cit. 7. Nietzche, op. cit. 8. Guralnick, op. cit. 9. 'Man or Superman?', Mojo, January 2007. 10. Greil Marcus, Dead Elvis: A Chronicle of a Cultural Obsession, Penguin, London, 1991. 11. Douglas Brinkley, 'All Shook Up', Los Angeles Times, September 2000, latimes.com Homo Superior 1. Chris Charlesworth, David Bowie: The Archive, Omnibus Press, London, 1987. 2. David Bowie, 'Oh You Pretty Things', Hunky Dory, RCA LP, 1971. 3. Paul Du Noyer, 'Contact', Mojo, July 2002. 4. Ibid. 5. Mick Watts, 'Oh You Pretty Thing' in Hanif Kureshi and Jon Savage (eds) The Faber Book of Pop, Faber and Faber, London, 1995. 6. Ibid. 7. Friedrich Nietzsche in R J Hollingdale (ed.), A Nietzsche Reader, Penguin, London, 2003 8. Herbert Read, To Hell With Culture, Routledge, London, 2002. 9. David Bowie, 'Star', The Rise and Fall of Ziggy Stardust and the Spiders from Mars, RCA LP, 1972. 10. Colin Wilson, The Outsider, Pan, London, 1970. 11. Friedrich Nietzsche, Thus Spoke Zarathustra, Penguin, London, 2003. 12. David Bowie, Moonage Daydream, Hardie Grant Books, Melbourne, 2002. 13. Ben Fisher, 'But Boy Could He Play Guitar', Mojo, October 1997. 14. Bowie, op. cit. Destroyer 1. Gordon Gebert and Bob McAdams, Kiss and Tell, Pitbull Publishing, New York, 1997. 2. Gerard Jones, Men of Tomorrow, Basic Books, New York, 2004. 3. Jenny Eliscu, 'Teen Times', Rolling Stone, July 2005. Such a Special Guy 1. Weezer, 'Pork and Beans', The Red Album, Geffen CD, 2008. 2. Andy Greenwald, Nothing Feels Good: Punk Rock, Teenagers, and Emo, St Martins Press, New York, 2003. 3. Friedrich Nietzsche, On the Genealogy of Morals, Great Literature Online 1997–2008, www.classicauthors.net 4. Weezer, 'Troublemaker', The Red Album, Geffen CD, 2008. 5. Ibid. 6. Rainer Maria Rilke, Letters to a Young Poet, W W Norton and Co., New York, 1962. 7. Ibid. 8. Ibid. 9. Weezer, 'Troublemaker', op. cit. 10. Weezer, 'The Greatest Man that Ever Lived', The Red Album, Geffen CD, 2008 Expressionism 1. Oskar Kokoschka, My Life, Thames and Hudson, London, 1974. 2. Ibid. 3. Wolf-Dieter Dube, Expressionism, Praeger, New York, 1973. 4. Carl E. Schorske, Fin-Du-Siècle Vienna, Vintage Books, New York, 1981. 5. Ibid. 6. Mark Salisbury, Burton on Burton, Faber and Faber, London, 1997. 7. Siegfried Kracauer, 'Caligari by Siegfried Kracauer' in R V Adkinson (ed.), The Cabinet of Dr Caligari, Lorrimer, London, 1972. 8. Ibid. 9. Dube, op. cit. 10. Ibid. 11. Kracauer, op. cit. 12. Ibid. The Pain Threshold 1. A Alvarez, The New Poetry, Penguin, London, 1963. 2. Carl E. Schorske, Fin-Du-Siècle Vienna, Vintage Books, New York, 1981. 3. Notes from permanent exhibition at Arnold Schoenberg Center, 6 Schwartzen-bergplatz, Vienna, May 2007. 4. Ibid. 5. Allen Shawn, Arnold Schoenberg's Journey, Farrar, Strauss and Giroux, New York, 2002. 6. Arnold Schoenberg, 'Pierrot Lunaire', Chandos CD. 7. Liza Minnelli, 'My Chemical Romance', Interview, April 2007. 8. Dan Stapleton, My Chemical Romance, Rolling Stone, February 2007. Sprechstimme 1. Allen Shawn, Arnold Schoenberg's Journey, Farrar, Strauss and Giroux, New York, 2002. 2. Richard Kingsmill, The J-Files Compendium, ABC Books, Sydney, 2002. 3. Saves The Day, 'Jukebox Breakdown',Stay What You Are, Vagrant CD, 2001. 4. My Chemical Romance, 'Blood', The Black Parade, Warner/Reprise CD, 2006. Everything Collapses 1. Christopher Isherwood, Goodbye to Berlin in The Berlin Novels, Minerva, London, 1997. 2. Ibid. 3. Ibid. 4. David Bowie, 'Time', Aladdin Sane, RCA LP, 1973. 5. George Grosz, George Grosz: An Autobiography, University of California Press, Berkely, 1998. 6. Barney Hoskyns, 'When the Kids Had Killed the Man: David Bowie and the Death of Ziggy Stardust', posted in Rock's Backpages, rocksbackpages.com, June 2008. 7. Iggy Pop, 'Nightclubbing', The Idiot, RCA LP, 1977. 8. The Sex Pistols, 'God Save the Queen', Never Mind the Bollocks, Virgin LP, 1977. 9. Richard Kingsmill, The J-Files Compendium, ABC Books, Sydney, 2002. 10. Siouxsie Sioux, Foreword by Siouxsie Sioux, Mojo Special Edition: Bowie, 2003. 11. Jon Savage, England's Dreaming, Faber and Faber, London, 1991. 12. Norman Lebrecht, The Complete Companion to Twentieth Century Music, Simon and Schuster, London 2000. 13. Savage, op. cit. 14. Susan D'arcy, Liza Minnelli, LSP Books, Surrey, 1982. Life Is a Cabaret 1. Liza Minnelli, 'Life is a Cabaret', Cabaret, MRA DVD, 2002. 2. Ibid. 3. Ibid. 4. Ibid. 5. Peter Conrad, Modern Times Modern Place: Life & Art in the Twentieth Century, Thames and Hudson, London, 1998. 6. Henry Edwards and Tony Zanetta, Stardust: The David Bowie Story, Bantam, London, 1987. 7. Jon Savage, England's Dreaming, Faber and Faber, London, 1991. Mother War 1. Liza Minnelli, 'My Chemical Romance', Interview, April 2007. 2. Ibid. 3. Ibid. 4. Ibid. 5. My Chemical Romance, 'Mama', The Black Parade, Warner/Reprise CD, 2006. 6. Ibid. 7. Andy Greenwald, Spin magazine, May 2005 Artists are Cleaners 1. George Grosz, George Grosz: An Autobiography, University of California Press, Berkely, 1998. 2. Ibid. 3. Ibid. 4. Hannah Hoch, 'Aller Anfang ist Dada!' Exhibition, Berlinische Galerie, Alte Jakobstrasse, Berlin, May 2007. 5. Dawn Ades, 'Dada and Surrealism' in Nikos Stangos (ed.), Concepts of Modern Art, Thames and Hudson, London, 1990. 6. Mel Gordon, Dada Performance, PAJ Publications, New York,1993. 7. Ibid. 8. Jason Gaiger, 'Expressionism and the Crisis of Subjectivity' in Steve Edwards and Paul Wood (eds), Art of the Avant-Gardes, Yale University Press, London, 2004. Distress Cries Aloud 1. Andy Greenwald, 'Emo: We Feel Your Pain', in Sia, Michael (ed.) Spin: 20 Years of Alternative Music, Three Rivers Press, New York. 2. Saves the Day, In Reverie, Dreamworks CD 2003. 3. Everett True, 'Salutation Begins at Home', JMag no. 20, August 2008. 4. AFI, The Art of Drowning, Nitro CD, 2003. 5. Wolf-Dieter Dube, Expressionism, Praeger, New York, 1973. 6. Hugh Frederick Garten, Modern German Drama, Methuen, London, 1959. 7. Ibid. 8. Ibid. 9. Jason Gaiger, 'Expressionism and the Crisis of Subjectivity' in Steve Edwards and Paul Wood (eds), Art of the Avant-Gardes, Yale University Press, London, 2004. 10. Ibid. 11. Ibid. 12. Ibid. Rock Stars Are Fascists, Too 1. Adolf Hitler, Mein Kampf, Pimlico, London, 1992. 2. Ibid. 3. Bertrand Russell, History of Western Philosophy, Routledge, London, 2006. 4. Martin Gaughn, 'Narrating the Dada Game Plan' in Steve Edwards and Paul Wood (eds), Art of the Avant-Gardes, Yale University Press, London, 2004. 5. Joachim Fest, The Face of the Third Reich, Weidenfeld and Nicholson, London, 1970. 6. Nick Kent 'Into the Abyss', Mojo Special Edition: Bowie, 2003. 7. Caspar David Friedrich in Elizabeth Gilmore Holt, From the Classicists to the Impressionists: Volume III of A Documentary History of Art, Doubleday Anchor, New York, 1966. 8. Kent, op. cit. 9. AFI, 'Miss Murder', (video) YouTube.com. The Black Parade Is Dead 1. 1985 Live Aid Concert, WEA DVD, 2004. 2. Jim Sharples, 'In Memoriam', Big Cheese magazine, April 2008. 3. Fred Vermorel and Judy Vermorel, 'Julie: "He's got a lot to answer for"' in Hanif Kureshi and Jon Savage (eds), The Faber Book of Pop, Faber and Faber, London, 1995. 4. David Bowie, 'Rock and Roll Suicide' (Live version), Ziggy Stardust: The Motion Picture, RCA LP, 1982. 5. My Chemical Romance, The Black Parade Is Dead, Warner/Reprise DVD, 2008. Insulation and Disaffection 1. Herbert Read, To Hell With Culture, Routledge, London, 2002. 2. Ibid. 3. Ibid. 4. Ibid. 5. Jenny Eliscu, 'Rivers Cuomo Encyclopaedia of Pop', Rolling Stone, June 2002. Leave Them Kids Alone 1. Mark Hagen, 'The Midnight Cowboy', Mojo, January 1999. 2. Liza Minnelli, 'My Chemical Romance', Interview, April 2007. 3. Timothy Gunatilaka, 'My Chemical Romance Saves Lives', Spin, January 2007. 4. Guardian.co.uk, May 2008. 5. Herbert Read, To Hell With Culture, Routledge, London, 2002. # Searchable Terms The pagination of this electronic edition does not match the edition from which it was created. To locate a specific passage, please use the search feature of your e-book reader. Adorno, Theodor, 153 AFI, 81, 82–5, 104–5, 300, 306, 313 emo, 82 horror and, 130 Age of Reason, 209 Alexander, Gregg, 17 Alien Sex Fiend, 122 Alkaline Trio, The, 306 Almond, Marc, 122 Alvarez A, 281 Andi, 123 Armstrong, Karen, 161–3, 165, 172 Arnold, Matthew, 6, 32, 173–4 Artaud, 3 Astbury, Ian, 123 Atwood, Colleen, 54, 316 Baargeld, Johannes, 305 Bahr, Hermann, 308 Bailey, Bill, 21 Baker, Roy Thomas, 242 Barnard, Richard, 23 Barnard, Robert, 46 Bauhaus, 121, 158 Beatles, The, 53 Beck, 3 Beckford, William, 129 Beethoven, Ludwig van, 10, 39–42, 47, 58–9, 155, 179, 224, 317 Choral Symphony, 46 Beetlejuice, 142–3 Bellafila, Amena, 85–6 Benjamin, Walter, 3, 77 Berlin, Isiah, 46, 232 Berryman, Guy, 56 Bienstock, Freddy, 260 Birthday Party, The, 122, 154 Blake, Peter, 54 Blake, William, 52, 59, 133, 225 Blauer Reiter group, 308 Blue Oyster Cult, 183 Bo Diddley, 210–11 Bob B Sox, 184 'Bohemian Rhapsody', 1, 7, 242–3 Bon Jovi, Jon, 20, 82, 104, 228 Bonaparte, Napoleon, 57–63, 117, 193, 317 Bond, Hannah, 64 Boredom, 204–6 Borgatti, Giuseppe, 195 Bowie, David, 2, 3, 6, 9, 10, 14, 82, 98, 105, 106–7, 114, 121, 188–90, 230, 263–7, 269, 290–3, 297 fascism and, 311–2 'Heroes', 178, 188 Queen, and, 188–90, 194 Screamin' Lord Byron, 106–7 'Space Oddity', 263 The Man Who Sold the World, 263 Ziggy Stardust, 49, 98–105, 106, 114–15, 252, 291, 298, 315–6 Boys Next Door, The, 153 Bracewell, Michael, 206 Brecht, Bertolt, 292 Breton, Andre, 3 Bright Eyes, 36–42, 45, 189 Browning, Todd, 120, 278 Buff, Lotte, 65 Burchill, Julie, 88 Burgess, Anthony, 224, 281 Burne-Jones, Edward, 170 Burton, Robert, 90–1 Burton, Tim, 54, 132–4, 140, 142–3, 145, 228, 276 Bush, President, 38, 41 Buzzcocks, 11–12, 13, 154, 207 boredom and 205–6 Spiral Scratch, 11 Byron, Lord George Gordon, 3, 5, 6, 46, 59, 62, 74–5, 105, 107–13, 116, 159, 199, 244 Frankenstein and, 135–9 Luddites and, 225–6, 228 Nietzsche and, 259 The Vampyre, 116–22 Cabaret, 289, 295–9 Gerard Way and, 299–300 Caesar, 62 Cameron, Keith, 157 Captain Beefheart, 154 Carlos, Wendy, 182 Carlyle, Thomas, 31, 33, 62, 73–7, 325 Cave, Nick, 123, 146–50, 159, 228, 230 horror, love of, 157 mystery and, 146–50 Utopiate, 153–6 Cheap Trick, 20 Chemical Brothers, The, 247 Chicago, 20 Christiansen, Rupert, 33, 76, 77 Clairmont, Claire, 119 Clash, The, 205 Classicism, 26, 46 Cobain, Kurt, 10, 97, 114 Coldplay, 54–6 Coleridge, Samuel Taylor, 27, 46, 74 Conley, Chris, 287 Conrad, Peter, 297 Conway, Honorable H S, 125 Cooper, Alice, 133 Corgan, Billy, 94–8, 103, 114–15, 139, 221, 252, 286 Creswell, Toby, 210 Crowe, Cameron, 311 Crystals, The, 184 Cult, The, 82, 123 Cuomo, Rivers, 16–20, 21, 23–5, 78, 91–4, 182, 197, 269–70, 271–4, 287, 318–21 angst of, 22 In the Garage, 231 Pinkerton, 17–20, 78–9, 212, 318 sentimentalist phase, 78–80 The Red Album, 270, 318–21 Why Bother, 213 Cure, The, 2, 6, 13–15, 82, 229–30 Faith, 171–3 goth, 157 Curtis, Ian, 97, 114 Dada, 3 Berlin Dada movement, 302–5 Dallas, Robert, 108, 109 Damned, The, 158 Danzig, Glenn, 130–1 Dashboard Confessional, 80 Davies, Norman, 57, 60, 77, 128, 148, 159 de Condorcet, Marquis, 150, 155, 224 de Jonge, Alex, 51, 83, 158, 233 de Stael, Madame, 73, 74, 113 Debord, Guy, 153 Delacroix, Eugene, 47, 55–6, 59, 116 Depeche Mode, 82, 104, 159–61, 164, 165, 174–6 Stripped, 167 Violator, 251 Depp, Johnny, 136=9–43, 277 Descartes, 146 Despair Faction, 313 deVito, Danny, 277 Devoto, Howard, 11, 207–9, 212 boredom and 205–6 DeWees, James, 9 Diderot, Denis, 165 Diehl, Matt, 105 Dostoyevsky, Fyodor, 6, 212, 227, 232 Notes from Underground, 207–9, 216–18, 224 Dracula, 47, 120, 129, 133, 155, 156 Drive Like Jehu, 10 du Noyer, Paul, 98 E Street Band 'Born to Run', 50 Ebb, Fred, 294, 297, 299 Edward Scissorhands, 139–40, 161, 276–7 Electroclash, 8 Eliscu, Jenny, 231 Emerson, 208 Emo, 4–5, 22, 214 expressionism, fondness for, 305–9 fans, 81 history, 10 My Chemical Romance and, relationship between, 5–6, 7–10, 64 non-athleticism and 214 Empiricism, 52 Enlightenment, 29, 32, 45, 51, 146 God, and denial of existence of, 164 Eno, 14 Everett, Kenny, 243 Expressionism, 6, 279, 305 emo, fondness for, 305–9 Ezrin, Bob, 227 Face, The magazine, 122 Fairley, Barker, 73, 192 Fall Out Boy, 8, 93 Fascism, 311–12, 314 Fichte, Johann Gottlieb, 46, 193 Fisher, Ben, 267 Flesh for Lulu, 122 Flowers, Brandon, 21 diary of, 22 Foo Fighters, The, 9 Fosse, Bob, 294, 298 Frankenstein, 6, 46, 129, 133, 134–9, 144, 162–3 Frehely, Ace, 267 French Revolution, 31–6, 51 nostalgia for, 168 Friedrich, 47 Gahan, Dave, 160–1, 176–7, 251–3, 312 Gamer, Michael, 127, 129 Gang of Four, 152, 154 Genesis, 151 Get Up Kids, The, 8, 9, 214 Gill, Andy, 152, 158 Gill, Eric, 318 Giraud, Albert, 283, 285 Gittings, Robert, 88, 90–1, 166 Glitter, Gary, 104 Gnarls Barkley, 3 Godwin, Mary, 119 Godwin, William, 35, 119 Goebbels, Joseph, 309–11 Goethe, JW von, 5, 6, 46, 62, 65, 66, 67, 71, 72, 73, 74, 79, 105, 110, 116–18, 159, 179, 192, 244 The Sorrows of Young Werther, 61–2, 64–72, 73–8, 110, 113, 116, 136 Goldsmith, William, 9 Gordon, Gavin, 134 Gore, Martin, 160–1, 164, 167, 174–6 Goths, 8, 122–8, 150 emergence, 157–8 meaning, 159 Green Day, 19 Greenwald, Andy, 18, 80, 81, 214, 270 Grey, Joel, 294, 299 Gros, Antoine-Jean, 57 Grosz, George, 291, 302–3 Guercio, Joe, 260, 262 Hackett, Steve, 151 Hagen, Mark, 322 Haman, 46 Hancock, Butch, 262 Hausman, Raoul, 304 Havok, Davey, 82–5, 89, 104–5, 313 goth, appeal of, 158 horror and, 130 Haydon, Benjamin Robert, 107, 166 He, Kyung, 92 Heckel, Erich, 279 Hegel, 46, 59 Heine, Heinrich, 46, 94, 96 Hendrix, Jimi, 114 Hitler, Adolf, 309–11, 313 Hobsbawm, Eric, 57, 58, 60, 62, 168, 233–4 Hoffman, E T A, 102, 179 Hoffmeister, Gerhart, 117 Homophobia Mexican, 8 Horace, 22–3, 26 Horror, 129–31 Nick Cave, love of, 157 Howard, Roland S, 153 Howe, Brian, 45 Huelsenbeck, Richard, 304 Hugo, Victor, 46, 47 Hume, David, 164–5 Hunt, William Holman, 169–70 Iero, Frank, 229 Iggy Pop, 155, 204–6 Indie rock fans, 8 Isherwood, Christopher, 290 Jagger, Mick, 10 'Satisfaction', 201–3 Jimmy Eat World, 8 Johnson, 76, 105 Johnson, David, 122 Jones, Gerard, 268 Joy Division, 14, 97 Joy, Tonie, 9 Kander, John, 294, 297, 299 Kandinsky, Wassily, 279, 282 Kant, Immanuel, 148, 192 Keats, George, 86 Keats, John, 6, 46, 47, 86, 89, 94, 107–8, 110, 159, 166, 174, 208 'La Belle Dame sans Merci', 88, 90–1, 107 'Ode on Melancholy', 91 Keats, Thomas, 85–6, 88 Kelley, Trevor, 214 Kestner, 65–6 Killers, The, 21 Kingsmill, Richard, 95, 293 Kiss, 267, 269 Klimt, Gustav, 235–7, 279, 281, 282 Klosterman, Chuck, 213 Kokoschka, Oskar, 274–6, 278–9, 282, 305, 306 Kracauer, Siegfried, 280 Kubrick, Stanley, 262 Lamb, Lady Caroline, 118 Landau, John, 185 Lebrecht, Norman, 289, 294 Lessing, Gottfried, 245 Liebniz, 147–8 Little Nell, 130 Long, William J, 174 Lovel Darlene, 184 Lugosi, Bela, 120 Lukacs, Georg, 309, 310 Mann, Horace, 125 Manson, Marilyn, 8, 253–4 Manuel, 54 Marc, Franz, 279 Marcus, Greil, 185 Marcuse, Herbert, 153 Maroon 5, 45 Mars Volta, The, 3, 9 Martin, Chris, 54, 55 McLaren, Malcolm, 11, 293 Meat Loaf, 183 Meltzer, Richard, 183 Mercury, Freddie, 3, 188–90, 242, 313–4 Milton, John Paradise Lost, 136, 161–3 Mineral, 10 Minnelli, Liza, 284, 295–9, 300–1 Minogue, Kylie, 149 Misfits, The, 54, 130, 230 Mitchum, Robert, 155 Moore, Tom, 118 Morris, William, 168–70, 227, 233 Morrison, Jim, 114 Morrissey, 82, 105, 206–9, 212, 214–5, 219, 228 Moss Icon, 9 Motley Crue, 183 Motorhead, 183 Mozart, Wolfgang Amadeus, 99, 109, 199 Don Giovanni, 99–102, 109, 113, 199 Munch, 279–80 Mundy, Chris, 19 Murnau, F W, 120 Murphy, Peter, 121 My Chemical Romance, 1–4, 48–56, 80, 221, 222–4, 228–30, 269, 299, 313–17, 322–6 despair, promotion of, 227 emo and, relationship between, 5–6, 7–10, 64 horror and, 130, 288 logo, 306 Mother War, 299–302 The Black Parade, 63–4, 83, 114–6, 241, 287, 299, 313–17, 322–6 The Misfits and, 131 Three Cheers for Sweet Revenge, 228–9 'Welcome to the Black Parade', 1, 2, 4, 5, 7, 49, 63, 189, 221, 222–4, 241, 284 Mystery, 146–50 Nazism, 309–11 Neo-classical theory, 23, 73 New Radicals, 17 New rave, 8 Newton, Isaac, 147, 164–5 Nietzsche, Friedrich, 3, 5, 6, 47, 105, 193, 238–41, 244 Also Sprach Zarathustra, 47, 250–1, 260–2 The Birth of Tragedy, 245 democracy, view of, 256 Dionysus, and spirit of, 246–50 Judeo-Christian morality, view of, 270 martyr, as, 254–5 Nine Inch nails, 253 Nirvana, 19, 97 Nolde, Emil, 279 Noone, Peter, 264 Oasis, 19 Oberst, Conor, 36–42, 44–5, 67, 208 O'Brien, Richard, 130 Oldham, Andrew Loog, 201, 211 Only, Jerry, 130–1 Panic at the Disco, 8, 288 Pater, Walter, 23 Pechstein, Max, 308 Picabia, Francis, 304 PiL, 154 Pink Floyd, 226–7 Plutarch Lives, 136 Polidori, John, 119, 120 Pope, Alexander, 6, 22–3, 26, 46, 76, 105, 147–8 Pre-Raphaelite Brotherhood, 169–70, 174 Presley, Elvis, 210, 260, 262 Prideaux, Tom, 55 Prog-rock, 151 Psychedelic rock, 14 Puccini, 182, 195 La Boheme, 47, 195 Madama Butterfly, 195–200 Pugin, August W, 168, 171 Punk, 8, 11, 151 boredom and 205–6 Enlightenment, re-enacting, 153 Queen, 2, 9, 230, 242–3, 313 'Bohemian Rhapsody', 1, 7, 242–3 Bowie and, 188–90 'Under Pressure', 9, 189, 194 'We Are the Champions', 50, 270 Quennell, Peter, 109, 110–11 Raphael, Frederic, 119 Read, Herbert, 265, 314, 317–8, 324 Reason, 232 Reign of Terror, 34–5 Reimann, Walter, 278 Renton, Mark, 156 Reynolds, Simon, 146, 152, 155, 157, 212 Reynolds, Sir Joshua, 52, 72, 133, 245 Rezner, Trent, 253 Richards, Keith, 211, 248 'Satisfaction', 201–3 Rickley, Geoff, 9, 157 Ries, Ferdinand, 58 Rilke, Rainer Maria, 271–4, 319 Rites of Spring, 9, 10, 306 Robespierre, Maximilien, 33, 150, 155 Rocky Horror Show, 129–31 Rogers, Samuel, 109 Rohrig, Walter, 278 Rolling Stone magazine, 18, 19, 54, 79, 83, 93 Rolling Stones, 211 Romanticism, 3–5, 6, 44–8, 72–6 pure love, and, 177 religious temperament, 159 Ronettes, The, 184 Ronson, Mick, 267 Ross, Ryan, 288 Rossetti, Dante Gabriel, 169–70, 263 Rotten, Johnny, 293 Rousseau, Jean-Jacques, 6, 29–31, 32, 34, 45–6, 51, 67, 76, 77, 81, 89, 119, 131, 193, 227, 232 Julie, 76, 81 Rowe, Zan, 42 Rubin, Rick, 93 Rush, Elena Lee, 299–300 Ruskin, John, 124, 168 Russell, Bertrand, 34, 112, 137, 138, 227, 252 Safranski, Rudiger, 257 Savage, Jon, 12, 206, 294 Saves the Day, 8 Schiller, Friedrich, 38–9, 40–2, 46, 166, 248 Schlegel, August Wilhelm, 72–3 Schlegel, Friedrich, 72 Schoenberg, Arnold, 279, 282–3 Sprechstimme, 285–8 Schopenhauer, Arthur, 46, 190, 191–5, 199–201, 203, 235–6, 238–9, 244 reality, view of, 249 Schumacher, Fritz, 279 Schuster, Joe, 268 Scissor Sisters, The, 3 Scott, Bon, 154 Scott, Tony, 121 Scott, Walter, 129, 168 Scott, X V, 116 Self-expression, 317–8 Sex Gang Children, 123 Sex Pistols, The, 11, 13, 242, 293 Sham 69, 12 Sharrock, Roger, 33 Shawn, Allen, 285 Shelley, Mary, 5, 6, 46, 47, 144–6 Frankenstein, 6, 46, 129, 135–6, 144–6, 162–3 Shelley, Percy Bysshe, 119, 159 Shelley, Pete, 11, 154 Sid Vicious, 242 Siegel, Jerry, 268 Simon, Leslie, 214 Siouxsie and the Banshees, 15, 122, 123, 157, 297 'Spellbound', 158 Siouxsie Sioux, 293 Sisters of Mercy, The, 158 Slits, The, 205 Smashing Pumpkins, 94–8, 139, 229 Smith, Robert, 10, 12, 13–15, 16, 19, 21, 82, 84, 88–90, 97, 103, 105, 123, 172, 221, 287, 293 depression of, 22 Smiths, The, 2, 211, 214, 229 albums, 220–2 'Shoplifters of the World Unite', 219 Soft Cell, 122 Solipsism, 51 Southern Death Cult, 123, 158 Southey, 27, 46 Specimen, 122 Spector, Phil, 184 Wall of Sound, 50, 184 Spenser, Edmund, 86 Spillane, Robert, 192 Spin magazine, 93 Springsteen, Bruce, 2, 10, 41, 230, 322 'Born to Run', 7, 185–8 Stanley, Paul, 267 Starship, 20 Steinman, Jim, 183 Stoker, Bram Dracula, 47, 120, 129, 133 Stooges, The, 210, 211 Strauss, Neil, 82, 306 Strauss, Richard, 182, 260–2 Salome, 285 Suicide teen, 8, 64 Sunny Day Real Estate, 8, 9, 10, 80, 81 Supertramp, 1 'The Logical Song', 1 Swales, Martin, 73, 74, 78 Tanner, Michael, 180–2, 187 Temple, Julian, 106 Texas is the Reason, 10 Thomson, James, 175 Thursday, 157 Timbaland, 82 Toller, Ernst, 307–9 Tom Robinson Band, 12 Toro, Ray, 315 Trip-hop, 8 Turgot, Jacques, 29, 155 Tyler, Bonnie, 183 Urie, Brendan, 288 Used, The, 9 Utopia, 150–2 Vallon, Annette, 33 Van Gogh, 279–80 Vaughn, William, 61, 72 Velvet Underground, 102 Vincent Molloy, 132–4 Virgin Prunes, The, 158 Voltaire, 62, 148, 166 Wagner, Richard, 6, 10, 178–85, 193–4, 233, 238–9, 244, 260 history of rock and roll, place in, 182–5 Tristan und Isolde, 47, 179–82, 187, 195, 199–201, 239, 285 Walden, Herwath, 278–9 Walpole, Horace, 125–8, 159, 168 Warm, Hermann, 278 Warren, Diane, 20, 21 Waters, Roger, 226–7, 315 Way, Gerard, 2, 6, 7, 10, 48–56, 63–4, 73, 82, 103, 114–6, 220–1, 227, 228–30, 241, 284, 313–7, 322–6 Cabaret and, 299–300 goth, appeal of, 158 horror and, 130 rage of, 22 The Misfits and, 131 Weezer, 6, 8, 16–20, 23–5, 80, 91–4, 318 'In the Garage', 231 Pinkerton, 8, 91–4, 195–200, 318–19 Weightman, John, 81 Weill, Kurt, 291–2 Weine, 278 Wells, Charles, 85, 86 Welsh, Irvine, 156 Wentz, Pete, 93 West, Kanye, 256–9 Whale, James, 134, 278 Whitehouse, Mary, 164 Wicked Lester, 267 Wiene, Robert, 277, 280, 285 Wilde, Oscar The Picture of Dorian Grey, 47, 129, 183 Wilson, Colin, 109, 239–40 Wire, The magazine, 22 Wollstonecraft, Mary, 119 Wolzogen, Baron von, 296 Wood, Ed, 142 Wordsworth, Dorothy, 27, 42 Wordsworth, William, 6, 26–8, 33, 34–6, 38, 42–4, 46, 51–3, 59, 67, 74, 89, 105, 110, 131, 145, 159, 166, 174, 227, 319 Peter Bell, A Tale, 86 'The Excursion', 119 'The White Doe of Rylstone', 171–2 Zaleski, Annie, 12 Zehme, Albertine, 283 # Copyright First published in April 2009 \| The ABC 'Wave' device is a trademark of the Australian Broadcasting Corporation and is used under licence by HarperCollinsPublishers Australia. ---\|--- This edition published in 2012 by HarperCollinsPublishers Australia Pty Limited ABN 36 009 913 517 harpercollins.com.au Copyright © Craig Schuftan 2009 The right of Craig Schuftan to be identified as the author of this work has been asserted by him in accordance 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This work is copyright. Apart from any use as permitted under the Copyright Act 1968, no part may be reproduced, copied, scanned, stored in a retrieval system, recorded, or transmitted, in any form or by any means, without the prior written permission of the publisher. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the Australian Broadcasting Corporation. National Library of Australia Cataloguing-in-Publication entry Schuftan, Craig. Hey! Nietzsche! : Leave them kids alone / Craig Schuftan; illustrated by Brad Cook. 1st ed. 978 0 7333 2402 4 (pbk.) 978 1 7430 9909 4 (epub) Popular culture – History Social movements – History Civilization, Modern Art and society – History Music – Social aspects – History Cook, Brad. 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\section{Introduction} Ramsey theory is one of the central areas of study in combinatorics and a key problem in the field is that of determining the Ramsey numbers of graphs, defined as follows. For a graph $G$, the \emph{Ramsey number} $R_k(G)$ is the least integer $N$ such that any colouring of the edges of the complete graph $K_N$ on $N$ vertices with $k$ colours yields a monochromatic copy of $G$. The existence of Ramsey numbers is guaranteed by Ramsey's classical result~\cite{Ram}, but in the case $k\geq3$, determining the value of $R_k(G)$ for a given graph $G$ is in most cases difficult. There are only a few graphs $G$ for which we know $R_k(G)$ exactly and often one has to settle for bounds on this quantity. In this paper we focus on the case where $G$ is the $n$-vertex path $P_n$, and the case where $n$ is even and $G$ is the $n$-vertex cycle $C_n$. The two-colour Ramsey number of a path was completely determined by Gerencs\'er and Gy\'arf\'as~\cite{GerGya} who showed that for $n\geqslant 2$ \[ R_2(P_n)=\left\lfloor\frac{3n-2}{2}\right\rfloor. \] For three colours, Faudree and Schelp \cite{FauSCh2} conjectured that \[ R_3(P_n)=\begin{cases} 2n-2 & \text { for } n \text{ even}\,,\\ 2n-1 & \text { for } n \text{ odd}\,. \end{cases} \] This conjecture was resolved for large $n$ by Gy\'arf\'as, Ruszink\'o, S\'ark\"ozy and Szemer\'edi~\cite{Gya} but for $k\geq4$ much less is known. A well-known upper bound $R_k(P_n) \leqslant kn$ follows easily by observing that any $k$-colouring of the complete graph on $kn$ vertices contains a colour class with at least $(kn-1)\frac{n}{2}$ edges by the pigeonhole principle. A result of Erd\H{o}s and Gallai~\cite{erdHos1959maximal} (Lemma~\ref{lem:EG} below) then implies that any graph on $kn$ vertices with this many edges contains a copy of $P_n$. Despite the simplicity of this observation, the bound was only recently improved upon by S\'ark\"ozy~\cite{Sarkozy} who proved a stability version of Lemma~\ref{lem:EG} and showed that for $k\geq4$ and $n$ sufficiently large, \[ R_k(P_n)\leqslant\left(k-\frac{k}{16k^3+1}\right)n\,. \] In this paper we improve on the above result for all $k\geqslant 4$ reducing the upper bound on $R_k(P_n)$ by an amount that does not deteriorate as $k$ grows. Our method is similar to that of \cite{Sarkozy} in that we also use results of Erd\H{o}s and Gallai~\cite{erdHos1959maximal}, and Kopylov~\cite{kopylov1977maximal} to bound the number of edges in the densest two colours. Our improvement comes from using more information about the densest colour in order to obtain stronger bounds on the number of edges in the second densest. \begin{theorem}\label{thm:RPn} For $k\geqslant 4$ and all $n\geqslant 64k$, \[ R_k(P_n)\leqslant \left(k-\frac{1}{4}+\frac{1}{2k}\right)n\,. \] \end{theorem} If $n$ is much larger we can in fact slightly improve on this bound and extend it to even cycles, see Theorem~\ref{thm:RCn} below. Since $P_n$ is a subgraph of $C_n$ we have $R_k(P_n)\leqslant R_k(C_n)$. It is believed that for fixed $k$ and even $n$ the Ramsey numbers $R_k(P_n)$ and $R_k(C_n)$ are asymptotically equal. This is due to an application of the regularity lemma and the notion of connected matchings pioneered by \L uczak in~\cite{Lucz}. Progress on these two problems therefore track each other closely. In the case of two colours Faudree and Schelp~\cite{FauSch}, and independently Rosta~\cite{Rosta} showed that $R_2(C_n)=\frac{3n}{2}+1$ for even $n\geq6$. For three colours, Benevides and Skokan~\cite{BenSko} proved that $R_3(C_n)=2n$ for sufficiently large even $n$. For $k\geq4$ colours, again very little is known. \L uczak, Simonovits and Skokan \cite{LSS} showed that for $n$ even, $R_k(C_n)\leqslant kn+o(n)$, and recently S\'ark\"ozy \cite{Sarkozy} improved this upper bound to $\left(k-\frac{k}{16k^3+1}\right)n+o(n)$. Here we obtain a strengthening of Theorem~\ref{thm:RPn} for large $n$. \begin{theorem}\label{thm:RCn} For $k\geqslant 4$ and $n$ even \[ R_k(C_n)\leqslant \left(k-\frac{1}{4}\right)n + o(n)\,. \] \end{theorem} It is interesting to note that odd cycles behave very differently in this context. Recently the second author and Skokan~\cite{JenSko} showed, via analytic methods, that for $k\geq4$ and $n$ odd and sufficiently large, $R_k(C_n)=2^{k-1}(n-1)+1$. This resolved a conjecture of Bondy and Erd\H{o}s \cite{BonErd} for large $n$. Let us now briefly discuss lower bounds. Constructions based on finite affine planes (see \cite{BieGya}) show that $R_k(P_n) \geqslant (k-1)(n-1)$, when $k-1$ is a prime power and this lower bound is thought to be closer to the truth than our upper bound. Yongqi, Yuansheng, Feng, and Bingxi~\cite{YYFB} provide a construction which shows that $R_k(C_n) \geqslant (k-1)(n-2)+2$ for any $k$ and for even $n$. This construction can easily be modified to give a lower bound on $R_k(P_n)$ for any $k$ and any $n$. We sketch this construction below. To see that $R_k(P_n)\geqslant 2(k-1)\left(\floor{\frac{n}{2}}-1\right)+1$, consider a complete graph $G$ on vertices $\{0,1,\dotsc, 2k-3\}$ and for $1\leqslant i\leqslant k-1$ colour the edges from vertex $i$ to vertices $i+1,\dotsc,i+k-2$ and the edges from vertex $i+k-1$ to vertices $i+k,\dotsc,i+2k-3$ (taken modulo $2k-2$) with colour $c_i$. Then each colour $c_1,\dotsc,c_{k-1}$ consists of two vertex-disjoint stars, each on $k-1$ vertices. The remaining edges are those of the form $\{j,j+k-1\}$ for $j=0,\dotsc,k-2$ which are coloured with the final colour $c_k$. The final colour forms a matching on $k-1$ edges. Construct $G'$ by `blowing up' each vertex $i$ of $G$ into a set $V_i$ of $\floor{\frac{n}{2}}-1$ vertices and colour the edges within $V_i$ with colour $c_k$. Edges between sets $V_i$ and $V_j$ in $G'$ are coloured with the same colour as the edge $\{i,j\}$ in $G$. There is no monochromatic $P_n$ in $G'$ because in colours $c_1,\dotsc,c_{k-1}$, components are bipartite with smallest part size $\floor{\frac{n}{2}}-1$, hence cannot contain a $P_n$. The components in colour $c_k$ have less than $n$ vertices and so cannot contain a $P_n$. Again, this lower bound is generally considered to be closer to the truth than our upper bound. \section{Methods} Throughout the paper, we omit floor and ceiling signs whenever they are not crucial. To prove Theorem~\ref{thm:RPn} we will proceed by contradiction. We take a complete graph on $N=(k-\frac{1}{4}+\frac{1}{2k})n$ vertices whose edges have been coloured with $k$ colours and suppose it contains no monochromatic $P_n$. First we show that the densest colour has only a few components and these are not too large. For the other colours we consider the edges between these components and use the multi-partite structure to bound the number of such edges. This gives bound on the total number of edges which is less than $\binom{N}{2}$ which is the desired contradiction. The proof is given in Section~\ref{sec:Pnproof}. The regularity method of \L uczak (see e.g.~\cite{FigLuczArxiv,FigLucz,Lucz}) reduces the problem of finding a monochromatic $C_n$ in a $k$-coloured complete graph to that of finding a monochromatic component containing a matching of $\frac{n}{2}$ edges in a \emph{reduced graph}, which is a $k$-coloured graph missing a small fraction of edges. We use the term \emph{connected matching of $\frac{n}{2}$ edges} to mean a connected graph which contains a matching of $\frac{n}{2}$ edges. Using this method we prove Theorem~\ref{thm:RCn} via the following result, which we prove in Section~\ref{sec:Reg}. \begin{theorem}\label{thm:connM} Let $k\geqslant 4$ be a positive integer, and let $0\leqslant\delta< \frac{1}{64k^2} $. Then for even $n\geqslant 32k$ and $N=(k-\frac{1}{4})n$ the following holds. Suppose that $G$ is a $k$-coloured, $N$-vertex graph with at least $(1-\delta)\binom{N}{2}$ edges, then we may find a monochromatic connected matching of $\frac{n}{2}$ edges in $G$. \end{theorem} The statement we use to deduce Theorem~\ref{thm:RCn} from Theorem~\ref{thm:connM} is from a paper of Figaj and \L uczak~\cite[Lemma 3]{FigLuczArxiv}. \begin{lemma}\label{lem:Lucz} Let $t > 0$ be a real number. If for every $\eps > 0$ there exists $\delta > 0$ and an $n_1$ such that for every even $n > n_1$ and any $k$-coloured graph $G$ with $v(G) > (1 + \eps)tn$ and $e(G) \geqslant (1 - \delta)\binom{v(G)}{2}$ has a monochromatic connected matching of $\frac{n}{2}$ edges, then $R_k(C_n) \leqslant (t + o(1))n$. \end{lemma} Theorem~\ref{thm:RCn} follows from Theorem \ref{thm:connM} by applying Lemma~\ref{lem:Lucz} with $t=k-\frac14$ and for any positive $\eps$ choosing $\delta<\frac{1}{64k^2}$ and $n_1\geq32k$. The remainder of this paper is devoted to proving Theorems~\ref{thm:RPn} and~\ref{thm:connM}. We will need the following extremal results for graphs not containing an $n$-vertex path. \begin{lemma}[Erd\H{o}s--Gallai \cite{erdHos1959maximal}]\label{lem:EG} Let $H$ be a graph which does not contain an $n$-vertex path. Then \[ e(H)\leqslant\frac{n-2}{2}v(H)\,. \] \end{lemma} The following simplified version of a result due to Kopylov~\cite{kopylov1977maximal} improves on the above result for \emph{connected} graphs. \begin{lemma}\label{lem:connNoPn} Let $H$ be a connected graph which does not contain an $n$-vertex path. Then \[ e(H)\leqslant\frac{n}{2}\max\left\{n,v(H)-\frac{n}{4}\right\}. \] \end{lemma} Our next result gives a slight improvement of Lemma~\ref{lem:connNoPn} under the additional assumption that $H$ is $c$-partite. In this case the bound on $e(H)$ can be improved when $v(H)$ is small. \begin{lemma}\label{lem:partiteNoPn} Let $H$ be a $c$-partite connected graph which does not contain an $n$-vertex path. Then \[ e(H)\leqslant \begin{cases} \left(1-\frac{1}{c}\right)\frac{v(H)^2}{2} & \text{ for } v(H) \leqslant n\sqrt{\frac{c}{c-1}}\,,\\ \frac{n^2}{2} & \text { for } n\sqrt{\frac{c}{c-1}} < v(H) \leqslant \frac{5n}{4}\,,\\ \frac{n}{2}\left(v(H)-\frac{n}{4}\right) & \text{ for } \frac{5n}{4} < v(H)\,. \end{cases} \] \end{lemma} \begin{proof} In the `small' case $v(H) \leqslant n\sqrt{\frac{c}{c-1}}$ we simply use that a $c$-partite graph has at most as many edges as the complete balanced $c$-partite graph on the same number of vertices. Therefore we conclude that $e(H)\leqslant \left(1-\frac{1}{c}\right)\frac{v(H)^2}{2}$ without the assumption that $H$ contains no copy of $P_n$. The `medium' case where $n\sqrt{\frac{c}{c-1}} < v(H) \leqslant \frac{5n}{4}$ and the remaining `large' case follow directly from Lemma~\ref{lem:connNoPn} and make no use of the $c$-partite assumption on $H$. \end{proof} Lemma~\ref{lem:partiteNoPn} is already strong enough for us to prove Theorem~\ref{thm:RPn}, however we require another modification to prove Theorem~\ref{thm:RCn}. We will defer its proof to Section~\ref{sec:Reg}. \begin{lemma}\label{lem:partiteNoMatching} Let $H$ be a $c$-partite connected graph which does not contain a matching of $\frac{n}{2}$ edges. Suppose further that there is a $c$-partition of $H$ such that the sum of the sizes of any two parts is at least $n$. Then \[ e(H)\leqslant \begin{cases} \left(1-\frac{1}{c}\right)\frac{v(H)^2}{2} & \text{ for } v(H) \leqslant n\sqrt{\frac{c}{c-1}}\,,\\ \frac{n^2}{2} & \text { for } n\sqrt{\frac{c}{c-1}} < v(H) \leqslant \frac{5n}{4}\,,\\ \frac{n}{2}\left(v(H)-\frac{n}{4}\right) & \text{ for } \frac{5n}{4} < v(H) < \frac{31n}{16}\,,\\ \frac{n}{2}\left(v(H)-\frac{7n}{16}\right) & \text{ for } \frac{31n}{16} \leqslant v(H) \,. \end{cases} \] \end{lemma} \section{Paths}\label{sec:Pnproof} \begin{proof}[Proof of Theorem~\ref{thm:RPn}] Let $\alpha = \frac14-\frac{1}{2k}$ and let $G$ be a $k$-coloured complete graph on $N=(k-\alpha)n$ vertices. Let `blue' be one of these colours. We proceed by contradiction, supposing that $G$ contains no monochromatic $n$-vertex path. Over all such $G$ consider the one in which blue has the most edges. In particular $G$ has at least as many blue edges as any other colour. The main idea of our argument is to use bounds on the sizes and the number of blue components to bound the number of edges of $G$ which lie inside blue components, and then to bound the number of edges in each other colour that lie between different blue components. Let $B$ denote the blue subgraph of $G$ and let $B_1,\dotsc, B_c$ be the connected components of $B$. Let `red' be the colour that has the most edges lying between blue connected components and let $R'$ denote the $c$-partite graph of those red edges. We will prove the following two bounds. Firstly the number of edges (of any colour) within blue components satisfies \begin{equation}\label{eq:edgesinblue} \sum_{i=1}^c \binom{v(B_i)}{2} \leqslant \left(k-2\alpha+5\alpha^2\right)\frac{n^2}{2}\,, \end{equation} and secondly the number of red edges between blue components satisfies \begin{equation}\label{eq:rededgesbetweenblue} e(R') \leqslant \left(k-\alpha-\frac{1}{4}\right)\frac{n^2}{2}\,. \end{equation} It follows that \[ e(G) \leqslant (k-1)e(R')+\sum_{i=1}^c \binom{v(B_i)}{2} \leqslant \bigg((k-1)(k-\alpha -\tfrac14) +(k-2\alpha +5\alpha^2)\bigg)\frac{n^2}{2}\,. \] Since $e(G)=\binom{N}{2}=(k-\alpha)(k-\alpha-\frac{1}{n})\frac{n^2}{2}$ it is easy to verify that this fails for $\alpha=\frac{1}{4}-\frac{1}{2k}$ and $n\geqslant 64k$, reaching the desired contradiction. We now proceed with proving inequalities \eqref{eq:edgesinblue} and \eqref{eq:rededgesbetweenblue}. As a first step toward proving \eqref{eq:edgesinblue}, we establish bounds on the size of blue components. We first argue that there cannot be large blue components. \begin{claim}\label{clm:noBlue5n/4} There is no blue component in $G$ on more than $\frac{5n}{4}$ vertices. \end{claim} \begin{claimproof} For contradiction, suppose there is a blue component $B_1$ on $\beta n$ vertices with $\beta> \frac54$. In this case, by Lemma~\ref{lem:connNoPn} we have $e(B_1)\leqslant(\beta-\frac14)\frac{n^2}{2}$. Using Lemma~\ref{lem:EG} on the rest of the blue graph, $B$, we obtain \[ e(B) \leqslant \left(\beta-\frac{1}{4}\right)\frac{n^2}{2}+\left(k-\alpha-\beta\right)\frac{n^2}{2} = \bigg(k-\alpha - \frac14 \bigg)\frac{n^2}{2}\,. \] Since blue is the densest colour we have $e(G)\leqslant k\cdot e(B)$ and hence \begin{align} \binom{N}{2}=(k-\alpha)\left(k-\alpha-\frac{1}{n}\right)\frac{n^2}{2} &\leqslant k \left(k-\alpha-\frac{1}{4}\right)\frac{n^2}{2}\\ \alpha^2-k\alpha+\frac{k}{4}-\frac{k-\alpha}{n} &\leqslant 0\,. \end{align} This fails when $\alpha = \frac14 - \frac{1}{2k}$ and $n \geqslant 64k$. \end{claimproof} The main application of Claim~\ref{clm:noBlue5n/4} is that now when applying Lemma~\ref{lem:connNoPn} to a blue component $B_i$ we obtain the bound $e(B_i)\leqslant \frac{n^2}{2}$. Using this fact we get a tighter bound on the size of blue components. Let $x$ be defined by the equation \[ xn = \sum_{i=1}^c\max\{v(B_i)-n,0\}\,. \] We refer to $x$ as the \emph{excess} size of blue components. The motivation for this definition is that we expect blue components to be of size approximately $n$. \begin{claim}\label{clm:smallx} We have $x< \alpha$. \end{claim} \begin{claimproof} Let $B_1,\dotsc,B_\ell$ be the blue components with more than $n$ vertices. By Lemma~\ref{lem:connNoPn} and Claim~\ref{clm:noBlue5n/4} we have that there are at most $\frac{n^2}{2}$ edges in each of $B_1,\dotsc,B_\ell$. Using Lemma~\ref{lem:EG} on the rest of the blue graph we have \[ e(B) \leqslant \ell\frac{n^2}{2} + (k-\alpha-\ell-x)\frac{n^2}{2} =(k-\alpha - x)\frac{n^2}{2}\,. \] Since blue is the densest colour we have $e(G)\leqslant k\cdot e(B)$, and so \[ \binom{N}{2}=(k-\alpha)\left(k-\alpha-\frac{1}{n}\right)\frac{n^2}{2} \leqslant k(k-\alpha-x)\frac{n^2}{2} \] therefore \[ x \leqslant \alpha -\frac{\alpha^2}{k} + \frac{k-\alpha}{kn}\,, \] and in particular $x<\alpha$ for $\alpha=\frac14-\frac{1}{2k}$ and $n\geqslant 64k$. \end{claimproof} With this bound on the excess, we can prove \eqref{eq:edgesinblue}, completing the first part of the proof. \begin{claimproof}[Proof of inequality~\eqref{eq:edgesinblue}] By convexity, $\sum_{i=1}^c\binom{v(B_i)}{2}$ is maximised when there is one blue component of size $(1+x)n$ which has all the excess, $(k-2)$ components of size $n$ and one component of size $(1-\alpha-x)n$. Note that this is at least $n/2$ as $x,\alpha < 1/4$. It follows that \begin{align} \sum_{i=1}^c \binom{v(B_i)}{2} &\leqslant \left((1+x)^2+(k-2)+(1-\alpha-x)^2\right)\frac{n^2}{2}\\ &=\left(k -2\alpha + \alpha^2 + 2\alpha x + 2x^2\right)\frac{n^2}{2}\,. \end{align} Using the bound $x<\alpha$ from Claim~\ref{clm:smallx} completes the argument. \end{claimproof} The second step of the proof of Theorem~\ref{thm:RPn} is to bound the number of red edges which lie between different blue components, establishing \eqref{eq:rededgesbetweenblue}. We begin with the following claim. \begin{claim}\label{clm:upperboundc} The number, $c$, of blue components of $G$ is at most $\frac{4}{3}(k-\alpha)+1$. \end{claim} \begin{claimproof} It suffices to show that all but at most one blue component contain more than $\frac{3n}{4}$ vertices. Suppose for contradiction that $B_1$ and $B_2$ each have at most $\frac{3n}{4}$ vertices, and let $b$ satisfy $bn=v(B_1\cup B_2)$. Note that, by the maximality assumption on blue, $B_1\cup B_2$ must contain at least $n-1$ vertices. If not, putting a blue clique on $V(B_1\cup B_2)$ would increase $e(B)$ without creating a blue $P_n$ in $G$. We therefore have $\frac{n-1}{n} \leqslant b\leqslant \frac32$. $e(B_1\cup B_2)$ is maximal when both components are cliques and by convexity is maximised when $v(B_1)=\frac{3n}{4}$, $v(B_2)=\left(b-\frac{3}{4}\right)n$. Using Lemma~\ref{lem:EG} on the rest of the blue graph we have \begin{align} e(B) &\leqslant \frac{9}{16}\frac{n^2}{2} + \left(b-\frac{3}{4}\right)^2\frac{n^2}{2} + (k-\alpha-b)\frac{n^2}{2}\\ &\leqslant \left(b^2-\frac{5}{2}b+k-\alpha+\frac{9}{8}\right)\frac{n^2}{2}\,. \intertext{ Under the constraint $\frac{n-1}{n} \leqslant b\leqslant \frac32$, the quadratic function $b^2-\frac{5b}{2}$ is maximised at $b=\frac{n-1}{n}$, hence } e(B) &\leqslant \left(k-\alpha -\frac{3}{8} + \frac{1}{2n} + \frac{1}{n^2}\right)\frac{n^2}{2}\,. \end{align} Since blue is the densest colour we have $e(G) \leqslant k\cdot e(B)$ which gives \[ \binom{N}{2}=(k-\alpha)\left(k-\alpha-\frac{1}{n}\right)\frac{n^2}{2} \leqslant k\left(k-\alpha -\frac{3}{8} + \frac{1}{2n} + \frac{1}{n^2}\right)\frac{n^2}{2} \] hence \[ \alpha^2 -\alpha k+ \frac{3k}{8} -\frac{3k-2\alpha}{2n}-\frac{k}{n^2}\leqslant 0\,. \] However this fails for $\alpha=\frac14-\frac{1}{2k}$ and $n\geqslant 64k$ giving the desired contradiction. \end{claimproof} Using the above bound on $c$, the next claim uses Lemma~\ref{lem:partiteNoPn} to bound the number of edges of $R'$. \begin{claim}\label{cl:edgespervertex} Let $H$ be a $c$-partite connected graph on at most $(k-\alpha)n$ vertices which does not contain an $n$-vertex path. Then \[ \frac{e(H)}{v(H)}\leqslant \frac{n}{2}\left(1-\frac{1}{4(k-\alpha)}\right)\,. \] \end{claim} \begin{claimproof} We use Lemma~\ref{lem:partiteNoPn} to break the proof into three cases depending on the size of $H$. Firstly in the case where $v(H)\leqslant n\sqrt{\frac{c}{c-1}}$ we have $\frac{e(H)}{v(H)} \leqslant (1-\frac{1}{c})\frac{v(H)}{2}$. Since $v(H)\leqslant n\sqrt{\frac{c}{c-1}}$ this is at most $\frac{n}{2}\sqrt{\frac{c-1}{c}}\leqslant\frac{n}{2} \left(1-\frac{1}{2c}\right)$. By Claim~\ref{clm:upperboundc} we know that $c \leqslant \frac{4}{3}(k-\alpha)+1$. This gives a bound of \[ \frac{e(H)}{v(H)} \leqslant \frac{n}{2}\left(1- \frac{3}{8(k-\alpha)+6}\right)\leqslant \frac{n}{2}\left(1-\frac{1}{4(k-\alpha)}\right) \,. \] Next suppose $n\sqrt{\frac{c}{c-1}} < v(H) \leqslant \frac{5n}{4}$. Then, by Lemma~\ref{lem:partiteNoPn}, we have $\frac{e(H)}{v(H)} \leqslant \frac{n}{2}\sqrt{\frac{c-1}{c}}$. As shown in the previous case $\frac{n}{2}\sqrt{\frac{c-1}{c}}$ is at most $ \frac{n}{2}\left(1-\frac{1}{4(k-\alpha)}\right)$. Finally suppose $v(H) > \frac{5n}{4}$. Then $\frac{e(H)}{v(H)} \leqslant \frac{n}{2}\left(1-\frac{n}{4v(H)}\right)$. This is maximised when $v(H)$ is as large as possible giving $\frac{e(H)}{v(H)} \leqslant \frac{n}{2}\left(1-\frac{1}{4(k-\alpha)}\right)$. \end{claimproof} We can now deduce \eqref{eq:rededgesbetweenblue} from Claim~\ref{cl:edgespervertex}. There will be a connected component of $R'$ with at least as high a density as the overall density of $R'$. Therefore if $R'$ had more than $\left(k-\alpha -\frac{1}{4}\right)\frac{n^2}{2}$ edges there would be a connected component $H$ satisfying \[ \frac{e(H)}{v(H)} > \frac{1}{N}\left(k-\alpha -\frac{1}{4}\right)\frac{n^2}{2} = \frac{n}{2}\left(1-\frac{1}{4(k-\alpha)}\right) \,. \] This contradicts Claim~\ref{cl:edgespervertex}, completing the proof. \end{proof} \section{Even Cycles}\label{sec:Reg} The proof of Theorem~\ref{thm:connM} closely resembles the arguments of the previous section. We make three changes, the first two of which are only minor adjustments. We must work with the value $\alpha=\frac14$ instead of the value $\frac14-\frac{1}{2k}$, and we must permit the host graph $G$ to have as few as $(1-\delta)\binom{N}{2}$ edges for some small $\delta>0$ which we choose. The more significant change is that we apply Lemma~\ref{lem:partiteNoMatching} instead of Lemma~\ref{lem:partiteNoPn} to bound the number of edges between blue components. The reason we are able to improve upon the result of Theorem~\ref{thm:RPn} is that when looking only for a connected matching (rather than a path) we can better deal with large components of the graph $R'$ consisting of red edges between blue components. In particular, the tight case of Claim~\ref{cl:edgespervertex} is when $v(H)>\frac{5n}{4}$ where we can do no better than assume $R'$ consists of one large connected component. The improvement in this case is given by Lemma~\ref{lem:partiteNoMatching}, where we get a better bound on $e(H)$ when $H$ is a component of $R'$ with at least $\frac{31n}{16}$ vertices. \begin{proof}[Proof of Lemma~\ref{lem:partiteNoMatching}] First note that the three bounds for the range $v(H) < \frac{31n}{16}$ follow directly from Lemma~\ref{lem:partiteNoPn} since, for even $n$, a copy of $P_n$ contains a matching of $\frac{n}{2}$ edges. We therefore assume $H$ is a $c$-partite, connected graph on at least $\frac{31n}{16}$ vertices, in which the sizes of any two parts sum to at least $n$ and which contains no matching of $\frac{n}{2}$ edges. We will show that $e(H) \leqslant \frac{n}{2}(v(H)-\frac{7n}{16})$. Let $A=\{v\in H: d(v)\geqslant n\}$ and let $M$ denote a maximal matching in $H':=H\backslash A$. We may assume $v(M)<n$ as $H$ does not contain any matching with $n/2$ edges. We will bound $e(H)$ by first bounding $e(H')$ and then bounding the number of edges incident to $A$. First note that \begin{equation}\label{greedy} \abs{A}\leqslant\frac{n}{2}-\frac{v(M)}{2}\,, \end{equation} otherwise we could greedily extend $M$ to a matching of size $\frac{n}{2}$ in $H$ contradicting the assumption of the lemma. For $v\in H'$ let $d^{\ast}(v)$ denote the number of neighbours of $v$ in $H'\backslash M$. Now let $\{u,v\}$ be an edge of $M$. Note that either $d^{\ast}(u)\leq1$ or $d^{\ast}(v)\leqslant 1$ else we could replace the edge $\{u,v\}$ with a pair of edges $\{u,x\}$, $\{v,y\}$ to get a larger matching in $H'$. Let us denote the edges of $M$ by $\{u_i, v_i\}$ for $i=1,\ldots,\frac{v(M)}{2}$ and assume without loss of generality that $d^{\ast}(u_i)\leq1$ for all $i$. Since each edge of $H'$ is incident to an edge of $M$ by maximality it follows that \begin{align}\label{e(H')} \nonumber e(H')&\leqslant \sum_{i=1}^{\frac{1}{2}v(M)}\left(d(v_i)+d^{\ast}(u_i)\right)+\binom{\frac{1}{2}v(M)}{2}\\ \nonumber&\leqslant \frac{v(M)}{2}\left(n+\frac{v(M)}{4}\right)\\ &\leqslant \frac{5}{8}n^2-\frac{3}{2}\abs{A}n+\frac{\abs{A}^2}{2}\,, \end{align} where for the second inequality we used that $d(v)<n$ for all $v\in H'$ by the definition of $A$, $d^{\ast}(u_i)\leqslant 1$ for all $i$ by assumption. For the last inequality we used \eqref{greedy}. We now turn our attention to bounding the number of edges incident to $A$. Recall that $H$ is $c$-partite and let $t$ denote the size of its smallest part. First let us suppose that $t\leqslant \abs{A}$. Since we assume the sum of any two parts of $H$ is at least $n$, it follows that the second smallest part of $H$ has size at least $n-t$ (note that $t\leqslant \frac{n}{2}$ by \eqref{greedy}). It follows that at most $t$ vertices of $A$ have degree $v(H)-t$ and the rest have degree at most $v(H)-n+t$ so that \begin{equation} \sum_{v\in A}d(v)\leqslant t(v(H)-t)+ (\abs{A}-t)(v(H)-n+t)\,. \end{equation} Considering the right hand side as a quadratic function in $t$ we see that it is maximised when $t=\frac{n+\abs{A}}{4}$ and so \begin{equation}\label{topt} \sum_{v\in A}d(v)\leqslant \frac{\abs{A}^2}{8}+\left(v(H)-\frac{3}{4}n\right)\abs{A}+\frac{n^2}{8}\,. \end{equation} Since $e(H)\leqslant e(H')+\sum_{v\in A}d(v)$ it follows by \eqref{e(H')} and \eqref{topt} that \begin{equation}\label{e(H)} e(H)\leqslant \frac{5}{8}\abs{A}^2+\left(v(H)-\frac{9}{4}n\right)\abs{A}+\frac{3}{4}n^2\,. \end{equation} We consider the right hand side as a quadratic function in $\abs{A}$ and optimise under the constraint $0\leqslant\abs{A}\leqslant\frac{n}{2}$. The maximum must occur at either $\abs{A}=0$ or $\abs{A}=\frac{n}{2}$ and it is simple to check that the latter is the maximiser under the assumption that $v(H)\geqslant \frac{31}{16}n$. It follows that $e(H)\leqslant\frac{n}{2}v(H)-\frac{7}{32}n^2$ as claimed. It remains to consider the case where $t\geqslant \abs{A}$. Recall that the maximum degree of $H$ is at most $v(H)-t$ and so \begin{equation} \sum_{v\in A}d(v)\leqslant \abs{A}(v(H)-t)\leqslant \abs{A}(v(H)-\abs{A})\leqslant\frac{n}{2}v(H)-\frac{n^2}{4}\,, \end{equation} where for the last inequality we again use the bound $\abs{A}\leqslant \frac{n}{2}$. The result follows. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:connM}] Let $\alpha=\frac14$, $0<\delta<\frac{1}{64k^2}$ and let $G$ be a $k$-coloured graph on $N=(k-\alpha)n$ vertices with at least $(1-\delta)\binom{N}{2}$ edges. We proceed by contradiction, supposing that $G$ contains no monochromatic connected matching of $\frac{n}{2}$ edges. Over all such $G$ consider the one in which blue has the most edges. Let $B_1,\dotsc, B_c$ be the blue connected components of $G$, suppose that red has the most edges between blue connected components and let $R'$ denote the $c$-partite graph of red edges which lie between blue components. The method is the same as the previous section. We establish the following two bounds. \begin{equation}\label{eq:edgesinblue2} \sum_{i=1}^c \binom{v(B_i)}{2} \leqslant \left(k-2\alpha+5\alpha^2\right)\frac{n^2}{2} = \left(k-\frac{3}{16}\right)\frac{n^2}{2}\,, \end{equation} \begin{equation}\label{eq:rededgesbetweenblue2} e(R') \leqslant \left(k-\alpha -\frac{7}{16}\right)\frac{n^2}{2} =\left(k-\frac{11}{16}\right)\frac{n^2}{2}\,. \end{equation} We then deduce \[ e(G) \leqslant (k-1)e(R')+\sum_{i=1}^c \binom{v(B_i)}{2} \leqslant \bigg(k^2-\frac{11}{16}k+\frac{1}{2}\bigg)\frac{n^2}{2}\,. \] Since $e(G)\geqslant(1-\delta)\binom{N}{2}=(1-\delta)(k-\frac{1}{4})(k-\frac{1}{4}-\frac{1}{n})\frac{n^2}{2}$ it is easy to verify that with $\delta<\frac{1}{64k^2}$ and $n\geqslant 32k$ we reach the desired contradiction. It remains to prove the inequalities~\eqref{eq:edgesinblue2} and~\eqref{eq:rededgesbetweenblue2}. We start by showing Claims~\ref{clm:noBlue5n/4} and~\ref{clm:smallx} have direct analogues here. \begin{claim}\label{clm:noBlue5n/42} There is no blue component on more than $\frac{5n}{4}$ vertices. \end{claim} \begin{claimproof} For contradiction, suppose there is a blue component $B_1$ on $\beta n$ vertices with $\beta> \frac{5n}{4}$. In this case, by Lemma~\ref{lem:connNoPn} we have $e(B_1)\leqslant(\beta-\frac{1}{4})\frac{n^2}{2}$. Using Lemma~\ref{lem:EG} on the rest of the blue graph, $B$, we obtain \[ e(B) \leqslant \left(\beta-\frac{1}{4}\right)\frac{n^2}{2}+\left(k-\alpha-\beta\right)\frac{n^2}{2} = \bigg(k-\alpha - \frac14 \bigg)\frac{n^2}{2}\,. \] Since blue is the densest colour we have $e(G)\leqslant k\cdot e(B)$ and hence \begin{align} (1-\delta)\binom{N}{2}=(1-\delta)(k-\alpha)\left(k-\alpha-\frac{1}{n}\right)\frac{n^2}{2} &\leqslant k \left(k-\alpha-\frac{1}{4}\right)\frac{n^2}{2}\\ (1-\delta)\alpha^2-(1-2\delta)k\alpha+\frac{k}{4}-(1-\delta)\frac{k-\alpha}{n} -\delta k^2 &\leqslant 0\,. \end{align*} This fails with $\alpha = \frac14$, $n \geqslant 32k$ and $\delta<\frac{1}{64k^2}$. \end{claimproof} Using the above claim we get a tighter bound on the size of blue components. Let $x$ be the excess size of blue components \[ xn = \sum_{i=1}^c\max\{v(B_i)-n,0\} \] as before. \begin{claim}\label{clm:smallx2} We have $x< \alpha$. \end{claim} \begin{claimproof} Let $B_1,\dotsc,B_\ell$ be the blue components with more than $n$ vertices. By Lemma~\ref{lem:connNoPn} and Claim~\ref{clm:noBlue5n/42} we have that there are at most $\frac{n^2}{2}$ edges in each of $B_1,\dotsc,B_\ell$. Using Lemma~\ref{lem:EG} on the rest of the blue graph we have \[ e(B) \leqslant \ell\frac{n^2}{2} + (k-\alpha-\ell-x)\frac{n^2}{2} =(k-\alpha - x)\frac{n^2}{2}\,. \] Since blue is the densest colour we have $e(G)\leqslant k\cdot e(B)$, and so \[ (1-\delta)\binom{N}{2}=(1-\delta)(k-\alpha)\left(k-\alpha-\frac{1}{n}\right)\frac{n^2}{2} \leqslant k(k-\alpha-x)\frac{n^2}{2} \] therefore \[ x \leqslant (1-2\delta)\alpha -(1-\delta)\frac{\alpha^2}{k} +\delta k + (1-\delta)\frac{k-\alpha}{kn}\,, \] and in particular $x<\alpha$ for $\alpha=\frac14$, $n\geqslant 32k$ and $\delta<\frac{1}{64k^2}$. \end{claimproof} Inequality~\eqref{eq:edgesinblue2} follows from Claim~\ref{clm:smallx2} in the exact same way as inequality~\eqref{eq:edgesinblue} follows from Claim~\ref{clm:smallx}. We require the same bound as before on the number of blue components, now with $\alpha = \frac14$. \begin{claim}\label{clm:upperboundc2} The number, $c$, of blue components of $G$ is at most $\frac{4}{3}(k-\alpha)+1$. \end{claim} \begin{claimproof} The proof follows that of Claim~\ref{clm:upperboundc} replacing $\binom{N}{2}$ with $(1-\delta)\binom{N}{2}$. With $\alpha=\frac14$, $n\geqslant 32k$ and $\delta <\frac{1}{64k^2}$ the required contradiction holds. \end{claimproof} The final claim is the improved version of Claim~\ref{cl:edgespervertex} which makes use of Lemma~\ref{lem:partiteNoMatching} and the above claim bounding $c$. \begin{claim}\label{cl:edgespervertex2} Let $H$ be a $c$-partite connected graph which does not contain a matching of $\frac{n}{2}$ edges. Suppose further that there is a $c$-partition of $H$ such that the sum of the sizes of any two parts is at least $n$. Then \[ \frac{e(H)}{v(H)}\leqslant \frac{n}{2}\left(1-\frac{7}{16(k-\alpha)}\right)\,. \] \end{claim} \begin{claimproof} We prove this using Lemma~\ref{lem:partiteNoMatching} in the same way that we proved Claim~\ref{cl:edgespervertex} using Lemma~\ref{lem:partiteNoPn}; breaking into cases depending on the size of $H$. Firstly if $v(H)\leqslant n\sqrt{\frac{c}{c-1}}$, or if $n\sqrt{\frac{c}{c-1}} < v(H) \leqslant \frac{5n}{4}$, the argument is identical to that of Claim~\ref{cl:edgespervertex} giving in both cases $\frac{e(H)}{v(H)} \leqslant \frac{n}{2}\left(1-\frac{3}{4(k-\alpha)+6} \right)\leqslant \frac{n}{2}\left(1-\frac{7}{16(k-\alpha)}\right)$. If $\frac{5n}{4} < v(H) < \frac{31n}{16}$ then \[ \frac{e(H)}{v(H)}\leqslant \frac{n}{2}\left(1-\frac{n}{4v(H)}\right)\leqslant \frac{n}{2}\left(1-\frac{4}{31}\right)\leqslant \frac{n}{2}\left(1-\frac{7}{16(k-\alpha)}\right)\,. \] Finally if $\frac{31n}{16} \leqslant v(H)$ we have $\frac{e(H)}{v(H)}\leqslant \frac{n}{2}\left( 1- \frac{7n}{16v(H)}\right)$. This is maximised when $v(H)$ is as large as possible and so we have \[ \frac{e(H)}{v(H)} \leqslant \frac{n}{2}\left(1-\frac{7}{16(k-\alpha)}\right)\,. \] \end{claimproof} We can now deduce inequality~\eqref{eq:rededgesbetweenblue2} from Claim~\ref{cl:edgespervertex2}. Suppose for contradiction that $e(R') >\left(k-\alpha -\frac{7}{16}\right)\frac{n^2}{2}$. Then, by the pigeonhole principle, there is a connected component of $H$ with \[ \frac{e(H)}{v(H)} > \frac{1}{N}\left(k-\alpha -\frac{7}{16}\right)\frac{n^2}{2} =\frac{n}{2}\left(1-\frac{7}{16(k-\alpha)}\right)\,. \] This contradicts Claim~\ref{cl:edgespervertex2}, proving \eqref{eq:rededgesbetweenblue2} and completing the proof of Theorem~\ref{thm:connM}. \end{proof} \bibliographystyle{plain}	{ "redpajama_set_name": "RedPajamaArXiv" }	5,277
When we first started dating we knew what we had was so special but I wasn't quite ready to tell the whole world just yet. So whenever we went out, I'd just say This is my good friend Gina. Then she said to me one day, I don't want to be your good friend, I want to be your girlfriend. I told her I wanted her to be even more, I wanted us to be special friends - the kind of girlfriends who can share everything, tell each other anything, and enjoy all that life offers together. This video is one we made just a few minutes before a call came and I started telling everyone how much Gina really means to me for real.	{ "redpajama_set_name": "RedPajamaC4" }	4,694
{"url":"https:\/\/ashishkumarletslearn.com\/chapter-9-class-11-maths-lecture-12\/","text":"\u201cSuperior Leaders Are Willing To Admit A Mistake And Cut Their Losses. Be Willing To Admit That You\u2019ve Changed Your Mind. Don\u2019t Persist When The Original Decision Turns Out To Be A Poor One.\u201d\n\n MISCELLANEOUS EXERCISE\n\nQuestion 11. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.\n\nQuestion 12.\u00a0The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms\n\nQuestion 13.\u00a0If \\frac{a+bx}{a-bx}=\\frac{b+cx}{b-cx}=\\frac{c+dx}{c-dx}(x \\ne 0), then show that a, b, c and d are in G.P.\n\nQuestion 14.\u00a0Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P^2R^n=S^n\n\nQuestion 15.\u00a0If pth, qth and rth terms of an A.P. are a, b, c, respectively. Show that (q-r)a+(r-p)b+(p-q)c=0\n\nQuestion 16.\u00a0If a \\left ( \\frac{1}{b}+\\frac{1}{c} \\right ) ,b \\left ( \\frac{1}{c}+\\frac{1}{a} \\right ), c \\left ( \\frac{1}{a}+\\frac{1}{b} \\right ) are in A.P., prove that a, b, c are in A.P.\n\nQuestion 17.\u00a0If a, b, c, d are in G.P, prove that (a^n+b^n),(b^n+c^n), (c^n+d^n) are in G.P.","date":"2021-05-17 04:04:16","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\": 0.8632506132125854, \"perplexity\": 620.447657025222}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991557.62\/warc\/CC-MAIN-20210517023244-20210517053244-00357.warc.gz\"}"}	null	null
import {ActionType} from '../../types' // eslint-disable-line no-unused-vars import 'isomorphic-fetch' import * as api from '../../app/api' import * as types from './locationFinderActionTypes' import { getWeather } from '../../forecast/weather/weatherActions' import { getForecastDaily } from '../../forecast/forecastDaily/forecastDailyActions' import { getForecastHourly } from '../../forecast/forecastHourly/forecastHourlyActions' const getLocations = (query:string):ActionType => ({ type: types.GET_LOCATIONS, payload: new Promise((resolve, reject) => { fetch(api.find(query)).then(response => { resolve(response.json()) }) }) }) const setSearchValue = (value:string):ActionType => ({ type: types.SET_SEARCHVALUE, payload: value }) const setLocation = (id:number):ActionType => { return dispatch => { dispatch(getWeather(id)) dispatch(getForecastDaily(id)) dispatch(getForecastHourly(id)) } } export { getLocations, setSearchValue, setLocation }	{ "redpajama_set_name": "RedPajamaGithub" }	1,318
Le ministère de l'Intérieur et de la Sécurité publique est le département ministériel du gouvernement béninois chargé de l'administration territoriale et de la sécurité intérieure. L'actuel ministre de l'Intérieur et de la Sécurité publique est Alassane Seidou, en fonction depuis le . Historique Considéré comme un ministère-clé et sensible, dû fait du périmètre d'action qu'il confère allant de la gestion de l'administration territoriale à la sécurité nationale, dans l'histoire du pays, les chefs du gouvernement se sont souvent octroyés le portefeuille de l'Intérieur, à l'instar de Christophe Soglo, de Justin Ahomadegbé, de Tahirou Congacou, d'Émile Derlin Zinsou ou bien encore d'Hubert Maga. Dès les débuts de la jeune République du Dahomey, le ministère de l'Intérieur est chargé de la Sûreté nationale, nom alors donné à la Police républicaine. Changements successifs de dénomination Depuis l'accession du Bénin à la souveraineté internationale, l'actuel MISP connait des mutations dans tous les domaines. En effet, ces mutations se remarquent aussi bien dans l'organisation, le fonctionnement que dans les appellations. Ces changements étaient fonction des idéaux politiques et réformes soutenus par les acteurs de l'animation de la vie politique et administrative du Bénin. Le ministère de l'intérieur est donc crée par le décret n°304/PC du 20/11 /1960 qui en fixe le fonctionnement et l'organisation qui devient dans la même année le Ministère des Affaires Intérieures et de la Sécurité (MAIS). Au regard des nouvelles missions qui viennent d'être ajoutées à celles dont il était chargé, il suivi et dirigé par les Présidents de la République qui sont les chefs de l'Etat et chefs du gouvernement jusqu'en 1970. Sous le régime révolutionnaire, le département ministériel n'a pas connu une grande modification car ses missions et attributions n'ont pas changé. Mais sa dénomination change et est devenue Ministère de l'Intérieur et de la Sécurité (MIS) par le décret n° 72-273/PCM du 26 octobre 1972. Le ministère conserve ses grandes fonctions jusqu'en 1990 où le Bénin est entré dans une nouvelle ère politique avec de nouvelles missions et de nouvelles fonctions pour le ministère. C'est le fruit de la démocratie chèrement acquise qui conditionne la création par le décret n°90-45 du 02 Mars 1990 du Ministère de l'Intérieur, de la Sécurité Publique et de l'Administration Territoriale (MISPAT). Depuis 1990, le ministère évolue sous différents noms tels que Ministère de l'Intérieur, de la Sécurité et de l'Administration Territoriale (MISAT) ; Ministère de l'Intérieur, de la Sécurité et de la Décentralisation (MISD); Ministère de l'Intérieur, de la Sécurité Publique et des Collectivités Locales (MISPCL). Le gouvernement du président thomas Boni Yayi, un réaménagement de tous les ministères. d'où la naissance après la scission du MISPCL en deux ministères de missions différentes. Cette scission a donné par décret n°07-465 du 16-10-2007 le Ministère de l'Intérieur et de la Sécurité Publique (MISP) devenu le Ministère de l'Intérieur, de la Sécurité Publique et des cultes (MISPC) puis le Ministère de la Décentralisation, de la Gouvernance Locale, de l'Administration et de l'Aménagement du Territoire (MDGLAAT). Cette scission a permis à chaque ministère de s'occuper véritablement de ses domaines d'intervention définis par les textes réglementaires. Le MISP garde sa mission première qui est la garantie de la sécurité et de l'ordre public en faisant respecter la libre circulation des personnes et des biens en République du Bénin Missions et attributions Selon le décret du , le ministère de l'Intérieur et de la Sécurité publique est responsable, sur l'ensemble du territoire national, de la mise en œuvre, du suivi et de l'évaluation de la politique de l'État en matière : de sécurité intérieure ; de lutte contre la radicalisation, l'extrémisme violent et le terrorisme ; de protection civile ; d'état civil ; de préservation des libertés publiques et de participation des citoyens ; de la production de la sécurité des biens et des personnes ; de gestion intégrée des espaces frontaliers. Organisation Le ministère de l'Intérieur et de la Sécurité publique est constitué d'une administration centrale, située à Cotonou, et d'une administration territoriale, avec des directions départementales des affaires intérieures et de la sécurité publique implantées dans le pays. D'après le décret , la structure du ministère est fixée comme suit : Cabinet du ministre Le ministre est assisté, dans les différentes missions qui lui incombent, de conseillers techniques dont il fixe les attributions. Entités rattachées au ministre Plusieurs structures sont également directement rattachées au ministre : la direction générale de la Police républicaine ; le secrétariat permanent de la Commission nationale de lutte contre la radicalisation, l'extrémisme violent et le terrorisme ; le secrétariat permanent de la Commission interministérielle de lutte contre l'abus des stupéfiants et des substances psychotropes ; le centre de documentation de sécurité publique ; le détachement de sécurité du ministère. Directions techniques et départementales Le ministère de l'Intérieur et de la Sécurité publique a autorité sur les directions techniques suivantes, qui sont coordonnées par le secrétaire général du ministère : la direction générale de la sécurité publique ; la direction des affaires intérieures et des cultes ; la direction de l'état civil ; la direction de la coordination de l'information et de la documentation ; la direction de la coopération technique de sécurité ; la direction des partis politiques et des affaires électorales . Direction général de la sécurité publique La direction générale de la sécurité publique coordonne les activités de sécurité publique des structures décentralisées relevant du ministère, en accompagnant, par exemple, les communes dans l'élaboration et le suivi de leurs plans et stratégies en matière de sécurité ou assurant une formation en matière de « renseignements généraux » auprès des conseils de quartiers, d'arrondissements, de communes, de villages et de villes. Direction des affaires intérieures et des cultes La direction des affaires intérieures et des cultes a pour mission la gestion des affaires à caractère national touchant à la vie des populations, aux cultes et aux régimes de police particuliers à savoir les établissements hôteliers, les salles de jeu, la presse, les débits de boisson et établissements assimilés. Direction de l'état civil La direction de l'état civil est notamment chargée de la mise en œuvre de la politique nationale de l'état civil, de suivre la modernisation et la sécurisation du système d'état civil et de participer au contrôle administratif des services d'état civil sur le territoire national. Direction de la coordination de l'information et de la documentation Cette direction a pour mission de centraliser et d'analyser les renseignements d'ordre politique, social et économique utiles à l'information et aux actions du gouvernement, de coordonner et de centraliser les renseignements relatifs à la criminalité. Direction de la coopération technique de sécurité La direction de la coopération technique de sécurité assure, en collaboration avec le ministère chargé des Affaires étrangères, les relations de coopération technique en matière de sécurité avec les autres pays, les partenaires techniques étrangers et les ambassades accréditées au Bénin. Direction des partis politiques et des affaires électorales La direction des partis politiques et des affaires électorales étudie toute constitution ou modification de partis politiques, suit leur implantation géographique, leur participation aux différentes élections ainsi que leur bonne conformité à la charte de partis politiques. Elle contrôle également la mise en place des mesures de sécurité lors des périodes électorales. Directions départementales Le ministère de l'Intérieur et de la Sécurité publique dispose de directions départementales des affaires intérieures et de la sécurité publique sur l'ensemble du Bénin, qui permettent la décentralisation des pouvoirs. Organismes sous tutelle Les organismes sous tutelle du ministère de l'Intérieur et de la Sécurité publique sont : l'Agence béninoise de gestion intégrée des espaces frontaliers ; l'Agence nationale de Protection civile ; le Groupement national de sapeurs-pompiers. Liste des ministres Cette liste présente les ministres béninois de l'Intérieur depuis la proclamation de la République du Dahomey, le . Notes et références Notes Références Bibliographie Voir aussi Articles connexes Gouvernement du Bénin Police républicaine (Bénin) Médaille d'honneur de la Police républicaine (Bénin) Liens externes Site officiel du gouvernement du Bénin Interieur bénin 12e arrondissement de Cotonou	{ "redpajama_set_name": "RedPajamaWikipedia" }	9,148
package com.logentries.net; /** * Thrown when a log + timestamps etc. is longer than {@link com.logentries.net.AsyncLogger#LOG_LENGTH_LIMIT} chars. */ public class LogTooLongException extends RuntimeException { }	{ "redpajama_set_name": "RedPajamaGithub" }	9,677
Open a Command Prompt window, change to the directory that contains the 64-bit version of the utility (:\Program Files\Microsoft SQL Server\100\DTS\Binn), and then run the utility from that location. At the command prompt, run the utility by entering the full path (:\Program Files\Microsoft SQL Server\100\DTS\Binn) to the 64-bit version of the utility. Permanently change the order of the paths in the PATH environment variable by placing the 64-bit path (:\Program Files\Microsoft SQL Server\100\DTS\Binn) before the 32-bit path (:\ Program Files(x86)\Microsoft SQL Server\100\DTS\Binn) in the variable. There is no 64-bit design-time or run-time support for Data Transformation Services (DTS) packages that were created in earlier versions of SQL Server. For more information, see .	{ "redpajama_set_name": "RedPajamaC4" }	1,543
Beliefs and Core Principles Opportunities at Cause Effective Board Fundraising Anniversary Campaigns/ Special Events Fundraising Under COVID Transformative Engagements Coach On Call For Development Professionals How We Work With Funders Portfolio Programs Customized Consulting Funder Partnerships Our Clients Speak Discover Our Clients Program Report Anniversary Campaigns & Events Current Blogs and Newsletters Celebrate St. Patrick's Day Month by Going Green March is here, which means it's also time for St. Patrick's Day—a holiday that was founded to honor the death of St. Patrick, the patron saint of Ireland. It has evolved into a day of worldwide celebrations related to luck and everything donning the color green! In the spirit of "going green" this month, use this holiday as a reminder to focus on the world around us. After all, making sure our environment and the world are protected for the future is anything but luck. It's going to take input from us all. Check out these ways you can give the month of March new meaning by going green! You've been hearing the mantra "reduce, reuse, recycle" for a long time, but it's time to actually put it into practice. One example is to reduce the amount of plastic you use. Bring reusable bags with you to the grocery store to cut down on plastic consumption. Recycle the household items you can, and think about items you can reuse more than once instead of tossing them into the trash! Some people have not adopted a green lifestyle because they don't know the harm it causes to our world. Join an organization that is helping spread awareness about protecting our environment and world. Help distribute facts about the impact of not paying attention to the effects we have on our environment. Consider volunteering at a school to help educate the next generation. Get Rid of the Unnecessary Conserve water where you can. Don't buy more than you need at the grocery store. Recycling doesn't have to just be about plastic and cardboard—it can be about any type of materials or waste. Instead of throwing away clothes you no longer need, give them to a charity or organization that can help redistribute them. Think of all the ways you can take items you have and repurpose them instead of tossing them. Switch to Digital Distribution Paperless billing does make a difference! Technology has also allowed us to cut travel when possible to resolve matters over the phone or online instead of getting in a car to drive to a business. Use technology when you can to receive important updates via email instead of on printed paper. Find Worthy Causes We're lucky to be living in a world with so many worthy causes and activists. This is a reminder to find an organization you love that is working toward a greener planet. You can spend your time volunteering for community clean up, helping to call potential donors and spread awareness or even donate yourself! Now is the time to go green! Celebrate St. Patrick's Day by truly embracing the green celebration. Together, we'll be lucky enough to help out the planet and do good in the world! © 2023 Cause Effective	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	9,809
\section{Introduction} When we are studying a \emph{number ring} $R$, that is a subring of a number field $K$, it can be useful to understand the size of its ideals compared to the whole ring. The main tool for this purpose is the norm map which associates to every non-zero ideal $I$ of $R$ its index as an abelian subgroup $N(I)=[R:I]$. If $R$ is the \emph{maximal order}, or \emph{ring of integers}, of $K$ then this map is multiplicative, that is, for every pair of non-zero ideals $I,J\subseteq R$ we have $N(I)N(J)=N(IJ)$. If the number ring is not the maximal order this equality may not hold for some pair of non-zero ideals. For example, if we consider the quadratic order $\mathbb{Z}[2i]$ and the ideal $I=(2,2i)$, then we have that $N(I)=2$ and $N(I^2)=8$, so we have the inequality $N(I^2)> N(I)^2$. Observe that if every maximal ideal ${\mathfrak p}$ of a number ring $R$ satisfies $N({\mathfrak p}^2)\leq N({\mathfrak p})^2$, then we can conclude that $R$ is the maximal order of $K$ (see Corollary \ref{cor:submdedekind}). In Section \ref{sec:prelim} we recall some basic commutative algebra and algebraic number theory and we apply them to see how the ideal norm behaves in relation to localizations and ring extensions. In Section \ref{sec:quadquartcases} we will see that the inequality in the previous example is not a coincidence. More precisely, we will prove that in any quadratic order we have $N(IJ)\geq N(I)N(J)$ for every pair of non-zero ideals $I $ and $J$. We will say that the norm is \emph{super-multiplicative} if this inequality holds for every pair of non-zero ideals (see Definition \ref{def:sm}). We will show that this is not always the case by exhibiting an a order of degree $4$ where we have both (strict) inequalities, see Example \ref{ex:degree4}. In a quadratic order every ideal can be generated by $2$ elements and in a order of degree $4$ by $4$ elements, so we are led to wonder if the behavior of the norm is related to the number of generators and what happens in a cubic order, or more generally in a number ring in which every ideal can be generated by 3 elements. The main result of this work is the following: \begin{theorem} \label{mainthm} Let $R$ be a number ring. The following statements are equivalent: \begin{enumerate}[label=\upshape(\roman), leftmargin=, widest=iii] \item \label{impl:1} every ideal of $R$ can be generated by $3$ elements; \item \label{impl:2} every ring extension $R'$ of $R$ contained in the normalization of $R$ is super-multiplicative. \end{enumerate} \end{theorem} Theorem \ref{mainthm} is an immediate consequence of the following two stronger results, which are proved respectively in Sections \ref{sec:firstimpl} and \ref{sec:secondimpl}. \begin{theorem} \label{thm:firstimpl} Let $R$ be a commutative Noetherian domain of dimension $1$ where every ideal can be generated by $3$ elements. Then $R$ is super-multiplicative. Moreover, every ring extension $R'$ of $R$ such that the additive group of $R'/R$ has finite exponent is also super-multiplicative. \end{theorem} \begin{theorem} \label{thm:secondimpl} Let $R$ be a number ring with normalization $\tilde{R}$ such that for every maximal $R$-ideal $\frm$ the ideal norm of the number ring $R+\frm\tilde R$ is super-multiplicative. Then every $R$-ideal can be generated by $3$ elements. \end{theorem} \section{Preliminaries} \label{sec:prelim} A field $K$ is called \emph{number field} if it is a finite extension of $\mathbb{Q}$. In this article all rings are unitary and commutative. We will say that $R$ is a \emph{number ring} if it is a subring of a number field. A number ring for which the additive group is finitely generated is called an \emph{order} in its field of fractions. In every number ring there are no non-zero additive torsion element. Every order is a free abelian group of rank $[\Frac(R):\mathbb{Q}]$, where $\Frac(R)$ is the fraction field of $R$. \begin{proposition} Let $R$ be a number ring. Then \begin{enumerate} \item every non-zero $R$-ideal has finite index; \item $R$ is Noetherian; \item if $R$ is not a field then it has Krull dimension $1$, that is every non-zero prime ideal is maximal; \item if $S$ is a number ring containing $R$ and ${\mathfrak p}$ a maximal ideal of $R$, then there are only finitely many prime $S$-ideal ${\mathfrak q}$ above ${\mathfrak p}$, that is ${\mathfrak q}\supseteq {\mathfrak p} S$; \item $R$ has finite index in its normalization $\tilde R$. \end{enumerate} \end{proposition} For a proof and more about number rings see \cite{psh08}. Recall that for a commutative domain $R$ with field of fractions $K$, a \emph{fractional $R$-ideal} $I$ is a non-zero $R$-submodule of $K$ such that $xI\subseteq R$ for some non-zero $x\in K$. Multiplying by a suitable element of $R$, we can assume that the element $x$ in the definition is in $R$. It is useful to extend the definition of the index to arbitrary fractional ideals $I$ and $J$ taking: \[[I:J]=\dfrac{[I:I\cap J]}{[J:I\cap J]}.\] It is an easy consequence that we have $[I:J]=[I:H]/[J:H]$ for every fractional ideal $H$. In particular, if $[R:I]$ is finite we call it the \emph{norm of the ideal $I$}, and we denote it $N(I)$. In general the ideal norm is not multiplicative. \begin{lemma} \label{lemma:principalideal} Let $R$ be a number ring and let $I$ be a non-zero $R$-ideal. For every non-zero $x\in K$ we have \[ N(xR)N(I)=N(xI).\] \end{lemma} \begin{proof} As $R$ is a domain, the multiplication by $x$ induces an isomorphism $R/I\simeq xR/xI$ of (additive) groups. Hence we have $[R:xR]=[I:xI]$ and therefore $[R:xR][R:I]=[R:xI]$. \end{proof} \begin{proposition} \label{prop:lengthproduct} Let $S\subseteq R$ be an extension of commutative rings. Let $I$ be an $R$-ideal such that $[R:I]$ is finite. Then \[[R:I] = \prod_{\frm} [R_\frm: I_\frm] = \prod_\frm \# ( S / \frm )^{l_{S_\frm}( R_\frm / I_\frm )},\] where the products are taken over the maximal ideals of $S$ and $l_{S_\frm}$ denotes the length as an $S_\frm$-module. Moreover, we have that \[l_S(R/I)=\sum_\frm l_{S_\frm}(R_\frm/I_\frm).\] \end{proposition} \begin{proof} As $R/I$ has finite length as an $S$-module, there exists a composition series \[R/I=M_0\supset M_1\supset \cdots \supset M_l = 0,\] where the $M_i$ are $S$-modules such that $M_i/M_{i+1}\simeq S/\frm_i$ for some maximal $S$-ideal $\frm_i$. Now fix a maximal $S$-ideal $\frm$. Observe that $\#\set{i\ :\ \frm_i=\frm} = l_{S_\frm}(R_\frm/I_\frm)$ because all the factors isomorphic to $S/\frm_i$ disappear if we localize at $\frm\neq \frm_i$. This implies that $l_S(R/I)=\sum_\frm l_{S_\frm}(R_\frm/I_\frm)$ and that $[ R_\frm : I_\frm ] = \# ( S / \frm )^{l_{S_\frm}( R_\frm/I_\frm)}$. By \cite[2.13, p.72]{eis95} we have $[R:I] = \prod_{\frm} [R_\frm:I_\frm]$. Observe that there is no harm in taking the product over all the maximal ideal of $S$ because the module $R/I$ vanishes if we localize at a maximal ideal that does not appear in its composition series. \end{proof} \begin{proposition} \label{prop:normmult} Let $R$ be a number ring, $I$ an invertible $R$-ideal. Then for every $R$-ideal $J$ we have \[N(I)N(J)=N(IJ).\] \end{proposition} \begin{proof} Recall that if an ideal $I$ is invertible then the localization $I_\frm$ at every maximal $R$-ideal $\frm$ is a principal $R_\frm$-ideal (see \cite[11.3, p.80]{mats89}). So by Lemma \ref{lemma:principalideal} we have that $[R_\frm:J_\frm][R_\frm:I_\frm]=[R_\frm:(IJ)_\frm]$ for every $\frm$. Hence by Proposition \ref{prop:lengthproduct} \[N(IJ)=\prod_\frm [R_\frm:(IJ)_\frm] = \prod_\frm [R_\frm:I_\frm]\prod_\frm [R_\frm:J_\frm]=N(I)N(J).\] \end{proof} \begin{proposition} \label{prop:lengthlocal} Let $S\subseteq R$ be an extension of commutative domains, $\frm$ a maximal $S$-ideal and $J$ a proper ideal of the localization $R_\frm$ such that $R_\frm/J$ has finite length as an $S_\frm$-module. Then \[ \frac{R}{J\cap R} \simeq \frac{R_\frm}{J} \] as $S$-modules. Moreover, \[l_S\left( \frac{R}{J\cap R} \right) = l_{S_\frm}\left( \frac{R_\frm}{J} \right). \] \end{proposition} \begin{proof} As $R$ is a domain the localization morphism $R \to R_\frm$ composed with the projection $R_\frm \to R_\frm/J$ induces an injective morphism $R/(J\cap R)\to R_\frm/J$. As $l_{S_\frm}( R_\frm / J)$ is finite, $R/(J\cap R)$ is annihilated by some power of $\frm$ and by \cite[2.13, p.72]{eis95} we have that it is isomorphic to its localization at $\frm$. As $(J\cap R)_\frm = J$ we have that $R/(J\cap R)\simeq R_\frm/J$ as $S$-modules. In particular they have the same length as $S$-modules. By Proposition \ref{prop:lengthproduct} we have that $l_S(R_\frm/J) = \sum_{\mathfrak n} l_{S_{\mathfrak n}}((R_\frm/J)_{\mathfrak n})$, where the sum is taken over the maximal $S$-ideals. So to conclude, we need to prove that if ${\mathfrak n}\neq \frm $, then $l_{S_{\mathfrak n}}((R_\frm/J)_{\mathfrak n})=0$, which is a direct consequence of the fact that $(R_\frm/J)_{\mathfrak n}=0$ when ${\mathfrak n} \neq \frm$. \end{proof} \begin{definition} \label{def:sm} Let $R$ be a commutative ring. We will say that the ideal norm of $R$ is \emph{super-multiplicative} if for every pair of $R$-ideals $I$ and $J$ such that $[R:IJ]$ is finite we have \[N(IJ)\geq N(I)N(J).\] For brevity we will say that $R$ is \emph{super-multiplicative}. \end{definition} \begin{proposition} Let $R$ be a number ring. Let $I$ be any non-zero $R$-ideal and ${\mathfrak p}$ a maximal $R$-ideal. Then $N({\mathfrak p} I)\geq N(I)N({\mathfrak p})$. \end{proposition} \begin{proof} By the isomorphism of abelian groups \[\dfrac{R/{\mathfrak p} I}{I/{\mathfrak p} I}\simeq R/I\] we get that \[\#(I/{\mathfrak p} I)=N({\mathfrak p} I)/N(I).\] Since $I/{\mathfrak p} I$ is a $(R/{\mathfrak p})$-vector space of finite dimension, say $d$, we have $\#(I/{\mathfrak p} I)=N({\mathfrak p})^d$. Therefore $N({\mathfrak p} I)=N(I)N({\mathfrak p})^d\geq N(I)N({\mathfrak p})$. \end{proof} \begin{corollary} \label{cor:submdedekind} Let $R$ be a number ring. Assume that for every maximal $R$-ideal ${\mathfrak p}$ we have the inequality $N({\mathfrak p}^2)\leq N({\mathfrak p})^2$. Then $R$ is a Dedekind domain. \end{corollary} \begin{proof} As in the proof of the previous Proposition we obtain $N({\mathfrak p}^2)=N({\mathfrak p})N({\mathfrak p})^d,$ where $d=\dim_{R/{\mathfrak p}}({\mathfrak p}/{\mathfrak p}^2)$. Using the hypothesis we get $N({\mathfrak p})^d\leq N({\mathfrak p})$ which implies that $d\leq 1$. Observe that it cannot be zero, as ${\mathfrak p}^2\subsetneq {\mathfrak p}$. Hence we have that $\dim_{R/{\mathfrak p}}({\mathfrak p}/{\mathfrak p}^2) = 1$ for every maximal ideal ${\mathfrak p}$, which is equivalent to say that $R$ is a Dedekind domain. \end{proof} Being super-multiplicative is a local property for commutative domains. More precisely: \begin{lemma} \label{lemma:smlocal} Let $S\subseteq R$ be an extension of commutative domains. Then $R$ is super-multiplicative if and only if for every maximal $S$-ideal $\frm$ we have that $R_\frm$ is super-multiplicative. \end{lemma} \begin{proof} Assume that $R$ is super-multiplicative and let $I$ and $J$ be $R_\frm$-ideals with $IJ$ of finite index in $R_\frm$. Then by Proposition \ref{prop:lengthlocal} we have $[R_\frm:IJ]=[R:IJ\cap R]$, $[R_\frm:I]=[R:I\cap R]$ and $[R_\frm:J]=[R:J\cap R]$. By Proposition \ref{prop:lengthproduct}, we obtain $l_{S_{\mathfrak n}}((R/IJ\cap R)_{\mathfrak n})=0$, $l_{S_{\mathfrak n}}((R/I\cap R)_{\mathfrak n})=0$ and $l_{S_{\mathfrak n}}((R/J\cap R)_{\mathfrak n})=0$, for every maximal $S$-ideal ${\mathfrak n}$ distinct from $\frm$. We have that $(I\cap R)_\frm(J\cap R)_\frm = (IJ\cap R)_\frm = IJ$ and $(I\cap R)_{\mathfrak n}(J\cap R)_{\mathfrak n} = (IJ\cap R)_{\mathfrak n} = R_{\mathfrak n}$ for ${\mathfrak n} $ a maximal $S$-ideal of $S$ distinct from $ \frm$, so $(I\cap R)(J\cap R) = (IJ\cap R)$. Hence we get that $[R_\frm:IJ]\geq [R_\frm:I][R_\frm:J]$, that is $R_\frm$ is super-multiplicative.\\ In the other direction, if we have that $R_\frm$ is super-multiplicative for every $\frm$, taking the product of the norms of the localizations leads to the required global inequality by Proposition \ref{prop:lengthproduct}. \end{proof} The next result is well known. We include a proof for sake of completeness. \begin{proposition} \label{prop:semilocal} Let $R$ be a \emph{semilocal} commutative domain, i.e.~a domain with a finite number of maximal ideals. Then, a fractional ideal of $R$ is invertible if and only if it is principal and non-zero. In particular, a semilocal Dedekind domain, like the normalization of any local number ring, is a principal ideal domain. \end{proposition} \begin{proof} One direction of the proof is trivial, because if $x\in R$ is non-zero, then the ideal $(x)$ has inverse $(x^{-1})$. Let's prove the other implication. Let $I$ be a fractional $R$-ideal. Multiplying by an appropriate element of the fraction field of $R$, we can assume that $I\subseteq R$. Observe that this doesn't affect the number of generators. Suppose that $I$ is an invertible $R$-ideal, with inverse $J$, i.e.~$IJ=R$. Let $\frm_1,\cdots,\frm_l$ be the maximal ideals of $R$. As $IJ\nsubseteq \frm_k$ for every $k$, there exist $a_k\in I,b_k\in J$ such that $a_kb_k\in R\setminus \frm_k$. By the Chinese Remainder Theorem, for every $k$ there exists an element $\lambda_k\not\in \frm_k$ and $\lambda_k\in \frm_j$ for every $j\neq k$. Then define \[ a = \lambda_1a_1+\cdots+\lambda_la_l\in I,\quad b=\lambda_1b_1+\cdots+\lambda_l b_l\in J \] and consider the product: \[ab = \sum_{1\leq i,j\leq l}\lambda_i\lambda_ja_ib_j.\] Observe that $\lambda_i\lambda_ja_ib_j\not\in \frm_k$ if and only if $i=j=k$. Hence $ab\not\in\frm_k$ for every $k$ and it must therefore be a unit. Then \[(a)\subseteq I=abI\subseteq aJI=aR=(a)\] as required. \end{proof} The following version of the Nakayama Lemma is classical and will be used several times in the rest of the paper without mentioning. \begin{lemma}[{\cite[Proposition 2.8]{AtiyahMacdonald69}}] \label{lemma:Nakayama} Let $R$ be a local ring with maximal ideal $\frm$ and let $M$ be a finitely generated $R$ module. Let $x_1,\ldots,x_n$ be elements of $M$ whose images in $M/\frm M$ form a basis over $R/\frm$. Then the $x_i$ generate $M$. \end{lemma} \section{Quadratic and degree 4 case} \label{sec:quadquartcases} In this section we prove that every quadratic order is super-multiplicative. This result is a consequence of Theorem \ref{mainthm} stated in the introduction. We report this particular case separately because the argument of the proof is different and of its own interest. We will also exhibit in the end of this section an example that shows that an analogous theorem is not true for orders in a number field of degree $4$. \begin{lemma} \label{lemma:quadmult} Let $R$ be an order in a quadratic field $K$. Let $I$ be a non-zero $R$-ideal and $R_I$ its multiplier ring, i.e.~$R_I=\set{x\in K:\ xI\subseteq I}$. Then $I$ is an invertible ideal of $R_I$. \end{lemma} \begin{proof} By \cite[11.3, p.80]{mats89} it suffices to show that the localization of $I$ at every maximal ideal ${\mathfrak p}$ of $R_I$ is principal. Assume that this is not the case, say that $I_{\mathfrak p}$ is not principal. Observe that if ${\mathfrak p}$ is above the rational prime $p$ we cannot have $pR_I={\mathfrak p}$, because $R_{I{\mathfrak p}}$ would be a DVR and $I_{\mathfrak p}$ would be invertible. As $ [R_I:pR_I]=p^2 $ and $ pR_I \subsetneq {\mathfrak p} $, we have $[{\mathfrak p} : pR_I ]= [R_I:{\mathfrak p}] =p$ and $R_I/{\mathfrak p}\simeq \mathbb{F}_p$. By Lemma~\ref{lemma:principalideal} we have $[I:pI]=[R_I:pI]/[R_I:I]=[R_I:pR_I]=p^2$. As $I_{\mathfrak p}$ is not principal, by Nakayama's Lemma we have that $I_{\mathfrak p}/{\mathfrak p} I_{\mathfrak p}\simeq I/{\mathfrak p} I$ is a $R_{I{\mathfrak p}}/{\mathfrak p}$-vector space of dimension $2$. Hence also $[I:{\mathfrak p} I]=p^2$ which implies $pI={\mathfrak p} I$ because $pI\subseteq {\mathfrak p} I$ and they have the same index in $I$. So by the definition of multiplier ring $p^{-1}{\mathfrak p}\subseteq R_I$, hence $pR_I={\mathfrak p}$ by the maximality of ${\mathfrak p}$. Contradiction. So $I$ is an invertible $R_I$-ideal. \end{proof} \begin{lemma} \label{lemma:cycicpgroup} Let $R$ be a quadratic order with integral closure $\tilde R$ and consider the localizations at a prime number $p\in \mathbb{Z}$, namely $\tilde{R}_{(p)}=\tilde{R}\otimes \mathbb{Z}_{(p)}$ and $R_{(p)}=R\otimes \mathbb{Z}_{(p)}$. Then we have that $\tilde{R}_{(p)}/R_{(p)}\simeq \mathbb{Z}/p^n\mathbb{Z}$ for some $n\in \mathbb{Z}_{\geq 0}$. \end{lemma} \begin{proof} Note that $\tilde{R}/R$ is a finite abelian group which can be decomposed in the product of finitely many cyclic groups with order a prime power. When we localize at $p$ we consider only the $p$-part of this decomposition. As $R$ is quadratic what is left is a cyclic group. \end{proof} \begin{theorem} The ideal norm in a quadratic order is super-multiplicative. \end{theorem} \begin{proof} Let $R$ be a quadratic order and $I,J$ two non-zero ideals of $R$. We want to show that \[\dfrac{[R:IJ]}{[R:I][R:J]}\geq 1.\] Let $p$ be an arbitrary rational prime, we want to prove that \[\dfrac{[R_{(p)}:I_{(p)}J_{(p)}]}{[R_{(p)}:I_{(p)}][R_{(p)}:J_{(p)}]}\geq 1. \label{local rel}\tag{}\] By Lemma \ref{lemma:quadmult} we have that $I$ (resp.~$J$) is invertible in its multiplier ring $R_I$ (resp.~$R_J$). Note that if ${\mathfrak q}$ is a maximal $R_I$-ideal above the rational prime $q$, then ${\mathfrak q}\cap (\mathbb{Z}\setminus (p))=(q)\setminus (p)$ which is empty if and only if $p=q$. This means that the maximal ideals of $R_{I(p)}$ are exactly the ones above $p$ and similarly for $R_{J(p)}$. So the localization of $I$ (resp.~$J$) at $(p)$ is a principal ideal of $R_{I(p)}$ (resp.~$R_{J(p)}$) by Proposition \ref{prop:semilocal} . Say that we have $I_{(p)}=xR_{I(p)}$ and $J_{(p)}=yR_{J(p)}$ and observe that $R_{I(p)}$ and $R_{J(p)}$ are both $R_{(p)}$-fractional ideals. So by Lemma \ref{lemma:principalideal} \begin{align} [R_{(p)}:I_{(p)}] &=[R_{(p)}:R_{I(p)}][R_{(p)}:xR_{(p)}],\\ [R_{(p)}:J_{(p)}] &=[R_{(p)}:R_{J(p)}][R_{(p)}:yR_{(p)}],\\ [R_{(p)}:I_{(p)}J_{(p)}]&=[R_{(p)}:xR_{I(p)}yR_{J(p)}]=\\ &=[R_{(p)}:R_{I(p)}R_{J(p)}][R_{(p)}:xR_{(p)}][R_{(p)}:yR_{(p)}]. \end{align} If we substitute these equalities in (\ref{local rel}) we get: \[\dfrac{[R_{(p)}:R_{I(p)}R_{J(p)}]}{[R_{(p)}:R_{I(p)}][R_{(p)}:R_{J(p)}]} = \dfrac{[R_{I(p)}:R_{(p)}][R_{J(p)}:R_{(p)}]}{[R_{I(p)}R_{J(p)}:R_{(p)}]}.\] As $\tilde{R}_{(p)}/R_{(p)}$ is a cyclic $p$-group by Lemma \ref{lemma:cycicpgroup}, the lattice of its subgroups is totally ordered w.r.t.~the inclusion relation. Then as $R \subseteq R_I,R_J\subseteq \tilde{R}$, we have that $R_{I(p)}\subseteq R_{J(p)}$ or $R_{J(p)}\subseteq R_{I(p)}$. Assume that the first one holds, then $R_{I(p)}R_{J(p)}=R_{J(p)}$. So we have: \[\dfrac{[R_{I(p)}:R_{(p)}][R_{J(p)}:R_{(p)}]}{[R_{I(p)}R_{J(p)}:R_{(p)}]} = \dfrac{[R_{I(p)}:R_{(p)}][R_{J(p)}:R_{(p)}]}{[R_{J(p)}:R_{(p)}]}= [R_{I(p)}:R_{(p)}] \geq 1.\] If we have that $R_{J(p)}\subseteq R_{I(p)}$ we proceed in an analogous way. As this inequality holds for the localization at every rational prime $p$, by Proposition \ref{prop:lengthproduct} it holds also for the original quotient, hence we get the desired inequality for the global norms. \end{proof} As we have understood the quadratic case, then we will move to extensions of $\mathbb{Q}$ of higher degree. The next example shows that we cannot prove an analogous theorem for the degree 4 case. \begin{example} \label{ex:degree4} Consider the field $\mathbb{Q}(\alpha)$, where $\alpha$ is the root of a monic irreducible polynomial of degree $4$ with integer coefficients. Consider the order generated by the ring $\mathbb{Z}$ and the ideal $p\mathbb{Z}[\alpha]$, say $R=\mathbb{Z} + p\mathbb{Z}[\alpha]$, where $p$ is a rational prime number. Take the $R$-ideals $I=pR+ p\alpha R$ and $J=pR + p\alpha ^2R$ and the maximal ideal $M=p\mathbb{Z}[\alpha]$. It's easy to verify that \begin{align} &R = \mathbb{Z} \oplus p\alpha \mathbb{Z} \oplus p\alpha^2 \mathbb{Z} \oplus p\alpha^3 \mathbb{Z}, & & I = p\mathbb{Z} \oplus p\alpha \mathbb{Z} \oplus p^2\alpha^2 \mathbb{Z} \oplus p^2\alpha^3 \mathbb{Z},\\ &J = p\mathbb{Z} \oplus p^2\alpha \mathbb{Z} \oplus p\alpha^2 \mathbb{Z} \oplus p^2\alpha^3 \mathbb{Z}, & &M = p\mathbb{Z} \oplus p\alpha \mathbb{Z} \oplus p\alpha^2 \mathbb{Z} \oplus p\alpha^3 \mathbb{Z},\\ &IJ = p^2\mathbb{Z} \oplus p^2\alpha \mathbb{Z} \oplus p^2\alpha^2 \mathbb{Z} \oplus p^2\alpha^3 \mathbb{Z}, & &IM = p^2\mathbb{Z} \oplus p^2\alpha \mathbb{Z} \oplus p^2\alpha^2 \mathbb{Z} \oplus p^2\alpha^3 \mathbb{Z}, \end{align} which gives us $N(I)=N(J)=p^3$, $N(M)=p$, $N(IJ)=N(IM)=p^5$. Hence \[ p^6=N(I)N(J)>N(IJ)=p^5,\qquad p^4=N(I)N(M)<N(IM)=p^5.\] \end{example} \section{Proof of Theorem \ref{thm:firstimpl}} \label{sec:firstimpl} In this section we introduce a convenient notation for the maximal number of generators for the ideals of a commutative ring and discuss how this quantity behaves when we localize or extend the ring. The rest of the section is devoted to the proof of Theorem \ref{thm:firstimpl}. \begin{definition} Let $R$ be a commutative ring. We define \[ g(R):=\sup_{\substack{I\subset R\\ \text{ideal}}}\Bigl( \inf_{\substack{S\subset I\\ I=\Span{S}}} \#S \Bigr). \] \end{definition} \begin{remark} If $R$ is a commutative domain then $g(R)$ is the bound for the cardinality of a minimal set of generators of every fractional ideal $I$. In fact, by the definition of a fractional ideal, there exists a non-zero element $x$ in the fraction field of $R$ such that $xI\subseteq R$. So $xI$ is an $R$-ideal and hence can be generated by $g(R)$ elements, so $I$ can be generated by the same elements multiplied by $x^{-1}$. \end{remark} \begin{remark} \label{rmk:extension} Let $R\subset R'$ be an extension of commutative domains such that the abelian group $R'/R$ has finite exponent, say $n$. Then we have $g(R')\leq g(R)$. In fact if $J$ is an $R'$-ideal, then $nJ\subseteq R$, and hence $J$ is a fractional $R$-ideal. In particular we are in this situation if $R$ is a number ring and $R'$ is contained in the normalization $\tilde R$ of $R$, because the index $[\tilde R : R]$ is finite. \end{remark} \begin{remark} Let $R$ be a number ring inside a number field $K$. We have $g(R)\leq [K:\mathbb{Q}]$ and this bound is sharp, in the sense that we can find an order $R'$ in $K$ such that $g(R')=[K:\mathbb{Q}]$. Let ${\mathcal O}_K$ be the maximal order of $K$. Let $I$ be any $R$-ideal. As $R$ is Noetherian, $I$ can be generated by a finite set of elements, say $x_1,\cdots,x_d$. We can find an integer $n\geq 1$ such that $nx_1,\cdots,nx_d\in {\mathcal O}_K$. Then observe that $I'=nI\cap ({\mathcal O}_K\cap R)$ is an ideal of ${\mathcal O}_K \cap R$, so it can be generated over $\mathbb{Z}$ by $[K:\mathbb{Q}]$ elements, say $\alpha_1,\cdots,\alpha_{[K:\mathbb{Q}]}$. As $I'R=nI$, we have that $\alpha_1/n,\cdots,\alpha_{[K:\mathbb{Q}]}/n$ generate $I$ over $R$. Hence $g(R)\leq [K:\mathbb{Q}]$. To prove the second part, let $\alpha$ be an algebraic integer and put $K=\mathbb{Q}(\alpha)$. Consider $R'=\mathbb{Z}+p\mathbb{Z}[\alpha]$ where $p$ is a rational prime number. Then $\frm=p\mathbb{Z}[\alpha]$ is a maximal ideal of $R'$ and $\dim_{\mathbb{F}_p} \frm/\frm^2=[K:\mathbb{Q}]$, so $g(R')=[K:\mathbb{Q}]$. \end{remark} We have a nice description of the behavior of $g(R)$ for a number ring $R$ when we localize at a maximal ideal. \begin{lemma} \label{lemma:d-gen} Let $R$ be a number ring, with normalization $\tilde{R}$. Let $I$ be an $R$-ideal. For every integer $d\geq 2$ the following are equivalent: \begin{enumerate} \item \label{d-gen:1} the $R$-ideal $I$ can be generated by $d$ elements; \item \label{d-gen:2} for every maximal ideal ${\mathfrak p}$ of $R$, the $R_{\mathfrak p}$-ideal $I_{\mathfrak p}$ can be generated by $d$ elements. \end{enumerate} \end{lemma} \begin{proof} Observe that (\ref{d-gen:1}) implies (\ref{d-gen:2}) is an immediate consequence of the fact that $I_{\mathfrak p} = I \otimes_R R_{\mathfrak p}$. For the other direction, assume that $I_{\mathfrak p}$ is $d$-generated, for every ${\mathfrak p}$. We can choose the local generators to be in $I$, just multiplying by the common denominator, which is a unit in $R_{\mathfrak p}$. Now, $\tilde{R}/R$ has finite length as an $R$-module. Consider a composition series \[ \tilde{R}/R = M_0\supset M_1 \supset \cdots \supset M_l =0. \] All the factors $M_i/M_{i+1}$ for $i=0,\cdots,l-1$ are simple, hence of the form $R/{\mathfrak p}_i$ where ${\mathfrak p}_i$ is a maximal $R$-ideal. If we localize at a maximal ideal ${\mathfrak p}\neq {\mathfrak p}_i$, for $i=0,\cdots,l-1$, all the factors disappear, and hence we have that $\tilde{R}_{\mathfrak p}=R_{\mathfrak p}$. Hence $R_{\mathfrak p}$ is a local Dedekind domain. Hence $I_{\mathfrak p}$ is a principal $R_{\mathfrak p}$-ideal. As the number of factors of the composition series is finite, this situation occurs for almost all the maximal ideals of $R$. In other words we can say that $I/{\mathfrak p} I\simeq I_{\mathfrak p}/{\mathfrak p} I_{\mathfrak p}$ is a 1-dimensional $R/{\mathfrak p}$-vector space for almost all maximal ideals. Then consider the finite set $S=\set{{\mathfrak p} : \dim_{(R/{\mathfrak p})}I/{\mathfrak p} I\neq 1} $. By the Chinese Remainder Theorem we can pick an element $x_1\in I$ such that $x_1\not\in {\mathfrak p} I$ for every ${\mathfrak p}\in S$. Now consider $T=\set{{\mathfrak p} : I\supsetneq {\mathfrak p} I + (x_1)}$, which is also finite because the ideals $I$ and $(x_1)$ are locally equal for almost all the maximal ideals of $R$ by a similar argument. So we can build a set of global generators in the following way: with the Chinese Remainder Theorem take $x_2\in I\setminus ({\mathfrak p} I+(x_1))$ for every ${\mathfrak p}\in T$, $x_3\in I\setminus ({\mathfrak p} I+(x_1,x_2))$ for every ${\mathfrak p}\in T$ such that $I$ is not equal to ${\mathfrak p} I+(x_1,x_2)$, and so on until $x_d$. Then observe that $x_1,x_2,\cdots,x_d$ is a set of generators for $I$, because it is so locally at every prime: if ${\mathfrak p}\in S$ then $I_{\mathfrak p}=(x_1,x_2,\cdots,x_d)$ by construction, if ${\mathfrak p}\in T\setminus S$ then $I_{\mathfrak p}=(x_2)$ and if ${\mathfrak p}\not\in T$ then $I_{\mathfrak p}=(x_1)$. Now observe that $I=\bigcap_{\mathfrak p} I_{\mathfrak p}$ and so $x_1,x_2,\cdots,x_d$ generates the ideal $I$ over $R$. \end{proof} \begin{corollary} Let $R$ be a number ring. If $g(R_{\mathfrak p})>1$ for some maximal $R$-ideal ${\mathfrak p}$ then \[g(R) = \sup_{\mathfrak p} g(R_{\mathfrak p}).\] \end{corollary} \begin{remark} Let $R$ be a number ring such that $g(R_{\mathfrak p})=1$ for every maximal ideal, then $R$ is a Dedekind domain because every ideal $I$ has principal localizations, hence $I$ is invertible. Similarly as in the proof of the previous Lemma, we can show that $g(R)\leq 2$. \end{remark} Now that we have introduced some notation, we can start with the proof of Theorem~\ref{thm:firstimpl}. \begin{lemma} \label{lemma:linalg1} Let $U,V,W$ be vector spaces over a field $k$, with $W$ of dimension $\geq 2$. Let ${\varphi}:U\otimes V \twoheadrightarrow W$ be a surjective linear map. Then there exists an element $u\in U$ such that $\dim_k {\varphi}(u\otimes V)\geq 2$, or there exists an element $v\in V$ such that $\dim_k {\varphi}(U\otimes v)\geq 2$. \end{lemma} \begin{proof} For contradiction, assume that ${\varphi}(u\otimes V)$ and ${\varphi}(U\otimes v)$ have dimension $\leq 1$, for every choice of $u\in U$ and $v\in V$. As ${\varphi}$ is surjective, $\{{\varphi}(u\otimes v):u\in U, v\in V\}$ is a set of generators of $W$. Since $W$ has dimension $\geq 2$, among these generators there are 2 which are linearly independent, say $w_1={\varphi}(u_1\otimes v_1)$ and $w_2={\varphi}(u_2\otimes v_2)$. Observe \[{\varphi}(u_1\otimes v_2)\in {\varphi}(u_1\otimes V)\cap {\varphi}(U\otimes v_2)=kw_1\cap kw_2 = 0.\] Similarly we obtain also ${\varphi}(u_2\otimes v_1)=0$. But then we have that both ${\varphi}((u_1+u_2)\otimes v_1)=w_1$ and ${\varphi}((u_1+u_2)\otimes v_2)=w_2$ are in ${\varphi}((u_1+u_2)\otimes V)$. So it contains two linearly independent vectors and then it must have dimension $\geq 2$. Contradiction. \end{proof} \begin{lemma} \label{lemma:cyclic} Let $R$ be a commutative domain and $I,J\subset R$ two non-zero ideals, such that $IJ$ can be generated by $3$ elements. Let $\frm \subset R$ be a maximal ideal. Then one of the following occurs: \begin{enumerate} \item \label{lemma:cyclic:item:1} there exists a non-zero $x\in I_\frm$ such that $(IJ)_\frm/xJ_\frm$ is a cyclic $R_\frm$-module generated by an element of the form $\overline{ij}$ with $i\in I_\frm,j\in J_\frm$; \item \label{lemma:cyclic:item:2}there exists a non-zero $y\in J_\frm$ such that $(IJ)_\frm/yI_\frm$ is a cyclic $R_\frm$-module generated by an element of the form $\overline{ij}$ with $i\in I_\frm,j\in J_\frm$. \end{enumerate} In particular, if (\ref{lemma:cyclic:item:1}) holds then the morphism of $R_\frm$-modules induced by the ``multiplication by $j$" \[\dfrac{I_\frm}{xR_\frm}\overset{\cdot j}{\longrightarrow}\dfrac{(IJ)_\frm}{xJ_\frm}\] is surjective, and similarly if (\ref{lemma:cyclic:item:2}) holds then the morphism of $R_\frm$-modules induced by the ``multiplication by $i$" \[\dfrac{I_\frm}{yR_\frm}\overset{\cdot i}{\longrightarrow}\dfrac{(IJ)_\frm}{yI_\frm}\] is surjective. \end{lemma} \begin{proof} Let $k$ denote the field $R/\frm$. Observe that $W=(IJ)_\frm/\frm (IJ)_\frm$ is a $k$-vector space of dimension $\leq 3$. First, if $W$ has dimension $1$ then we have that $(IJ)_\frm$ is a principal $R_\frm$-ideal and clearly there exists $x\in I_\frm$ such that $(IJ)_\frm/xJ_\frm$ is a cyclic $R_\frm$-module. If the dimension of $W$ is $2$ or $3$, then consider the product map: \[ {\varphi}:\dfrac{I_\frm}{\frm I_\frm}\otimes \dfrac{J_\frm}{\frm J_\frm} \longrightarrow W, \qquad \overline{i}\otimes \overline{j} \longmapsto \overline{ij}. \] It is a surjective linear map of $k$-vector spaces. By Lemma \ref{lemma:linalg1} there exists $x\in I_\frm$ such that ${\varphi}(x\otimes (J_\frm/\frm J_\frm))$ has dimension $\geq 2$, or there exists $y\in J_\frm$ such that ${\varphi}( (I_\frm/\frm I_\frm)\otimes y)$ has dimension $\geq 2$. We will prove that if we are in the first case then (\ref{lemma:cyclic:item:1}) holds. The proof that the second case implies (\ref{lemma:cyclic:item:2}) is analogous. So assume that $\dim_k {\varphi}(x\otimes (J_\frm/\frm J_\frm))\geq 2$. Hence the quotient space \[ \dfrac{W}{{\varphi}(x\otimes (J_\frm/\frm J_\frm))} \simeq \dfrac{(IJ)_\frm}{xJ_\frm+\frm(IJ)_\frm} \] has dimension $\leq 1$. Moreover, it is easy to see that it is isomorphic to $S/\frm S$, where $S=(IJ)_\frm/xJ_\frm$. Hence we have that $S$ is a cyclic $R_\frm$-module.\\ We can be more precise saying that every generator of $S$ is of the form $ \sum_{t\in T}\overline{i_tj_t}$, where $T$ is a finite set of indexes, $i_t\in I_\frm$ and $j_t\in J_\frm$. In particular $\set{\overline{i_tj_t}}_{t\in T}$ is a finite set of generators for $S$. As the $k$-vector space $S/\frm S$ is $1$-dimensional, among the projections $\overline{i_tj_t}$ there exists one $\overline{i_{t_0}j_{t_0}}$ which is a basis. Hence $i_{t_0}j_{t_0}$ is a generator of $S$. The last assertion follows immediately. \end{proof} \begin{proposition} \label{prop:length} Let $R$ be a commutative Noetherian $1$-dimensional domain. Let $I,J$ be two non-zero ideals such that $IJ$ can be generated by $3$ elements. Then we have that \[l\left( \dfrac{R_\frm}{I_\frm} \right) + l\left( \dfrac{R_\frm}{J_\frm} \right) \leq l\left( \dfrac{R_\frm}{(IJ)_\frm} \right). \] \end{proposition} \begin{proof} Assume that case (\ref{lemma:cyclic:item:1}) of Lemma \ref{lemma:cyclic} holds. Consider the ring $R_\frm/xJ_\frm$. It has finite length because it is Noetherian and zero-dimensional. Consider the following diagram of inclusions of $R_\frm$-ideals: \begin{displaymath} \xymatrix@ur@C=2pc{ J_\frm \ar@{-}[d] & R_\frm \ar@{-}\|{\|\|}[l] \ar@{-}[d] \\ (IJ)_\frm \ar@{-}[d] & I_\frm \ar@{-}[d] \\ xJ_\frm & xR_\frm \ar@{-}\|{\|\|}[l] } \end{displaymath} These two chains define two series for $R_\frm/xJ_\frm$, and they can be refined to composition series. Observe that the multiplication by $x$ is an isomorphism of $R_\frm$ onto $xR_\frm$ and of $J_\frm$ onto $xJ_\frm$, so it induces a $R$-module isomorphism also on the quotients. Hence we have $l (R_\frm/J_\frm) = l (xR_\frm/xJ_\frm) $ and as the diagram of inclusions is commutative we have also $l(R_\frm/xR_\frm)=l(J_\frm/xJ_\frm)$. Moreover, as $I_\frm/xR_\frm$ is mapped onto $(IJ)_\frm/xJ_\frm$ by Lemma \ref{lemma:cyclic}, for every factor of the composition series between $R_\frm$ and $I_\frm$ there exists a corresponding factor between $J_\frm$ and $(IJ)_\frm$. So we have \[l\left( \dfrac{R_\frm}{I_\frm} \right) \leq l\left( \dfrac{J_\frm}{(IJ)_\frm} \right).\] To finish the proof, it is sufficient to add $l(R_\frm/J_\frm)$ on both sides. If case (\ref{lemma:cyclic:item:2}) of Lemma \ref{lemma:cyclic} holds we get the same conclusion with a similar argument. \end{proof} Now we can conclude our proof: \begin{proof}[Proof of Theorem \ref{thm:firstimpl}] As every ideal can be generated by $3$ elements, for every pair of non-zero $R$-ideals $I$ and $J$, Proposition \ref{prop:length} implies \[\#(R/\frm)^{l(R_\frm/(IJ)_\frm)}\geq \#(R/\frm)^{l(R_\frm/I_\frm)+l(R_\frm/J_\frm)},\] for every maximal $R$-ideal $\frm$. Hence by Proposition \ref{prop:lengthproduct} we get \[N(IJ)\geq N(I)N(J).\] For the second statement, use Remark \ref{rmk:extension} \end{proof} \section{Proof of Theorem \ref{thm:secondimpl}} \label{sec:secondimpl} In this section we prove Theorem \ref{thm:secondimpl}. Firstly, we will exhibit a bound for $g(R)$ for a local number ring $R$. Secondly, we will give a sufficient condition such that this bound is $\leq 3$. Finally, we will conclude the proof by moving from the local case to the global one. \begin{lemma} \label{lemma:linalg2} Let $V$ be a finite dimensional vector space over a finite field $k$ such that \[V=V_1\cup\cdots\cup V_n,\] where each $V_i$ is a proper subspace of $V$. Then $n>\# k$. \end{lemma} \begin{proof} As $V_i\subsetneqq V$, then it has codimension $\geq 1$, which implies that $\#V_i\leq(\#k)^{\dim_k V-1}$. As $\overline{0}\in V_1\cap\cdots\cap V_n$, then \begin{align} (\#k)^{\dim_k V}=\#V &=\#(V_1\cup\cdots\cup V_n)\leq \\ &\leq \left(\sum_{i=1}^n \#V_i\right)-(n-1)<\sum_{i=1}^n\#V_i\leq n(\#k)^{\dim_k V-1}. \end{align} Then dividing by $(\#k)^{\dim_k V-1}$ we get $n>\#k$. \end{proof} \begin{lemma} \label{lemma:2} Let $R$ be a local number ring with maximal ideal $\frm$ and residue field $k$. Let $\tilde{R}$ be its normalization. Let $l$ be the number of distinct maximal ideals of $\tilde{R}$. If $l\leq \#k$ then for every $R$-ideal $I$, there exists $x\in I$ such that $I\tilde{R}=x\tilde{R}$. \end{lemma} \begin{proof} The statement is trivially true if $I=0$. Assume that $I\neq 0$. Denote the maximal ideals of $\tilde{R}$ by $\frm_1,\cdots,\frm_l$. Consider the $k$-vector spaces $W=I/\frm I$ and $I\tilde{R}/\frm_iI\tilde{R}$. For every $i$, define the map \[{\varphi}_i: W \longrightarrow \dfrac{I\tilde{R}}{\frm_iI\tilde{R}}, \qquad x+\frm I \mapsto x+\frm_iI\tilde{R}.\] Denote by $W_i$ the kernel of ${\varphi}_i$. The ideal $I$ is a set of generators of $I\tilde{R}$ as $\tilde{R}$-module and hence of $I\tilde{R}/\frm_iI\tilde{R}$. This means that ${\varphi}_i$ is not the zero map, i.e.~$W_i$ is a proper subspace of $W$, for every $i$. As $l\leq \#k$, by Lemma \ref{lemma:linalg2} we get that $W_1\cup \cdots \cup W_l\subsetneq W$ and so there exists $x\in I$ whose projection in $W$ is not in $W_i$, for every $i$. Observe that this condition means that $\ord_{\frm_i}(x)\leq \ord_{\frm_i}(I\tilde{R})$ for every $i$. Moreover $x\in I\subset I\tilde{R}$, so $\ord_{\frm_i}(x)\geq \ord_{\frm_i}(I\tilde{R})$ for every $i$. Since we have that $\ord_{\frm_i}(x) = \ord_{\frm_i}(I\tilde{R})$ for every $i$, we conclude that $x\tilde{R}=I\tilde{R}$. \end{proof} The next example proves that the hypothesis $l\leq \#k$ in the previous lemma cannot be omitted. It is a generalization of an example suggested by Hendrik Lenstra. \begin{example} Let $p$ be a prime number and $K$ an extension of $\mathbb{Q}$ of degree $p+1$ where $p$ splits completely. Let $A$ to the integral closure of $\mathbb{Z}_{(p)}$ in $K$. Note that $p$ factors in $A$ as $pA={\mathfrak q}_1{\mathfrak q}_2\cdots {\mathfrak q}_{p+1}$ and $A/pA$ is isomorphic to the product of $p+1$ copies of $k=\mathbb{F}_p$. Let $R=\mathbb{Z}_{(p)}+pA$. It is a local subring of $A$ with integral closure $A$ and unique maximal ideal $pA$. Consider the surjective morphism ${\varphi}:A\to A/pA \tilde\to k^{p+1}$. As $R$ contains the kernel of ${\varphi}$ and ${\varphi}(R)\simeq k$ we see that $R={\varphi}^{-1}(\{(r,r,\ldots,r):r\in k\})$. This implies that the preimage under ${\varphi}$ of any additive subgroup of $k^{p+1}$ is a fractional $R$-ideal. Define $J$ as the preimage of the additive subgroup generated by the elements $(1,0,1,1,...,1)$ and $(0,1,1,2,3,...,p-1)$. Observe that every element of $J \mod pA$ has the form $(x,y,x+y,x+2y,x+3y,...,x+(p-1)y)$ for some $0\leq x,y\leq p-1$ and hence it has a coordinate equal to $0$. Moreover, for every index $i=1,...,p+1$ there exists an element with the $i$-th coordinate non-zero. In particular we have $JA=A$ and hence $JA$ is a pricipal $A$-ideal generated by any unit, say $u$. Observe that the coordinates of ${\varphi}(u)$ are all non-zero and so $u$ cannot be in $J$. We conclude that $J$ is a fractional $R$-ideal whose extension to $A$ cannot be generated by an element of $J$. \end{example} \begin{lemma} \label{lemma:4} Let $R$ be a local number ring with maximal ideal $\frm$, residue field $k$ and normalization $\tilde{R}$. Let $l$ be the number of distinct maximal $\tilde R$-ideals. If $l\leq \#k$ then for every $R$-ideal $I$ we have that \[\dim_k\dfrac{I}{\frm I}\leq \dim_k \dfrac{\tilde{R}}{\frm \tilde{R}}.\] \end{lemma} \begin{proof} By Lemma \ref{lemma:2} we know that $\frm\tilde{R}=x\tilde{R}$ for some $x\in \frm$. Since the additive groups of $\tilde R/I$ and $x\tilde R/xI$ are isomorphic, we have \[[\tilde{R}:\frm\tilde{R}]=[\tilde{R}:x\tilde{R}]=[I:xI]=[I:\frm I][\frm I:xI].\] In particular, $[I : \frm I]$ divides $[\tilde{R}:\frm\tilde{R}]$, and as $I/\frm I$ and $\tilde{R}/\frm \tilde{R}$ are both $k$-vector spaces we get our statement on their $k$-dimensions. \end{proof} Now we would like to drop the hypothesis on the size of the residue field. The construction described in the proof of the next theorem allows us to enlarge the residue field without losing information on the number of generators of any ideal. Compare with \cite{delcdvor00}[Section 3.1]. \begin{theorem} \label{thm:local3gp} Let $R$ be a local number ring with maximal ideal $\frm$, residue field $k$ and normalization $\tilde{R}$. Then for every $R$-ideal $I$ we have that \[\dim_k\dfrac{I}{\frm I}\leq \dim_k \dfrac{\tilde{R}}{\frm \tilde{R}}.\] \end{theorem} \begin{proof} We want to apply Lemma \ref{lemma:4}. Let $\frm_1,\cdots,\frm_l$ be the distinct maximal ideals of $\tilde{R}$ which are above $\frm$. Choose $\overline{f}$, a monic irreducible polynomial in $\mathbb{F}_p[X]$ of degree $d$ coprime with $[(\tilde{R}/\frm_i):\mathbb{F}_p]$ for every $i=1,\cdots,l$ and such that $(\#k)^d\geq l$. Observe that such $d$ is also coprime with $[k:\mathbb{F}_p]$ because each $\tilde{R}/\frm_i$ is a field extension of $k$. Let $f$ be a monic lift of $\overline f$ to $\mathbb{Z}[X]$. Note that $f$ is irreducible and of degree $d$. Consider the order $S=\mathbb{Z}[X]/(f)$. We know that as $f$ is irreducible modulo $p$ the prime $p$ is inert in $S$. In particular, \[ \frac{S}{pS}\simeq \frac{\mathbb{F}_p[X]}{(\overline f)}\simeq \mathbb{F}_{p^d}.\] Define $T=R\otimes_{\mathbb{Z}} S$ and observe that $ T\simeq R[X]/(f) $. We have \[ \dfrac{S}{pS} \otimes_{\mathbb{F}_p} k \simeq \dfrac{k[X]}{(\tilde f)} \simeq \dfrac{R[X]}{(\frm,f)}, \] where $\tilde f$ is the image of $f$ in $k[X]$. Since the degrees of $S/pS$ and $k$ over $\mathbb{F}_p$ are coprime, we deduce that $\tilde f$ is irreducible in $k[X]$ and $(\frm,f)$ is a maximal ideal of $R[X]$. Since the maximal ideals of $T$ are in bijection with the maximal ideals of $R[X]$ containing $(\frm,f)$, we deduce that $T$ is a local domain. We will denote its unique maximal ideal by ${\mathfrak M}$. Let $\tilde{T}$ be normalization of $T$. With a similar argument we can show that $\tilde T$ is semilocal with maximal ideals corresponding to the maximal ideals $(\frm_i,f)$ of $\tilde R[X]$, for $i=1,\ldots,l$. Observe that $T/{\mathfrak M}$ has $(\# k)^d$ elements, which is bigger than $l$. Then we can apply Lemma~\ref{lemma:4} and we get \[\dim_{(T/{\mathfrak M})}\dfrac{I\otimes_\mathbb{Z} S}{{\mathfrak M}(I\otimes_\mathbb{Z} S)}\leq \dim_{(T/{\mathfrak M})}\dfrac{\tilde{T}}{{\mathfrak M}\tilde{T}}.\] Now observe that $I\otimes_\mathbb{Z} S= I\otimes_R T$ and using the canonical isomorphisms of tensor products we get \[\dfrac{I\otimes_R T}{{\mathfrak M}(I\otimes_R T)}\simeq (I\otimes_R T)\otimes_T \dfrac{T}{{\mathfrak M}}\simeq I\otimes_R \dfrac{T}{{\mathfrak M}}\simeq I\otimes_R k\otimes_k \dfrac{T}{{\mathfrak M}}\simeq \dfrac{I}{\frm I}\otimes_k \dfrac{T}{{\mathfrak M}},\] so \[\dim_{(T/{\mathfrak M})}\dfrac{I\otimes_\mathbb{Z} S}{{\mathfrak M}(I\otimes_\mathbb{Z} S)}=\dim_{(T/{\mathfrak M})} \left(\dfrac{I}{\frm I}\otimes_k \dfrac{T}{{\mathfrak M}}\right) = \dim_k \dfrac{I}{\frm I}. \] Similarly we have that \[\dfrac{\tilde{T}}{{\mathfrak M}\tilde{T}}\simeq \tilde{T}\otimes_T\dfrac{T}{{\mathfrak M}}\simeq (\tilde{R}\otimes_R T)\otimes_T \dfrac{T}{{\mathfrak M}}\simeq \tilde{R}\otimes_R \dfrac{T}{{\mathfrak M}} \simeq \tilde{R}\otimes_R k\otimes _k \dfrac{T}{{\mathfrak M}} \simeq \dfrac{\tilde{R}}{\frm\tilde{R}}\otimes_k \dfrac{T}{{\mathfrak M}},\] so also \[\dim_{(T/{\mathfrak M})}\dfrac{\tilde{T}}{{\mathfrak M}\tilde{T}} = \dim_k \dfrac{\tilde{R}}{\frm\tilde{R}}. \] Finally, we conclude that \[ \dim_k \dfrac{I}{\frm I} \leq \dim_k \dfrac{\tilde{R}}{\frm\tilde{R}}. \] \end{proof} \begin{corollary} \label{cor:weakmainthm} Let $R$ be a local number ring with maximal ideal $\frm$, residue field $k$ and normalization $\tilde{R}$, then $g(R)=\dim_k (\tilde{R}/\frm \tilde{R})$. \end{corollary} \begin{proof} Let $r=\dim_k (\tilde{R}/\frm \tilde{R})$ and let $I$ be any $R$-ideal. By Theorem \ref{thm:local3gp} we obtain that $\dim_k (I/\frm I)\leq r$. As every number ring is Noetherian, we have that $I$ is finitely generated and hence we can apply Nakayama's Lemma to get that $I$ is generated by at most $r$ elements. Hence $g(R)\leq r$. Moreover observe that $\tilde{R}$ is a fractional $R$-ideal and we know that it is generated by exactly $r$ elements, so $g(R)=r$. \end{proof} The next theorem is due to Hendrik Lenstra. \begin{theorem} \label{thm:lenstra} Let $k$ be a field and $A$ a $k$-algebra with $\dim_k A\geq 4$. Then exactly one of the following holds: \begin{enumerate}[label=\upshape(\roman), leftmargin=, widest=iii] \item \label{item:1} there exist $x,y\in A$ such that $\dim_k(k1 + kx + ky +kxy)\geq 4;$ \item \label{item:2} there exists a $k$-vector space $V$ with $A=k\oplus V$ and $V\cdot V=0;$ \item \label{item:3} there exists a $k$-vector space $V$ with $A\simeq \begin{bmatrix} k & V\\ 0 & k \end{bmatrix}$, that is $A=ke\oplus kf\oplus V$, with $V\cdot V=eV=Vf=0, e^2=e, f^2=f, ef=fe=0, e+f=1$. \end{enumerate} \end{theorem} \begin{proof} Suppose that \ref{item:1} does not hold, which means that for every $x,y\in A$ such that $x\not\in k$ and $y\not\in k+kx$ we have that $xy\in k1 + kx + ky$. First we claim that for every $x\in A$ we have $x^2\in k+kx$. Pick $y\not\in k+kx$. We have $xy\in k1 + kx + ky $ and $x(y+x)\in k1 + kx + k(y+x)=k1 + kx + ky;$ hence $x^2\in k1 + kx + ky$. We can use the same argument for $z\not\in k1 + kx + ky\supset k+kx$ (which exists because the dimension of $A$ over $k$ is $\geq 4$) and we get that $x^2\in k1 + kx + kz$, so $x^2\in (k1 + kx + ky) \cap (k1 + kx + kz)=k + kx$. From these considerations we get that every subspace $W\subset A$ containing $1$ is closed under multiplication, hence it is a ring. Observe that each $x\in A$ acts by multiplication on the left on $A/(k+kx)$ and each vector is an eigenvector. This means that there is one eigenvalue and hence the action of $x$ is just a multiplication by a scalar. This means that there exists a unique $k$-linear morphism $\lambda: A\longrightarrow k$, such that $xy \equiv \lambda(x) y \mod (k+kx)$ for every $y\in A$. We can use the same argument for the action of $y$ on $A/(k+ky)$ and the action of $xy$ on $A/(k+kx+ky)$, which has dimension $>0$, by hypothesis. As all the actions are scalar on $A/(k+kx+ky)$ we get that $\lambda(x)\lambda(y)=\lambda(xy)$. As this works for every $x,y\in A$ then $\lambda:A\rightarrow k$ is a $k$-algebra morphism. We can use the same argument for the multiplication on the right, to get that there is a unique ring homomorphism $\mu:A\rightarrow k$ such that for every $x,z\in A$ we have $zx\equiv \mu(x)z \mod (k+kx)$. Then we get that $A=k +\ker \lambda = k+\ker \mu$, which also implies that the dimension over $k$ of the kernels is $\geq 3$. Now we want to prove that $\ker\lambda \cdot \ker \mu =0$. For $x\in \ker\lambda$ and $y\in \ker \mu$ we have $xA\subset k+kx$ and $Ay\subset k+ky$. Observe that $xy\in xA\cap Ay$. If $k+kx\neq k+ky$ then $xA\cap Ay \subseteq k$ and as both $\lambda$ and $\mu$ are the identity on $k$ then $xy=\lambda(xy)=\lambda(x)\lambda(y)=0$. Otherwise if $k+kx=k+ky$, pick $z\in \ker\mu \setminus (k+kx)$, which is possible because $\dim_k \ker \mu\geq 3$. Then observe that $(k+kx)\cap (k+kz) = k$, so $xz\in xA\cap Az \subseteq k$. As $\mu$ is the identity on $k$, we have $xz=\mu(xz)=\mu(x)\mu(z)=0$. Similarly $x(y+z)\in xA\cap A(y+z)\subset (k+kx)\cap (k+k(y+z)) = (k+kx)\cap (k+kz) = k$, so also $x(y+z)=\mu(x(y+z))=\mu(x)(\mu(y)+\mu(z))=0$. Hence we get that $xy=0$. Now we have to distinguish two cases. If $\ker \mu=\ker \lambda$ then, as $\lambda$ and $\mu$ agree on $k$, they coincide on the whole $A$. So we are in case \ref{item:2} with $V=\ker \mu = \ker \lambda$. If $\ker \mu\neq \ker \lambda$, then call $V=\ker \mu \cap \ker \lambda$ which has exactly codimension $2$: as the kernels are different it must be strictly bigger than $1$ and it is strictly smaller than $3$ because $\ker \mu, \ker \lambda$ have codimension $1$. So the projections of $1,\ker\lambda,\ker\mu $ are 3 distinct lines in $A/V$. Hence: $\ker \lambda =k\cdot e + V$ where we choose $e$ with $\mu(e)=1$ (it can be done as $\mu$ maps surjectively onto $k$), $\ker \mu = k\cdot f + V$ where $f=1-e$. Observe that $ef=e(1-e)=(1-f)f=0$, because $e\in \ker \lambda$ and $f\in \ker \mu$. Then we obtain $e^2=e,f^2=f,fe=0$. Also $eV=Vf=0$. From this conditions we get that $A=ke\oplus kf \oplus V$, because $\ker \lambda= ke\oplus V$ has codimension $1$ and $f\not\in \ker \lambda. $ Then \begin{align} A & \longrightarrow \begin{bmatrix} k & V\\ 0 & k \end{bmatrix}\\ ae+bf+v & \longmapsto \begin{pmatrix} b & v\\ 0 & a \end{pmatrix} \end{align} is a well defined morphism and clearly it is bijective. So we are in case \ref{item:3}.\\ To conclude, observe that if \ref{item:2} holds then $A$ is a commutative algebra and in case \ref{item:3} $A$ is not. If $A$ has \ref{item:2} then it has not \ref{item:1}, because the subspace $k1+kx+ky$ is a ring and so $\dim_k(k1+kx+ky+kxy)\leq 3$. If $A$ has \ref{item:3} then it cannot have \ref{item:1}, because if $x= \begin{pmatrix} a & u\\ 0 & b \end{pmatrix}$ and $y= \begin{pmatrix} c & v\\ 0 & d \end{pmatrix}$ then we have $(x-a)(y-d)=0$ and so $xy\in k+kx+ky$. \end{proof} \begin{proposition} \label{prop:3} Let $R$ be a local number ring, with maximal ideal $\frm$ and residue field $k$. Assume that $R'= \frm\tilde{R}+R$ is super-multiplicative, where $\tilde{R}$ is the normalization of $R$. Then \[\dim _k \dfrac{\tilde{R}}{\frm \tilde{R}}\leq 3.\] \end{proposition} \begin{proof} Put $A=\tilde{R}/\frm\tilde{R}$. Observe that $A$ is an $R$-module annihilated by the maximal ideal $\frm$, so it is a finite dimensional $k$-algebra. Assume by contradiction that $\dim_k A \geq 4$, so we are in one of the three cases of Theorem \ref{thm:lenstra}. As $\tilde{R}$ is commutative, then $A$ is the same, so we cannot be in case \ref{item:3}. Assume that we are in case \ref{item:2}, that is $A=k\oplus V$, with $V$ a $k$-vector space such that $V^2=0$. Consider the projection $\tilde{R}\twoheadrightarrow A$ and let $\tilde{\frm}$ be the pre-image of $V$. Observe that $k=A/V\simeq \tilde{R}/\tilde{\frm}$, hence $\tilde{\frm}$ is a maximal ideal of $\tilde{R}$. The ring $\tilde{R}$ is integrally closed so we have that $\dim_k (\tilde{\frm}/\tilde{\frm}^2)=1$. Therefore also $\dim_k (V/V^2)=\dim_k V =1$ as $V^2=0$. This implies that $\dim_k A=2$. Contradiction. Assume that we are in case \ref{item:1}. Then there exist $\overline{x},\overline{y}\in A$ such that $\dim_k(k1+k\overline{x}+k\overline{y}+k\overline{xy})\geq 4$. Let $x$ and $y$ be the preimages in $\tilde{R}$ of $\overline{x}$ and $\overline{y}$. Now consider the $R'$-fractional ideals $I=(1,x,\frm \tilde{R})$ and $J=(1,y,\frm\tilde{R})$. Observe that $\tilde{R}/R'\simeq A/k$ and inside it we have $I/R'$ and $J/R'$ which are generated by the images of $x$ and $y$, respectively, so they corresponds to subspaces of dimension $1$ over $k$. The image of the product $IJ/R'$ is generated by the projections of $x,y$ and $xy$. Therefore it has dimension $\leq 3$ over $k$. Recalling our convention on the index of fractional ideals, we have \[ (\#k)^{3}\geq [IJ:R']>[I:R'][J:R']=(\#k)^{2}.\] But this contradicts the hypothesis that $R'$ is super-multiplicative. Therefore we must have $\dim_k A \leq 3$. \end{proof} Now to conclude the proof of Theorem \ref{thm:secondimpl} stated in the introduction, we need to return to the non-local case. \begin{proof}[Proof of Theorem \ref{thm:secondimpl}] Observe that by Lemma \ref{lemma:smlocal} we have that the localization of $R+\frm \tilde R$ at every maximal ideal $\frm$ is super-multiplicative. Then by Proposition \ref{prop:3} and Corollary \ref{cor:weakmainthm} we get that every $R_\frm$-ideal is generated by 3 elements, for every $\frm$. Then by Lemma \ref{lemma:d-gen} we have that every $R$-ideal is generated by 3 elements. \end{proof} Let us summarize what we proved: let $R$ be a number ring with normalization $\tilde R$ and consider the ring extensions of $R$ given by $R'(\frm) = R+\frm\tilde R$, where $\frm$ is a maximal ideal of $R$. Then \begin{displaymath} \xymatrix{ g(R)\leq 3 \ar@2{->}[d]\ar@2{->}[r] & g(R'(\frm))\leq 3\ (\forall \frm) \ar@2{->}[d] \\ R \text{ super-mult. } & R'(\frm)\text{ super-mult.} \ (\forall \frm) \ar@2{->}[ul] } \end{displaymath} We cannot say that all the statement are equivalent because if $R$ is super-multiplicative then it is possible that there exists an extension $R'$ (of the required form) which is not, as we show in the next example, which was communicated by Hendrik Lenstra. \begin{example} Let $p$ be a prime number. Let $\alpha$ be a root of a monic polynomial of degree $4$ with coefficients in $\mathbb{Z}_{(p)}$ which is irreducible modulo $p$. Let $A=\mathbb{Z}_{(p)}[\alpha]$. Observe that $A$ is a local domain with maximal ideal $pA$. Moreover $A$ is Noetherian and has Krull dimension $1$. Therefore $A$ is a discrete valuation ring and so it is integrally closed. Put $R'=\mathbb{Z}_{(p)}\oplus pA$ and $R=\mathbb{Z}_{(p)}\oplus p\alpha\mathbb{Z}_{(p)}\oplus p\alpha^2\mathbb{Z}_{(p)}\oplus p^2\alpha^3\mathbb{Z}_{(p)}$. Observe that $R'$ is the ring of Example \ref{ex:degree4} tensored with $\mathbb{Z}_{(p)}$, hence not super-multiplicative. Moreover, $R$ is a local subring of $R'$ with maximal ideal $\frm=p\mathbb{Z}_{(p)}\oplus p\alpha\mathbb{Z}_{(p)}\oplus p\alpha^2\mathbb{Z}_{(p)}\oplus p^2\alpha^3\mathbb{Z}_{(p)}$, normalization $A$ and residue class field $k=\mathbb{F}_p$. Notice that $R'$ can be described also as $R'=R+\frm A$. We will prove now that $R$ is super-multiplicative. First we look at the quotient $R/pR$. Let $x$ and $y$ be the images of $p\alpha$ and $p\alpha^2$ under the quotient map. Then $R/pR$ is a $k$-algebra of dimension $4$ with basis $1,x,y$ and $xy$. Moreover $R/pR$ is a local ring with maximal ideal $(x,y)$ and, from the relations $x^2=xy^2=0$, we see that the annihilator of $x$ in $R/pR$ is $kx+kxy$ and the annihilator of $(x,y)$ is $kxy$. Pulling back this statement to $R$, we obtain that $R\cap ((pR):\frm) = pR+p^2\alpha^3\mathbb{Z}_{(p)}$. But $((pR):\frm)$ is contained in $((pR):(pR))=R$, so $((pR):\frm)=pR+p^2\alpha^3\mathbb{Z}_{(p)}$. Dividing by $p$ we get $(R:\frm)=R+p\alpha^3\mathbb{Z}_{(p)}=R'$. In particular $(R:\frm)$ is a ring and $[(R:\frm):R]=p$. Now take two non-zero fractional $R$-ideals $I$ and $J$. We want to prove that $N(IJ)\geq N(I)N(J)$. Observe that multiplying by non-zero principal ideals of $R$ does not change the problem. By Lemma \ref{lemma:2} there exists $s$ in $I$ such that $IA=sA$. Then $R\subseteq (1/s)I \subseteq (1/s)IA = A$ so we can assume that $I$ contains $R$ and is contained in $A$, and similarly for the ideal $J$. If $I$ or $J$ equals $R$ the inequality holds (with equality). So we assume that both $I$ and $J$ properly contain $R$. In particular $N(I)$ and $N(J)$ are at most $1/p$. Then $I/R$ and $J/R$ are finite non-zero $R$-modules and have therefore a non-trivial piece annihilated by $\frm$. Hence $I\cap (R:\frm)$ contains $R$ properly and using the fact that $[(R:\frm):R]=p$ we obtain that $I\supset (R:\frm)$. The same holds for $J$. Suppose first that $N(I)=1/p$, then $I=(R:\frm)$ and so $IJ=J$. Then the inequality is valid: $N(IJ)=N(J)>N(J)/p=N(I)N(J)$. Likewise if $N(J)=1/p$. It remains to check the case when both $I$ and $J$ have norm at most $1/p^2$. In this case the inclusion $IJ\subset A$ implies $N(IJ)\ge N(A) = 1/[A:R]=1/p^4 \ge N(I)N(J)$, as required. \end{example} \section{Acknowledgement} This article is the result of the work done for my Master Thesis written at the University of Leiden under the supervision of Bart de Smit. I thank him for his guidance and suggestions. I am grateful to Hendrik Lenstra for his help in some key passages and for his comments. I would also like to thank Jonas Bergstr\"om for his help in revising previous versions of the article. The author would also like to express his gratitude to the Max Planck Institute for Mathematics in Bonn for their hospitality.	{ "redpajama_set_name": "RedPajamaArXiv" }	1,078
{"url":"http:\/\/magic10percent.net\/watkins-extract-sevo\/page.php?tag=8ebf58-bond-order-of-n22%2B","text":"How many resonance structures exist for the formate ion, HCO2\u2212? If the distribution of electrons in the molecular orbitals between two atoms is such that the resulting bond would have a bond order of zero, a stable bond does not form. According to molecular orbital theory, what is the bond order of N 2 2-? Use \u2026 Still have questions? For example, sulfur dioxide has three bonds: a single bond for sulfur oxide in one Lewis structure and a double bond for sulfur oxide in a different Lewis structure within the molecule. Top. Just put one more electron into the lowest unoccupied MO (LUMO) of C2 to get C2-. Provide your answer as the numerical value in decimal form. Bond order in Oxygen molecule (O 2-2) =\u00bd [(Number of bonding electrons) \u2013 (number of anti-bonding electrons)] = \u00bd [10 \u2013 8] = 1 With the increase in number of electrons in anti-bonding orbital the stability and bond order of the molecule decreases. Source(s): Class XI. based on molecular orbital theory, the bond order of the n-n bond in n2 is. Using molecular orbital theory, the N-N bond order of the N-N in the N22+ is _____. Solution for Although the peroxide ion, O22-, and the acetylide ion, C22-, have long been known, the diazenide ion N22- has only been prepared much more\u2026 So, the bond order is zero. We\u2019re being asked to determine the bond order of NO 2 + For this, we need to do the following steps: Step 1: Draw the Lewis Structure. Answer: antibonding 56. The order of a covalent bond is a guide to its strength; a bond between two given atoms becomes stronger as the bond order increases (Table 1). Each number of bond is one bond order. Bond order is 3 in N2 since it has a triple bond. The reason the potential energy was not lowered by the full amount is that the presence of two electrons in the same orbital gives rise to a repulsion that acts against the stabilization. Using the VSEPR model, the electron-domain geometry of the central atom in ClO2 - is _____ (a) tetrahedral (b) trigonal bipyramidal (c) octahedral (d) linear (e) trigonal planar. It contains 1 unpaired electron and is paramagnetic. Favorite Answer. Bond order of N22+ N2\u00b2- and Molecular orbital diagram 2 See answers anjanibahuguna245 anjanibahuguna245 Explanation: sorry I don't know . Create\u00a0your\u00a0account. N2- has 5e- in antibonding while N2+ has 4. Bond order, as introduced by Linus Pauling, is defined as the difference between the number of bonds and anti-bonds.. Bond order is a measurement of electrons that participate in bond formation. Bond order for C2- is 2.5. similar elements tend to react similarly, but may do so at different reaction rates. Each hydrogen atom contributes one 1s atomic orbital, and thus, the orbitals overlap according to MO theory to form one sigma_(1s) and one sigma_(1s)^\"\" MO by conservation of orbitals. Question: According To Molecular Orbital Theory, What Is The Bond Order Of N22-? Relevance. HARD aipmt. In a typical multiple bond results from overlap of _____ orbitals and the \u03c0 bond(s) result from overlap of _____ orbitals. Join Yahoo Answers and get 100 points today. We next look at some specific examples of MO diagrams and bond orders. Question: What is the bond order of N 2 +?. 3. N22+ \u03c31s2 \u03c31s2 \u03c32s2 \u03c32s2 \u03c02p2x = \u03c02p2y. You will find an MO diagram of C2 about halfway down the page (scroll down). Thus Decreasing order of stability is: O 2 > O 2-> O 2-2. pranavkumbhar66 pranavkumbhar66 Answer:, Explanation: .. ..... New questions in Science. HARD. Bond Order: In molecular orbital theory, if the bond order is greater than zero, stability is gained between the two atoms forming a bond. Which two species have the same bond order? I'm looking for a piece of glassware from France? Again, Ne2-Here There are more number of Atoms then the Orbitals.. What is the bond order for N 2 2+? View Answer. An antibonding MO_____ the corresponding bonding Mo. 2. Consider the molecule below. Question: What Is The Bond Order For N22+ ? According to MOT bond order is given by the formula: B. O. This will form one sigma bond by head-to-head overlap of one sp orbital from both N atoms, and two pi bonds formed by sideways overlap of two p orbitals from each Ni atom. Presented below is a partial amortization schedule... Molecular Orbital Theory: Tutorial and Diagrams, Effective Nuclear Charge & Periodic Trends, Lattice Energy: Definition, Trends & Equation, Bond Length: Definition, Formula & Calculation, Diamagnetism & Paramagnetism: Definition & Explanation, Coordinate Covalent Bond: Definition & Examples, The Octet Rule and Lewis Structures of Atoms, Bond Enthalpy: Definition, Calculations & Values, Calculating Formal Charge: Definition & Formula, Covalent Bonds: Predicting Bond Polarity and Ionic Character, Valence Bond Theory of Coordination Compounds, What is the Steric Effect in Organic Chemistry? 0. Question: Is H2SO3 an ionic or Molecular bond ? Higher bond order = shorter bond length so, since 2.5 is lesser it has greater bond lenght. Hint: Write the molecular GSEC first. Based on molecular orbital theory, the bond order of the N \u2014 N bond in the N22+ ion is _____ (a) 3 (b) 0 (c) 1\/2 (d) 2 (e) 1. CO+ CATION. Answer: p 55. By MOT, the bond orders of both are 2.5. Here we can take help from bond order Bond Order=1\/2( nb-na )Bond order Of N2:Total number of electrons=14EC:- \u03c31s\u00b2 \u03c31s\u00b2 \u03c32s\u00b2 \u03c32s\u00b2 \u03c02py\u00b2 [\u03c02pz\u00b2 \u03c32px\u00b2 ]T\u2026 deeksha64621 deeksha64621 05.01.2019 Chemistry Secondary School Compare the relative stability of following species N2, N2+, N2- 1 See answer deeksha64621 is waiting for your help. See the answer. 5. See the answer. (a) N2 +(13 e-): \u03c32 1s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c312p N2 2+(12 e-): \u03c32 1s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p N2 (14 e-): \u03c32 1s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c322p N2-(15 e-): \u03c321s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c322p\u03c012p N2 2-(16 e-): \u03c321s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c322p\u03c012p\u03c012p (b) Bond orders are: N2 + = 2.5 ; N 2 2+ = 2.0 ; N 5 while that in N O + is 3. Provide Your Answer As The Numerical Value In Decimal Form. Therefore, Bond order of C2+ = 1\/2 (5 - 2) = 3\/2 = 1.5 Bond order of C2- = 1\/2 (7 - 2) = 5\/2 = 2.5 Bond order of C2 = 1\/2 (6 - 2) = 2 Highest bond order means highest bond energy and shortest bond length. C2 is unstable \"diatomic carbon\" and if it exists, it will have a double bond and a bond order of 2. Bond order for N2+= 9-4\/2 = 2.5. 5. Now acc to MOT only, the compound having more electrons in antibonding orbitals will form a weaker bond and a longer bond., IF their bond orders are equal. Thus, O 2+ is more stable than O 2-33. Electrons in _____ bonds can \u2026 If the value of a bond order is high, the atom contains a strong bond. For example, in diatomic nitrogen N\u2261N the bond number is 3, in ethyne H\u2212C\u2261C\u2212H the bond number between the two carbon atoms is also 3, and the C\u2212H bond order is 1. When N 2 goes to N 2 + , the N \u2212 N bond distance increase, and when O 2 goes to O 2 + , the O \u2212 O bond distance decrease. Cd advphishing. Solved: Given N22-, using molecular orbital and valence bond theory: a) Write the molecular orbital configuration. 0. (4) The electronic configuration of O 2 2 \u2212 ion is K K (\u03c3 2 s) 2 (\u03c3 \u2217 2 s) 2 (\u03c3 2 p z ) 2 (\u03c0 2 p x ) 2 (\u03c0 2 p y ) 2 (\u03c0 \u2217 2 p x ) 2 (\u03c0 \u2217 2 p y ) 2 Its bond order is 2 8 \u2212 6 = 1. Solution: In BrF5 the central atom \u2026 The bond order in N O is 2. Thus Ne2 molecule . It shows a chemical bond is stable. View Answer. Experimentally, one finds that it takes only 452 kJ to break apart a mole of hydrogen molecules. Bond order for N2-= 10-5\/2 = 2.5. Expert Answer . Using the VSEPR model, the electron-domain geometry of the central atom in ClO2 - is __________ (a) tetrahedral (b) trigonal bipyramidal (c) octahedral (d) linear (e) trigonal planar 3. Use Mo diagram with sp mixing: raises energy of \u03c33> \u03c01; s,p labels changed to numerical labels: N2: 10 e\u207b = \u03c31(2e\u207b) \u03c32 (2e\u207b) \u03c01(4e\u207b) \u03c33(2e\u207b) \u03c02(0e\u207b) \u03c34(0 e\u207b), Bond Order = \u00bd[\u03a3 (bonding e-) - \u03a3 (antibonding e-)], bo = \u00bd[\u03a3 \u03c31(2e\u207b)\u03c01(4e\u207b)\u03c33(2e\u207b) - \u03c32 (2e\u207b)] = 3.0, [N2]^2+ : 8e\u207b = \u03c31(2e\u207b) \u03c32 (2e\u207b) \u03c01(4e\u207b) \u03c33(0e\u207b) \u03c02(0e\u207b) \u03c34(0 e\u207b), bo = \u00bd[\u03a3 \u03c31(2e\u207b)\u03c01(4e\u207b) - \u03c32 (2e\u207b)] = 2.0. b) Determine the bond order. In valence bond theory, a \u03c0 bond is described as the sideways overlap of two unhybridized _____ orbitals. To determine the bond order between two covalently bonded atoms, follow these steps: CARBONATE ANION By considering its resonance delocalization, the bond order on this molecular ion can be determined. -1 ...as medium difficulty. 2. New questions in Chemistry. Bond Order: Bond order is a measurement of electrons that participate in bond formation. Go for N2+. We should notice that for the pi bond order, there will be 2 pi electrons delocalized onto three bonds, making each bond on average having 2\/3 of a pi electron, thereby giving a pi bond order of 1\/3. View Answer. all electrons in the MO electron configuration of F2 are paired. Using molecular orbital and valence bond theory, give the following for {eq}N_2^{2-} {\/eq}. All rights reserved. \u2026 Is H2SO3 an ionic or Molecular bond ? According to Molecular Orbital Theory, the greater the bond order greater is the bond energy. N2 has largest bonding strength, but N2+ and N2- has the same number of bonding order. It shows a chemical bond is stable. If you need more Bond Order practice, you can also practice Bond Order practice problems. true or false. For diatomic species are listed below, identify the correct order in which the bond order is increasing in them. The highest bond order you can get is for an sp hybridized N atom in each example. So, bond Order: 10 - 10 \/2 = 0. we cant say anything about it. Electrons in _____ bonds remain localized between two atoms. Is always \u2026 All other trademarks and copyrights are the property of their respective owners. Bond order for O-2 = 10-7\/2 = 1.5. Lewis structures are a bit sketchy for these. 4 minutes ago. Chemmunicator. Not sure that any of these (apart from the peroxide) would actually exist for much more than a microsecond, but assuming the question is theoretical, you just have to count the \u2026 Base on MO theory. 1 Answer. For this, we need to do the following steps: Step 1: Draw the Lewis Structure Step 2: Count the # of single bonds and double bonds in the molecule Step 3: Get the ratio of for every single bond and double bond for N-O; then add together to get the bond \u2026 Each hydrogen atom contributes one electron, and thus, \"H\"_2^(-) has three electrons while \"H\"_2^(+) has one. Answered By . Chem_Mod Posts: 19060 Joined: Thu Aug 04, 2011 8:53 pm Has upvoted: 770 times. Bond order = 1\/2 (2-2) = 0 ---> no bond formation. What is the bond order of {eq}N_2^+{\/eq}, {eq}N_2^-{\/eq}, and {eq}N_2^{2-}{\/eq}? Diagram \|\| Bond Order \|\| Magnetic Property \|\| Stability Order Bond order = \u00bd[electrons in BMO \u2013 electrons in ABMO] For N2 = 10-4\/2 = 3 . (2.00000) BD ( 1) N 1- N 2 ( 50.00%) 0.7071* N 1 s( 0.00%)p 1.00( 99.74%)d 0.00( 0.24%) f 0.00( 0.02%) ( 50.00%) 0.7071* N 2 s( 0.00%)p 1.00( 99.74%)d 0.00( 0.24%) f 0.00( 0.02%) 2. View Answer. Is H2SO3 an ionic or Molecular bond ? Related Questions. Q. If the distribution of electrons in the molecular orbitals between two atoms is such that the resulting bond would have a bond order of zero, a stable bond does not form. - Definition & Examples, Disproportionation: Definition & Examples, Cyclohexane Conformations: Chair, Boat & Twist-Boat, Spectrochemical Series: Definition & Classes of Ligands, Regioselectivity, Stereoselectivity & Chemoselectivity: Definitions & Examples, Electron Affinity: Definition, Trends & Equation, Oxidation Number: Definition, Rules & Examples, NY Regents Exam - Living Environment: Help and Review, High School Chemistry Syllabus Resource & Lesson Plans, GACE Chemistry (528): Practice & Study Guide, Holt Science Spectrum - Physical Science: Online Textbook Help, ScienceFusion The Dynamic Earth: Online Textbook Help, ScienceFusion Earth's Water & Atmosphere: Online Textbook Help, ScienceFusion Space Science: Online Textbook Help, ScienceFusion Ecology and the Environment: Online Textbook Help, Organic & Inorganic Compounds Study Guide, Biological and Biomedical The order of a covalent bond is a guide to its strength; a bond between two given atoms becomes stronger as the bond order increases (Table 1 in Chapter 8.1 Valence Bond Theory). View Answer. \u00a9 copyright 2003-2020 Study.com. a) The bond order of O2 is 2.5 and it is paramagnetic b) The bond order of O2 is 1.5 and it is paramagnetic MEDIUM. Cirlce The Exact Answer. Bond order corresponds usually to the number of bonds. If the value of a bond order is high, the atom contains a strong bond. Get your answers by asking now. Calculate its bond order and predict its magnetic behaviour simran9396 simran9396 28.03.2019 Chemistry Secondary School write the molecular orbital configuration of N2. The hybridization of nitrogen in the H-C\u2261N: molecule is_____. Problem: The bond order of N22+is 2.5 2 1.5 3 FREE Expert Solution Show answer. So, bond order: 10-9 \/2 = 0.5. This can be verified by the usual equation: BO = 1 2(Bonding - Antibonding) = 1 2\u239b \u239c\u239d\u23a1 \u23a2\u23a3 2\u03c3 2 + 1\u03c0 2(2) + 3\u03c3 2 \u23a4 \u23a5\u23a6 \u2212\u23a1 \u23a2\u23a3 2\u03c3* 2 \u23a4 \u23a5\u23a6\u239e \u239f\u23a0 = 3. The hybrid orbitals used for bonding by the sulfur atom in the SF4 molecule are _____ orbitals. Which of the following statements is true for these two species? a) Write the molecular orbital configuration. sp. In the equilibrium+ HO HA+OH(K. - 1.0x10\"). = 2 N B \u2212 N A where N A , N B are the number of electrons in antibonding and bonding molecular orbital respectively, Bond order for O 2 = 2 1 0 \u2212 6 = 2. Each atom has a remaining sp hybrid non-bonding orbital on the N atom, and only \u2026 The correct order of increasing N - N bond stability of N22-, N2, N2\u2295, N2s is (a) N22- > N2 > N2s > N2\u2295 (b) N2 > N2\u2295 = N2s > N22- (c) N22- > N2s = N2\u2295 > N2 Ne2 = S1s(2),S1s(2),S2s(2),S2s(2),S2pz(2),P2px(2)=P2py(2),P2px(2)=P2py(2), S2pz(2). F, Cl, Br, I F, Cl, Br, I asked Sep 28 in Periodic Classification of Elements by Rajan01 ( 46.3k points) Bond order for O2+ = 10-5\/2 = 2.5. toppr. Wiberg bond order in the NAO basis for the $\\ce{N-N}$ bond was found to be 2.2511. Add your answer and earn points. The atomic number of fluorine is 9, so a (neutral) F2 molecule has a total of 18 electron, or 14 valence electrons (excluding the four 1s electrons). Based on molecular orbital theory, the bond order of the N \u2014 N bond in the N22+ ion is _____ (a) 3 (b) 0 (c) 1\/2 (d) 2 (e) 1. Bond order for O 2 \u2212 = 2 1 0 \u2212 7 = 1. Question: What Is The Bond Order For N22+ ? Post by Chem_Mod \u00bb Sun Sep 11, 2011 7:32 am Answer: For the purposes of this class, N2+ and N2- will be considered equal as they both have a bond order of 2.5. In molecular orbital theory, the bond order is defined as 1\/2(the number of electrons in bonding orbitals \u2013 the number of electrons in _____ orbitals). ANSWERS TO MOLECULAR ORBITALS PROBLEM SET 1. ANSWERS TO MOLECULAR ORBITALS PROBLEM SET 1. (a) N2 +(13 e-): \u03c32 1s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c312p N2 2+(12 e-): \u03c32 1s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p N2 (14 e-): \u03c32 1s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c322p N2-(15 e-): \u03c321s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c322p\u03c012p N2 2-(16 e-): \u03c321s\u03c321s\u03c322s\u03c322s\u03c022p\u03c022p\u03c322p\u03c012p\u03c012p (b) Bond orders are: N2 + = 2.5 ; N 2 2+ = 2.0 ; N You can figure this from the formula: bond order = 1\/2 x [# of e- in bonding MO's - # of e- in antibonding MO's] C2- has 5 more electrons in bonding than in antibonding MO's. Q. The order of a covalent bond is a guide to its strength; a bond between two given atoms becomes stronger as the bond order increases (Table 1). Services, Bond Order: Definition, Formula & Examples, Working Scholars\u00ae Bringing Tuition-Free College to the Community. It contains all paired electrons and is diamagnetic. sp3d. What is the difficulty of this problem? If the value of a bond order is high, the atom contains a strong bond. doesn't exist. Provide Your Answer As The Numerical Value In Decimal Form. The bond order of any molecule containing equal numbers of bonding and antibonding electrons is_____. Re: Determining bond strength for N2, N2+, N2-, N2. Bond order = 1\/2 (2-2) = 0 ---> no bondformation. Calculate its bond order and predict its magnetic behaviour 2 See answers merielalex merielalex BO = [Nb-Na] = [10-4] = 3 Since all the electrons in nitrogen are paired, it is diamagnetic molecule. Bond Order: In molecular orbital theory, if the bond order is greater than zero, stability is gained between the two atoms forming a bond. Among the following, metal carbonyls, the C-O bond is strongest in: MEDIUM. b) Determine the bond order. What is the effect of the following processes on the bond order in N2 and O2? N2 O2 , F2 , Cl2 (Increasing order of bond dissociation energy) 2. So you\u2019re just supposed to know that a carbon atom exists every where the lines meet in a line drawing of a compound\u00a0? Bond order is calculated by the following formula: Bond Order = {eq}\\dfrac{1}{2}\\left[ {{n_b} - {n_a}} \\right]{\/eq}......{eq}({\\rm{I}}) {\/eq},... Our experts can answer your tough homework and study questions. In order to test whether there may be $\\ce{s}$-$\\ce{p}$ mixing and also to check whether there may be actually a bound state, I optimised this molecule (at PBE0-D3\/def2-TZVPD) using NWChem 6.6 and analysed it using natural bonding orbitals (NBO 5.9).I found following minimum geometry: Figure 1: Minimum geometry found for $\\ce{N2^{-2}}$ (distance in angstrom) CO+ has one electron removed from a bonding MO, so its bond order decreases by 1 2 to become 2.5. Step 3: Get the ratio of for every single bond and double bond for N-O; then add together to get the bond \u2026 Bond order for N 2 2+= 8-4\/2 = 2. To get a negative charge on O2, you would have to have a single bond, so bond order is 1. Explanation: The order of stability is directly proportional to the bond order. Using a molecular orbital (MO) energy diagram, indicate the bond order of the following: - Liz MO Diagram B22+ MO Diagram MO Diagram Bond Order Bond Order Bond Order 6. 94% (34 ratings) Problem Details. Based on molecular orbital theory, the bond order of the Be-Be bond in the Be2 molecule is_____. we can choose our subjects from 9 class in new education policy like arts and all. 2.5; 2; 1.5; 3; All Chemistry Practice Problems Bond Order Practice Problems. Thus, this molecule doesn't exist. See the reference. So, obiously, it will not be Exist! veeralpunia. Question: Write the following molecules in order of increasing bond strength: N2, N2+, N2-. In a covalent bond between two atoms, a single bond has a bond order of one, a double bond has a bond order of two, a triple bond has a bond order of three, and so on. Hint: Write The Molecular GSEC First. Sciences, Culinary Arts and Personal Two bonding NBOs were found: (Occupancy) Bond orbital\/ Coefficients\/ Hybrids ----- 1. 2. The bond order of a molecule that has multiple Lewis structures is calculated as the average of these Lewis structures. again. 1 decade ago. Bond order of N22+ N2\u00b2- and Molecularorbital diagram - Brainly.in. Thus, this molecule doesn't exist. Therefore, the correct order of stability is N22- < N2\u2013 = N2+ < N2. 1. Step 2: Count the # of single bonds and double bonds in the molecule . The bond order of N 2 2+ is. This is the answer that is wanted but it is not quite correct as \u03c32 is not completely antibonding. This gives you have a triple bond for N2+, N2, and N2 2+. Problem: The bond order of N22+is 2.5 2 1.5 3 Based on our data, we think this question is relevant for Professor Dixon's class at UCF. On the basis of molecular orbital theory, select the most appropriate option. N2 would have a bond order of 3 while N2+ have 2.5. Calculate the bond order of :nitrogen, oxygen, oxide ion and peroxide ion. Bond order is 3, because it has a triple bond. A bond order of unity corresponds to a conventional \"single bond\". Bond Order: Bond order is a measurement of electrons that participate in bond formation. Its bond order is 2 8 \u2212 5 = 1. Become a Study.com member to unlock this It is Paramagnetic. Which of the following statements are correct? The (F2)- ion has one more valence electron, or 15. Hint: Write The Molecular GSEC First. hybrid, atomic. 5. a.N22- and O22-b.F22+ and N22-c.F22+ and O22-d.O22- and B22+ e.N22- and B22+ Answer Save. Thus, the order of stability is: N2> N2- > N2+> N 2 2+ 40. Science. Bond order for O 2 + = 2 1 0 \u2212 5 = 2. I'm assuming you mean \"H\"_2^(-) vs. \"H\"_2^(+). Q. Well, build the molecular orbital (MO) diagram. 0 1. This problem has been solved! We\u2019re being asked to determine the bond order of NO 2 + . Our tutors rated the difficulty ofWhat is the bond order for the NO bond in NO 3 \u2013 ?A. can we study arts subjects with non medical from 9 in nep 2020 if you are given \u2026 answer is 2 HOW W W. Expert Answer 100% (2 ratings) Previous question Next question Get more help from Chegg. Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. 3. molecular orbital theory predicts diamagnetism of f2 bc. Lv 5. veeralpunia. Question: Is H2SO3 an ionic or Molecular bond ? If the distribution of electrons in the molecular orbitals between two atoms is such that the resulting bond would have a bond order of zero, a stable bond \u2026 5. Bond Order. Secondary School. Bond order for O 2 2 \u2212 = 2 1 0 \u2212 8 = 1. 1. Cirlce The Exact Answer. Bond Order = \u00bd[\u03a3 (bonding e-) - \u03a3 (antibonding e-)] bo = \u00bd[\u03a3 \u03c31(2e\u207b)\u03c01(4e\u207b)\u03c33(2e\u207b) - \u03c32 (2e\u207b)] = 3.0 [N2]^2+ : 8e\u207b = \u03c31(2e\u207b) \u03c32 (2e\u207b) \u03c01(4e\u207b) \u03c33(0e\u207b) \u03c02(0e\u207b) \u03c34(0 e\u207b) The bond number itself is the number of electron pairs (bonds) between a pair of atoms. It shows a chemical bond is stable. So, the highest bond order with highest bond energy and the shortest bond length is found in C2-. ... What is the bond order of N2+, N2-, and N22-? Write the above statement is true or false: HARD. Because the antibonding ortibal is filled, it destabilizes the structure, making the \"molecule\" H 22- very non-stable. Bond order is the number of bonding pairs of electrons between two atoms. answer! 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This video and our entire Q & a library the hybrid orbitals for... Has a triple bond ; 2 ; 1.5 ; 3 ; all practice! Kj to break apart a mole of hydrogen molecules problem: the bond order of:,... High, the atom contains a strong bond question: what is the of... No bond formation in them to react similarly, but may do so at different rates! Answers anjanibahuguna245 anjanibahuguna245 Explanation:....... New questions in Science 2 2+= 8-4\/2 = 2 since 2.5 lesser. And O2 found to be 2.2511 in Science MO diagram of C2 to get C2- bond.... Making the molecule '' H 22- very non-stable many resonance structures exist for the formate ion,?. May do so at different reaction rates bonds ) between a pair of.... Mot, the order of the Be-Be bond in NO 3 \u2013? a: a write. Need more bond order greater is the bond order practice, you would have to have a double and... Order with highest bond energy and the shortest bond length so, bond order as. 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Bond is strongest in: MEDIUM peroxide ion & get Your Degree, get access to this and! Ortibal is filled, it will have a triple bond only 452 kJ break! Order of: nitrogen, oxygen, oxide ion and peroxide ion is more stable than O 2-33 statements. = 1\/2 ( 2-2 ) = 0 -- - > NO bondformation sp hybridized N atom each... Other trademarks and copyrights are the property of their respective owners + = 2 be 2.2511 10 10! May do so at bond order of n22+ reaction rates bond ( s ) result from overlap of _____ orbitals has more! 2 is not quite correct as \u03c3 * 2s2 \u03c02p2x = \u03c02p2y all in! \u2212 7 = 1 [ electrons in _____ bonds can \u2026 Explanation: i! Single bond, so its bond order of increasing bond strength for N2 = 10-4\/2 3... \u2026 bond order of increasing bond strength: N2, and N22- for these two species: i... Has one more electron into the bond order of n22+ unoccupied MO ( LUMO ) C2! By MOT, the bond number itself is the bond order: 10 - 10 =. 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N2 since it has greater bond lenght and copyrights are the property of respective! We find out which one has larger bonding strength than the other one: molecule is_____ two atoms =... Determining bond strength: N2, N2+, N2-, and N22- N2 has largest bonding strength but!, using molecular orbital configuration: sorry i do n't know C2 to get.... Are paired configuration of F2 bc tend to react similarly, but N2+ and N2- has 5e- in while! Posts: 19060 Joined: Thu Aug 04, 2011 8:53 pm has upvoted 770! Our subjects from 9 class in New education policy like arts and all 2 1 \u2212! 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extern void __VERIFIER_error() __attribute__ ((__noreturn__)); typedef long unsigned int size_t; typedef int wchar_t; union wait { int w_status; struct { unsigned int __w_termsig:7; unsigned int __w_coredump:1; unsigned int __w_retcode:8; unsigned int:16; } __wait_terminated; struct { unsigned int __w_stopval:8; unsigned int __w_stopsig:8; unsigned int:16; } __wait_stopped; }; typedef union { union wait __uptr; int __iptr; } __WAIT_STATUS __attribute__ ((__transparent_union__)); typedef struct { int quot; int rem; } div_t; typedef struct { long int quot; long int rem; } ldiv_t; __extension__ typedef struct { long long int quot; long long int rem; } lldiv_t; extern size_t __ctype_get_mb_cur_max (void) __attribute__ ((__nothrow__ , __leaf__)) ; extern double atof (__const char __nptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))) ; extern int atoi (__const char __nptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))) ; extern long int atol (__const char __nptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))) ; __extension__ extern long long int atoll (__const char __nptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))) ; extern double strtod (__const char __restrict __nptr, char __restrict __endptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern float strtof (__const char __restrict __nptr, char *__restrict __endptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern long double strtold (__const char __restrict __nptr, char *__restrict __endptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern long int strtol (__const char __restrict __nptr, char *__restrict __endptr, int __base) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern unsigned long int strtoul (__const char __restrict __nptr, char *__restrict __endptr, int __base) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; __extension__ extern long long int strtoq (__const char __restrict __nptr, char *__restrict __endptr, int __base) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; __extension__ extern unsigned long long int strtouq (__const char __restrict __nptr, char *__restrict __endptr, int __base) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; __extension__ extern long long int strtoll (__const char __restrict __nptr, char *__restrict __endptr, int __base) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; __extension__ extern unsigned long long int strtoull (__const char __restrict __nptr, char *__restrict __endptr, int __base) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern char l64a (long int __n) __attribute__ ((__nothrow__ , __leaf__)) ; extern long int a64l (__const char __s) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))) ; typedef unsigned char __u_char; typedef unsigned short int __u_short; typedef unsigned int __u_int; typedef unsigned long int __u_long; typedef signed char __int8_t; typedef unsigned char __uint8_t; typedef signed short int __int16_t; typedef unsigned short int __uint16_t; typedef signed int __int32_t; typedef unsigned int __uint32_t; typedef signed long int __int64_t; typedef unsigned long int __uint64_t; typedef long int __quad_t; typedef unsigned long int __u_quad_t; typedef unsigned long int __dev_t; typedef unsigned int __uid_t; typedef unsigned int __gid_t; typedef unsigned long int __ino_t; typedef unsigned long int __ino64_t; typedef unsigned int __mode_t; typedef unsigned long int __nlink_t; typedef long int __off_t; typedef long int __off64_t; typedef int __pid_t; typedef struct { int __val[2]; } __fsid_t; typedef long int __clock_t; typedef unsigned long int __rlim_t; typedef unsigned long int __rlim64_t; typedef unsigned int __id_t; typedef long int __time_t; typedef unsigned int __useconds_t; typedef long int __suseconds_t; typedef int __daddr_t; typedef long int __swblk_t; typedef int __key_t; typedef int __clockid_t; typedef void __timer_t; typedef long int __blksize_t; typedef long int __blkcnt_t; typedef long int __blkcnt64_t; typedef unsigned long int __fsblkcnt_t; typedef unsigned long int __fsblkcnt64_t; typedef unsigned long int __fsfilcnt_t; typedef unsigned long int __fsfilcnt64_t; typedef long int __ssize_t; typedef __off64_t __loff_t; typedef __quad_t __qaddr_t; typedef char __caddr_t; typedef long int __intptr_t; typedef unsigned int __socklen_t; typedef __u_char u_char; typedef __u_short u_short; typedef __u_int u_int; typedef __u_long u_long; typedef __quad_t quad_t; typedef __u_quad_t u_quad_t; typedef __fsid_t fsid_t; typedef __loff_t loff_t; typedef __ino_t ino_t; typedef __dev_t dev_t; typedef __gid_t gid_t; typedef __mode_t mode_t; typedef __nlink_t nlink_t; typedef __uid_t uid_t; typedef __off_t off_t; typedef __pid_t pid_t; typedef __id_t id_t; typedef __ssize_t ssize_t; typedef __daddr_t daddr_t; typedef __caddr_t caddr_t; typedef __key_t key_t; typedef __clock_t clock_t; typedef __time_t time_t; typedef __clockid_t clockid_t; typedef __timer_t timer_t; typedef unsigned long int ulong; typedef unsigned short int ushort; typedef unsigned int uint; typedef int int8_t __attribute__ ((__mode__ (__QI__))); typedef int int16_t __attribute__ ((__mode__ (__HI__))); typedef int int32_t __attribute__ ((__mode__ (__SI__))); typedef int int64_t __attribute__ ((__mode__ (__DI__))); typedef unsigned int u_int8_t __attribute__ ((__mode__ (__QI__))); typedef unsigned int u_int16_t __attribute__ ((__mode__ (__HI__))); typedef unsigned int u_int32_t __attribute__ ((__mode__ (__SI__))); typedef unsigned int u_int64_t __attribute__ ((__mode__ (__DI__))); typedef int register_t __attribute__ ((__mode__ (__word__))); typedef int __sig_atomic_t; typedef struct { unsigned long int __val[(1024 / (8 * sizeof (unsigned long int)))]; } __sigset_t; typedef __sigset_t sigset_t; struct timespec { __time_t tv_sec; long int tv_nsec; }; struct timeval { __time_t tv_sec; __suseconds_t tv_usec; }; typedef __suseconds_t suseconds_t; typedef long int __fd_mask; typedef struct { __fd_mask __fds_bits[1024 / (8 * (int) sizeof (__fd_mask))]; } fd_set; typedef __fd_mask fd_mask; extern int select (int __nfds, fd_set __restrict __readfds, fd_set __restrict __writefds, fd_set __restrict __exceptfds, struct timeval __restrict __timeout); extern int pselect (int __nfds, fd_set __restrict __readfds, fd_set __restrict __writefds, fd_set __restrict __exceptfds, const struct timespec __restrict __timeout, const __sigset_t __restrict __sigmask); __extension__ extern unsigned int gnu_dev_major (unsigned long long int __dev) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)); __extension__ extern unsigned int gnu_dev_minor (unsigned long long int __dev) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)); __extension__ extern unsigned long long int gnu_dev_makedev (unsigned int __major, unsigned int __minor) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)); typedef __blksize_t blksize_t; typedef __blkcnt_t blkcnt_t; typedef __fsblkcnt_t fsblkcnt_t; typedef __fsfilcnt_t fsfilcnt_t; typedef unsigned long int pthread_t; typedef union { char __size[56]; long int __align; } pthread_attr_t; typedef struct __pthread_internal_list { struct __pthread_internal_list __prev; struct __pthread_internal_list __next; } __pthread_list_t; typedef union { struct __pthread_mutex_s { int __lock; unsigned int __count; int __owner; unsigned int __nusers; int __kind; int __spins; __pthread_list_t __list; } __data; char __size[40]; long int __align; } pthread_mutex_t; typedef union { char __size[4]; int __align; } pthread_mutexattr_t; typedef union { struct { int __lock; unsigned int __futex; __extension__ unsigned long long int __total_seq; __extension__ unsigned long long int __wakeup_seq; __extension__ unsigned long long int __woken_seq; void __mutex; unsigned int __nwaiters; unsigned int __broadcast_seq; } __data; char __size[48]; __extension__ long long int __align; } pthread_cond_t; typedef union { char __size[4]; int __align; } pthread_condattr_t; typedef unsigned int pthread_key_t; typedef int pthread_once_t; typedef union { struct { int __lock; unsigned int __nr_readers; unsigned int __readers_wakeup; unsigned int __writer_wakeup; unsigned int __nr_readers_queued; unsigned int __nr_writers_queued; int __writer; int __shared; unsigned long int __pad1; unsigned long int __pad2; unsigned int __flags; } __data; char __size[56]; long int __align; } pthread_rwlock_t; typedef union { char __size[8]; long int __align; } pthread_rwlockattr_t; typedef volatile int pthread_spinlock_t; typedef union { char __size[32]; long int __align; } pthread_barrier_t; typedef union { char __size[4]; int __align; } pthread_barrierattr_t; extern long int random (void) __attribute__ ((__nothrow__ , __leaf__)); extern void srandom (unsigned int __seed) __attribute__ ((__nothrow__ , __leaf__)); extern char initstate (unsigned int __seed, char __statebuf, size_t __statelen) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2))); extern char setstate (char __statebuf) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); struct random_data { int32_t fptr; int32_t rptr; int32_t state; int rand_type; int rand_deg; int rand_sep; int32_t end_ptr; }; extern int random_r (struct random_data __restrict __buf, int32_t __restrict __result) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int srandom_r (unsigned int __seed, struct random_data __buf) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2))); extern int initstate_r (unsigned int __seed, char __restrict __statebuf, size_t __statelen, struct random_data __restrict __buf) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2, 4))); extern int setstate_r (char __restrict __statebuf, struct random_data __restrict __buf) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int rand (void) __attribute__ ((__nothrow__ , __leaf__)); extern void srand (unsigned int __seed) __attribute__ ((__nothrow__ , __leaf__)); extern int rand_r (unsigned int __seed) __attribute__ ((__nothrow__ , __leaf__)); extern double drand48 (void) __attribute__ ((__nothrow__ , __leaf__)); extern double erand48 (unsigned short int __xsubi[3]) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern long int lrand48 (void) __attribute__ ((__nothrow__ , __leaf__)); extern long int nrand48 (unsigned short int __xsubi[3]) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern long int mrand48 (void) __attribute__ ((__nothrow__ , __leaf__)); extern long int jrand48 (unsigned short int __xsubi[3]) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern void srand48 (long int __seedval) __attribute__ ((__nothrow__ , __leaf__)); extern unsigned short int seed48 (unsigned short int __seed16v[3]) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern void lcong48 (unsigned short int __param[7]) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); struct drand48_data { unsigned short int __x[3]; unsigned short int __old_x[3]; unsigned short int __c; unsigned short int __init; unsigned long long int __a; }; extern int drand48_r (struct drand48_data __restrict __buffer, double __restrict __result) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int erand48_r (unsigned short int __xsubi[3], struct drand48_data __restrict __buffer, double __restrict __result) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int lrand48_r (struct drand48_data __restrict __buffer, long int __restrict __result) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int nrand48_r (unsigned short int __xsubi[3], struct drand48_data __restrict __buffer, long int __restrict __result) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int mrand48_r (struct drand48_data __restrict __buffer, long int __restrict __result) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int jrand48_r (unsigned short int __xsubi[3], struct drand48_data __restrict __buffer, long int __restrict __result) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int srand48_r (long int __seedval, struct drand48_data __buffer) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2))); extern int seed48_r (unsigned short int __seed16v[3], struct drand48_data __buffer) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int lcong48_r (unsigned short int __param[7], struct drand48_data __buffer) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern void malloc (size_t __size) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__malloc__)) ; extern void calloc (size_t __nmemb, size_t __size) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__malloc__)) ; extern void realloc (void __ptr, size_t __size) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__warn_unused_result__)); extern void free (void __ptr) __attribute__ ((__nothrow__ , __leaf__)); extern void cfree (void __ptr) __attribute__ ((__nothrow__ , __leaf__)); extern void alloca (size_t __size) __attribute__ ((__nothrow__ , __leaf__)); extern void valloc (size_t __size) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__malloc__)) ; extern int posix_memalign (void *__memptr, size_t __alignment, size_t __size) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern void abort (void) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__noreturn__)); extern int atexit (void (__func) (void)) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern int on_exit (void (__func) (int __status, void __arg), void __arg) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern void exit (int __status) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__noreturn__)); extern void _Exit (int __status) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__noreturn__)); extern char getenv (__const char __name) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern char __secure_getenv (__const char __name) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern int putenv (char __string) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern int setenv (__const char __name, __const char __value, int __replace) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2))); extern int unsetenv (__const char __name) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern int clearenv (void) __attribute__ ((__nothrow__ , __leaf__)); extern char mktemp (char __template) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern int mkstemp (char __template) __attribute__ ((__nonnull__ (1))) ; extern int mkstemps (char __template, int __suffixlen) __attribute__ ((__nonnull__ (1))) ; extern char mkdtemp (char __template) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern int system (__const char __command) ; extern char realpath (__const char __restrict __name, char __restrict __resolved) __attribute__ ((__nothrow__ , __leaf__)) ; typedef int (__compar_fn_t) (__const void , __const void ); extern void bsearch (__const void __key, __const void __base, size_t __nmemb, size_t __size, __compar_fn_t __compar) __attribute__ ((__nonnull__ (1, 2, 5))) ; extern void qsort (void __base, size_t __nmemb, size_t __size, __compar_fn_t __compar) __attribute__ ((__nonnull__ (1, 4))); extern int abs (int __x) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)) ; extern long int labs (long int __x) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)) ; __extension__ extern long long int llabs (long long int __x) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)) ; extern div_t div (int __numer, int __denom) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)) ; extern ldiv_t ldiv (long int __numer, long int __denom) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)) ; __extension__ extern lldiv_t lldiv (long long int __numer, long long int __denom) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)) ; extern char ecvt (double __value, int __ndigit, int __restrict __decpt, int __restrict __sign) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4))) ; extern char fcvt (double __value, int __ndigit, int __restrict __decpt, int __restrict __sign) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4))) ; extern char gcvt (double __value, int __ndigit, char __buf) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3))) ; extern char qecvt (long double __value, int __ndigit, int __restrict __decpt, int __restrict __sign) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4))) ; extern char qfcvt (long double __value, int __ndigit, int __restrict __decpt, int __restrict __sign) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4))) ; extern char qgcvt (long double __value, int __ndigit, char __buf) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3))) ; extern int ecvt_r (double __value, int __ndigit, int __restrict __decpt, int __restrict __sign, char __restrict __buf, size_t __len) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4, 5))); extern int fcvt_r (double __value, int __ndigit, int __restrict __decpt, int __restrict __sign, char __restrict __buf, size_t __len) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4, 5))); extern int qecvt_r (long double __value, int __ndigit, int __restrict __decpt, int __restrict __sign, char __restrict __buf, size_t __len) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4, 5))); extern int qfcvt_r (long double __value, int __ndigit, int __restrict __decpt, int __restrict __sign, char __restrict __buf, size_t __len) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (3, 4, 5))); extern int mblen (__const char __s, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) ; extern int mbtowc (wchar_t __restrict __pwc, __const char __restrict __s, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) ; extern int wctomb (char __s, wchar_t __wchar) __attribute__ ((__nothrow__ , __leaf__)) ; extern size_t mbstowcs (wchar_t __restrict __pwcs, __const char __restrict __s, size_t __n) __attribute__ ((__nothrow__ , __leaf__)); extern size_t wcstombs (char __restrict __s, __const wchar_t __restrict __pwcs, size_t __n) __attribute__ ((__nothrow__ , __leaf__)); extern int rpmatch (__const char __response) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))) ; extern int getsubopt (char __restrict __optionp, char __const __restrict __tokens, char __restrict __valuep) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2, 3))) ; extern int getloadavg (double __loadavg[], int __nelem) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern void memcpy (void __restrict __dest, __const void __restrict __src, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern void memmove (void __dest, __const void __src, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern void memccpy (void __restrict __dest, __const void __restrict __src, int __c, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern void memset (void __s, int __c, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern int memcmp (__const void __s1, __const void __s2, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern void memchr (__const void __s, int __c, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))); extern char strcpy (char __restrict __dest, __const char __restrict __src) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern char strncpy (char __restrict __dest, __const char __restrict __src, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern char strcat (char __restrict __dest, __const char __restrict __src) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern char strncat (char __restrict __dest, __const char __restrict __src, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern int strcmp (__const char __s1, __const char __s2) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern int strncmp (__const char __s1, __const char __s2, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern int strcoll (__const char __s1, __const char __s2) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern size_t strxfrm (char __restrict __dest, __const char __restrict __src, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2))); typedef struct __locale_struct { struct __locale_data __locales[13]; const unsigned short int __ctype_b; const int __ctype_tolower; const int __ctype_toupper; const char __names[13]; } __locale_t; typedef __locale_t locale_t; extern int strcoll_l (__const char __s1, __const char __s2, __locale_t __l) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2, 3))); extern size_t strxfrm_l (char __dest, __const char __src, size_t __n, __locale_t __l) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2, 4))); extern char strdup (__const char __s) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__malloc__)) __attribute__ ((__nonnull__ (1))); extern char strndup (__const char __string, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__malloc__)) __attribute__ ((__nonnull__ (1))); extern char strchr (__const char __s, int __c) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))); extern char strrchr (__const char __s, int __c) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))); extern size_t strcspn (__const char __s, __const char __reject) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern size_t strspn (__const char __s, __const char __accept) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern char strpbrk (__const char __s, __const char __accept) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern char strstr (__const char __haystack, __const char __needle) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern char strtok (char __restrict __s, __const char __restrict __delim) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2))); extern char __strtok_r (char __restrict __s, __const char __restrict __delim, char *__restrict __save_ptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2, 3))); extern char strtok_r (char __restrict __s, __const char __restrict __delim, char *__restrict __save_ptr) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2, 3))); extern size_t strlen (__const char __s) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))); extern size_t strnlen (__const char __string, size_t __maxlen) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))); extern char strerror (int __errnum) __attribute__ ((__nothrow__ , __leaf__)); extern int strerror_r (int __errnum, char __buf, size_t __buflen) __asm__ ("" "__xpg_strerror_r") __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (2))); extern char strerror_l (int __errnum, __locale_t __l) __attribute__ ((__nothrow__ , __leaf__)); extern void __bzero (void __s, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern void bcopy (__const void __src, void __dest, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern void bzero (void __s, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern int bcmp (__const void __s1, __const void __s2, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern char index (__const char __s, int __c) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))); extern char rindex (__const char __s, int __c) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1))); extern int ffs (int __i) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__const__)); extern int strcasecmp (__const char __s1, __const char __s2) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern int strncasecmp (__const char __s1, __const char __s2, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__pure__)) __attribute__ ((__nonnull__ (1, 2))); extern char strsep (char __restrict __stringp, __const char __restrict __delim) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern char strsignal (int __sig) __attribute__ ((__nothrow__ , __leaf__)); extern char __stpcpy (char __restrict __dest, __const char __restrict __src) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern char stpcpy (char __restrict __dest, __const char __restrict __src) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern char __stpncpy (char __restrict __dest, __const char __restrict __src, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); extern char stpncpy (char __restrict __dest, __const char __restrict __src, size_t __n) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1, 2))); static void calloc_model(size_t nmemb, size_t size) { void ptr = malloc(nmemb size); return memset(ptr, 0, nmemb * size); } extern int __VERIFIER_nondet_int(void); struct L4 { struct L4 next; struct L5 down; }; struct L3 { struct L4 down; struct L3 next; }; struct L2 { struct L2 next; struct L3 down; }; struct L1 { struct L2 down; struct L1 next; }; struct L0 { struct L0 next; struct L1 down; }; static void* zalloc_or_die(unsigned size) { void ptr = calloc_model(1U, size); if (ptr) return ptr; abort(); } static void l4_insert(struct L4 list) { struct L4 item = zalloc_or_die(sizeof item); item->down = zalloc_or_die(119U); item->next = list; list = item; } static void l3_insert(struct L3 list) { struct L3 item = zalloc_or_die(sizeof item); do l4_insert(&item->down); while (__VERIFIER_nondet_int()); item->next = list; list = item; } static void l2_insert(struct L2 list) { struct L2 item = zalloc_or_die(sizeof item); do l3_insert(&item->down); while (__VERIFIER_nondet_int()); item->next = list; list = item; } static void l1_insert(struct L1 list) { struct L1 item = zalloc_or_die(sizeof item); do l2_insert(&item->down); while (__VERIFIER_nondet_int()); item->next = list; list = item; } static void l0_insert(struct L0 list) { struct L0 item = zalloc_or_die(sizeof item); do l1_insert(&item->down); while (__VERIFIER_nondet_int()); item->next = list; list = item; } static void l4_destroy(struct L4 list) { do { free(list->down); struct L4 next = list->next; free(list); list = next; } while (list); } static void l3_destroy(struct L3 list) { do { l4_destroy(list->down); struct L3 next = list->next; free(list); list = next; } while (list); } static void l2_destroy(struct L2 list) { do { l3_destroy(list->down); struct L2 next = list->next; free(list); list = next; } while (list); } static void l1_destroy(struct L1 list) { do { l2_destroy(list->down); struct L1 next = list->next; free(list); list = next; } while (list); } static void l0_destroy(struct L0 list) { do { l1_destroy(list->down); struct L0 next = list->next; free(list); list = next; } while (list); } int main() { static struct L0 list; do l0_insert(&list); while (__VERIFIER_nondet_int()); l0_destroy(list); }	{ "redpajama_set_name": "RedPajamaGithub" }	6,623
{"url":"https:\/\/www.physicsforums.com\/threads\/rank-of-the-jacobian-matrix.889507\/","text":"I Rank of the Jacobian matrix\n\n1. Oct 17, 2016\n\nspaghetti3451\n\nLet the matrix of partial derivatives $\\displaystyle{\\frac{\\partial y^{j}}{\\partial y^{i}}}$ be a $p \\times p$ matrix, but let the rank of this matrix be less than $p$.\n\nDoes this mean that some given element of this matrix, say $\\displaystyle{\\frac{\\partial y^{1}}{\\partial u^{2}}}$, can be written as\n\n$\\displaystyle{\\frac{\\partial y^{1}}{\\partial u^{2}}=A_{1k_{1}}M_{k_{1}2}}$,\n\nwhere $A$ is a $p\\times p$ matrix of rank less than $p$ and $M$ is an arbitrary matrix?\n\n2. Oct 17, 2016\n\nandrewkirk\n\nThe answer to that is trivially yes. Given the element $\\frac{\\partial y^1}{\\partial u^2}$ we define the matrix $A$ to have all zero entries except for\n$A_{1k_1}=\\frac{\\partial y^1}{\\partial u^2}$ and the matrix $M$ is all zeros except that $M_{k_12}=1$.\n\nIt sounds like you were trying to ask something different and less trivial, but it's not clear what that is.\n\n3. Oct 17, 2016\n\nspaghetti3451\n\nIndeed, I am trying to ask something different and less trivial.\n\nConsider the following expression:\n\n$\\displaystyle{\\alpha_{j_{1}\\dots j_{p}}\\frac{\\partial y^{j_{1}}}{\\partial u^{i_{1}}} \\dots \\frac{\\partial y^{j_{p}}}{\\partial u^{i_{p}}}du^{i_{1}}\\wedge \\dots \\wedge du^{i_{p}}}$.\n\nMy goal is to show that this is zero, if the matrix of partial derivatives is of rank less than the dimension $p$ of the matrix. My approach is to try and rewrite every factor of partial derivatives in the form\n\n$\\displaystyle{\\frac{\\partial y^{j_{1}}}{\\partial u^{i_{2}}}=A_{j_{1}k_{1}}M_{k_{1}i_{2}}}$\n\nand play around with indices.","date":"2017-08-21 22:31:30","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\": 0.9364305734634399, \"perplexity\": 128.35884959877083}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886109670.98\/warc\/CC-MAIN-20170821211752-20170821231752-00564.warc.gz\"}"}	null	null
{"url":"https:\/\/ncatlab.org\/nlab\/show\/nGrpd","text":"# nLab nGrpd\n\nDefinition\n\n### Context\n\n#### $(\\infty,1)$-Category theory\n\n(\u221e,1)-category theory\n\nBackground\n\nBasic concepts\n\nUniversal constructions\n\nLocal presentation\n\nTheorems\n\nExtra stuff, structure, properties\n\nModels\n\n#### Higher category theory\n\nhigher category theory\n\n## Definition\n\nThe (n+1,1)-category $n Grpd$ is the collection of all n-groupoids. It is the full sub-(\u221e,1)-category on the $n$-truncated objects in \u221eGrpd.\n\ncategory: category\n\nLast revised on October 26, 2010 at 00:04:02. See the history of this page for a list of all contributions to it.","date":"2023-02-07 21:59:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 4, \"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\": 0.27882739901542664, \"perplexity\": 8737.970483021023}, \"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-2023-06\/segments\/1674764500641.25\/warc\/CC-MAIN-20230207201702-20230207231702-00199.warc.gz\"}"}	null	null
#include "config.h" #include "core/editing/iterators/SimplifiedBackwardsTextIterator.h" #include "core/dom/FirstLetterPseudoElement.h" #include "core/editing/EditingUtilities.h" #include "core/editing/iterators/TextIterator.h" #include "core/html/HTMLElement.h" #include "core/html/HTMLTextFormControlElement.h" #include "core/layout/LayoutTextFragment.h" namespace blink { static int collapsedSpaceLength(LayoutText* layoutText, int textEnd) { const String& text = layoutText->text(); int length = text.length(); for (int i = textEnd; i < length; ++i) { if (!layoutText->style()->isCollapsibleWhiteSpace(text[i])) return i - textEnd; } return length - textEnd; } static int maxOffsetIncludingCollapsedSpaces(Node* node) { int offset = caretMaxOffset(node); if (node->layoutObject() && node->layoutObject()->isText()) offset += collapsedSpaceLength(toLayoutText(node->layoutObject()), offset); return offset; } template <typename Strategy> SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::SimplifiedBackwardsTextIteratorAlgorithm(const PositionAlgorithm<Strategy>& start, const PositionAlgorithm<Strategy>& end, TextIteratorBehaviorFlags behavior) : m_node(nullptr) , m_offset(0) , m_handledNode(false) , m_handledChildren(false) , m_startNode(nullptr) , m_startOffset(0) , m_endNode(nullptr) , m_endOffset(0) , m_positionNode(nullptr) , m_positionStartOffset(0) , m_positionEndOffset(0) , m_textOffset(0) , m_textLength(0) , m_singleCharacterBuffer(0) , m_havePassedStartNode(false) , m_shouldHandleFirstLetter(false) , m_stopsOnFormControls(behavior & TextIteratorStopsOnFormControls) , m_shouldStop(false) , m_emitsOriginalText(false) { ASSERT(behavior == TextIteratorDefaultBehavior \|\| behavior == TextIteratorStopsOnFormControls); Node* startNode = start.anchorNode(); if (!startNode) return; Node* endNode = end.anchorNode(); int startOffset = start.deprecatedEditingOffset(); int endOffset = end.deprecatedEditingOffset(); init(startNode, endNode, startOffset, endOffset); } template <typename Strategy> void SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::init(Node* startNode, Node* endNode, int startOffset, int endOffset) { if (!startNode->offsetInCharacters() && startOffset >= 0) { // NodeTraversal::childAt() will return 0 if the offset is out of range. We rely on this behavior // instead of calling countChildren() to avoid traversing the children twice. if (Node* childAtOffset = NodeTraversal::childAt(startNode, startOffset)) { startNode = childAtOffset; startOffset = 0; } } if (!endNode->offsetInCharacters() && endOffset > 0) { // NodeTraversal::childAt() will return 0 if the offset is out of range. We rely on this behavior // instead of calling countChildren() to avoid traversing the children twice. if (Node childAtOffset = NodeTraversal::childAt(endNode, endOffset - 1)) { endNode = childAtOffset; endOffset = lastOffsetInNode(endNode); } } m_node = endNode; m_fullyClippedStack.setUpFullyClippedStack(m_node); m_offset = endOffset; m_handledNode = false; m_handledChildren = !endOffset; m_startNode = startNode; m_startOffset = startOffset; m_endNode = endNode; m_endOffset = endOffset; #if ENABLE(ASSERT) // Need this just because of the assert. m_positionNode = endNode; #endif m_havePassedStartNode = false; advance(); } template <typename Strategy> void SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::advance() { ASSERT(m_positionNode); if (m_shouldStop) return; if (m_stopsOnFormControls && HTMLFormControlElement::enclosingFormControlElement(m_node)) { m_shouldStop = true; return; } m_positionNode = nullptr; m_textLength = 0; while (m_node && !m_havePassedStartNode) { // Don't handle node if we start iterating at [node, 0]. if (!m_handledNode && !(m_node == m_endNode && !m_endOffset)) { LayoutObject layoutObject = m_node->layoutObject(); if (layoutObject && layoutObject->isText() && m_node->nodeType() == Node::TEXT_NODE) { // FIXME: What about CDATA_SECTION_NODE? if (layoutObject->style()->visibility() == VISIBLE && m_offset > 0) m_handledNode = handleTextNode(); } else if (layoutObject && (layoutObject->isLayoutPart() \|\| TextIterator::supportsAltText(m_node))) { if (layoutObject->style()->visibility() == VISIBLE && m_offset > 0) m_handledNode = handleReplacedElement(); } else { m_handledNode = handleNonTextNode(); } if (m_positionNode) return; } if (!m_handledChildren && m_node->hasChildren()) { m_node = m_node->lastChild(); m_fullyClippedStack.pushFullyClippedState(m_node); } else { // Exit empty containers as we pass over them or containers // where [container, 0] is where we started iterating. if (!m_handledNode && canHaveChildrenForEditing(m_node) && m_node->parentNode() && (!m_node->lastChild() \|\| (m_node == m_endNode && !m_endOffset))) { exitNode(); if (m_positionNode) { m_handledNode = true; m_handledChildren = true; return; } } // Exit all other containers. while (!m_node->previousSibling()) { if (!advanceRespectingRange(m_node->parentOrShadowHostNode())) break; m_fullyClippedStack.pop(); exitNode(); if (m_positionNode) { m_handledNode = true; m_handledChildren = true; return; } } m_fullyClippedStack.pop(); if (advanceRespectingRange(m_node->previousSibling())) m_fullyClippedStack.pushFullyClippedState(m_node); else m_node = nullptr; } // For the purpose of word boundary detection, // we should iterate all visible text and trailing (collapsed) whitespaces. m_offset = m_node ? maxOffsetIncludingCollapsedSpaces(m_node) : 0; m_handledNode = false; m_handledChildren = false; if (m_positionNode) return; } } template <typename Strategy> bool SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::handleTextNode() { int startOffset; int offsetInNode; LayoutText* layoutObject = handleFirstLetter(startOffset, offsetInNode); if (!layoutObject) return true; String text = layoutObject->text(); if (!layoutObject->firstTextBox() && text.length() > 0) return true; m_positionEndOffset = m_offset; m_offset = startOffset + offsetInNode; m_positionNode = m_node; m_positionStartOffset = m_offset; ASSERT(0 <= m_positionStartOffset - offsetInNode && m_positionStartOffset - offsetInNode <= static_cast<int>(text.length())); ASSERT(1 <= m_positionEndOffset - offsetInNode && m_positionEndOffset - offsetInNode <= static_cast<int>(text.length())); ASSERT(m_positionStartOffset <= m_positionEndOffset); m_textLength = m_positionEndOffset - m_positionStartOffset; m_textOffset = m_positionStartOffset - offsetInNode; m_textContainer = text; m_singleCharacterBuffer = 0; RELEASE_ASSERT(static_cast<unsigned>(m_textOffset + m_textLength) <= text.length()); return !m_shouldHandleFirstLetter; } template <typename Strategy> LayoutText* SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::handleFirstLetter(int& startOffset, int& offsetInNode) { LayoutText* layoutObject = toLayoutText(m_node->layoutObject()); startOffset = (m_node == m_startNode) ? m_startOffset : 0; if (!layoutObject->isTextFragment()) { offsetInNode = 0; return layoutObject; } LayoutTextFragment* fragment = toLayoutTextFragment(layoutObject); int offsetAfterFirstLetter = fragment->start(); if (startOffset >= offsetAfterFirstLetter) { ASSERT(!m_shouldHandleFirstLetter); offsetInNode = offsetAfterFirstLetter; return layoutObject; } if (!m_shouldHandleFirstLetter && offsetAfterFirstLetter < m_offset) { m_shouldHandleFirstLetter = true; offsetInNode = offsetAfterFirstLetter; return layoutObject; } m_shouldHandleFirstLetter = false; offsetInNode = 0; ASSERT(fragment->isRemainingTextLayoutObject()); ASSERT(fragment->firstLetterPseudoElement()); LayoutObject* pseudoElementLayoutObject = fragment->firstLetterPseudoElement()->layoutObject(); ASSERT(pseudoElementLayoutObject); ASSERT(pseudoElementLayoutObject->slowFirstChild()); LayoutText* firstLetterLayoutObject = toLayoutText(pseudoElementLayoutObject->slowFirstChild()); m_offset = firstLetterLayoutObject->caretMaxOffset(); m_offset += collapsedSpaceLength(firstLetterLayoutObject, m_offset); return firstLetterLayoutObject; } template <typename Strategy> bool SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::handleReplacedElement() { unsigned index = m_node->nodeIndex(); // We want replaced elements to behave like punctuation for boundary // finding, and to simply take up space for the selection preservation // code in moveParagraphs, so we use a comma. Unconditionally emit // here because this iterator is only used for boundary finding. emitCharacter(',', m_node->parentNode(), index, index + 1); return true; } template <typename Strategy> bool SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::handleNonTextNode() { // We can use a linefeed in place of a tab because this simple iterator is only used to // find boundaries, not actual content. A linefeed breaks words, sentences, and paragraphs. if (TextIterator::shouldEmitNewlineForNode(m_node, m_emitsOriginalText) \|\| TextIterator::shouldEmitNewlineAfterNode(m_node) \|\| TextIterator::shouldEmitTabBeforeNode(m_node)) { unsigned index = m_node->nodeIndex(); // The start of this emitted range is wrong. Ensuring correctness would require // VisiblePositions and so would be slow. previousBoundary expects this. emitCharacter('\n', m_node->parentNode(), index + 1, index + 1); } return true; } template <typename Strategy> void SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::exitNode() { if (TextIterator::shouldEmitNewlineForNode(m_node, m_emitsOriginalText) \|\| TextIterator::shouldEmitNewlineBeforeNode(m_node) \|\| TextIterator::shouldEmitTabBeforeNode(m_node)) { // The start of this emitted range is wrong. Ensuring correctness would require // VisiblePositions and so would be slow. previousBoundary expects this. emitCharacter('\n', m_node, 0, 0); } } template <typename Strategy> void SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::emitCharacter(UChar c, Node* node, int startOffset, int endOffset) { m_singleCharacterBuffer = c; m_positionNode = node; m_positionStartOffset = startOffset; m_positionEndOffset = endOffset; m_textOffset = 0; m_textLength = 1; } template <typename Strategy> bool SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::advanceRespectingRange(Node* next) { if (!next) return false; m_havePassedStartNode \|= m_node == m_startNode; if (m_havePassedStartNode) return false; m_node = next; return true; } template <typename Strategy> Node* SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::startContainer() const { if (m_positionNode) return m_positionNode; return m_startNode; } template <typename Strategy> int SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::endOffset() const { if (m_positionNode) return m_positionEndOffset; return m_startOffset; } template <typename Strategy> PositionAlgorithm<Strategy> SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::startPosition() const { if (m_positionNode) return PositionAlgorithm<Strategy>::createLegacyEditingPosition(m_positionNode, m_positionStartOffset); return PositionAlgorithm<Strategy>::createLegacyEditingPosition(m_startNode, m_startOffset); } template <typename Strategy> PositionAlgorithm<Strategy>SimplifiedBackwardsTextIteratorAlgorithm<Strategy>::endPosition() const { if (m_positionNode) return PositionAlgorithm<Strategy>::createLegacyEditingPosition(m_positionNode, m_positionEndOffset); return PositionAlgorithm<Strategy>::createLegacyEditingPosition(m_startNode, m_startOffset); } template class CORE_TEMPLATE_EXPORT SimplifiedBackwardsTextIteratorAlgorithm<EditingStrategy>; template class CORE_TEMPLATE_EXPORT SimplifiedBackwardsTextIteratorAlgorithm<EditingInComposedTreeStrategy>; } // namespace blink	{ "redpajama_set_name": "RedPajamaGithub" }	3,094
{"url":"https:\/\/physics.stackexchange.com\/questions\/46318\/is-there-a-small-enough-planet-or-asteroid-you-can-orbit-by-jumping\/46325","text":"# Is there a small enough planet or asteroid you can orbit by jumping?\n\nI just had this idea of orbiting a planet just by jumping and then flying upon it on its orbit kind of like superman. So,\n\nWould it be theoretically possible or is there a chance of that small body to be & remain its unity?\n\n\u2022 I want to see an answer! \u2013\u00a0m0nhawk Dec 8 '12 at 23:00\n\u2022 Since asteroid comes in all sizes right down to grains of dust the answer is trivially \"yes\". Now, is there a particular one that has been identified will take a little calculation and going through catalogs. \u2013\u00a0dmckee Dec 8 '12 at 23:04\n\u2022 THIS PICTURE. +1 \u2013\u00a0Dylan O. Sabulsky Dec 8 '12 at 23:13\n\u2022 Of course running on surface of that thing is going to be difficult, especially as you approach the orbital speed getting any friction against the ground is going to be hard. \u2013\u00a0SF. Dec 8 '12 at 23:22\n\u2022 Other problem: finding a perfectly spherical asteroid. Irregular shapes will make this a lot more difficult ! \u2013\u00a0FrenchKheldar Dec 9 '12 at 2:40\n\nLet's assume mass of the person plus spacesuit to be $m_1$=100kg\n\nAsteroid density: $\\rho=$2g\/cm$^3$ (source) that is 2 000kg\/m$^3$\n\n15km\/hour is a good common run. That's roughly v=4m\/s\n\nThe orbital height is negligible comparing to the radius, assume 0 over surface.\n\nLinear to angular velocity (1): $$\\omega = {v \\over r }$$ Centripetal force (2): $$F = m r \\omega ^2$$ Gravity force (3): $$F= G \\frac{m_1 m_2}{r^2}$$ Volume of a sphere (4): $$V = \\frac{4}{3}\\pi r^3$$ Mass of a sphere (5): $$m_2 = V \\rho = \\frac{4}{3}\\pi r^3 \\rho$$ Combining (1),(2),(3), reducing: $${ m_1 r v^2 \\over r^2 } = G { m_1 * m_2 \\over r^2 }$$ $$r v^2 = G m_2$$ Combining with (5) $$r v^2 = G \\frac{4}{3}\\pi r^3 \\rho$$\n\n$$r^2 = \\frac{v^2}{\\rho G \\frac{4}{3}\\pi}$$\n\n$$r = v ({\\frac{4}{3}\\pi G \\rho})^{-{1 \\over 2}}$$ Substituting values: $$r = 4 ({1.333333.14159 6.6730010^{-11} 2000})^{-{1 \\over 2}}$$\n\nThat computes to roughly 5.3 kilometers\n\nMore interestingly, the radius is directly proportional to the velocity,\n\n$$r[m] = 1337[s] * v [m\/s] = 371.51[h\/1000] * v[km\/h] = 597[mh\/mile] v[mph]$$\n\nSo, a good walk on a 2km radius asteroid will get you orbiting.\n\nSomething to fit your bill would be Cruithne, a viable target for a space mission thanks to a very friendly orbit.\n\nNote, while in rest on Cruithne, the astronaut matching the m_1=100kg would be pulled down with force of 4.5N while not in motion. That is like weighing about 450g or 1lbs on Earth.\n\n\u2022 While this is a great answer (+1), it should be pointed out that walking would be very difficult (perhaps impossible?) on a body with such low gravity, and that makes it hard to imagine how you could get up to 15km\/hour in order to make the jump. \u2013\u00a0Nathaniel Dec 9 '12 at 0:52\n\u2022 @Nathaniel: Probably with help of a small (handheld?) jet propulsion engine. You'd want some means of exiting the orbit anyway. \u2013\u00a0SF. Dec 9 '12 at 2:17\n\u2022 Shouldn't it be 371.51 mph and 597 km\/h? \u2013\u00a0Eugene Seidel Dec 9 '12 at 10:50\n\u2022 @SF.: According to the \"Shell Theorem\", any spherically symmetrical body is equivalent to a point source of the same mass, if you are outside of it. (Inside, the apparent mass of the point source changes.) \u2013\u00a0Dietrich Epp Dec 9 '12 at 14:09\n\u2022 @Beska: You can always use it to keep yourself pressed to the surface instead, and just cut it off when you're running fast enough. \u2013\u00a0SF. Dec 9 '12 at 16:01\n\nNo, not by jumping. Jumping gives you an acceleration only from the location on the surface. As soon as you leave the surface, you have no way of adjusting your orbit. Either you reach escape velocity, or you will return to your initial location after exactly one orbit.\n\nThe only way to prevent this would be to have an additional acceleration once you have departed from the surface. Spacecraft use rockets to do this. A tiny acceleration may be enough \u2014 though I wouldn't like approaching a planet with high speed only to move 5 cm over its surface with high speed!\n\nEdit: A different way would be jump from a ladder, as Claudius pointed out in the other answer.\n\n\u2022 Another approach: Take a rock with you, and throw it directly backwards when you're half way around. This should give you enough extra velocity so your orbit doesn't intersect the surface at your starting point. \u2013\u00a0Keith Thompson Dec 8 '12 at 23:57\n\u2022 You don't necessarily come back to your original location after your jump. Any other ballistic trajectory is possible if your speed is below the escape velocity. \u2013\u00a0FrenchKheldar Dec 9 '12 at 2:39\n\u2022 @FrenchKheldar - true, but I think that's a quibbling objection. Assuming it's a central force you would return to your starting point if the planet didn't get in the way. \u2013\u00a0John Rennie Dec 9 '12 at 9:47\n\u2022 \"though I wouldn't like approaching a planet with high speed\" \u2014 but if you ran\/jumped to start on your orbit, then that's how fast you'll touch back down, so it's not like you'll actually be moving at a very high speed upon touchdown. It's physically impossible to end up moving faster than that if the surface is a perfect sphere. \u2013\u00a0Roman Starkov Feb 2 '15 at 22:48\n\u2022 @romkyns Right, by space standards that's not very fast, but suppose I can sprint at 20 km\/h \u2014 it wouldn't kill me but I'd want to make sure I touchdown feet first and\/or wear a helmet. \u2013\u00a0gerrit Feb 2 '15 at 23:30\n\nOK, I tried to do the math here. Something remotely resembling maths, at least.\n\n## Assumptions:\n\n\u2022 It is possible to reach an orbital\/horizontal speed of $v_O = 5\\textrm{ ms}^{-1}$, for example by running.\n\u2022 The density of the object to orbit is similar to Earth's density, i.e. $\\rho = 5500\\textrm{ kgm}^{-3}$.\n\u2022 We want to orbit at a height of $2\\textrm{ m}$ above the ground. You can get there with a ladder (Yes, you will have to start running on that ladder or something like that....how about stilts?).\n\u2022 No atmosphere or other source of friction.\n\n## Layout:\n\nThe basic idea is to link the orbital velocity $v_O$ to the radius $r$ of the object. The mass is given by $M = \\frac{4}{3} \\pi r^3 \\rho$ (God I hope I remembered this formula correctly).\n\n## Calculation:\n\nWe have\n\n\\begin{eqnarray} & v_O & = \\sqrt{\\frac{G M}{r+2\\textrm{ m}}} = 5\\textrm{ ms}^{-1} \\\\ \\Rightarrow & M & = \\frac{25\\frac{\\textrm{m}^2}{\\textrm{s}^2} \\left( r + 2\\textrm{ m} \\right)}{G} \\\\ \\Rightarrow & 25 \\frac{\\textrm{m}^2}{\\textrm{s}^2} r + 50 \\frac{\\textrm{m}^3}{\\textrm{s}^2} & = \\frac{4}{3} \\pi G r^3 5500 \\frac{\\textrm{kg}}{\\textrm{m}^3} \\end{eqnarray}\n\nwhich then should give us $r$. I used Mathematica for this because it is half past eleven in the evening and I don\u2019t want to guess solutions to get a starting point for polynomial division, getting:\n\nIn: Solve[-4\/3 * Pi * 6.6738410^(-11) x^3 * 5500 + 25 x + 50 == 0, x]\nOut: {{x -> -4031.33327417391}, {x -> -2.00000049201392}, {x -> 4033.33327466592}}\n\n\nThat is, if you found an asteroid of $r \\approx 4\\textrm{ km}$, your dream might come true. However, if it is mostly ice (rather than molten iron, which I imagine would be a pretty good reason to stay in orbit), you will have to correct the 5500 up there to the density of ice, say, 930, and would then need an asteroid of $r \\approx 9.8\\textrm{ km}$.\n\nNote that the assumption that $m_{\\textrm{Human}} \\ll m_{\\textrm{Object}}$, encoded in the expression for orbital velocity, is fulfilled relatively well in these cases (five orders of magnitude).\n\nNevertheless, feel free to point out mistakes :)\n\n\u2022 Are any asteroids of this size remotely spherical? \u2013\u00a0gerrit Dec 8 '12 at 23:37\n\u2022 Beat me to the answer. I get a similar result (about 3.5 km) phrasing the question as \"can a single jump from a typical person impart enough energy to reach escape velocity). \u2013\u00a0user10851 Dec 8 '12 at 23:37\n\u2022 @gerrit No, but asphericity helps if you start on the bulge. \u2013\u00a0user10851 Dec 8 '12 at 23:38\n\u2022 @gerrit Just build a spherical one. 4 km is not that much :-) \u2013\u00a0Claudius Dec 8 '12 at 23:38\n\u2022 @Claudius well, that would be awesome! :D if they could place it on of the lagrangian points between moon and earth, it would also be easy to get there for a superman-like vacation :D \u2013\u00a0Ahmet Yildirim Dec 8 '12 at 23:56\n\nSince the calculations are already in others' answers, I'll just refer to this great, classic xkcd. Deimos and Phobos, the two small moons of Mars, match (or almost match) the criteria SF and Claudius derive.\n\nAs Munroe points out,\n\n(The diagram is a representation of the gravity wells of both moons, represented by their height at constant Earth surface gravity.)\n\nBased on that I think you really should be able to run yourself into orbit using a smallish ramp and a fire extinguisher to stabilize your orbit on the other side (to avoid the pitfall gerrit mentions).\n\nDeimos is between 10 and 15 km across and its escape velocity is about 20 km\/h. At low altitudes, and since circular-orbit velocities are lower by $\\sqrt{2}$ than escape velocities, you'd need to run up to some 15 km\/h to orbit. Thus you'd do about one lap every three hours, whizzing along this ~city sized object at about bicycle speeds.\n\nOn the other hand, it's unlikely that you will last very long in that orbit. The reason for this is that orbits are elliptical only around perfectly spherical planets, and any irregularities in the body you're orbiting will tend to perturb and even destabilize your orbit. Even on the Moon, low orbits are unstable and end up crashing into the surface, as was the fate of a subsatellite deployed during Apollo 16, which lasted only a month in orbit. With something as lumpy as the Martian moons, you would probably want to stay well away!\n\n\u2022 Thanks for the answer and graphics :) I'm also a game developer,After I see that this is physically quite possible,I'm quite impressed. Do you think a game based on travelling on asteroids by jumping & collecting stuff etc. would be fun? \u2013\u00a0Ahmet Yildirim Dec 10 '12 at 18:53\n\u2022 Of course that would depend on the game and how smooth the handling was, but I'd definitely be quite willing to try it. It reminds me of A Slower Speed of Light somewhat. \u2013\u00a0Emilio Pisanty Dec 10 '12 at 18:57\n\u2022 I'm thinking, the part you collect bonuses etc. would be in 3d TPS-like view on asteroid. But jumping to another asteroid would need to switch to 2d to make it easier to interact for player. I would ask user to make the jump on a specific position for maximum jump. And after jump it would switch back to 3d and user would control yaw\/pan\/tilt to land. \u2013\u00a0Ahmet Yildirim Dec 10 '12 at 19:05\n\u2022 Unless you have some sort of rocket (cf. the fire extinguisher), there isn't anything you can do to alter your orbital motion. (Flailing around won't help!) \u2013\u00a0Emilio Pisanty Dec 10 '12 at 19:07\n\u2022 Also, if you're thinking of jumping from one asteroid to another, then the details of the initial condition are crucially important and if you choose your problem wrong you'll have exponential dependence on them (i.e. meaning chaos, and a spectacularly hard gameplay where a hair's breadth in jump-off point lands you on the other side of the target asteroid). That is one reason the Apollo programme was as much about developing the onboard computer as it was about the rockets. \u2013\u00a0Emilio Pisanty Dec 10 '12 at 19:10\n\nIf you want an idea what this might actually be like have a look at Kerbal Space Program. This is a game currently in development by Squad. So not real life, but the orbital physics is accurately modelled (atmospheric flight not so much, yet). There are several small moons and asteroids in the Kerbin system where you can do essentially this jump to orbit maneuvre using nothing but EVA suit thrusters. You can see examples in some of Scott Manley's videos. Here is a video featuring an interplanetary trip with an EVA suit - a 49 day space walk!\n\n(I'm not affiliated with KSP, Squad or Scott Manley in any way, and since the question has been properly answered already I thought this might just be a fun thing to share. Also, KSP and the similar game Orbiter are good ways of building intuition for orbital mechanics. :) Hope this doesn't break the rules.)\n\n## protected by Qmechanic\u2666Apr 22 '14 at 19:09\n\nThank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).","date":"2019-02-19 13:26:28","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\": 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\": 0.6610516905784607, \"perplexity\": 1047.6706870953933}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247490107.12\/warc\/CC-MAIN-20190219122312-20190219144312-00366.warc.gz\"}"}	null	null
Q: Jquery changes position of div/img I have found a workaround to center a large image (no background) in a smaller div. #pic { width: 60%; height: 349px; float:right; overflow:hidden; position:relative; } #pic img { left:50%; margin-left: -100%; position:relative; } I am happy with the positioning. However, once I use the Jquery cycle plugin, it completely messes up the positioning and size of the images. As when I inspect the element (With DOM inspector), it the following values are striked through: left:50%; margin-left: -100%; This is causing the repositioning. Why are they all in the sudden not valid anymore after using Jquery and how can I fix this? Thank you for any help! Scott A: You could try to add !important in the css #pic { width: 60% !important; height: 349px; float:right; overflow:hidden; position:relative; } #pic img { left:50%; margin-left: -100% !important; position:relative; }	{ "redpajama_set_name": "RedPajamaStackExchange" }	5,406
\section{Introduction} \label{s:intro} Spectroscopy of Earth-sized planets around the so-called habitable zones \citep[HZs; e.g.,][]{Kasting+1993} is one of the key near-future targets in astronomical observations. Several approaches have been put forward so far. The first major approach is direct (high-contrast) imaging. Since the late 20th century, space-based direct imaging possibilities---with coronagraphs in the visible and near-infrared domains, or using nulling interferometers in the mid-infrared (MIR) domain---have been examined and were encapsulated in the proposed missions like TPF-C, TPF-I, and Darwin \citep[e.g.,][]{Levine+2009,Beichman+1999,Leger+1996}. These studies are inherited by the ongoing discussions for the future space missions for direct imaging of scattered light of Earth-sized planets (e.g., Starshade with Roman Space Telescope, LUVOIR, HabEx, LIFE). In the meanwhile, the successful atmospheric characterization of Hot Jupiters through transmission spectroscopy, eclipse spectroscopy, and detection of phase variations has encouraged its application to smaller Earth-sized planets around late-type stars---this is the second appraoch \citep[see a review by][and references therein]{DemingSeager2017}. It has actually been applied to the Hubble Space Telescope (HST) observations of TRAPPIST-1 planets \citep{Gillon+2016,Gillon+2017}, the Earth-sized transiting planets around the habitable zone of an M8-type star, while only the upper limit of atmospheric features have been obtained \citep{deWit+2018,Zhang+2018}. The upcoming James Webb Space Telescope (JWST) will be used for more sensitive transit spectroscopy of smaller planets in a wider wavelength range \citep{Beichman+2014,Greene+2016}. The high-dispersion spectrographs on the next-generation ground-based telescopes (i.e., extremely large telescopes) may also be used for transit transmission spectroscopy to identify the atmospheric features of Earth-sized exoplanets buried in a number of telluric absorption lines \citep[e.g.,][]{Snellen+2013}. The third approach to study potentially habitable exoplanets is to combine high-contrast imaging and high-resolution spectroscopy with these next-generation ground-based telescopes \citep[e.g.,][]{Snellen+2015}. Such observations using ground-based telescopes are planned mainly in the near-infrared wavelengths, so as to avoid thermal background that is inevitable for ground-based observations at longer wavelengths, while achieving the spatial resolution necessary for high-contrast imaging. Lastly, the atmospheric features of potentially habitable planets may be identified in the combined spectrum of the star and the planet in the MIR range, without a planetary transit or a starlight-suppressing instrument \citep[e.g.,][]{KreidbergLoeb2016, Snellen+2017}. Indeed, the planet-to-star flux ratio of potentially habitable planets is significantly improved in the MIR domain ($\sim $a few tens of parts-per-million or larger) compared with that in the shorter visible domain ($\sim 10^{-7}$ or less) as shown in the upper panel of Figure \ref{fig:contrast-photoncount}, and it is not unrealistic to detect planetary signal in the combined spectra. Approximately estimating the wavelength-dependence of the signal-to-noise ratio for the planet signal per wavelength resolution element by $\mathcal{N}_p \Delta \lambda /\sqrt{\mathcal{N_{\star }}\Delta \lambda } = \sqrt{\mathcal{N}_p C \Delta \lambda } = \sqrt{\mathcal{N}_p C \lambda / \mathcal{R}}$ ($\mathcal{N}_p$ and $\mathcal{N}_{\star }$ are the photon spectrum of the planet and the star, respectively, $\Delta \lambda$ is the wavelength width of the resolution element, $C$ is the planet-to-star flux ratio, i.e., $C\equiv \mathcal{N}_p / \mathcal{N}_{\star }$, and $\mathcal{R}$ is the fixed spectral resolution), the MIR domain, specifically around 13-100~$\mu $m, is most useful (the lower panel of Figure \ref{fig:contrast-photoncount}). The unique advantages of this MIR approach are that (1) unlike high-contrast technique it does not require specialized instruments to occult stellar flux, and that (2) unlike transmission or eclipse spectroscopy it can be applied to both transiting and non-transiting planets. The latter is critical to increase the number of targets, as the transit probability of habitable-zone planets is up to about 5\%. \begin{figure}[tb!] \includegraphics[width=1.0\hsize]{contrast_photon_midMstar_planets_v2.pdf} \caption{Upper panel: A schematic figure of the photon count per wavelength resolution element assuming a black body planet spectrum with 288~K (i.e. $\mathcal{N}_{\rm p}\Delta \lambda$; red dashed) and the planet-to-star flux ratio assuming a black body spectrum mimicking a mid-M host star (i.e. $C$; blue solid). The wavelength dependence of the planet-to-star flux ratio does not depend on the spectral types of the star as long as the stellar spectrum can be approximated by Rayleigh-Jeans law. Lower panel: Signal-to-noise ratio per wavelength resolution element as a function of wavelength, estimated by the product of the planet photon count and the planet-to-star flux ratio. } \label{fig:contrast-photoncount} \end{figure} Along this line, \citet{KreidbergLoeb2016} proposed low-resolution spectroscopy of Proxima Centauri systems to try to detect 9.6 $\mu $m O$_3$ features originated from the atmosphere of Proxima Centauri b, the nearest non-transiting potentially habitable planets \citep{Anglada-Escude+2016}. \citet{Snellen+2017} proposed that medium-resolution spectroscopy (MRS) in the MIR range can be used to identify high-frequency features due to planetary atmospheric molecules in the combined spectra, and they estimated that CO$_2$ features of Proxima Centauri b would be detected after 5 days of observations with MRS mode of JWST/MIRI. Although these are encouraging possibilities, such observations rely on the precise knowledge of the stellar spectrum as well as the sensitivity of the detector elements. While the above studies have in mind the upcoming JWST that have only low-resolution and medium-resolution capabilities in the MIR, technologies for high-dispersion spectrograph in the MIR have been recently developed for future cryogenic space telescopes, including SPICA \citep[e.g.,][]{Sarugaku+2012}. The technologies are expected to largely reduce the size of the MIR spectrograph and enhance its throughput. In the further future, Origins Space Telescope (OST) is also contemplating high-resolution spectrograph with a larger aperture \citep{Sakon+2018}. With high-resolution spectroscopy ($\mathcal{R} \sim 30,000$), not only the fine structures of absorption bands of the planetary spectrum can be identified, but also the Doppler shift of a HZ planet can be resolved (unless its orbit is close to face-on). Thanks to the Doppler-shift, it is possible to identify planetary features even without the assumption for the stellar spectrum, and in theory the orbital inclination can also be constrained. Although these new opportunities will be useful for characterizing atmospheres of potentially habitable planets, its potential remains largely unexplored yet. In this paper, we examine the utility of High-Resolution Spectroscopy (HRS) in the MIR for characterizing atmospheres of nearby potentially habitable planets. In Section \ref{s:theoretical_spectra}, we begin by demonstrating the MIR spectral features of potentially habitable planets using a simplified 1-dimensional atmospheric model. We then move to detectability assessment in Section \ref{s:detectability}. We propose to observe the target system at two quadrature phases, and discuss how this method could constrain the presence of molecular features and the orbital inclination under the realistic assumptions on the observational noise, highlighting the difference between the two analysis strategies that reflect the possible uncertainty in the prior knowledge of the stellar spectrum. The dependence on the spectral resolution, bandpass, and the abundance of molecules are examined in Section \ref{s:Detecability_Resolution_Abundance}. Section \ref{s:discussion} discusses the range of the targets and the synergies with other techniques to characterize potentially habitable planets (Section \ref{ss:targets}), the effects of other noise (Section \ref{ss:other_noise}), the effect of different thermal profiles of the planet (\S\ref{ss:dependence_on_atmosphere}), and the comparison with the previously studied medium-resolution spectroscopy (\S\ref{ss:comp_MRS}). Finally, Section \ref{s:summary} summarizes our results. \section{Atmospheric features in MIR high-resolution spectra} \label{s:theoretical_spectra} In this section, we model high-resolution planetary spectra of an Earth-like planet and discuss the characteristics of the spectral features of molecules of interest. These modeled spectra will be used as input for the detectability analyses in Section \ref{s:detectability}. \subsection{Model Atmosphere} \label{ss:method_spectra} \begin{figure}[tb!] \includegraphics[width=1.0\hsize]{simple_profiles_dryadiabat_v2.pdf} \caption{The assumed vertical profiles of the mixing ratios of the molecules considered (left) and temperature (right). The mixing ratio of molecules are based on ``US standard'' model. The temperature profile is determined by the dry adiabatic lapse rate (9.8~K/km) in the lower atmospheres below 0.1~bar, above which an isothermal profile is assumed. } \label{fig:simple_TP_profile} \end{figure} Our model atmosphere is based on Earth's atmosphere except for the temperature profile. We consider four molecules that are present in Earth's atmosphere and are radiatively active in the mid infrared range: CO$_2$, H$_2$O, N$_2$O, and O$_3$. The vertical profiles of the mixing ratios of these molecules are taken from the ``US standard'' model and are shown in the left panel of Figure \ref{fig:simple_TP_profile}. The surface temperature is set at 288~K, again referring to the ``US standard'' model. The vertical temperature profile is, however, replaced by a simplified one comprised of a troposphere with a constant lapse rate ($\Gamma =g/C_p$ where $g$ is the gravity and $C_p$ is the specific heat capacity) and an isothermal stratosphere. The removal of stratospheric thermal inversion is motivated by the fact that O$_3$, even if it exists, does not lead to a strong thermal inversion under the irradiation of M-type stars due to the reduced near-UV flux. The effects of the atmospheric profile on the detectability of molecules are discussed in Section \ref{sss:thermal_inversion}. For simplicity, the effects of clouds are ignored. The surface pressure is fixed at 1~bar, and the tropopause is set at 0.1~bar. This tropopause pressure is consistent with the observations of Solar system planets \citep[e.g.,][]{RobinsonCatling2014} and 3D climate simulations for habitable planets around M-type stars \citep[e.g.,][]{Fujii+2017}. \begin{table}[] \centering \begin{tabular}{lcc}\hline \hline description & symbol & value\\ \hline surface pressure & $P_{\rm surf} $ & 1~bar \\ surface temperature & $T_{\rm surf} $ & 288~K \\ tropopause & $P_{\rm tp} $ & 0.1~bar \\ surface gravity & $g $ & 9.8~m/s$^2$ \\ tropospheric lapse rate & $\Gamma (=g/C_p)$ & 9.8~K/km \\ \hline \end{tabular} \caption{Assumptions for atmospheric profiles. } \label{tbl:parameters} \end{table} Given a vertical profile, the top-of-atmosphere outgoing radiance at wavelength $\lambda $ and the cosine of the zenith angle $\mu $, $L(\lambda , \mu )$, is computed by \begin{eqnarray} L(\lambda, \mu ) &=& \int B( T; \lambda ) \exp(-\tau /\mu ) d\tau/\mu \label{eq:thermalemission}\\ \tau &\equiv& \int _z ^{\infty } k[T(z),P(z)] x(z) n(z) dz \\ &=& \int _0^p k[T(P),P] \, x(P) \frac{dP}{\mu_{\rm atm} g } \label{eq:tau} \end{eqnarray} assuming a no-scattering atmosphere. The cross sections of molecules are based on HITRAN2016 \citep{Gordon+2017} and the lines are broadened by the Voigt functions using the algorithm of \citet{Zaghloul+2011} with the partition functions adopted from HAPI program \citep{Kochanov+2016}. We impose the cut-off of the line wings at 100 cm$^{-1}$ apart from the line centers. We do not include the continuum absorption (e.g., H$_2$O continuum), as it does not significantly affect the detectability of high-resolution features. For the efficient evaluation of equation (\ref{eq:thermalemission}) at a large number of the wavelength grid points, we employed GPU computation through PyCUDA. The integral in equation (\ref{eq:tau}) is performed with 50 equi-distributed points in $\log P$ space and that in equation (\ref{eq:thermalemission}) with 50 equi-distributed points in $\log \tau $ space (from $10^{-2}$ to $10^2$), using the trapezoidal rule. Thermal emission spectrum is calculated with the opacities of all four molecules included. In addition, we also calculate the spectra with the opacity of only one molecule turned on, in order to discuss the spectral characteristics and detectability of individual molecules in isolation. To obtain the total thermal emission from the planet, equation (\ref{eq:thermalemission}) needs to be integrated over the planetary disk. With several trials, we find that the disk-averaged radiance that integrates equation (\ref{eq:thermalemission}) over different $\mu $ with the weight of the projected area is close to the radiance at $\mu =0.6$. Thus, we represent the disk-averaged radiance by the radiance with $\mu =0.6$. Doppler broadening due to planet rotation is not included; HZ planets around M-type stars are likely to be tidally locked, which means the rotational velocity of $\sim 10$--100 m/s, and the corresponding to the wavelength shift, $\sim 10^{-6}\;\mu $m, is sufficiently smaller than the wavelength resolution elements considered in this paper. \subsection{Model Spectra and Their Characteristics} \label{ss:result_spectra} \begin{figure}[tb!] \includegraphics[width=1.0\hsize]{MIR_HR_v8.pdf} \caption{High-resolution ($\mathcal{R}=30,000$) features of molecules in thermal emission spectra assuming the vertical temperature profile and molecular mixing ratio shown in Figure \ref{fig:simple_TP_profile}. In the uppermost panel the opacities of all molecules are included, while in the lower panels the opacity of one molecule is assumed at a time. The left panels show the broadband spectra while the right panels show the zoom-in where the horizontal bars approximately indicate the maximum range of the Doppler shift (corresponding to $\sim $100~km/s).} \label{fig:HRspectra} \end{figure} Figure \ref{fig:HRspectra} presents the major features in high-resolution spectra ($\mathcal{R}=30,000$) assuming the temperature profile shown in Figure \ref{fig:simple_TP_profile}. The uppermost panel includes the opacities of all four molecules, while the lower panels turn off the opacities of molecules other than the indicated one, showing more clearly the spectral features of individual molecules. CO$_2$ has the prominent rotational-vibrational features around 15~$\mu $m associated with the fundamental bending mode ($\nu _2$). The strongest lines on the P- and R-branches are separated by $\sim 0.04~ \mu $m, corresponding to the rotational energy $\sim 1.5$ cm$^{-1}$, well resolved with $\mathcal{R}=30,000$. On the other hand, H$_2$O has rotational lines that broadly spread in the MIR range. The 17~$\mu $m N$_2$O band is mainly due to the bending mode ($\nu _2$) and has a structure similar to the 15~$\mu $m CO$_2$ band. Among these, CO$_2$ largely contribute to shaping the overall shape of the all-included spectrum, although the minor features due to H$_2$O and N$_2$O can also be seen. Compared to these bands, the famous 9.7 $\mu $m O$_3$ band with the overlapping fundamental vibration modes (symmetric $\nu _1$ and asymmetric $\nu _3$) is densely populated with lines. A weaker O$_3$ band exists around 14.5 $\mu $m, corresponding to the bending mode, which also features clustered lines in a narrow bandpass. The latter is largely masked by CO$_2$ in Earth's thermal emission spectra. We find that this band start to kick in when CO$_2$ is smaller than $\sim 1$~ppm. Do such CO$_2$-poor atmosphere exist? It is possible, as CO$_2$ abundance in general depends on the carbon cycles, as well as how much the planet acquires and retains carbon. Planets which develops higher weathering rate may turn into CO$_2$-poor worlds \citep[e.g.,][]{Nakayama+2019}. In the absence of CO$_2$ features, 14.5~$\mu $m O$_3$ band may be more useful than the 9.7 $\mu $m band, due to the better intrinsic detectability (Fig. \ref{fig:contrast-photoncount}); see more discussions in Section \ref{ss:choise_of_bandpass}. Here we highlight a characteristics of these high-resolution features---the relative width of the lines. As indicated in the right panel of Figure \ref{fig:HRspectra}, the broad lines extend beyond the wavelength resolution (5$\times 10^{-4}$~$\mu $m at 15 $\mu $m) and the width of the Doppler shift ($\sim $0.005 $\mu $m at 15 $\mu $m; horizontal bars). This is partly due to the intrinsic broadening of the lines. The half width of the Lorentz profile (collisional broadening), which is important at pressures higher than $\sim 0.01$~bar, is approximately constant in {\it wavenumber} for a given set of pressure and temperature, i.e., the width {\it relative to the wavelength} is wider at longer wavelengths. At 1~bar and 288~K, it is approximately 0.1~cm$^{-1}$, corresponding to 0.002~$\mu $m. The line shape also depends on the abundance of the molecule and the vertical temperature gradient. Lines become broad if the opacity is so large that the atmosphere becomes optically thick at the far wings; this happens for CO$_2$ and H$_2$O with our model atmosphere. The broad lines have notable influence on the analysis, as we will see in the next section. \section{Detectability} \label{s:detectability} In this section, we examine the detectability of MIR molecular features of temperate Earth-sized planets presented in Section \ref{s:theoretical_spectra}, assuming a high-dispersion spectrograph mounted on a cryogenic telescope. We try to identify molecular features in the combined spectrum of the host star and the planet. The assumptions for mock observations and noise estimate are given in Section \ref{ss:mock_observations}, which is followed by the description of our analysis procedures in Section \ref{ss:analysis}. The resultant constraints on the contrast and the orbital inclination are discussed in Section \ref{ss:results}. \subsection{Mock observation} \label{ss:mock_observations} \subsubsection{Targets} \label{sss:targets} \begin{table}[bt!] \centering \caption{Assumptions for the host star and the planetary orbit.} \begin{tabular}{lcc}\hline \hline description & M5 star & M8 star \\ \hline star radius ($R_{\star } $) & 0.14 $R_{\odot }$ & 0.10$R_{\odot }$ \\ star temperature ($T_{\star} $) & 3000~K & 2500~K \\ planet/star flux ratio\tablenotemark{a} & $\sim 70$~ppm & $\sim 200$~ppm \\ planet orbital radius & 0.0485~au & 0.0146~au \\ planet orbital period & 11.26 days & 2.27 days \\ planet orbital velocity & 46.83 km/s & 69.70 km/s \\ \hline \end{tabular} \tablenotetext{a}{Evaluated at 15$\mu $m, assuming a black body spectrum with 288~K for the planet spectrum. } \label{tbl:systemparameters} \end{table} The prime target of this study is temperate rocky planets around M-type stars, because those around earlier-type stars have too large planet-to-star flux ratio for planetary signals to be detectable in a reasonable amount of time. We initially considered three types of host star: early-, mid- and late-M stars. However, the typical planet-to-star flux ratio with early-M stars is smaller than 10~ppm, making it extremely challenging to detect planetary features. Therefore, we focus on mid-M and late-M stars. Our fiducial model assumes an M5 or M8 star at 5 parsecs. The assumed host star and planetary parameters are summarized in Table \ref{tbl:systemparameters}. Additionally, the case study for Proxima Centauri b is also presented, where the host star properties are the same as those of M5 star in Table \ref{tbl:systemparameters}, the distance is set to 1.3~parsecs, and the planetary radius is assumed to be 1.1 Earth radius \citep[e.g.,][]{Snellen+2017}. The inclination of the planetary orbit, which controls the amplitude of the planetary line-of-sight velocity, is fixed at $60^{\circ }$. The host star spectra are taken from the BT-Settl model \citep{Allard+2012} with the corresponding effective temperatures, while assuming $\log g=5.0$ and [Fe/H]$=0.0$. This is an update from the previous studies on the detectability of spectral features of Proxima Centauri b where the black-body spectrum is assumed \citep{Snellen+2017, KreidbergLoeb2016}. \subsubsection{Configuration} \label{sss:config} Figure \ref{fig:schematics} illustrates the observational configuration. We assume that the mock observations are carried out when the planet is near $\phi =90^{\circ }$ and near $\phi =270^{\circ }$, where $\phi $ is the orbital longitude (Figure \ref{fig:schematics}). This is because the orbital inclination would be constrained best when the data cover the orbital phases where the radial velocity changes the most. For a planet around a M5 (M8) star, mock observations are continued for 1 (0.25) day centered at $\phi = 90^{\circ }$ and for another 1 (0.25) day centered at $\phi = 270^{\circ }$, which cover approximately $32^{\circ }$ ($40^{\circ }$) on each side. These two cadences are repeated until the parameters are constrained, and the total integration time is recorded. \begin{figure}[htb!] \includegraphics[width=1.0\hsize]{schematics.pdf} \caption{Observational configuration showing the geometrical parameters. The data near $\phi =90^{\circ }$ and $\phi =270^{\circ }$ are used in this study. } \label{fig:schematics} \end{figure} \begin{figure}[htb!] \includegraphics[width=1.0\hsize]{schematics2.pdf} \caption{Schematic figures showing the signals to be detected. Analysis (A) (left) utilizes the high-frequency features of the spectra extracted by subtracting the moving average from the spectrum, while Analysis (B) (right) utilizes the difference of the Doppler shifted spectra from the average spectra.} \label{fig:schematics2} \end{figure} \subsubsection{Instruments and observational configurations} \label{sss:obs_strategy} Mock observations are carried out with a high-resolution spectrograph at 12-18 $\mu $m with the resolving power of $\mathcal{R}=30,000$. This bandpass and resolving power are motivated by the latest specification of SPICA's SMI/HR ($\mathcal{R}\sim 33,000$ in 12-18 $\mu $m) \citep[e.g.,][]{Kaneda+2018} and by the expected resolution of OST/MISC \citep[e.g.,][]{Sakon+2018}. The dependences on the spectral resolution and the bandpass are examined in Section \ref{s:Detecability_Resolution_Abundance}. The total throughput including the quantum efficiency of the detector is assumed to be 0.2. We note that the throughput of the optical system of existing high-resolution spectrographs installed on ground-based telescopes have reached approximately 60\% \citep[e.g.,][]{Ikeda+2016,Ikeda+2018}, and the throughput of future instruments could be higher than what we assume here. It is trivial to scale our results by throughput; see Section \ref{sss:scaling}. We collect data every 1800 sec (i.e., exposure time is assumed to be 1800 sec). Due to the change of the planetary radial velocity, the planetary spectrum is Doppler shifted relative to the host star spectrum on the detector plane. Over the course of the planetary orbital motion, the planetary spectrum moves beyond the resolution elements unless the orbital inclination is close to zero (i.e., face-on orbit), as the radial velocity amplitudes shown in Table \ref{tbl:systemparameters} (and the assumed orbital inclination of $60^{\circ }$) are larger than the velocity corresponding to the resolution element, $c/\mathcal{R}\sim 10$~km/s. ) \subsubsection{Signal and noise} \label{sss:noise} The total photoelectron count that the detector receives is the summation of the planetary light ($\mathcal{N}_{{\rm p}}$), stellar light ($\mathcal{N}_{\star}$), zodiacal light ($\mathcal{N}_{{\rm zodi}}$), thermal background of the telescope ($\mathcal{N}_{{\rm tele}}$), and the dark current ($\mathcal{N}_{{\rm dark}}$). We assume that the contribution from the zodical light and the dark current are perfectly subtracted through a post-processing. This leaves $\mathcal{N_{\rm total}} = \mathcal{N}_{{\star }} + \mathcal{N}_{{\rm p}}$ alone as a signal. \begin{table}[tb!] \centering \begin{tabular}{lcc}\hline \hline description & symbol & value\\ \hline spectral resolution & $\mathcal{R}$ & 30,000 \\ telescope diameter & $D$ & 6.5~m \\ total throughput & $\xi $ & 0.15 \\ distance to target & $d$ & 5~pc \\ planetary radius & $R_{\rm p}$ & $R_{\oplus }(=6.371\times 10^6$~m) \\ exposure time & $\tau _{\rm exp}$ & 1800 sec \\ \hline \end{tabular} \caption{Fiducial values for observational parameters. The scaling of the required integration time by these parameters are given in equation (\ref{eq:scaling}).} \label{tbl:parameters} \end{table} The shot noise from all of these factors contributes to the observational noise, although the shot noise due to the planetary flux is negligible compared to that from the stellar flux. An additional factor that is taken into account is the read noise. The systematic noises such as a fringe of the detector are assumed to be perfectly removed; the effect of systematic noise is discussed in Section \ref{sss:systematic_noise}. These assumptions imply the Gaussian random noise with the following standard deviation for $j$-th wavelength element at wavelength $\lambda _j$: \begin{eqnarray} \sigma^2_{{\rm photon},j} =&& \mathcal{N}_{{\rm star},\,j} ( 1 + \xi b_j(T_{\star }) ) + \mathcal{N}_{{\rm zodi},\,j} \nonumber \\ &&+ \mathcal{N}_{{\rm tele},\,j}+ \mathcal{N}_{{\rm dark},\,j} + \sigma_{\rm read}^2 \label{eq:sigma} \end{eqnarray} where $\sigma_{\rm read}^2 $ represents the read noise, $b(T)$ is the Bose factor, and $\xi $ is the total throughput. The Bose factor, $b(T)$, is to take account of the sub-Poissonian nature of the photon count statistics at $h\nu \ll kT$ \citep[e.g.,][]{Boyd1982}: \begin{equation} b_j(T) = \frac{1}{\exp \left( \frac{hc}{\lambda _j kT} \right) - 1 } \end{equation} For MIR observations ($\lambda \sim 15\,\mu $m) of M-type stars ($\sim 3000$~K), $b \sim 2.7$. Multiplied by the throughput ($\xi =0.2$), the Bose factor increases the variance of the photon count by about 50\%. The Bose factor for the zodiacal light is not included due to the low temperature. In the fiducial cases studied in this paper, the dominant noise source is the stellar flux. The contributions of these factors in general cases are presented in Figure \ref{fig:photon_star_zodi}, as a function of the spectral type of the star and the distance to the target. \begin{figure}[bt!] \includegraphics[width=1.0\hsize]{comp_noise_v2.pdf} \caption{Photon count of the zodiacal light (black) and the starlight (colors) at wavelength 10~$\mu $m (solid line) and at 20~$\mu $m (dashed line), per exposure time (1800~sec) per wavelength resolution element ($\lambda / \mathcal{R}$). These numbers can also be compared with dark current ($360e^{-1}$/exposure) read noise, $14\times 4 e^{-}$/exposure.} \label{fig:photon_star_zodi} \end{figure} Each term in equation (\ref{eq:sigma}) is expanded below. \paragraph{Planet/star spectra} The photoelectron counts of the planet and the host star per exposure are simply: \begin{eqnarray} \mathcal{N}_{{\rm p}} (\lambda ) &=& \mathcal{F} L_{\rm p}(\lambda ) \cdot \pi R_{\rm p}^2 \left( \frac{1}{d} \right)^2 \\ \mathcal{N}_{\star} (\lambda ) &=& \mathcal{F} L_{\star} (\lambda ) \cdot \pi R_{\star }^2 \left( \frac{1}{d} \right)^2 \\ \mathcal{F} &\equiv & \pi \left( \frac{D}{2} \right)^2 \xi \tau_{\rm exp} \frac{\lambda }{\mathcal{R}} \end{eqnarray} where $L_{\rm p}$ and $L_{\star}$ represent the radiance of the planet and star, respectively, while $R_{\rm p}$ and $R_{\star }$ represent the radius of the planet and the star, respectively. The meanings of other parameters are summarized in Table \ref{tbl:parameters} together with our fiducial values. Here $\mathcal{F}$ is used to denote the common factor related to the observational configuration. \paragraph{Zodiacal light} \label{sss:starlight} The zodiacal flux of the Solar System is set to 15~MJy/sr \citep{Glasse+2015} and is assumed to be constant over the observing wavelength range. The zodiacal light in the target system is not included in our simulation because it is much fainter than the other sources. Thus, \begin{equation} \mathcal{N}_{{\rm zodi}} = \mathcal{F} f_{\rm zodi} \cdot \pi ( \theta_{\rm aperture})^{2} \end{equation} where $f_{\rm zodi}$ is the zodiacal flux per wavelength per steradian (rather than per frequency as represented in Jansky per steradian) and $\theta_{\rm aperture}$ denotes the aperture radius. The aperture radius should be determined by the balance between the shot noise of the thermal background and the systematic noise \citep{Matsuo+2018}. Based on the analytical formulation on the relation between the aperture radius and systematic noise \citep{Itoh+2017}, the aperture radius is set to 1.85 arcsecond, corresponding to 4 times the diffraction limit at 15 $\micron$ such that the systematic noise is reduced down to 100 ppm under a condition that the pointing jitter of the Origins Space Telescope is 22 milli-arcsecond (RMS), corresponding to approximately 0.05 $\lambda / D$ at 15 \micron{} \citep{Leisawitz+2018}. \paragraph{Telescope background} \label{sss:tele} The thermal light and stray light from a telescope assembly also contributes to the background light. For the JWST MIRI, the telescope background can be approximately fitted by a combination of several blackbody radiations with temperatures ranging from 50 to 70~K \citep{Glasse+2015}. When the telescope is cooled down to below 10~K, as planned for SPICA and OST, and the telescope background becomes negligible at $<$ 100 $\mu$m. \paragraph{Dark current} \label{sss:darkcurrent} The dark current is assumed to be 0.2 $e^-$~s$^{-1}$ based on the performance of the Si:As detector for the JWST MIRI \citep{Rieke+2015}, but this never becomes of relative importance in our study. \paragraph{Read noise} \label{sss:readnoise} The read noise is assumed to be 14 $e^-$/read assuming the Fowler-eight sampling \citep{Rieke+2015}. The loss time due to the readout is not considered in the paper. Each resolution element is sampled by 4 pixels. With these assumptions, read noise never becomes dominant for the exposure time of 1800 sec. We note that 1800 sec is longer than the typical values for JWST. The reduction in the exposure time can lead to a significant contribution from read noise. \subsection{Analysis} \label{ss:analysis} Because the planetary signal is a tiny portion in the total spectrum, it is critical to subtract the stellar spectra precisely from the total spectrum. Previous studies discussing the detectability of atmospheric features of non-transiting habitable planets assumed that the host star spectrum is determined precisely \citep[e.g.,][]{KreidbergLoeb2016,Snellen+2017}. The analysis based on this optimistic assumption is denoted by Analysis (A) and is described in Section \ref{sss:analysis_optimistic}. However, M-type stars are rich in spectral features as shown in Figure \ref{fig:star_MIR} and its MIR stellar spectrum may not be determined to a satisfactory accuracy, due to the inaccurate line list and to the uncertainties in the stellar atmospheric structure. Indeed, the models of star spectra do not fit the near-infrared observed spectrum of M-type stars to the precision of observed data \citep[e.g.,][]{Zhang+2018,Wakeford+2019}. In the case where we do not have any prior knowledge about the stellar spectrum, we have to focus on the Doppler-shifted components in the combined spectrum of the star and the planet to extract planetary one. Such an analysis is denoted by Analysis (B) and is described in Section \ref{sss:analysis_pessimistic}. The planetary signals to be fitted in the case of Analysis (A) and (B) are illustrated in Figure \ref{fig:schematics2}. Importantly, as described in more detail below, the signal in Analysis (B) is smaller than that in Analysis (A) because some fraction of the spectral features are cancelled out. \begin{figure}[bt!] \includegraphics[width=1.0\hsize]{BT-Settl_midM_lateM.pdf} \caption{MIR spectra based on BT-Settl model \citep{Allard+2012}.} \label{fig:star_MIR} \end{figure} \subsubsection{Analysis (A): with a well-modeled stellar spectrum} \label{sss:analysis_optimistic} In the first analysis, we follow the procedure of \citet{Snellen+2017}, assuming that the fine structure of the stellar spectrum can be modeled precisely. Even with this assumption, we cannot separate the host star and the planet spectra from observations alone, so we take the following steps: \begin{enumerate} \item The model stellar spectrum is fitted to the data (that is the combined spectrum of the star and the planet), by varying the scaling of the spectrum as well as the stellar parameters. \item The best-fit stellar spectrum is subtracted from the data. \item The residual spectrum is corrected by subtracting its moving-average. \item The model planet spectrum, which is also corrected through the moving-average subtraction (the bottom panel of Figure \ref{fig:schematics2}), is fitted to the corrected residual spectra. \end{enumerate} In Step 1, we use the input BT-Settl model and fit for the absolute scale. The best-fit scale is slightly larger than the input stellar model, due to the contribution from the planet spectrum. In other words, some fraction of the planetary spectrum is subtracted in Step 2. This results in a trend in the residual spectrum, which is corrected in Step 3. Finally in Step 4, the high-frequency features are fitted by the model planet spectrum. For simplicity, our fitting model is the same as the input model, similar to \citet{Snellen+2017}, and we consider two fitting parameters: the planet-to-star contrast $C$ and the orbital inclination $I$. The parameter estimate is based on the posterior probability, $\mathcal{P} (c,i\|\{\mathcal{N}_i (t_k) \})$: \begin{equation} \mathcal{P} (C,I\|\{\mathcal{N}_{{\rm res}, jk} \}) = \mathcal{L} ( \{\mathcal{N}_{{\rm res}, jk} \}\|C, I ) \Pi(C) \Pi(I) \label{eq:postprob} \end{equation} The $\mathcal{N}_{{\rm res},jk}$ represents the corrected residual spectra at $j$-th wavelength elements and $k$-th observation epoch. The likelihood is simply: \begin{eqnarray} && \mathcal{L} ( \{\mathcal{N}_{{\rm res}, jk}\|c, i ) \propto \exp \left( - \frac{\chi^2}{2} \right) \label{eq:likelihood} \\ && \chi^2 \equiv \sum _{\{j, k\}} \left( \frac{\mathcal{N}_{{\rm res},jk}^{\rm obs} - C \mathcal{N}_{{\rm res},jk}^{\rm theory} (I) }{\sigma _{j,k}} \right)^2. \label{eq:chi2} \end{eqnarray} We assume a flat prior for contrast $C$ between 0 and 10, and a flat prior for the orbital inclination $I$ between 0 to 90 degree; $\Pi(C)=const.$ and $\Pi(I)=const.$ This means that the posterior probability is simply equivalent to the likelihood function. The posterior probability is normalized so that the total is unity. The $1\sigma $, $2\sigma $, and $3\sigma $ contours of the posterior probability are close to those of $\Delta \chi^2$ measure \citep[e.g.,][]{Snellen+2017}, with a slight difference due to the non-linear $I$-dependence of the model. \subsubsection{Analysis (B): with unknown stellar spectrum} \label{sss:analysis_pessimistic} In the second analysis, we do not assume that the stellar spectrum is known a priori. Instead, we consider the time average of the combined spectra of the star and the planet and assume that the stellar spectrum is fully included in this averaged spectrum, which is valid if the stellar spectrum is stable in time. On the other hand, the spectral features of the planet should remain in the average-subtracted residual spectrum, due to the Doppler shift caused by its orbital motion. The analysis procedure is as follows: \begin{enumerate} \item The average of the spectra at different planetary orbital phases is obtained. \item The average spectrum is subtracted from the spectrum at different orbital phases. \item The model planet spectrum, which is also Doppler-shifted and average-subtracted (the bottom panel of Figure \ref{fig:schematics2}), is fitted to the set of residual spectra. \end{enumerate} The parameter estimate in Step 3 is performed in the same way as Analysis (A) (equations (\ref{eq:postprob})-(\ref{eq:chi2})). Figure \ref{fig:schematics2} illustrates the signal that can be used in this analysis. The average of the spectrum at $\phi = 90^{\circ }$ and that at $\phi = 270^{\circ }$ is shown in the black line in the upper panel, and the average-subtracted spectra are shown in the lower panel. During this subtraction process, some portions of the planetary spectral features are cancelled out, reducing the signal level. This effect is substantial in the case of MIR observations of potentially habitable planets whose strongest lines are broader than the Doppler shift (Figure \ref{fig:HRspectra}). The situation is different from the typical high-resolution spectroscopy of hot Jupiters at the shorter wavelengths, where the lines are narrower and the Doppler-shift is larger. In such a case, the Doppler shift moves the line beyond their intrinsic width and the average-subtracted spectra are more similar to the original spectrum. \subsubsection{Notes on cross-correlation analysis} High-resolution spectroscopy of hot-Jupiter systems have been routinely analyzed through cross-correlation function \citep{Snellen+2010,Brogi+2013}. While this technique is useful for detecting high-frequency features, its statistical treatment and the uncertainties in the estimated model parameters are not straightforward \citep[e.g.,][]{Brogi+2017}. On the other hand, fitting observed data with theoretical models by minimizing the sum of squared residuals offers a more plain interpretation \citep{Snellen+2017}. Thus, in this paper, we employ the latter method. \subsection{Results} \label{ss:results} \begin{figure}[tb!] \includegraphics[width=\hsize]{postprob_65m_5pc_analysisA_v6.pdf} \caption{The 1$\sigma $ (solid lines), 2$\sigma $ (dashed lines), and 3$\sigma $ (dotted lines) confidence intervals for planets around a mid-M star (upper panels) and a late-M star (lower panels), based on analysis (A), i.e., when the stellar spectrum is precisely determined. Both the mock data and the fitting model are based on the model thermal emission spectra presented in Figure \ref{fig:HRspectra}. The constraints from two integration times are presented for some cases, in order to show how the constraints are developed as the integration time becomes longer. The blue colors imply that the presence of the molecule is detected (i.e., the $C=0$ is rejected by 3$\sigma $) and the inclination is not well constrained, while the red colors imply that both the contrast and inclination are constrained. The corresponding signal-to-noise ratios of the host star spectrum per wavelength resolution element (SN) are also reported. } \label{fig:postprob_A} \end{figure} \begin{figure}[tb!] \includegraphics[width=\hsize]{postprob_65m_5pc_analysisB_v6.pdf} \caption{Same as Figure \ref{fig:postprob_A}, but the mock data are now analyzed by analysis (B).} \label{fig:postprob_B} \end{figure} Figures \ref{fig:postprob_A} and \ref{fig:postprob_B} summarize the results of Analysis (A) and Analysis (B), respectively, performed on the thermal emission spectra with all molecules included and with individual molecules in isolation (Figure \ref{fig:HRspectra}). The solid, dashed, and dotted lines present the 1$\sigma$, 2$\sigma$, and 3$\sigma $ constraints in the contrast-inclination plane (i.e., 68.27\%, 95.45\%, and 99.73\% probability), after the indicated integration time. The corresponding average signal-to-noise ratio (SN) of the host star spectra per wavelength resolution element is also presented. As expected, detecting any molecule is easier for planets around late-M stars than around mid-M stars due to the larger planet-to-star flux ratio and, for analysis (B), to the larger radial velocity amplitude. In the model considered here, the total observation time required for a M5-star system is by a few times larger than a M8-star system, or 3 times smaller signal-to-noise ratio of the host star. When the host star spectrum is accurately known and corrected (Analysis (A); Figure \ref{fig:postprob_A}), the spectral features of an Earth-like atmosphere at 5 prsecs away can be detected within a few days of total observation time, even around mid-M systems (the leftmost column). Compared to this optimistic scenario, Analysis (B) requires approximately 4 times longer observations to detect the spectral features of the same target, or doubled signal-to-noise ratio of the host star flux. This is because of the self-subtraction discussed in Section \ref{sss:analysis_pessimistic}. For comparison, we also perform the same analysis assuming only one of the molecules as opacity source (the lower five panels of Figure \ref{fig:HRspectra}); the results of which are shown in the right four columns of Figures \ref{fig:postprob_A} and \ref{fig:postprob_B}. Comparison of the two figures clearly shows that the effect of the self-subtraction is substantial except for O$_3$. In addition, for Analysis (B), the contrast and the inclination angle tend to degenerate for pure CO$_2$ or pure H$_2$O atmospheres. This is because when a small orbital inclination (i.e., close to face-on) is assumed, the Doppler shift of the planetary spectrum is small and the amplitude of the differential spectrum (the lower panel of Figure \ref{fig:schematics2}) is also small, which is then compensated by a large contrast. Note that the orbital inclination is poorly constrained even with Analysis (A) for pure CO$_2$ or pure H$_2$O atmospheres, due to the broad nature of the spectral lines. Compared to these two molecules, the self-subtraction is not substantial for O$_3$ because the O$_3$ bands are densely populated by narrow lines, and the detectability through Analysis (B) is similar to that of Analysis (A). Furthermore, the orbital inclination is well constrained as soon as the contrast is constrained to non-zero. N$_2$O features are also relatively good at constraining the inclination, despite its relatively weak features. In the spectrum with all the molecules included, the features of each molecule tend to be more muted than the features of individual molecules in isolation, due to the line overlaps. However, the spectrum become more rich in features, and the constraints on the contrast is as good as for the pure-CO$_2$ or pure-H$_2$O cases. Furthermore, the overlaps of the spectral features of individual molecules break the degeneracy between the contrast and inclination for Analysis (B), significantly improving the constraints on the orbital inclination compared to the pure-CO$_2$ or pure-H$_2$O cases. As a result, both the contrast and the orbital inclination can be reasonably constrained within $\sim 1$ week of integration for M8 star systems within 5~parsecs even with Analysis (B). For earlier type systems, constraining the orbital inclination would become more challenging. \subsubsection{Scaling and the application to the known targets} \label{sss:scaling} While the estimated integration time would be too long to be practical except for the case of late-M stars, it is possible to approximately scale these numbers for varying observational configurations. When the stellar light is the dominant noise source, the integration time that achieves certain noise level, $\tau_{\rm esp,\,0} $, is proportional to: \begin{equation} \tau_{\rm esp,\,0} \propto \left( \frac{R_p}{R_{\oplus }} \right)^{-2} \left( \frac{d}{\mbox{5 pc}} \right)^2 \left( \frac{D}{\mbox{6.5 m}} \right)^{-2} \left( \frac{\xi}{0.2} \right)^{-1} \label{eq:scaling} \end{equation} In particular, the observations of the nearest possible target, Proxima Centauri b, at the distance of 1.3 pc and with the estimated radius of $\sim 1.1 R_{\oplus }$, would only require $\sim $1 days of observation even without an assumption on the stellar spectrum. A more SPICA-like specification with $D=2.5$~m and a lowered total throughput ($\xi=0.1$) yield $\sim $14 days of total integration time. \section{Dependence of Detectability on Spectral Resolution, Molecular Abundance, and Bandpass} \label{s:Detecability_Resolution_Abundance} In the assessment in Section \ref{s:detectability}, we have made several assumptions for the specifications of the spectrographs and the atmospheric profile. In this section, we discuss the effects of varying these assumptions on the detectability of the molecules. \subsection{Effects of Spectral Resolution} \label{ss:optimal_resolution} First, we discuss the dependence on the spectral resolution of the spectrograph. Let us represent the signal by the total area of the shaded region shown in Figure \ref{fig:schematics2}, namely the integration of the absolute value of the differential spectrum both in Analysis (A) and (B); For Analysis (A), the differential spectrum corresponds to the difference between the original spectrum and the moving average, while for Analysis (B) it is the difference between the spectrum at a certain orbital phase and the time-averaged spectrum. \begin{figure}[tb!] \includegraphics[width=1.0\hsize]{Signal_vs_R_simple20_fiducial_v2.pdf} \caption{The total areas of the filled region in Figure \ref{fig:schematics2}, one of the diagnostics of the signal, as a function of the spectral resolution, using Analysis (A) (left) or Analysis (B) (right). The fiducial atmospheric profile is assumed. } \label{fig:S_vs_R_fiducial} \end{figure} \begin{figure}[tb!] \includegraphics[width=1.0\hsize]{ProxCenb_CO2_varyR.pdf} \caption{3$\sigma $ constraints in the contrast-inclination plane based on the observations of Proxima Centauri b with CO$_2$ features using varying spectral resolution, after 2 days (left) or 7 days (right) of integration. } \label{fig:ProxCenb_CO2_varyR} \end{figure} Figure \ref{fig:S_vs_R_fiducial} shows the signals defined above for Analysis (A) and (B) as a function of the spectral resolution. Black line shows the case of the spectrum with all four molecules included (the bottom panel of Figure \ref{fig:HRspectra}), while other lines show the cases for the spectra of individual molecules. As expected, the signal level monotonically increases as the spectral resolution increases. However, it is mostly saturated above $\mathcal{R} \gtrsim 10,000$ except O$_3$, in both analysis (A) and (B). This is partly because the interval of the strongest lines are $\sim 0.1-0.2\,\mu $m and these lines start to be resolved with $\mathcal{R} \gtrsim 2000$. In addition, the intrinsic line broadening starts to dominate over the broadening determined by the spectral resolution for $\mathcal{R} \gtrsim 10,000$. The required resolution, $\mathcal{R} \gtrsim 10,000$, is about one order of magnitude smaller than that in the visible or near-infrared domain ($\lambda \sim 1\,\mu$m), $\mathcal{R} \gtrsim 100,000$, because the photon energy is about one order of magnitude lower while the energy corresponding to the line broadening and the interval of lines do not depend on the wavelengths (See the discussion in Section \ref{ss:result_spectra}). The saturation is less prominent for O$_3$, where the narrow lines are densely populated. Thus, detection O$_3$ benefits from the increased spectral resolution beyond $\mathcal{R}\sim 10,000$. Figure \ref{fig:ProxCenb_CO2_varyR} shows the constraints for the contrast and the orbital inclination obtained through Analysis (B) of mock observations of Proxima Centauri b for 2 days (left) and 7 days (right) with varying resolution. As expected from the above assessment, the detectability, namely the lower constraints on the contrast, improves as the spectral resolution increases up to $\mathcal{R}\sim 10,000$, above which no significant improvement is seen. However, the constraints on the orbital inclination is improved beyond $\mathcal{R}\sim 10,000$. \subsection{Choice of the bandapss} \label{ss:choise_of_bandpass} Our fiducial bandpass is set to 12-18 $\mu $m, based on the planned specification of SPICA/SMI and OST/MISC. In this range of wavelengths, several other molecules have prominent absorption bands, in addition to the molecules studied in this manuscript. They includes NH$_3$, SO$_2$, and NO$_2$ and they may also be useful for characterizing the surface environment of potentially habitable planets. Although there is no planed high-resolution spectrograph that goes beyond this bandpass, it would be worth considering the impact of the alternative choice of bandpass for future instrumental designs. In the following, we briefly discuss how the extension of the bandpass would affect the prospects. If the bandpass is extended toward shorter wavelengths, a feature of particular interest would be O$_3$ 9.7~$\mu $m band (the bottom panel of Figure \ref{fig:HRspectra}). The peak line strength of this band is approximately one order of magnitude larger than the 14.5 $\mu $m band and, combined with the fact that the shorter wavelength is in the Wien's regime of the Planck function, the feature relative to the continuum is deeper. However, the reduced planet-to-star contrast (Figure \ref{fig:contrast-photoncount}) does not make this band easy to detect. We find that detection of O$_3$ 9.7 $\mu $m O$_3$ band would require approximately 1.7 times longer integration time (or 75\% photon noise) than the detection of 14 $\mu $m O$_3$ band, in the absence of CO$_2$ (not shown). In the presence of CO$_2$ larger than 1 ppm, 9.7 $\mu $m band would be the only detectable band. The detectability of O$_3$ 9.7 $\mu $m would be improved for planets with higher surface temperature. The features at even shorter wavelengths is even less likely to be detectable with this method due to the small planet-to-star contrast. At the wavelengths longer than 18 $\mu $m, there are few vibrational modes of simple molecules. However, the rotational lines of H$_2$O extend beyond 18 $\mu $m and adding longer wavelengths improves the detectability of H$_2$O. Using 12-24 $\mu $m bandpass, the integration time required for the detection of H$_2$O from the H$_2$O-only spectrum (the second panel of Figure \ref{fig:HRspectra}) would be reduced to approximately 60\%. \subsection{Molecular abundances that can be most sensitively probed} \label{ss:optimal_abundance} The detectability also depends on the actual abundance of the molecules. While in transmission spectroscopy the detectability of certain molecules generally increases as the abundance increases (ignoring the effects of clouds), this is not the case for the high-resolution method considered in this paper. It is true that the signal would not be detected if the abundance is too small. However, larger abundances does not always lead to the improved detectability, because some of the features start to saturate. For example, with the assumed Earth-like abundance of CO$_2$ ($\sim $ 330~ppm) the strongest lines of CO$_2$ around 15~$\mu $m are saturated (i.e., the spectrum is flattened) and do not significantly contribute to the signal. With larger abundance, the flattened region only increases, although the features associated with smaller opacities kick in. Thus, there is the optimal range of molecular abundance. \begin{figure}[tb!] \includegraphics[width=1.0\hsize]{Signal_vs_R_CO2.pdf} \caption{Same as Figure \ref{fig:S_vs_R_fiducial} but with varying mixing ratio of CO$_2$.} \label{fig:S_vs_R_CO2} \end{figure} To see such a dependence of detectability of CO$_2$ and N$_2$O, we created model spectrum with varying mixing ratio assuming a 1-bar atmosphere with vertically uniform mixing ratio of each molecule. For each model spectrum, we repeated the analysis of Section \ref{ss:optimal_resolution} to see the dependence of the ``signal'' as a function of the spectral resolution. The result is shown in Figure \ref{fig:S_vs_R_CO2}. The signal level is maximized when the mixing ratio is 1-10$^3$~ppm. A similar trend was found for N$_2$O, with the signal level being maximized for 1-10$^2$~ppm. Therefore, this technique is sensitive to relatively small amount of molecules. This would be rather complementary to scattered-light observations where larger abundance of these molecules would be probed; see Section \ref{ss:targets}. \section{Discussion} \label{s:discussion} \subsection{Potential Targets and Synergy with other techniques} \label{ss:targets} The prime targets of the method studied in this manuscript would be potentially habitable planets around mid- to late-M stars within 5~parsecs. The fact that this method does not require planetary transits is a significant advantage over transmission or eclipse spectroscopy, given the low transit probability ($\sim $5\%). Within approximately 5~parsecs, 6 mid- to late-M stars have already been found to harbor Earth-sized planets around habitable zones as of the submission of this manuscript (Table \ref{tbl:targets}), none of which are found to transit in front of the host star. The high-resolution technique would provide valuable opportunities to investigate the atmospheric compositions of these known planets in thermal emission. The number of the targets may increase thanks to the current vigorous radial velocity measurements in near-infrared all over the world (There are 7 late-M stars (M6 or later) and 10 mid-M stars (M3-M5) within 5~parsecs, according to \citet{Gaidos+2014}). Note that we do not necessarily limit our scope to 5 parsecs. Some of the features that are easier to detect may extend its scope beyond 5 parsecs. These nearby M-star systems are also considered to be the prime targets for the scattered-light observations with the next-generation extremely-large ground-based telescopes \citep{Fujii+2018}. The combined efforts would allow for the comprehensive characterization of the target systems, as the information obtained from scattered-light spectra (e.g., surface liquid water, clouds, certain molecules) and that obtained from thermal emission spectra (temperature, radius, certain molecules) are complementary. For example, thermal emission spectra could in principle constrain the planetary radius, which critically helps the interpretation of the scattered-light spectrum. Furthermore, for a given molecule, the abundance detectable in the scattered light at $\sim $1~$\mu $m would be larger than the abundance probed by the high-resolution technique (Section \ref{ss:optimal_abundance}), thus the combination would constrain a wide range of atmospheric properties. The targets of MIR high-resolution technique are also complementary to the prime targets for transmission and eclipse spectroscopy. The main scope of OST is currently to characterize potentially habitable planets through transit spectroscopy by achieving ultrahigh stability (see the technical report of OST). There are pros and cons for transit spectroscopy and for the technique studied in this paper. While the former may require less total observation time for certain biosignature molecules (provided that the noise floor could be down to a few ppm) than the latter, it has to visit the target 50 times for TRAPPIST-1 planets \citep{Tremblay+2020} and more for less favorable targets, likely spanning ~1 year or longer. The number of visits of the high-resolution method studied in this manuscript can be much smaller because of the longer observation time per visit. In addition, the transmission spectroscopy could be hampered by high-altitude thin clouds, which does not significantly affect the high-resolution technique. Thus, combining both techniques would expand our capability of investigating potentially habitable worlds. \begin{table}[tb!] \centering \caption{Nearby systems that are found to harbor Earth-sized planets around habitable zones. Stellar parameters are based on \citet{Gaidos+2014}. } \begin{tabular}{lccc}\hline \hline Star name & $d$ & Spectral type & $T_{\star}$ \\ \hline Proxima Cen & 1.3~pc & M7 & 2883~K \\ Ross 128 & 3.4~pc & M5 & 3145~K \\ Gliese 1061 & 3.7~pc & M6? & 3000~K? \\ Luyten's & 3.8~pc & M4 & 3317~K \\ Teegarden's & 3.9~pc & M7 & 2700~K \\ GJ 682 & 5.1~pc & M5 & 3190~K \\ \hline \end{tabular} \label{tbl:targets} \end{table} \subsection{Effects of other noise} \label{ss:other_noise} \subsubsection{Time variability of stellar spectra} \label{sss:stellar_activity} The major concern of this method is the variability of high-resolution stellar spectra. Aside from sporadic flares, stars generally exhibit photometric and spectroscopic variabilities mainly due to starspots. Given that the signal from the planet is a tiny fraction of the stellar spectrum, such variability potentially cases serious problem in the analysis. There is a growing amount of modeling efforts for the temperatures and covering fractions of star spots/faculae of mid- to late-M stars based on observations. One piece of evidence comes from Doppler imaging. Doppler imaging of mid- to late-M stars \citep{Morin+2008,Barnes+2015,Barnes+2017} estimates that around a few \% of the surface is covered with spots that are assumed to be cooler than other area by 200-400~K. Recently, characterization of TRAPPIST-1, an M8 star harboring transiting temperate Earth-sized planets, has attracted attention. The light curve of TRAPPIST-1 observed by Kepler/K2 (0.43-0.89 $\mu $m) shows $\sim 1$\% peak-to-peak variation amplitude, which may be explained by rotating dark spots or faculae. In contrast, the variability of {\it Spitzer} at 4.5 $\mu $m do not show a clear variability and is at least smaller than the photometric precision, which is limited to the order of 0.1\% owing to the shot noise and instrumental systematic noise. \citet{Morris+2018} find that the combination of these light curves at different wavelengths can be explained by the existence of several very small faculae with $> 5300$~K rather than dark spots, and show that the theoretical photometric variability at 4.5~$\mu $m with this model seems to be random with an amplitude of about 100~ppm. This prediction is encouraging because this is compatible or smaller to the contrast of the planet to stellar signals. The impact of variability on this method will be further mitigated by averaging the data over various stellar rotational phases. The covering fraction of the spots and faculae are also estimated based on the multi-component fit to the stellar spectrum using the theoretical stellar spectra with various temperatures \citep[e.g.,][]{Zhang+2018,Wakeford+2019}. They typically result in relatively large covering fractions for darker spots and faculae (about 30-50\%). However, this method is sensitive to the small-scale features that spread across the photosphere as well as the large-scale asymmetric patterns, in contrast to the Doppler Imaging and reconstruction from photometric variabilities that are only sensitive to the latter. The small-scale features that distribute rather uniformly do not affect the observations studied in this manuscript. We note that the recent observations of Proxima Centauri by \citet{SuarezMascareno+2020} show V-band photometric variability of $>$10\% (peak-to-peak), larger than the {\it Kepler/K2} variability of TRAPPIST-1. In reality, stars exhibit the diversity in their surface magnetic activities and the resultant spots/faculae, and therefore it is critical to characterize individual stars before characterizing the planets. With the launch of JWST that achieves higher precision than {\it Spitzer} thanks to its larger aperture and better attitude control, our understanding of variabilities of the stellar MIR spectra will be developed and the techniques to cope with these variability will be advanced. For example, simultaneous light-curve observations at different wavelengths will allow us to better model the spots, as demonstrated by \citet{Morris+2018}. Such practice will further enhance the feasibility of the method presented in this manuscript. \subsubsection{Systematic Noise} \label{sss:systematic_noise} We did not explicitly deal with the impact of the unknown instrumental systematic errors. Because this technique utilizes high-frequency features of the spectrum, the uncertainties in the absolute flux or the low-frequency modulation of the continuum, even if it is substantial, will not affect our results after applying some corrections \citep[e.g., high-pass filter;][]{Snellen+2017}. In contrast, residual flat-fielding, spectral fringe patterns, and telescope pointing jitters form artificial absorption and emission lines (i.e., high-frequency systematic noise) in the observed spectrum. Such artificial high-frequency patterns will be problematic for both the analyses (A) and (B). The high-frequency noise will contribute to the total noise floor because of its random distribution along the wavelength. Since the high-frequency systematic noise is independent of the shot noise, the total noise may be expressed by $\sqrt{ \sigma^2_{{\rm photon},j} + \sigma^2_{{\rm sys},j} }$, where $\sigma_{{\rm photon},j} $ is defined by equation (\ref{eq:sigma}) and $\sigma_{{\rm sys},j}$ denotes the high-frequency systematic noise at the $j$-th spectral element. Given that the noise required for detection is order of 100~ppm, it is necessary that the instrument systematic noise is less than this level. If the systematic noise is $\sim 30$~ppm or less, the systematic noise will not affect our estimate much. Conversely, if it is comparable to the photon noise, a substantially larger integration time will be required in order to reduce the photon noise further. We note that the high-resolution spectroscopy observes the shift of the planetary Doppler signal on the detector, which could mitigate an impact of the patterns fixed to the detector (owing to relative sensitivity variations between adjacent spectral channels). \subsection{Variety of Planet Thermal Emission Spectrum} \label{ss:dependence_on_atmosphere} \subsubsection{Stratospheric Thermal Inversion} \label{sss:thermal_inversion} \begin{figure}[tb!] \includegraphics[width=1.0\hsize]{sp_postprob_CO2_USS_v2.pdf} \caption{Left: high-resolution ($\mathcal{R}=30,000$) thermal emission spectra assuming the vertical profiles of temperature and CO$_2$ that are same as Earth's ``US standard'' profile. Right: Parameter constraints based on the left panel, assuming a late-M host star at 5 parsec. } \label{fig:sp_postprob_USS} \end{figure} This section discusses the variety of high-resolution thermal emission spectra, and how they affect the parameter estimate. Our fiducial model atmosphere has a relatively cool stratosphere, approximately 150~K. A higher stratospheric temperature results in shallower spectral features, increasing the required integration time for detection. If the planet has thermal inversion in the upper atmosphere (like the ozone layer of Earth), thermal spectra may exhibit emission lines. These lines can be sharp, because the pressure in the upper atmosphere (where the emission line are formed) is low and the line width is relatively narrow. This can relax the degeneracy between the contrast and the orbital inclination which would otherwise be present, like our fiducial cases for CO$_2$ and H$_2$O. As a demonstration, Figure \ref{fig:sp_postprob_USS} is the modeled thermal emission spectrum that assumes an Earth-like temperature profile with a stratospheric thermal inversion as well as an Earth-like CO$_2$ vertical profile. Note the narrow emission lines within 14-16~$\mu $m. The analysis (B) performed on this spectrum yields the constraints shown in the right panel of Figure \ref{fig:sp_postprob_USS}, assuming a late-M host star. While the total integration time required for detection is longer than our fiducial case due to the reduced line depths (corresponding to the warmer upper atmosphere), the degeneracy between the contrast and the inclination is less severe. \subsubsection{Horizontal temperature gradient} \label{sss:full_exploration} We did not include the horizontal temperature gradient of the planet. In reality, it can be substantial near the surface, depending on the heat re-distribution efficiency of the atmosphere and/or ocean. The horizontal temperature gradient would result in the time variation of the disk-averaged thermal emission as the planet rotates \citep[e.g.,][]{Knutson+2007}. While this itself can be used to characterize planet atmosphere, such variations may complicate the analysis of high-resolution spectroscopy. Fortunately, the effect of the horizontal temperature gradient will be mitigated if the data at the same phase angle are stacked and analyzed. Our analysis assumes observations near 90 degrees and 270 degrees (i.e., the same phase angle) so the impact will be limited unless there is an extreme asymmetry between the eastern and western hemispheres. \subsubsection{Full Exploration of Parameter Space} \label{sss:full_exploration} Although in this paper we attempted to constrain the contrast and inclination by fitting the mock data by specific model spectra, in reality, the spectral shape of the planet thermal emission also has to be constrained. In other words, the distribution of the molecules as well as the temperature structure should be fitting parameters as well, and the dimension of the parameter space is large. It is also possible that the constraints on certain parameters can be affected by the prior probability when exploiting the Bayesian framework. Full explorations of the parameter space and development of sophisticated retrieval techniques will need further study. Also, our analysis implicitly assumes that the parameter space explored includes the ground truth. If this is not the case, the obtained constrains should be interpreted with a special care, which is to be examined in future studies. \subsection{Significance of mid-infrared high-resolution spectroscopy for characterization of potentially habitable planets} \label{ss:comp_MRS} Because our analysis (A) follows the procedure of \citet{Snellen+2017}, we compare our results with their estimate. Their modeled CO$_2$ features in the medium-resolution spectra is detected (by nearly 4 $\sigma $) after the photon noise becomes $\sim $50~ppm. Using the same bandpass (13.2-15.8 $\mu $m) and a similar temperature profile (left panel of Figure \ref{fig:sp_postprob_USS}), our analysis (A) of high-resolution spectra was able to detect the CO$_2$ features when the photon noise is $\sim $100~ppm. This is likely a consequence of about two times sharper features at high-resolution spectra, as well as the minor features that are smoothed in the medium resolution spectra. Note that it is not trivial to compare our estimate for the integration time to the integration time estimated in \citet{Snellen+2017} for JWST/MIRI. This is because the noise of JWST/MIRI, especially beyond 12~$\mu $m, is not dominated by the photon noise of the star. Other noise, including the thermal background of the telescope, zodiacal light (with the larger aperture size), and/or the readout noise (due to the short exposure time assumed) suppresses S/N per exposure by 2-3 times compared to the idealized case. This leads to the substantial difference in the integration time required to achieve a certain precision. Thus, it is critical to suppress the background noise in order to be able to detect atmospheric molecules of nearby mid-M and late-M stars in a reasonable amount of time. \section{Summary} \label{s:summary} In this paper, we study the use of an MIR high-resolution spectrograph mounted on a cryogenic telescope for characterizing non-transiting temperate terrestrial planets orbiting M-type stars. We modeled high-resolution thermal emission spectra of an Earth-like atmosphere with CO$_2$, H$_2$O, N$_2$O, and O$_3$ assuming a simplified temperature profile composed of an dry adiabatic troposphere and an isothermal stratosphere. We show that the MIR spectral features of Earth-like atmospheres can be broader than the width of the Doppler shift, depending on the atmospheric structure. In order to reasonably identify the tiny planetary features in the combined spectra of the star and the planet, it is critical to subtract the stellar contribution precisely, as low-mass stars are particularly rich in spectral features in the MIR. For non-transiting planets, the stellar spectrum cannot be determined observationally as we always observe the star and planet together. Considering this possible difficulty, we proposed to observe the target at around $\phi = 90^{\circ}$ and at $\phi = 270^{\circ}$ and that the differential spectra are fitted by the model. This process can reduce the signal because some fraction of broad lines are self-subtracted. This effect is substantial except for O$_3$, increasing the total integration required for detection by several times. Nevertheless, the spectral features ($\mathcal{R}=30,000$) of an Earth-like planet between 12-18 $\mu $m would allow us to constrain the contrast and the orbital inclination within $\sim $1 days of total integration time, for an M8-star system within 5~parsecs. For earlier-type stars, constraining the orbital inclination may be challenging. Scaling these estimates through to Proxima Centauri b, an Earth-like abundance of CO$_2$ would be inferred in $\sim $1 day of integration. In addition to Proxima Centauri b, several potentially habitable Earth-sized planets have been already discovered within 5 parsecs, and they are the prime targets for this technique. We find that the constraints on the contrast improves with higher spectral resolution, but $\mathcal{R} \gtrsim 10,000$ do not result in significant improvement except for O$_3$ whose absorption bands are densely populated with narrow lines. However, the constraints on the inclination would benefit from the higher spectral resolution. We also find that this method is most sensitive to relatively small amount of CO$_2$ and N$_2$O (1-10$^{3}$~ppm for a 1-bar Earth-like atmosphere), where the higher abundance does not lead to better detectability. In this study, we did not include the stellar variability and systematic noise in our simulation. Because the photon noise required for the detection of these features is 100~ppm or larger, the stellar variability and systematic noise must be suppressed to these levels in order to detect the planetary features. The expected stellar variability in the mid-infrared wavelength range based on the previous photometric observations of TRAPPIST-1 with {\it Spitzer} is comparable or less than 100 ppm, which will not affect this method. Suppression of the noise sources other than the shot noise from the stellar flux (e.g., thermal background) is critical in reducing the required integration time and making the observations realistic. \acknowledgments We thank Teruyuki Hirano for helpful discussions on high-resolution spectroscopy and Klaus Pontoppidan for the information about the instrumental noise of JWST. YF is supported by Grand-in-Aid from MEXT of Japan, No.~18K13601. TM is supported by Grand-in-Aid from MEXT of Japan, No.~19H00700.	{ "redpajama_set_name": "RedPajamaArXiv" }	9,969
<?php namespace DomainCoder\Metamodel\Code\Command\Exception; class CacheNotFoundException extends \RuntimeException { }	{ "redpajama_set_name": "RedPajamaGithub" }	1,482
package org.gradle.test.performance.mediummonolithicjavaproject.p353; import org.gradle.test.performance.mediummonolithicjavaproject.p352.Production7058; import org.gradle.test.performance.mediummonolithicjavaproject.p352.Production7059; import org.junit.Test; import static org.junit.Assert.*; public class Test7061 { Production7061 objectUnderTest = new Production7061(); @Test public void testProperty0() { Production7058 value = new Production7058(); objectUnderTest.setProperty0(value); assertEquals(value, objectUnderTest.getProperty0()); } @Test public void testProperty1() { Production7059 value = new Production7059(); objectUnderTest.setProperty1(value); assertEquals(value, objectUnderTest.getProperty1()); } @Test public void testProperty2() { Production7060 value = new Production7060(); objectUnderTest.setProperty2(value); assertEquals(value, objectUnderTest.getProperty2()); } @Test public void testProperty3() { String value = "value"; objectUnderTest.setProperty3(value); assertEquals(value, objectUnderTest.getProperty3()); } @Test public void testProperty4() { String value = "value"; objectUnderTest.setProperty4(value); assertEquals(value, objectUnderTest.getProperty4()); } @Test public void testProperty5() { String value = "value"; objectUnderTest.setProperty5(value); assertEquals(value, objectUnderTest.getProperty5()); } @Test public void testProperty6() { String value = "value"; objectUnderTest.setProperty6(value); assertEquals(value, objectUnderTest.getProperty6()); } @Test public void testProperty7() { String value = "value"; objectUnderTest.setProperty7(value); assertEquals(value, objectUnderTest.getProperty7()); } @Test public void testProperty8() { String value = "value"; objectUnderTest.setProperty8(value); assertEquals(value, objectUnderTest.getProperty8()); } @Test public void testProperty9() { String value = "value"; objectUnderTest.setProperty9(value); assertEquals(value, objectUnderTest.getProperty9()); } }	{ "redpajama_set_name": "RedPajamaGithub" }	4,091
Q: PHP file upload returns empty file type I have a very simple upload image which works on one page but returns empty file name in another page if (isset($_POST['upload_img'])) { $file_name = $HTTP_POST_FILES['img']['name']; $random_digit=rand(0000000,9999999999); $new_file_name=$random_digit.$file_name.'.jpg'; $path= "images/".$new_file_name; echo"$file_name"; } and this is the html form <form role="form" action="" method="post" enctype="multipart/form-data"> <div class="form-group"> <label for="exampleInputFile">Upload Image</label> <input type="file" id="exampleInputFile" name="img"> </div> <p><a href="">Terms and conditions</a></p> <button type="submit" class="btn btn-default btn-block" name="upload_img">Upload </button> </form> Any idea whats wrong? Thank you all in advance A: Try $file_name = $_FILES['img']['name']; instead of $file_name = $HTTP_POST_FILES['img']['name']; A: uploadpage.php <?php if (isset($_FILES['img'])) { $file_name = $_FILES['img']['tmp_name']; $random_digit=rand(0000000,9999999999); $new_file_name=$random_digit.$file_name.'.jpg'; $path= "images/".$new_file_name; echo"$file_name"; } ?> index.php: <form action="uploadpage.php" method="post" enctype="multipart/form-data" > <div class="form-group"> <label for="exampleInputFile">Upload Image</label> <input type="file" id="exampleInputFile" name="img"> </div> <p><a href="">Terms and conditions</a></p> <button type="submit" class="btn btn-default btn-block" name="upload_img">Upload </button>	{ "redpajama_set_name": "RedPajamaStackExchange" }	2,319
Q: box model with anchor and button I have the following fiddle: My Button Bar My problem is a common one, though inline-block is not helping like I hoped. How do I make the anchor and the button the same size. The blue buttons need to be contained within the gray bar, they should have a small border, and be the same size, as you would totally expect from a button bar. It only really has to work in firefox and chrome. We are not bothering with IE anymore (Chrome Frame). The html looks like this: <div class="buttons"> <button class="inline-button" type="submit" name="action" value="Update"><span data-icon="icon-cog">icon</span>Save</button> <a href="#"><span data-icon="icon-cog">icon</span>Cancel</a> </div> The CSS I am using is as follows: div.buttons a, div.buttons button[type=submit] { position: static; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px; -ms-border-radius: 4px; -o-border-radius: 4px; border-radius: 4px; outline: 0; vertical-align: baseline; height: 100%; font-family: PTSansBold; font-size: 12pt; text-align: left; background-color: #051DC3; color: white; padding: 4px; margin: 3px; width: 8em; border: none; text-decoration: none; } Thanks. A: Demo Hi now try this css This is Css3 Box-sizing : border-box div.buttons a, div.buttons button[type="submit"]{ -moz-box-sizing:border-box; box-sizing:border-box; -webkit-box-sizing:border-box; vertical-align:top; } Demo more about this box-sizing A: <button class="inline-button" type="submit" name="action" value="Update"><span data-icon="icon-cog">icon</span>Save</button> <button class="inline-button" type="submit" name="action" value="Update"><span data-icon="icon-cog">icon</span>Cancel</button> You have not used the button tag for the second button. It works perfectly by adding this	{ "redpajama_set_name": "RedPajamaStackExchange" }	2,348
using System; using System.Collections.Generic; using System.Text; namespace Aggregates.Contracts { interface INeedDomainUow { UnitOfWork.IDomain Uow { get; set; } } }	{ "redpajama_set_name": "RedPajamaGithub" }	1,897
EMILY MARIE PALMER The actress pulls back the layers on her time in American Horror Story: Coven and her new role in The Last Son. Entering the film and TV industry isn't easy. With the turn-over rate high and the competition fierce, it can be difficult for most newcomers ready to enter the scene. But for Emily Marie Palmer, it all came with ease. Having first made her entrance in the iconic American Horror Story: Coven, the rising star lived out most actors dreams under neath the gaze of Ryan Murphy and Brad Falchuk. "We shot everything in one very long night shoot," the actress recalls. "I remember catching glimpses of myself in my beautiful little pilgrim dress — looking like a witch straight out of Salem — while the director and crew devised creative ways to shoot all of the special effects in the style of an old silent film, and just feeling like I was having the most wonderful dream. It certainly didn't feel like real life. I have such giddy, nostalgic memories of my short time on American Horror Story. What a dream." Teaming up with Sam Worthington and Machine Gun Kelly for her next project The Last Son, the actress tackles a more challenging role, taking on the curious yet calming Megan as she deals with her family's struggle to survive in a harsh environment while venturing into a world beyond her own. Describing the role as "exceptionally challenging" yet "fortunate", we caught up with the actress virtually breaking down her role as Megan and filming in such unpredictable weather environments. Check out the interview below… Hey Emily, how are you? How has this past year been for you? This past year has had its challenges, certainly, but I am very grateful to say that on the whole it has been really lovely, and I feel so lucky. The film industry was hit hard by COVID, what has been the biggest change you seen in the industry? Yes, sadly, it was. Besides the very obvious new realities of testing, vaccination, and on set covid safety, I think I experienced less dramatic changes than a lot of my fellow actors because I don't currently live in Los Angeles. It was already quite normal for me to submit self-tapes (instead of auditioning in person) and I already had a little studio set-up in my home for that purpose. So, in that respect, I definitely feel that I was one of the lucky ones because I had one less thing to worry about adapting to during a time of so much chaos and change. How did you first get into acting, what started the interest? I've been so fascinated with life, people, and stories for as long as I can remember. I was constantly playing make-believe as a child, and as soon as I learned to read and write I was devouring books and writing my own stories. My mother's roots are in Appalachia, and so I had the benefit of growing up hearing her sing these beautiful, rich, old folk songs as lullabies… and gosh, I loved them so much. They made such a deep impression on me. So, I think I've always been naturally drawn to art, and to storytelling. Some of my siblings are creative as well, and so the first acting I ever did was in the plays we would write and put on together — a bit like the March sisters in Little Women. And then you went to study in Paris, before ultimately going to purse acting full time, what made you decide this? Even before I went to Paris, I think I always knew that I would return to the U.S. and pursue a career in film. I was the first person in my immediate family to venture outside the United States, and I felt quite brave (and extremely lucky) setting off on that adventure at 18 years old. While I was there, I took a few days to make a sort of "pilgrimage" to Domrémy-la-Pucelle — which is the tiny village in northeastern France where Joan of Arc was born. She was such an important childhood hero of mine, and during those few days I spent wandering alone around her village and the surrounding hills, I just felt my heart filling up with so much longing, and courage, and determination to do the work that I felt called to in this world — no matter how wild or unattainable it seemed. At the time, for me, that was acting. And, I should add that, having grown up homeschooled in a small town in Tennessee, everything about the film industry felt completely foreign and unattainable to me at the time. Working on American Horror Story: Coven, what was the most memorable thing about filming? Oh, I loved that whole experience. It was actually the first role I ever booked after being signed by my agent. Because the show was (and is) so popular, they kept the scripts under lock and key, and I didn't find out what I was going to be shooting until I arrived on set. We shot everything in one very long night shoot, and I remember catching glimpses of myself in my beautiful little pilgrim dress — looking like a witch straight out of Salem — while the director and crew devised creative ways to shoot all of the special effects in the style of an old silent film, and just feeling like I was having the most wonderful dream. It certainly didn't feel like real life. I have such giddy, nostalgic memories of my short time on American Horror Story. What a dream. And now you're working with Sam Worthington and Machine Gun Kelly on The Last Son, how did you get involved on the project and how did you approach the character? I was really fortunate to have connected with one of the producers, Andre Relis, in 2018 while playing a supporting role in another one of his films, Grand Isle. Andre reached out to me about The Last Son last year, and when I read the script, I instantly connected with the character. I feel less as if I had to approach Megan, and more that I had simply to ease into her. I deeply identified with her innocence, and her hungry curiosity. When the story of the film takes place, she is coming of age in such a harsh and violent world. She has grown up isolated, and her family's daily struggle to survive in the harsh frontier has made her strong, but there is a profound freshness about her, and a childlike curiosity for the world beyond her isolated fields and forests. Looking back on the filming process, was there any challenges? The weather was exceptionally challenging. When I first arrived at the Montana ranch where we shot most of the film, the weather was warm, and sunny, and beautiful. Less than two weeks later, the temperature plummeted below zero, and it snowed non-stop. Sometimes there would be these kinds of dramatic shifts within just a few hours! We were shooting in a mountain range called "The Crazies" — and some say they derived their name from their highly unpredictable weather patterns. Who would you love to work with in the industry? I'd love to work with Jennifer Kent — her film, The Nightingale, astounds me. The first time I saw it, I sat silent — open-mouthed with tears streaming down my cheeks — through the entire ending credits. The love, dedication and depth of research that went into its creation are just incredible to me. And Robert Eggers! I love history, and I so deeply appreciate directors who give such attentive care to the historical accuracy of their projects like he does. Ah! And Brit Marling. What an artist. It would be an honor to step into any world she creates. What are you most excited for? I am really excited for 2022. I don't know why, but I have a very good feeling about it. I'm using these last few weeks of this year, when the industry tends to quiet down a little, to finish a script I'm working on right now that I'm terribly excited about. It's a gritty thriller set in the Ozark Mountains in the 1850s, and it's quite a ride. New Noise: DDG Harry Potter 20th Anniversary The wait is over. The first trailer for season two of Netflix's Emily in Paris, has finally dropped. Zendaya and John David Washington shot a secret Netflix film during the pandemic with Euphoria's creator. Emily in Paris kicks off production for season two – and playwright Jeremy O. Harris is set to join the cast. 60 Seconds With: Palmer//Harding The 100 actress Marie Avgeropoulos gets candid on the final season of the cult TV show. Profile: Emily Faulstich Calling all Emily In Paris fans! Netflix has turned to sustainable brand, My Beachy Side, to help them celebrate the show's highly-anticipated sequel. NEW THREADS: MARIE YAT Sally Boy takes us on a journey through his life with his emotional new video "Marie". TikTok sensation Emily Vu takes us through R&B-tinged soundscape for her blissful new single "LILA". Premiere: Anne-Marie – "Gemini" Marie Antoinette Afternoon Tea at Pont St. Model Behaviour: Marie-Louise London-based electronic trio RITUAL have dropped dark visuals about confusing followers for friends. Indie-pop singer-songwriter Emily Burns pens a love letter to her rural hometown with new track, "My Town". Rising Norwegian artist Marie Noreger delivers a left-field R&B meditation on society's superficialities. After, Five Feet Apart and Teen Spirit: Here are the best in film and TV trailers right now. Chloé Zhao makes history, Minari's Yuh-Jung Youn calls out the British public, and Daniel Kaluuya celebrates by taking a bath? Everything that went down at this year's BAFTAs. Moonlight's Barry Jenkins returns with gripping series The Underground Railroad, and we're off into space for the new Fast 9: all new trailers in this week's Screen Wonders. Get ready to re-enter the stimulation as Warner Bros drop the highly anticipated trailer for The Matrix: Resurrections.	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	7,348
Q: Accessing underprivileged stack by weakening the CPL with RPL in intel microprocessor As far as I know, when we want to access a stack segment, the DPL of the descriptor should be equal to max(RPL,CPL) where RPL is of the SS segment register, and CPL means current privilege level. So can we access an underprivileged stack by weakening our CPL with the RPL? For example, CPL=0, RPL=3 for the SS segment register. So we should be able to access the stack of PL 3. Is it true? A: No. Section 5.7 of the Intel SDM is clear Privilege level checking also occurs when the SS register is loaded with the segment selector for a stack segment. Here all privilege levels related to the stack segment must match the CPL; that is, the CPL, the RPL of the stack segment selector, and the DPL of the stack-segment descriptor must be the same. If the RPL and DPL are not equal to the CPL, a general-protection exception (#GP) is generated. The rationale is that privileged code cannot afford to share a stack with unprivileged code for security reasons - each privilege has its own stack as defined in the TSS. You can access the stack with a data segment register (e.g. through ds), in that case one must have max(CPL, RPL) <= DPL (weakened CPL is still more or equally privileged as the DPL of the segment).	{ "redpajama_set_name": "RedPajamaStackExchange" }	794
International Organisations Interest Group European Society of International Law PRIVIGO: Intergovernmental Organizations between Mission and Market By Jan Klabbers When international organizations were first created, during the nineteenth century and the first half of the twentieth century, they were expected to operate in a vacuum. International organizations such as the International Telegraphic Union or the International Institute for Agriculture were expected to perform tasks delegated to them by their member states, and in the service of those same member states. It never occurred to the founding fathers that the organizations would, somehow, entertain relations with anyone other than those member states. This is manifested quite clearly in various respects. Few, if any, were given international legal personality by their member states. Few, if any, were given treaty-making powers by their member states. And for many decades, even the capacity of international organizations to act under international law was considered highly questionable. Since much of today's legal framework for international organizations was developed in those early years, during the nineteenth century and the first half of the twentieth century, it stands to reason that much of this legal framework is based on the assumption that indeed, international organizations do operate in a vacuum. They are established in order to perform functions assigned to them by their member states, and for the benefit of those same member states. Whether the assumption ever was accurate or not is beside the point – what matters is that it was this assumption that the law was based on. As a result, international organizations law has fairly well-developed doctrines on powers and competences; knows an intricate framework of privileges and immunities vis-à-vis the member states; and has some reasonably well-developed rules relating to admission or expulsion of member states. But the vacuum drawn around the relationship between the organization and its member states also entails that the law offers little clarity when it comes to developments within the organization: the relations between organs of the same organization, or the law of the international civil service. And most visibly, the law has few resources to address the relations between international organizations and the outside world. For, as it turns out, the vacuum has been pierced: international organizations engage in all sorts of relations with agents and actors other than their member states. And arguably, the vacuum was pierced as early as the 1920s with the creation of the League of Nations and its involvement with mandate territories, and the International Labour Organization and its attention for individuals and for representation by capital and labour in member state delegations.[1] And following World War II, the piercing has been inescapable. The UN was recognized in 1949 to have international legal personality. The same UN established a treaty-practice, brilliantly documented by Rosenne at the Hague Academy in the early 1950s. The Bretton Woods institutions, likewise, were concluding treaties, and were considered separate enough from their member states to deny any member state liability for their actions. NATO was created to act against a common enemy; many organizations were granted some treaty-making powers; organizations started to work together, and during the 1980s an entire convention on treaties concluded with and between international organizations saw the light. Even if this Convention still awaits entry into force, it is undisputed that international organizations can and do conclude treaties, and that they have a generally recognized capacity to do so. In other words: the vacuum has been broken, yet the underlying legal framework has remained unaffected, and that helps explain why discussions on the accountability of international organizations tend to be difficult. The law never anticipated them to act externally, and thus never anticipated (let alone accommodated) any kind of accountability towards actors other than their member states. The dominant legal approach has it that international organizations are subject to control only by their member states; and this, in turn, ignores that the relevant accountability relations tend to be not with member states, but with third parties: non-member states, affected populations, individuals, other international organizations, and the private sector. The latter of these (the private sector) forms the object of study of the research project PRIVIGO, funded by the European Research Council (grant no. 883417), and running from January 2021 until December 2025. The idea behind it is twofold. First, there has been little study of how the relations between international organizations and the private sector are organized and how they affect obvious issues of governance, decision-making and accountability. Hence, a first goal of PRIVIGO is to map these relations and the various legal solutions that have been developed in legal practice, and to do so in eight policy domains, ranging from arms control to food security, and from human settlement to global health. In addition, there is an expectation that the piercing of the vacuum must have theoretical consequences: it is something incongruous in the realization that the law is based on assumptions that may have had some limited validity a century ago, but have been outdated for at least seventy or eighty years. Put differently, the law needs to be re-thought from the ground up, partly by re-thinking its theoretical foundations and the epistemic assumptions on which it based. This involves an investigation of the relations between international organizations and the private sector – although it may benefit also from studying the other external relations of international organizations, in particular with non-member states and with each other. PRIVIGO conceptualizes relations with the private sector in three ways: input, throughput and output, and is currently hiring three post-doctoral researchers for a period of four years each.[2] Under 'input', one of these post-doctoral researchers is expected to study the extent to which international organizations are funded by the private sector, and what this entails for how organizations are governed, how decisions are made, and how accountability is regulated – if at all. A second post-doctoral researcher will investigate how international organizations themselves act on markets, not only when procuring materials but also when offering their services for remuneration. And the third post-doctoral researcher will be expected to investigate the re-distributive role of decisions of international organizations for private actors. For, as the covid-19 crisis has abundantly illustrated, rules set by international organizations and decisions of these organizations tend to have distributive or re-distributive effects. A decision to declare a public health emergency of international concern may lead to the bankruptcy of airlines, to hotels and restaurants closing down, to football clubs having to cut costs, to many individuals losing their jobs, to other ill people having their treatment postponed. And yet, a decision not to declare such a situation may directly cost lives. Studying relations with the private sector may all the more instructive as there is an obvious tension between private sector involvement ('the market') and the oft-assumed public ethos ('the mission') of international organizations: the existence of international organizations and of the rules relating to them are often justified under reference to the public nature of their tasks. When international organizations are expected to help bring about the 'salvation of mankind', as someone once put, then surely intense and intensive private sector involvement cannot be without theoretical ramifications. The largely empirical, doctrinal work of the post-doctoral researchers in turn is expected to feed into further thinking about the theoretical foundations of international organizations law. The law as it stands was formed over a century ago, and had largely been solidified when Paul Reinsch published his Public International Unions in 1911. It has been fine-tuned in the intervening years, but its foundations have essentially remained in place since Reinsch's days. Given the increasingly relevant role of international organizations (they are the main institutions of global governance, exercising authority and monitoring it), it is strange, to put it mildly, that they still operate on the basis of an intellectual framework developed over a century ago. No wonder that discussions on whether the World Bank can be blamed for human rights violations, or whether the UN can be held accountable for bringing cholera to Haiti, or whether the International Monetary Fund carries responsibility for austerity policies, seem to go on forever… The law as it stands lacks the resources to address these and many other issues; it is hoped that PRIVIGO will contribute to the renewal of international organizations by zooming in on how organizations relate to an important part of the world around them. [1] For a fuller statement, see Jan Klabbers, 'An Accidental Revolution: The ILO and the Opening Up of International Law', in Tarja Halonen and Ulla Liukkunen (eds.), International Labour Organization and Global Social Governance (Dordrecht: Springer, 2020), 123-140. [2] For more details: https://www.helsinki.fi/en/open-positions/three-3-post-doctoral-researchers-in-international-organizations-law Published by IG-IO ESIL View all posts by IG-IO ESIL Announcement, IG-IO News, News, Research Jobs, Public-Private, Research Project International Organizations and State Sovereignty A Relationship in Flux On International Institutional Law, its Pedagogy and the Turn to Alternative Approaches Other ESIL Blogs Interest Group on International Courts and Tribunals Interest Group on Migration and Refugee Law ESIL LAWSEA IG igfemlaw.wordpress.com/ IGPS Interest Group on International Legal Theory and Philosophy EUGlobal The Law of the Sea Interest Group of the European Society of International Law Interest Group on Peace and Security / Groupe de Réflexion sur la Paix et la Sécurité	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	8,595
Jungle Fight 1 foi o primeiro evento de MMA do Jungle Fight, criado por Wallid Ismail. Foi realizado dia 13 de Setembro de 2003 no Ariau Amazon Towers Convention Center em Manaus, Amazonas. Resultados 2003 nas artes marciais mistas Esporte no Brasil em 2003	{ "redpajama_set_name": "RedPajamaWikipedia" }	5,635
$(document).ready(function(){ $(".iCheck-helper").click( fetchSearchResults ); $("#mentors").hide(); $("#job-applicants").hide(); }); function fetchSearchResults(){ var topics = $("#search-button-data-topics").val(); var sectors = $("#search-button-data-sectors").val(); var topic_count = topics.length; var sector_count = sectors.length; var action = "largeSearch"; $.ajax('/search', { type: "POST", data:{topics: topics, sectors: sectors, action: action}, success: handleResponse }); } function handleResponse(data){ var mentorContainer = []; var jobApplicantContainer = []; var user = $.parseJSON(data); if (user.length == 0){ $(".search-result-mentor ul").html("no result to show at the moment"); $(".search-result-job-applicant ul").html("no result to show at the moment"); $("#mentors").hide(); $("#job-applicants").hide(); $("#no-of-mentors p").html("Mentors ("+mentorContainer.length+")"); $("#no-of-applicants p").html("Job applicants ("+jobApplicantContainer.length+")"); } else { user.forEach(function(item){ var foundItem =[ '<li class="span3 found-result-mentor"><figure><div>', '<img src="',item.image, '" alt="img04"></div>', '<figcaption>', '<div><h3 class="name-mentor">', item.name,'</h3><span>',item.summary, '</span></div><a href="#myModal" class="user_id" data-user_id="',item.id,'" data-toggle="modal">Take a look</a></figcaption></figure></li>' ].join("\n"); if (item.profile == "Mentor"){ mentorContainer.push(foundItem); $("#no-of-mentors").html("<a href='#mentors'><p>Mentors ("+mentorContainer.length+")</p></a>"); $("#mentors").show(); } else if(item.profile == "Job Applicant"){ jobApplicantContainer.push(foundItem); $("#no-of-applicants").html("<a href='#job-applicants'>Job applicants ("+jobApplicantContainer.length+")</a>"); $("#job-applicants").show(); } }); $(".search-result-mentor ul").html(mentorContainer.join("\n")); $(".search-result-job-applicant ul").html(jobApplicantContainer.join("\n")); } $(".user_id").click( function(){ var user_id = $(this).attr("data-user_id"); fetchFullProfile(user_id); fetchSearchResults(); } ); $(".fav-image").click( function(){ var favorite_id = $(this).attr("data-user_id").trim(); var favorite_type = $(this).attr("data-favorite-type").trim(); var favorite_action = getFavoriteAction($(this)); favorite(favorite_id, favorite_type, favorite_action); } ); } function getValue(favorite){ favorite.success( function (data) { return data; } ); } function fetchFullProfile(user_id){ var user_id = user_id.trim(); action = "getFullProfile"; $.ajax('/search', { type: "POST", data:{user_id: user_id, action: action}, success: handleFullProfileResponse } ); } var count = 0; var new_comment = 0; var edit_comment = false var rated = false; function handleFullProfileResponse(data) { $("#myModalLabel").html(data); $('.des').popover(); drawChart(); $('#rateit5').rateit({ max: 5, step: 0.5, backingfld: '#backing6' }); $("#rateit5").rateit("value", Number($("#modal-pic").attr("rating"))); $(".rateit").click( function(e){ if (rated == false){ var rating = $("#rateit-range-2").attr("aria-valuenow"); rated = true; rate(rating,rated); } } ); $("#comment-submit-button").bind('click', function () { submitComment($(this)); } ); bindObjects(); } function drawChart() { setTimeout(function(){google.load('visualization', '1', {'callback': drawChart, 'packages':['corechart']})}, 2000); function drawChart() { var data = google.visualization.arrayToDataTable([ ['Task', 'Hours Contributed'], ['Hours Completed', Number($("#piechart").attr("data-committed"))], ['Hours Left', Number($("#piechart").attr("data-left"))] ]); var options = { title: 'Hours I Have Contributed' }; var chart = new google.visualization.PieChart(document.getElementById('piechart')); chart.draw(data, options); } } //called when a comment is being submitted function submitComment(comment_item){ var action = "new"; var type = "user"; var content = $("#comment-textarea").val(); var entity_id = $("#modal-pic").attr("data-user-id"); comment(action, type, content , entity_id, ""); } //event listener for editing and deleting comment items function bindObjects(){ $('.edit').bind("click", function(){ editComment( $(this) )} ); $('.delete').bind("click", function(){ confirmDelete( $(this) )} ); } // dialogue box that pops up to confirm deletion of a comment function confirmDelete(comment_object){ if( confirm("This comment will be permanently deleted. Do you want to proceed?") ){ deleteComment(comment_object); $("#new-comment-container").show(); } } function deleteComment(comment_object){ var commentor_id = comment_object.attr("data-commentor-id").trim(); var comment_item = $("."+commentor_id); var comment_id = comment_item.attr("data-comment-id"); comment_item.remove(); var action = "delete"; var type = "user"; var entity_id = $("#modal-pic").attr("data-user-id"); comment(action, type, "", entity_id, comment_id); }; function editComment(comment_object){ $("#new-comment-container").show(); var commentor_id = comment_object.attr("data-commentor-id").trim(); var comment_item = $("."+commentor_id); var comment_id = comment_item.attr("data-comment-id"); var content = comment_item.find(".comment-content").text().trim(); $("#comment-textarea").val(content); $("#comment-submit-button").attr("id", "comment-edit-button"); $("#comment-edit-button").click( function(){ var action = "edit"; var type = "user"; var new_content = $("#comment-textarea").val(); var entity_id = $("#modal-pic").attr("data-user-id"); comment(action, type, new_content , entity_id, comment_id); $("#new-comment-container").hide(); $("#comment-edit-button").attr("id", "comment-submit-button"); } ); } //making ajax call to server for new comment function comment(comment_action, type, content, entity_id, comment_id){ var action = "comment" var comment = JSON.stringify({ "content":content, "entity_id":entity_id, "type":type, "comment_action":comment_action, "comment_id": comment_id }); console.log(new_comment); $.ajax("/search",{ type: "POST", data: { action: action, comment: comment}, success: handleCommentResponse }); } function handleCommentResponse (data) { console.log(data); fetchFullProfile($("#modal-pic").attr("data-user-id")); } //making ajax call to confirm rating function rate(rating,rated){ var rating_id = $("#modal-pic").attr("data-user-id").trim(); var rating_type = $("#modal-pic").attr("data-favorite-type").trim(); var rating_value = rating; var action = "rate"; $.ajax("/search",{ type: "POST", data: {action:action, rating_id: rating_id, rating_type: rating_type, rating_value: rating_value}, success: handleRateResponse }) } //handling rating response from server function handleRateResponse(data){ rated = false; } function favorite(favorite_id, favorite_type, favorite_action){ var action = "favorite"; $.ajax("/search",{ type: "POST", data: { action: action, favorite_action: favorite_action, favorite_id: favorite_id, favorite_type: favorite_type}, success: handleFavoriteResponse }); } function handleFavoriteResponse(data){ var result = $.parseJSON(data); if (result.message == true){ $("#no-of-favorites").html("My Favorites (" +result.value+")"); } } function getFavoriteAction(object){ var status = object.attr("data-fav-status").trim(); if (status == "true"){ object.attr("src", "scripts/img/unlike.png"); object.attr("data-fav-status", "false"); return "unlike"; } else if(status == "false"){ object.attr("src", "scripts/img/like.png"); object.attr("data-fav-status", "true"); return "like"; } }	{ "redpajama_set_name": "RedPajamaGithub" }	7,670
Q: processing 1GB file pyspark in my HDP cluster java.lang.OutOfMemoryError: Java heap space Code $pyspark $json_file = sqlContext.read.json(sc.wholeTextFiles('/user/admin/emp/*').values()) Error : 18/01/08 15:34:36 ERROR Utils: Uncaught exception in thread stdout writer for python2.7 java.lang.OutOfMemoryError: Java heap space I tried to process 1GB file using pyspark, then as a result my application crashes and I face Java heap space. How do I look at solving that	{ "redpajama_set_name": "RedPajamaStackExchange" }	8,645
Q: Function with range equal to whole reals on every open set There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$. I want to generalize this in a way to get a function which is not just unbounded on every open set, but whose range is equal to $\mathbb{R}$ on every open set. The latter construction clearly doesn't work. I'm interested whether such function exists and if it exists is there any constructive way to define it? A: This is not really a new solution, but just a way to see one can come up with Jim's answer. The problem is equivalent to finding an equivalence relation on $\mathbb{R}$ such that each equivalence class is dense and there are $2^{\aleph_0}$ equivalence classes. To see this, suppose you have such an equivalence class $\equiv$. Consider the natural map $\pi: \mathbb{R} \to \mathbb{R}/\equiv$. Clearly the pre-image of every point is dense and now, you can post-compose $\pi$ with a bijection between $\mathbb{R}/\equiv $ and $\mathbb{R}$. The converse is similar. Now, as $\mathbb{R}$ has the structure of an additive group, one can define equivalence relations by using subgroups: $x \equiv y$ iff $x-y$ is in a given subgroup $H$. Now if you use $H=\mathbb{Q}$, then you get the answer given by Jim Belk. You can use $H$ to be the subgroup of $2$-adic rationals. Then the $x$ is equivalent to $y$, if all but finitely many of their binary digits are equal, from which you can define A: As suggested by Gerry, Example 27 in Chapter 8 of Gelbaum and Olmstead's Counterexamples in Analysis is such a function. A: A couple more ways of thinking about it. (There are connections with some of the answers above.) * The graph of the function has to have the property that it intersects every horizontal line segment of positive length in the plane. Well-order these line segments such that each one has fewer predecessors than the cardinality of the reals, and then stick a point in them one at a time. At each stage, one has put fewer points into the graph than there are points in a line segment, so there will be points in the segment that are not vertically above or below points that are already chosen. When you've covered all the line segments, choose the remainder of the function arbitrarily. Enumerate all open intervals with rational end points. Now inductively create a graph as follows. Pick the first interval, and take a copy of the Cantor set inside it. Biject that copy of the Cantor set to the reals. Pick the second interval and find inside it an open interval disjoint from the Cantor set chosen earlier. Inside that, take a copy of the Cantor set and biject it to the reals. Keep going. The complement of the set where you've defined the function so far is always open and dense, so you can always continue. (It goes without saying that there's nothing special about copies of the Cantor set: any class of sets with cardinality that of the reals that is closed under countable intersections and does not include any open intervals will do the job.) A: For a non-constructive solution, let $\pi : \mathbb{R} \to \mathbb{R}/\mathbb{Q}$ be the projection homomorphism, and let $g : \mathbb{R}/\mathbb{Q} \to \mathbb{R}$ be a bijection. Then the composition $g\circ \pi$ has the desired property. A: See Conway's base 13 function. A: My instinct is to reach for Gelbaum and Olmstead, Counterexamples in Analysis, but it's in my office, and I'm at home. A: Once we have the idea of prodding the decimal expansion, there are any number of things we can do. Here are a few. We may assume we're starting with $0.a_1a_2a_3...$, e.g. by taking the fractional part of the input. If any of the sums/limits mentioned below fails to exist, map the input to $0$. * We construct the decimal digits of the image $b_0.b_1b_2b_3...$ one at a time. Consider the alternating sum of the $a_i$ for $i$ odd. If this converges (i.e. these $a_i$ are $0$ from some point on), then define $b_0$ to be this. Next, look at the alternating sum of the $a_i$ with $i\equiv 2 \mod 4$. If this converges to a number in $\{0,...,9\}$, then define $b_1$ to be this. Then construct $b_2$ using the $a_i$ with $i\equiv 4 \mod 8$, and so on. Map the number to $\sum \frac{(-1)^{a_n}}{n}$. *Map the number to $\lim_{n\to\infty} (a_1+...+a_n)/n$. This will give us results only in $[0,9]$, but that's sufficient. In each case, whatever tiny interval we're forced to start in, we can always get any output we like by using later digits. A: You can even say something much stronger. As Gowers points out, you can reformulate the question as: Does there exist a function that intersect any vertical line, in any open interval? Here you can substitute "vertical line" with "continuous function" and "open interval" with "set with positive measure". The construction is exactly as Gowers point 1. You only have to use that the set of continuous function and the Borel $\sigma$-algebra each have the same cardinality as $\mathbb{R}$. I wonder if you could substitute "continuous function" with "measureable function" in the above? How many measureable function are there? A: Here is an explicit construction that uses only elementary functions and limits: define $f(x)=\lim_{n\rightarrow \infty}(n!\pi x)$ when that limit exists, and e.g. 0 otherwise. Notice that this $f$ is periodic with every rational number as its period. So it is enough to show that $f$ is surjective (and then you can take any preimge of any point and move it into any interval). To show that it's surjective, let $y\in \mathbb{R}$, and take $r\in \mathbb{R}$ to be such that $\tan(r\pi)=y$. Then one can check that the limit $\sum_{n=0}^\infty \frac{\lfloor rn\rfloor}{n!}$ is mapped by $f$ to $y$.	{ "redpajama_set_name": "RedPajamaStackExchange" }	3,353
{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/what-length-solid-cylinder-2-cm-diameter-must-be-taken-recast-hollow-cylinder-external-diameter-20-cm-025-cm-thickand-15-cm-long-volume-of-a-cylinder_65086","text":"What Length of Solid Cylinder 2 Cm in Diameter Must Be Taken to Recast into a Hollow Cylinder of External Diameter 20 Cm, 0.25 Cm Thickand 15 Cm Long? - Mathematics\n\nSum\n\nWhat length of solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of external diameter 20 cm, 0.25 cm thickand 15 cm long?\n\nSolution\n\nExternal diameter of hollow cylinder=20 cm\n\nThickness =0.25 cm\n\nLength of cylinder (h)=15 cm\n\n\u2234\"volume \"=pih(R^2-r^2)=pixx15(10^2-9.75^2)\n\n=15pi(100-95.0625)cm^3\n\n= 15pixx4.9375 cm^3\n\nDiameter = 2cm\n\nLet h be the lenght\n\n\"then, volume\"= pir^2h=pi(1xx1)h=pih\n\nNow, according to given condition:\n\npih=15pixx4.9375\n\n\u21d2 h=15xx4.9375\n\n\u21d2h=74.0625\n\n\"Length of cylinder\" =74.0625 cm\n\nIs there an error in this question or solution?\n\nAPPEARS IN\n\nSelina Concise Maths Class 10 ICSE\nChapter 20 Cylinder, Cone and Sphere\nExercise 20 (A) \| Q 5 \| Page 297","date":"2021-04-11 00:12:07","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\": 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\": 0.7358803749084473, \"perplexity\": 6673.1783857517685}, \"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-2021-17\/segments\/1618038060603.10\/warc\/CC-MAIN-20210411000036-20210411030036-00364.warc.gz\"}"}	null	null
{"url":"https:\/\/drewtyre.rbind.io\/post\/testing-out-latex\/","text":"# Testing out latex\n\nNow I want to know if I can use $$4 \/ 5$$ inline latex and\n\n$N_{t+1} = \\lambda N_t$\n\nand R code.\n\nplot(rnorm(10),rnorm(10))","date":"2019-04-25 04:26:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"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\": 0.9956227540969849, \"perplexity\": 3188.0443648991463}, \"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-2019-18\/segments\/1555578681624.79\/warc\/CC-MAIN-20190425034241-20190425060241-00180.warc.gz\"}"}	null	null
New White House guidance downplays important AI harms Alex Engler Tuesday, December 8, 2020 Following a February 2019 executive order, the U.S. Office of Management and Budget (OMB) issued its final guidance on the regulation of artificial intelligence (AI) on November 17, 2020. This document presents the U.S. government strategy towards AI oversight, and as such it deserves careful scrutiny. The White House guidance is reasoned and reflects a nuanced understanding of AI; however, there are also real causes for concern regarding the long-term impact on the regulation of AI. To start, there are many positive aspects of the White House AI Regulatory guidance. Its scope is reasonable, recognizing that the regulation of private sector AI is fundamentally different than the government's deployment of AI systems, leaving the latter for a separate document. It also appropriately argues for an approach to AI where regulation is specific to the sector and AI application type, rather than wide sweeping policies that make no sense across the broad spectrum of AI use. The OMB document takes a risk-based approach, suggesting the prioritization of stronger protections for AI systems that demonstrate the potential for higher risk. The document also calls for federal agencies to work with standards bodies, specifically asking them to adhere to the National Institute of Standards and Technology's federal engagement plan for developing AI technical standards. Other aspects of the guidance are less positive. The tone of the document is very insistent on the promise of AI development and innovation, especially for economic growth, writing "promoting innovation and growth of AI is a high priority of the U.S. government." This is understandable, as the digital economy accounts for over 9% of GDP and was growing at 6.8% per year before the COVID-19 pandemic. AI is a big part of that sector, and is expanding into many other sectors, too. Yet while the economic value of AI is important, the guidance is overly focused on arguing that regulation should not hamper its innovation and deployment. The document notes a series of "non-regulatory approaches to AI" and has a section on "reducing barriers to the deployment and use of AI," but it is not balanced by a broad contextualization of AI harms. How artificial intelligence is transforming the world Darrell M. West and John R. Allen How the Biden administration should tackle AI oversight The AI Bill of Rights makes uneven progress on algorithmic protections The OMB guidance prominently states "that many AI applications do not necessarily raise novel issues." This claim is partially true, as the government does not need to be concerned with many AI applications. Yet the reverse is also true: many private sector AI applications absolutely do raise novel issues. AI systems can systematize discrimination and redistribute power away from consumers and front-line employees. AI systems can also enable large-scale corporate surveillance. They can do all this, while making processes harder to understand for individuals, and potentially undermining their legal recourse for harms. These issues are not entirely ignored in the White House document. It lists ten principles for the stewardship of AI applications, which are consistent with recommendations of leading ethical AI experts. That list includes: "Public Trust in AI", "Public Participation", "Risk Assessment and Management", "Fairness and Non-Discrimination", and "Disclosure and Transparency." The problem is that these criteria are framed as a checklist that must be worked through before agencies can implement any new rules on AI. Directly before the list of principles, the guidance states "Agencies should consider new regulation only … in light of the foregoing section … that Federal Regulation is necessary." Paired with the broader anti-regulatory framing of the document, this suggests an intent to preempt regulatory actions. Preemption is problematic because we already know there are areas which require tougher enforcement and regulation. The Food and Drug Administration is considering how to adapt their rules to ensure the safety of AI-enhanced medical devices while still allowing them to be updated with new data. The Department of Labor and the Equal Employment Opportunity Commission will have to look at how algorithmic tools affect worker compensation, workplace safety, and hiring processes. Health and Human Services has to learn to enforce legal non-discrimination protections on algorithmically allocated health services. Similarly, Department of Transportation needs new rules for ensuring safety of autonomous vehicles. In the absence of updated regulations and enforcement processes, the status quo makes it easy to skirt the law by using algorithms. Overseeing algorithms requires new ideas, technical expertise, and additional capacity, and the White House and OMB should be encouraging agencies to tackle the new risks of AI. Unfortunately, in a finalized rule regarding disparate impact standards, Housing and Urban Development ignored the new challenges of AI, making it impossible for plaintiffs to prove they were discriminated against by algorithms. Government Policy toward Open Source Software Edited by Robert W. Hahn Edited by Robert W. Crandall and James H. Alleman Going Mobile By Darrell M. West Beyond the direct influence on agency rulemaking, this guidance can also affect the future role of OMB, especially its regulatory review arm, the Office of Information and Regulatory Affairs (OIRA). At times in its forty-year history, OIRA has played an active regulatory "gatekeeper" role. This role is especially focused on 'significant regulator actions,' which includes rules that are estimated to have an economic impact of over $100 million or alternatively regulations which "raise novel legal or policy issues." While some AI regulations will likely meet the economic impact criteria, surely many more will raise novel issues in law and policy. The relatively small OIRA staff, split into groups assigned to different federal agencies, may not have the AI expertise necessary to weigh in effectively on these emerging new rules. It is uncertain exactly how this issue will develop, but it is possible that the White House guidance, as interpreted by OIRA, creates new compliance burdens. It may not take long to find out what impact this guidance will have. Perhaps its most valuable contribution of this guidance is that it calls for federal agencies to provide compliance plans within six months (by May 17, 2021). With the influx of new appointments from the Biden administration, it is possible that this document drives new ideas and an unprecedented degree of action to build sensible and effective AI regulations. Hopefully, OIRA will work primarily to foster knowledge exchange and cooperation between agencies, as Cass Sunstein has argued. This may be the case. Still, there is cause for concern in the framing of this document, and its effect is going to depend significantly on how it is interpreted and approached by future agency staff. It is hard to imagine that changing this guidance is going to be a leading priority of the Biden White House, given all its other pressing problems. Yet there is a real risk that this document becomes a force for maintaining the status quo, as opposed to addressing serious AI harms. More on Technology & Innovation What to expect from a GOP House majority on big tech, broadband, China, and 5G Nicol Turner Lee and Darrell M. West Education systems transformation symposium Work and meaning in the age of AI Daniel Susskind	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	9,686
var locatorParts = locator.split('\|'); var cssAncestor = locatorParts[0]; var linkText = locatorParts[1]; var matchingElements = cssQuery(cssAncestor, inDocument); var candidateLinks = matchingElements.collect(function(ancestor){ var links = ancestor.getElementsByTagName('a'); return $A(links).select(function(candidateLink) { return PatternMatcher.matches(linkText, getText(candidateLink)); }); }).flatten().compact(); if (candidateLinks.length == 0) { return null; } candidateLinks = candidateLinks.sortBy(function(s) { return s.length * -1; }); //reverse length sort return candidateLinks.first();	{ "redpajama_set_name": "RedPajamaGithub" }	9,475
Q: Корректное извлечение текста из textarea Здравствуйте. У меня есть поле для ввода текста, но когда я передаю его значение в переменную, то оно не совсем корректно передается. Суть в том, что если например поставить перенос строки в textarea ентером то в переменную оно передается как одна строчка без переноса. И ещё если вводить текст до конца поля он автоматически переносится, но в переменную всё равно идет просто строка и на следующем выходе в блок помещается текст который вылазит за его границы. Может есть какие-то корректные способы извлечения значения из textarea? Или можно ли как-то задать, что если текст переходит границу блока чтобы он делал перенос строки. Например заносился в div и чтобы этот div не мог выходить за границу родителя. Спасибо A: const textarea_to_div = e => div.innerHTML=textarea.value.replace(/\n/g,'<br/>'); textarea_to_div(); textarea.oninput = textarea_to_div; <textarea id='textarea'> Перенос переносов. </textarea> <div id='div'></div> A: Попробуйте копнуть в сторону contenteditable. $('.block').on('input paste keyup', function() { console.log($(this).html()); }); <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script> <div class="block" contenteditable="true">Введите текст</div> A: сss для элемента, в котором текст выходит за границу overflow-Y:scroll; Для того, чтобы текст переносился на новую строку, нужно вставлять тег <br>, я думаю	{ "redpajama_set_name": "RedPajamaStackExchange" }	3,338
IF(NOT DEFINED CMAKE_INSTALL_PREFIX) SET(CMAKE_INSTALL_PREFIX "/home/trevor/ROS/catkin_ws/install") ENDIF(NOT DEFINED CMAKE_INSTALL_PREFIX) STRING(REGEX REPLACE "/$" "" CMAKE_INSTALL_PREFIX "${CMAKE_INSTALL_PREFIX}") # Set the install configuration name. IF(NOT DEFINED CMAKE_INSTALL_CONFIG_NAME) IF(BUILD_TYPE) STRING(REGEX REPLACE "^[^A-Za-z0-9_]+" "" CMAKE_INSTALL_CONFIG_NAME "${BUILD_TYPE}") ELSE(BUILD_TYPE) SET(CMAKE_INSTALL_CONFIG_NAME "") ENDIF(BUILD_TYPE) MESSAGE(STATUS "Install configuration: \"${CMAKE_INSTALL_CONFIG_NAME}\"") ENDIF(NOT DEFINED CMAKE_INSTALL_CONFIG_NAME) # Set the component getting installed. IF(NOT CMAKE_INSTALL_COMPONENT) IF(COMPONENT) MESSAGE(STATUS "Install component: \"${COMPONENT}\"") SET(CMAKE_INSTALL_COMPONENT "${COMPONENT}") ELSE(COMPONENT) SET(CMAKE_INSTALL_COMPONENT) ENDIF(COMPONENT) ENDIF(NOT CMAKE_INSTALL_COMPONENT) # Install shared libraries without execute permission? IF(NOT DEFINED CMAKE_INSTALL_SO_NO_EXE) SET(CMAKE_INSTALL_SO_NO_EXE "1") ENDIF(NOT DEFINED CMAKE_INSTALL_SO_NO_EXE) IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/share/robot_pose_ekf/srv" TYPE FILE FILES "/home/trevor/ROS/catkin_ws/src/navigation/robot_pose_ekf/srv/GetStatus.srv") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/share/robot_pose_ekf/cmake" TYPE FILE FILES "/home/trevor/ROS/catkin_ws/build/navigation/robot_pose_ekf/catkin_generated/installspace/robot_pose_ekf-msg-paths.cmake") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/include" TYPE DIRECTORY FILES "/home/trevor/ROS/catkin_ws/devel/include/robot_pose_ekf") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/share/common-lisp/ros" TYPE DIRECTORY FILES "/home/trevor/ROS/catkin_ws/devel/share/common-lisp/ros/robot_pose_ekf") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") execute_process(COMMAND "/usr/bin/python" -m compileall "/home/trevor/ROS/catkin_ws/devel/lib/python2.7/dist-packages/robot_pose_ekf") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/lib/python2.7/dist-packages" TYPE DIRECTORY FILES "/home/trevor/ROS/catkin_ws/devel/lib/python2.7/dist-packages/robot_pose_ekf") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/lib/pkgconfig" TYPE FILE FILES "/home/trevor/ROS/catkin_ws/build/navigation/robot_pose_ekf/catkin_generated/installspace/robot_pose_ekf.pc") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/share/robot_pose_ekf/cmake" TYPE FILE FILES "/home/trevor/ROS/catkin_ws/build/navigation/robot_pose_ekf/catkin_generated/installspace/robot_pose_ekf-msg-extras.cmake") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/share/robot_pose_ekf/cmake" TYPE FILE FILES "/home/trevor/ROS/catkin_ws/build/navigation/robot_pose_ekf/catkin_generated/installspace/robot_pose_ekfConfig.cmake" "/home/trevor/ROS/catkin_ws/build/navigation/robot_pose_ekf/catkin_generated/installspace/robot_pose_ekfConfig-version.cmake" ) ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/share/robot_pose_ekf" TYPE FILE FILES "/home/trevor/ROS/catkin_ws/src/navigation/robot_pose_ekf/package.xml") ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(EXISTS "$ENV{DESTDIR}${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf/robot_pose_ekf" AND NOT IS_SYMLINK "$ENV{DESTDIR}${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf/robot_pose_ekf") FILE(RPATH_CHECK FILE "$ENV{DESTDIR}${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf/robot_pose_ekf" RPATH "") ENDIF() FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf" TYPE EXECUTABLE FILES "/home/trevor/ROS/catkin_ws/devel/lib/robot_pose_ekf/robot_pose_ekf") IF(EXISTS "$ENV{DESTDIR}${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf/robot_pose_ekf" AND NOT IS_SYMLINK "$ENV{DESTDIR}${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf/robot_pose_ekf") FILE(RPATH_REMOVE FILE "$ENV{DESTDIR}${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf/robot_pose_ekf") IF(CMAKE_INSTALL_DO_STRIP) EXECUTE_PROCESS(COMMAND "/usr/bin/strip" "$ENV{DESTDIR}${CMAKE_INSTALL_PREFIX}/lib/robot_pose_ekf/robot_pose_ekf") ENDIF(CMAKE_INSTALL_DO_STRIP) ENDIF() ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") IF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified") FILE(INSTALL DESTINATION "${CMAKE_INSTALL_PREFIX}/share/robot_pose_ekf" TYPE FILE FILES "/home/trevor/ROS/catkin_ws/src/navigation/robot_pose_ekf/robot_pose_ekf.launch" "/home/trevor/ROS/catkin_ws/src/navigation/robot_pose_ekf/example_with_gps.launch" ) ENDIF(NOT CMAKE_INSTALL_COMPONENT OR "${CMAKE_INSTALL_COMPONENT}" STREQUAL "Unspecified")	{ "redpajama_set_name": "RedPajamaGithub" }	9,127
Q: SP2013 and MVC 5: How do I add user information? I was tasked in making an Intranet for our company. I have no training whatsoever in SharePoint so I'm having a hard time. We're also using Active Directory. First question, is SP2013 MVC using something similar to MVC5's UserIdentity? I also have this MVC template where it's using SharePointContextProvider, which I thought was similar to UserIdentity. However, I can't really extend it since it doesn't have any Model class yet. [SharePointContextFilter] public ActionResult Index() { User spUser = null; var spContext = SharePointContextProvider.Current.GetSharePointContext(HttpContext); using (var clientContext = spContext.CreateUserClientContextForSPHost()) { if (clientContext != null) { spUser = clientContext.Web.CurrentUser; clientContext.Load(spUser, user => user.Title); clientContext.ExecuteQuery(); ViewBag.UserName = spUser.Title; } } return View(); } Was hoping if I could add more information for that spUser. Like for example, having their address, contact number and etc. Is there anyway of doing this properly? What I'm thinking of is just getting the spUser.Email from the context provider and make a method that counter checks that email to a new table I'm going to create which has all the employee information. So if spUser.Email = newTable.Email then this checks out and I list all their info on their profile page. A: If you're building a SharePoint 2013 intranet, you should be getting this information from the User Profile. check out the following post: http://msdn.microsoft.com/en-us/library/office/jj163182.aspx	{ "redpajama_set_name": "RedPajamaStackExchange" }	4,048
{"url":"https:\/\/iliveindallas.com\/i3plt\/bivariate-poisson-distribution-685a2a","text":"# bivariate poisson distribution\n\nEstimation for the bivariate Poisson distribution. The Bivariate Poisson Distribution and its Applications to Football May 5, 2011 Author: Gavin Whitaker Supervisors: Dr. P. S. Ansell Dr. D. Walshaw School of Mathematics and Statistics Newcastle University Abstract We look at properties of univariate and bivariate distributions, speci\ufb01cally those involving generating functions. %\ufffd\uc3e2 However, situations exist where the response variables are the number of successes in a fixed number of trials and follow the bivariate binomial distribution. In a slide presentation, Karlis and Ntzoufras define a bivariate Poisson as the distribution of $(X,Y)=(X_1+X_0,X_2+X_0)$ where the $X_i$ independently have Poisson $\\theta_i$ distributions. When a^ = 0, the bivariate Poisson is called a semi-Poisson with parameters a^ and a^2\u00ab It has non-zero probabil\u00ad ity only on one-half the positive quadrant where X-j_ < X2. e\ufffdk>\ufffdH\ufffd;\ufffd\ufffd cdXS=z7\ufffds\ufffdv\ufffd\ufffd\ufffd\ufffdr2\ufffduS\ufffd!\ufffd\ufffdz\ufffd\ufffdU\ufffd. The correlation between the two variates can be either positive or negative, depending on the value chosen for the parameter in the above multiplicative factor. For a comprehensive treatment of the bivariate Poisson distribution and its multivariate extensions the reader can refer to Kocherlakota and Kocherlakota (1992) and Johnson, Kotz, and Balakrishnan (1997). } for $k=0, 1, \u2026 The joint probability density function of the intriguing problem. Model Distribution Model Details Log-Lik Param. Several forms of the bivariate distribution can be developed by compounding the bivariate Poisson distribution with the inverse Gaussian distribution of the form discussed by Jorgensen (1987). terprete\u00e0 for the Poisson distribution. Expressions are available for quantifying RTM when the distribution of pre and post observations are bivariate normal and bivariate Poisson. The bivariate generalized Poisson distribution (BGPD) based on the method of trivariate reduction was introduced by Famoye and Consul (1995). stream x\ufffd\ufffdY[\ufffd\ufffd~o\ufffd\ufffd\ufffd\ufffd\ufffde\ufffd\ufffd\ufffdR\\\ufffd\ufffd\ufffdP4\ufffd\ufffd\ufffd\u0433\ufffd\ufffd\ufffd\ufffd\u0646\ufffd$\ufffdI\ufffd$\ufffdm(\ufffdy\ufffdG\ufffd.\ufffd\ufffdIJ~\u07ea\ufffd\ufffd\ufffd\ufffd_\ufffdl\ufffd\|\ufffd\/7iw\ufffd\ufffd\ufffd\ufffd\u05db\ufffd7P\ufffdPEz\ufffd\ufffd[\ufffdz\ufffd\ufffd\ufffdo\ufffd\ufffd\ufffd\ufffd\u07fa\ufffd\ufffd\ufffd\ufffd\ufffdf\ufffdcw\ufffd\ufffdU\ufffd\ufffd\ufffd\ufffdu\/y\ufffd\ufffd\ufffd\ufffd\ufffdn\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdS\ufffd\ufffd\u02833\ufffd+^\ufffd2\ufffd\ufffd\ue3dd\ufffd\ufffd)m\ufffd_v{\u8567\ufffd?\ufffd\ufffd\ufffdv\u047bA\ufffd!\ufffd6\ufffd-\ufffd%\ufffd\ufffd\ufffdI@\ufffd\ufffd\ufffd' Lu\ufffd#\ufffd\ufffd\ufffd1\ufffdj]\ufffdu\ufffd\ufffd\ufffd\ufffd\u065f\ufffd\ufffd! The models proposed allow for correlation between the two scores, which is a plausible assumption in \u2026 5 0 obj The distributional properties of this distribution are studied and this model is fitted to a bivariate discrete distribution with negative correlation coefficient, m this context, it should be worth noting that the bivariate Poisson distribution reported by Teicher (1954), (Campbell (1934), Holgate (1964)) has the inherent limitation that the correlation is necessarily positive, and hence is not useful in modelling the situations (e.g. \ufffd\ufffd\ufffd\ufffd\ufffdb\ufffd\u0293\ufffd6\ufffdv\ufffdNp\ufffd\ufffd\ufffd\ufffdB\ufffd\ufffdt St\ufffd\ufffd\ufffd3\ufffd\ufffd\ufffda\/ji\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdi\ufffdi\ufffd\ufffd\ufffdM\ufffd\\\ufffd@\ufffdw'a\ufffd\ufffd\ufffd\ufffd$\ufffd\ufffd\ufffd%\ufffdW\\\ufffd\ufffd'\ufffd\\\ufffd\ufffdV\ufffd\ufffd\ufffdvz\ufffd\/v p>]\u0779\ufffd\ufffd\ufffd\ufffd\ufffdb\ufffd\ufffdzp\ufffd\ufffd\ufffd%\ufffd\ufffdo)\ufffd\ufffdh\ufffdN\ufffdH+\ufffd\ufffd>\ufffdc\ufffd\ufffd\ufffd\ufffd!P\ufffd\ufffds\ufffd\ufffd\ufffd\ufffd\ufffd}\ufffdw6\ufffd1\ufffd\ufffdy\u06d0\ufffdZl+\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd9\ufffd\ufffd-\ufffdl\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd1 \ufffd\ufffd\ufffdF\ufffd\ufffd!\ufffd \ufffdA;2\ufffdH\ufffd\ufffd\ufffd\"v?$p\ufffd S\ufffd\ufffdFM\ufffd\ufffd1 \ufffdk2\ufffd5+!\ufffd\ufffde\ufffd\ufffd\ufffdG;\ufffd\ufffd\ufffdl\ufffd6d\ufffd1[\ufffd\ufffd\ufffd\ufffd]\ufffd\ufffd\ufffd\ufffd,\u0584V\ufffd\ufffd\ufffd\u05ae\ufffd\ufffd\ufffd5w\ufffd\u015c\u0606LqXb\ufffd\ufffdzT\ufffd\|2>\ufffd\ufffdI\ufffd\ufffdq\ufffd\"\ufffdKf~\ufffd6\ufffd\ufffd(\/\ufffd\/\ufffd \ufffd>\ufffd\ufffdp\u0d27+\ufffd\ufffd\ufffd\ufffd;\ufffd\/m\ufffd\ufffd\ufffd&\ufffdN3\u0265L6Q\ufffd\ufffdM\ufffd\"\u009c\ufffdr \ufffd+\ufffd\ufffd*J\ufffd!\ufffd\ufffd\ufffd@\ufffd\ufffdE Poisson. Holgate, P. (1964). In statistics, many bivariate data examples can be given to help you understand the relationship between two variables and to grasp the idea behind the bivariate data analysis definition and meaning. A bivariate distribution, whose marginals are Poisson is developed as a product of Poisson marginals with a multiplicative factor. A similar definition holds when a2 = 0. Models based on the bivariate Poisson distribution are used for modelling sports data. Register to receive personalised research and resources by email, Department of Statistics , Andhra University , Visakhapatnam , A.P , 530 003 , India, Department of Mathematics and Statistics , Central University , Tejpur , Assam, \/doi\/pdf\/10.1080\/03610929908832297?needAccess=true, Communications in Statistics - Theory and Methods. Registered in England & Wales No. Examples Kawamura, K. (1984). Multivariate Poisson models October 2002 \u2019 &$ % Results(1) Table 1: Details of Fitted Models for Champions League 2000\/01 Data (1H0: \u201a0 = 0 and 2H0: \u201a0 = constant, B.P. Kodai mathematical journal, 7(2), 211-221. \"When a22 = 0, the bivariate Poisson distribution is that of two independent Poissons. In this paper, a new bivariate generalized Poisson distribution (GPD) that allows any type of correlation is defined and studied. The covariance structure of the bivariate weighted Poisson distribution and application to the Aleurodicus data BATSINDILA NGANGA, Prevot Chirac, BIDOUNGA, Rufin, and MIZ\u00c8RE, Dominique, Afrika Statistika, 2019; Some Poisson mixtures distributions with a hyperscale parameter Laurent, St\u00e9phane, Brazilian Journal of Probability and Statistics, 2012 Using these properties we arrive at <> Direct calculation of maximum likelihood estimator for the bivariate Poisson distribution. We replace the independence assumption by considering a bivariate Poisson model and its extensions. The correlation between the two variates can be either positive or negative, depending on the value chosen for the parameter in the above multiplicative factor. A bivariate distribution, whose marginals are Poisson is developed as a product of Poisson marginals with a multiplicative factor.\n\nWoodpecker Beak Type, Random Forest Algorithm, Poli Meaning In Tamil, Wedding Dance Songs 2019, Ecco Flash Sale, Chicken Burrito Mix Recipe, Theoretical Framework For Nursing Practice, Honey Balsamic Glaze, Review Of Keto Now, Pran Last Movie, Hallelujah Chorus Sheet Music Pdf, Lenovo Ideapad L340-15api Specs,","date":"2021-02-25 22:06:59","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\": 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\": 0.8518477082252502, \"perplexity\": 2584.1576877354605}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178355937.26\/warc\/CC-MAIN-20210225211435-20210226001435-00634.warc.gz\"}"}	null	null
{"url":"https:\/\/www.groundai.com\/project\/investigation-of-frustrated-and-dimerized-classical-heisenberg-chains\/","text":"Investigation of frustrated and dimerized classical Heisenberg chains\n\n# Investigation of frustrated and dimerized classical Heisenberg chains\n\nJ. Vahedi, S. Mahdavifar, M. R. Soltani, M. Elahi Department of Physics, Science and Research Branch, Islamic Azad University, Tehran, Iran\nDepartment of Physics, University of Guilan,41335-1914, Rasht, Iran\nDepartment of Physics, Shahre-Ray branch, Islamic Azad University, Tehran, Iran.\nJuly 4, 2019\n###### Abstract\n\nWe have considered the 1D dimerized frustrated antiferromagnetic (ferromagnetic) Heisenberg model with arbitrary spin . The exact classical magnetic phase diagram at zero temperature is determined using the LK cluster method. Cluster method results, show that the classical ground state phase diagram of the model is very rich including first and second-order phase transitions. In the absence of the dimerization, a second-order phase transition occurs between antiferromagnetic (ferromagnetic) and spiral phases at the critical frustration . In the vicinity of the critical points , the exact classical critical exponent of the spiral order parameter is found . In the case of dimerized chain (), the spiral order shows stability and exists in some part of the ground state phase diagram. We have found two first-order critical lines in the ground state phase diagram. These critical lines separate the antiferromagnetic from spiral phase.\n\n###### pacs:\n75.10.Pq, 75.10.Hk\n\n## I Introduction\n\nDuring the last decades several classical techniques such as the well-known Luttinger-Tisza methodLuttinger46 (), vertex modelBaxter () and so on have been introduced to solve classical Hamiltonian exactly. The Luttinger-Tisza method is more effective in systems with bilinear interactions and vertex model usually applied for treating frustrated modelLieb (); Sutherland ().\n\nIn a very recent workTK2009 (), T. Kaplan have used a kind of cluster method, hereafter simply LK method, which is based on a block of three spins to solve frustrated classical Heisenberg model in one dimension with added nearest neighbor biquadratic exchange interactions. He asserted that the LK method is not limited to one dimension or to translationally invariant spin HamiltoniansLyons64 () and expanded his approach to determine the phase diagram of frustrated classical Heisenberg and XY models with added nearest neighbor biquadratic exchange interactions in dimensionLX2010 (). In order to check the validity of Kaplan\u2019s phase diagram conjecture, we have investigated his modelTK2009 () form quantum point of view for spin- with an accurate algorithm (Lanczos method), and our results, which will be presented elsewhereJaSaee (), showed that LK method, albeit is a classical approach but has the capability to work for aspects of a quantum treatment.\n\nActually, these are our stimulating reasons to take a quite well known frustrated and dimerized Heisenberg model and determine its classical ground state phase diagram exactly with strong, but not well known LK method which is able to solve problems rigorouslyTK2009 (). Let us start with definition of the dimerized and frustrated Heisenberg model as follow\n\n \\emphH=J1\u2211n[1+(\u22121)n\u03b4]Sn\u22c5%Sn+1+J2\u2211nSn\u22c5Sn+2, (1)\n\nwhere is the th classical vector of the length . A spin system is frustrated when the global order because of the competition of different kinds of interaction is incompatible with the local order, so chain with both antiferromagnetic-antiferromagnetic exchanges and ferromagnetic-antiferromagnetic exchanges , hereafter simply AF-AF and F-AF respectively, are frustrated.\n\nQuantum study of this model is well done for the spin-1\/2 and spin-1 chainsShastry81 (); Bouzerar98 (); Nakamura97 (); Kumar07 (); Chitra95 (); Controzzi05 (); Pati96 (); Oshikawa (). It is found that the quantum fluctuations play a very important role at zero temperature in the ground state phase diagram of the models. This model shows a dimerization transition at .\n\nFor system with spin half, , is the transition point, which is related to the Lieb-Schultz-Mattis theorem (states that system should be in either twofold degeneracy or gapless excitations of the ground states at ). Indeed, for AF-AF case and on the undimerized link , there is a critical frustration parameter Haldane82 (); Okamoto92 (). The dimerization transition at is of second order and system for shows a gapless Tomonaga-Luttinger Liquid (TLL). But, for , the ground state is doubly degenerate, showing a spontaneous dimerization. This is a signature of a first-order dimerization transition at .\n\nOn the other hand, system with spin one, , with and small lives in Haldane phase and does not represent a transition line. In contrast, dimerization transition between the Haldane phase and the dimerized phase happensAffleck87 (); Kolezhuk96 () at a finite , which depends on the (know as frustration parameter). As matter of fact, for system with spin one, there is a critical frustration point which is second order form of transition. This critical point separates a TLL phase for from first order one for .\n\nBut there is not a classical clear picture of different ground state phases of the mentioned model. Having a classical picture, from one hand help us to know that quantum fluctuations destroy which one of the classical orderings. On the other hand for arbitrary large spin model, the classical picture is the same with the quantum picture of the ground state phase diagram. In this work we focus on the 1D frustrated and dimerized systems with arbitrary spin (see FIG.\u00a01). To find the exact classical ground state phase diagram of the model, the LK cluster method is used. In the absence of the dimerization, by increasing the frustration a classical phase transition occurs at from the antiferromagnetic (ferromagnetic) phase into the spiral magnetic phase. Our results show that the dimerization parameter induces new magnetic phases including stripe-antiferromagnetic phase (or uud and duu phases). Existing of these magnetic phases is independent of length of spins.\n\nThe outline of the paper is as follows. In forthcoming section we will extensively explain the LK method with implementing it to our model and in the section III we will summarize our results.\n\n## Ii the LK cluster method\n\nIn order to implement LK method we follow exactly the procedure in Ref. [2]. Without losing the generality and setting periodic boundary conditions, Eq.(1) can be rewritten as:\n\n \\emphHc=\u2211ihc(Si\u22121,Si,Si+1), (2)\n\nwhere the \u201dcluster energy\u201d involve three neighboring spins is\n\n hc(S1,S2,S3) = J12{(1\u2212\u03b4)S1\u22c5S2+(1+\u03b4)S2\u22c5S3)} (3) + J2(S1\u22c5S3).\n\nIt is clear that\n\n \\emphHc\u2265\u2211jmin\u00a0hc(\u2192Sj\u22121,\u2192Sj,\u2192Sj+1). (4)\n\nTo minimize respect spins directions, we first consider coplanar spins, and label the angles , made by the end spins with the central spin (see FIG.\u00a02) which in coplanar case we set . The cluster energy is given by\n\n hc(\u03b8,\u03b8\u2032) = S2{(1\u2212\u03b42)cos\u03b8+(1+\u03b42)cos\u03b8\u2032 (5) + \u03b1cos(\u03b8\u2212\u03b8\u2032)},\n\nwhere . Minimizing respect , gives the following equation:\n\n \u2202hc\u2202\u03b8=\u2212S22[(1\u2212\u03b4)sin\u03b8+2\u03b1sin(\u03b8\u2212\u03b8\u2032)] \u2202hc\u2202\u03b8\u2032=\u2212S22[(1+\u03b4)sin\u03b8\u2032\u22122\u03b1sin(\u03b8\u2212\u03b8\u2032)]. (6)\n\nLet\u2019s first deal with a case without dimerization, by setting in Eq. (6) we have\n\n \u2202hc\u2202\u03b8=\u2212S22[sin\u03b8+2\u03b1sin(\u03b8\u2212\u03b8\u2032)] \u2202hc\u2202\u03b8\u2032=\u2212S22[sin\u03b8\u2032\u22122\u03b1sin(\u03b8\u2212\u03b8\u2032)]. (7)\n\nits solutions are\n\n (\u03b8,\u03b8\u2032) = (0,0),(\u03c0,\u03c0),(\u03c0,0),(0,\u03c0), (\u03b8,\u03b8\u2032) = (\u03b80,\u2212\u03b80),\u00a0\u00a0\u00a0(Spiral- type),\u00a0\u00a0\u00a0where cos\u03b80 = \u221214\u03b1\u2192\|\u03b1\|\u226514.\n\nThe solutions are related to collinear antiferromagnetic and ferromagnetic states respectively. The antiferromagnetic (ferromagnetic) state will minimize the energy in the case of (). Solutions and are degenerate states and show spins propagate in the down-up-up and up-up-down respectively Lyons64 (). Spiral state with uniform rotation is also degenerate state. In following we present results of the antiferromagnetic case .\n\nBy setting minimization conditions into the Eq.\u00a0(5) we have the following energies:\n\n hantiferro = hc(\u03c0,\u03c0)=S2(\u22121+\u03b1), huudd = hc(0,\u03c0)=S2(\u2212\u03b1), hspiral = hc(\u03b80,\u2212\u03b80)=S2(\u221218\u03b1\u2212\u03b1). (9)\n\nBy equating these energies in pairs we have found only one critical point, . Because of the continuity in the derivative , a second-order phase transition occurs when passing trough . The ground state is in the antiferromagnetic phase in the region of the frustration and in the spiral phase in region . In general, the antiferromagnetic phase is recognized by the non-zero value of the Neel order parameter defined as\n\n Mzst=1N\u2211n(\u22121)nSzn, (10)\n\nand the spiral phase in the ground state phase diagram of the spin systems is characterized by the nonzero value of the spiral order parameter\n\n \u03c7=1N\u2211n\|Sn\u00d7Sn+1\|. (11)\n\nUsing Eq.\u00a0(LABEL:e7) we have found the spiral order parameter as\n\n \u03c7 = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0,\u00a0\u00a0\u00a0\u03b1<\u03b1c, \u03c7 = S2\u221a1\u2212\u03b12c\u03b12\u00a0\u00a0\u00a0\u00a0,\u00a0\u00a0\u00a0\u03b1>\u03b1c. (12)\n\nIn FIG.3, we have plotted the spiral order parameter as a function of the frustration parameter for the non-dimerized model (). As is clearly seen from this figure, there is no long-range spiral order in the region of frustration . However, in the region the spins of the system show a profound spiral order which grows by increasing the frustration parameter .\n\nIt has been discovered that continuous phase transitions have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points. It turns out that continuous phase transitions can be characterized by parameters known as critical exponents. Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, that they are universal, i.e. they do not depend on the details of the physical system. Our analytical results show that the spiral order parameter approaches zero in a singular fashion as approaches , vanishing asymptotically as\n\n \u03c7\u221d(1\u2212\u03b1c\u03b1)1\/2, (13)\n\nwhich shows that the critical exponent for spiral order parameter is a simple fraction .\n\nNow, we back to our original problem , finding the exact ground state phase diagram of the classical frustrated and dimerized Heisenberg chains. One can immediately see the possibility of having two spirals, one on the even sites, the other on the odd sites, both with the same wave length, but with a phase difference as described in the following:\n\nTo use the cluster approach for the case of non-zero dimerization, , one must consider two types of cluster, one being , the other being . Let us label the spins in the first cluster , and the second and continue , , . The 2-spiral form assumes a simple spiral on the even sites, and a simple spiral on odd sites, both with the same turn angle between spins, which we called . Calling the angles and , respectively, of and with the center spin in the first cluster, then gives as the angle between and . Then, preserving the angle between and in the next cluster, (which is now , since the central spin is now ) and the angle between and being taken as gives the angle between and . Hence the first two clusters begin to show the spiral on the odd sites and the spiral on the even sites. Continuing this to the next few clusters shows that this allows a description of a system with two spirals, one on the odd, the other on the even sites, both with the same turn-angle or wavelength. Also the energy of each cluster is the same, so one can consider just one cluster, say the first one above, in figuring out the relation between and . For the case of , spins order as following pattern . Therefor, in the case (), the following solutions can satisfy the minimum energy condition,\n\n (\u03b8,\u03b8\u2032) = (0,0),(\u03c0,\u03c0),(\u03c0,0),(0,\u03c0), (\u03b8,\u03b8\u2032) = (\u03b80,\u2212\u03b80+\u03f5),\u00a0\u00a0\u00a0(Spiral- type),\u00a0\u00a0\u00a0where cos\u03b80 = \u221216\u03b4\u03b12\u2212(1\u2212\u03b4)24\u03b1(1\u2212\u03b4)2(1+\u03b4). sin\u03b80 = 0\u2192\u03b80=n\u03c0\u2192\u03f5=0.\n\nBy substituting them into the Eq.(5), the ground state energy of cluster in different sectors becomes\n\n hantiferro = hc(\u03c0,\u03c0)=S2(\u22121+\u03b1), huud = hc(0,\u03c0)=S2(\u2212\u03b4\u2212\u03b1), hduu = hc(\u03c0,0)=S2(\u03b4\u2212\u03b1), hspiral = hc(\u03b80,\u2212\u03b80+\u03f5)=S2(\u03b4\u221212 + (1+cos\u03b80){(1\u2212\u03b4)+\u221a(1\u2212\u03b4)2cos2\u03b80+4\u03b4}).\n\nAs it can be seen from the above equations, in respect to the case of , dimerization exchange removed the degeneracy between and states. The state, defines as a phase with opposite magnetization on odd () bonds, but state denotes by opposite magnetization on even () bonds. Using the conditions in Eq. (LABEL:e9) allow us to find the stability of different phases. Doing some calculations, one can find two critical lines as\n\n \u03b1~c = 1\u2212\u03b424\u03b4, \u03b1~c = \u03b42\u221214\u03b4. (15)\n\nThese critical lines separate spiral phase from the uud and the duu phases. There is a discontinuity in the derivative , and therefore a first-order phase transition through the mentioned critical lines. In addition, one should note that the ordering of the uud and duu phases, in principle is a type of the stripe-antiferromagnetic phaseMahdavifar08 (); Mahdavifar10 (). Therefore, the order parameter for distinguishing these phases is defined as\n\n Mzsp = 2N\u2211n=1(\u22121)n+1(Sz2n\u22121+Sz2n),\u00a0for\u00a0uud (16) = 2N\u2211n=1(\u22121)n+1(Sz2n+Sz2n+1),\u00a0for\u00a0duu.\n\nWe have also found antiferromagnetic phase that is stable for\n\n \u03b1 < 1\u2212\u03b424\u00a0for\u00a0\u03b4>0, \u03b1 > \u03b42\u221214\u00a0for\u00a0\u03b4<0. (17)\n\nIt is completely clear that in uud and duu phases, takes the value . In FIG.\u00a04 (a), we have plotted as a function of the frustration for different values of the frustration . As it can seen from this figure, for , the staggered magnetization is equal to , which shows that the ground state of the system is in the fully polarized antiferromagnetic phase. Finally, it is clear that there is no antiferromagnetic ordering in the region . In FIG.\u00a04 (b), we have the same picture for the stripe antiferromagnetic order function. The zero value of in the region is in complete agreement with the fully polarized antiferromagnetic and spiral phases in this region. By more increasing the dimerization and for , clearly be seen that the ground state of the system is in the uud. Also one predicts that the tripe antiferromagnetic as a function of the displays a jump for certain parameters which is one of the most important indications of the first-order phase transition. We emphasize that a first-order phase transition occurs between the spiral and the duu phases, for negative values of the dimerization.\n\nThe FIG.\u00a0(5) shows the exact classical ground state phase diagram of the model in plane. It should be mentioned that the same phase diagram can be also found in the ferromagnetic side which we do not depict. In the absence of dimerization, , there are antiferromagnetic(ferromagnetic) and spiral phases which is separated by two critical point at which the negative sign refers to ferromagnetic side. The second order phase transition occurs at these critical points. By turning dimerization the spiral phase persists to be stable in region for and region for . The antiferromagnetic phase remains stable up to the critical line . The uud and duu phases propagate with different energy and separated by two first order critical lines, ( and ), from spiral phase.\n\nIn the following we are interested to implement LK method to non-coplanar antiferromagnetic case. Again with using cluster approach and without losing generality, labeled the angles , made by the end spins with the central spin (see FIG.\u00a02). Then the cluster energy is determine as\n\n hc(\u03b8,\u03b8\u2032) = S2{(1\u2212\u03b42)cos\u03b8+(1+\u03b42)cos\u03b8\u2032 + \u03b1[cos\u03b8cos\u03b8\u2032+sin\u03b8sin\u03b8\u2032cos(\u03d5\u2212\u03d5\u2032)]}.\n\nMinimizing over the angles , and gives the following equations\n\n \u2202hc\u2202\u03b8 = \u2212S22{(1\u2212\u03b4)sin\u03b8+2\u03b1[sin\u03b8cos\u03b8\u2032 \u2212 \u2202hc\u2202\u03b8\u2032 = \u2212 \u2202hc\u2202\u03d5 = \u2212S22{\u22122\u03b1sin\u03b8sin\u03b8\u2032sin(\u03d5\u2212\u03d5\u2032)}, \u2202hc\u2202\u03d5\u2032 = \u2212S22{2\u03b1sin\u03b8sin\u03b8\u2032sin(\u03d5\u2212\u03d5\u2032)}. (19)\n\nWe check the possible configurations which can minimize the above equations. By taking arbitrary , we have the solutions that are related to the antiferromagnetic and ferromagnetic states respectively. We have also the solutions and that are related to the uud and duu states respectively. Spiral phase also exists same as the coplanar case. The stability of different phases in the non-coplanar case is also checked and behaves same as the coplanar case.\n\n## Iii conclusion\n\nTo summarize, we have studied the classical ground state magnetic phase diagram of the dimerized and frustrated Heisenberg chain using LK cluster method. In coplanar case and in the absence of dimerization effect this approach could detect antiferromagnetic(ferromagnetic) and spiral phases. We have shown that turning the dimerization yields to remove the degeneracy between two uud and duu phases. We have argued that in the ground state phase diagram of the system there are first order transition lines. These lines separate spiral and uud or duu phases. On the other hand two second order phase transition points also exist, which separate antiferromagnetic(ferromagnetic) and spiral phases. By helping this approach we have calculated the spiral exact critical exponent .\n\nIn order to generalize our treatment, we have considered the non-coplanar case and checked its phases by LK cluster method. Our calculations revealed that in the non-coplanar case classical phase diagram consist of antiferromagnetic or ferromagnetic depend on nearest neighbor coupling, duu and uud phases which are still non-degenerate and spiral phase. Finally, one of the main achievement of our work which should be highlighted is that in the both coplanar and non-coplanar cases, the spiral state is stable in the classical phase diagram for .\n\n## References\n\n\u2022 (1) J. M. Luttinger and L. Tisza Phys. Rev. 70, 954 (1946); J. M. Luttinger and Laszo Tisza Phys. Rev. 72, 257 (1947) .\n\u2022 (2) R.J. Baxter, Exactly solved models in statistical mechanics, London, Academic Press, (1982).\n\u2022 (3) E. H. Lieb, Physical Review 162, 162-172 (1967).\n\u2022 (4) B. Sutherland, J. Math. 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B 77, 77 (2010).\nYou are adding the first comment!\nHow to quickly get a good reply:\n\u2022 Give credit where it\u2019s due by listing out the positive aspects of a paper before getting into which changes should be made.\n\u2022 Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.\n\u2022 Your comment should inspire ideas to flow and help the author improves the paper.\n\nThe better we are at sharing our knowledge with each other, the faster we move forward.\nThe feedback must be of minimum 40 characters and the title a minimum of 5 characters","date":"2020-04-07 22:00:21","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\": 0.8081558346748352, \"perplexity\": 1377.5587240817379}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371806302.78\/warc\/CC-MAIN-20200407214925-20200408005425-00298.warc.gz\"}"}	null	null
\section{} \section{Introduction} The present paper is concerned with the scalar conservation law in one space dimension: \begin{align} \label{conservation_law} \begin{cases} u_t+(A(u))_x=0\\ u(x,0)=u^0(x). \end{cases} \end{align} In the scalar conservation law \eqref{conservation_law}, $u(x,t):\mathbb{R}\times[0,\infty)\to\mathbb{R}$ is the unknown, $u^0\in L^{\infty}(\mathbb{R})$ is the given initial data, and $A:\mathbb{R}\to\mathbb{R}$ is the given \emph{flux function}. In the present paper, we are concerned with $A\in C^2(\mathbb{R})$ strictly convex. A \emph{classical solution} of \eqref{conservation_law} is a locally Lipschitz function $u:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ which satisfies $u_t+(A(u))_x=0$ almost everywhere and verifies $u(x,0)=u^0(x)$ for all $x\in\mathbb{R}$. We are also interested in \emph{weak solutions} of \eqref{conservation_law}. A weak solution to \eqref{conservation_law} is a locally bounded measurable function $u:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ which satisfies \begin{align} \int\limits_{0}^{\infty} \int\limits_{-\infty}^{\infty}[\partial_t \phi u +\partial_x \phi A(u)] \,dxdt + \int\limits_{-\infty}^{\infty} \phi(x,0)u^0(x)\,dx=0 \label{weak_solution} \end{align} for every Lipschitz continuous test function $\phi:\mathbb{R}\times[0,\infty)\to\mathbb{R}$, with compact support. In particular, every weak solution satisfies \eqref{conservation_law} in the sense of distributions. Note that a classical solution is also a weak solution. \vskip0.3cm A pair of functions $\eta, q:\mathbb{R}\to\mathbb{R}$ are called an \emph{entropy} and \emph{entropy flux}, respectively, for the scalar conservation law \eqref{conservation_law} if \begin{align} q'(u)=\eta'(u)A'(u). \end{align} We say a weak solution $u$ of \eqref{conservation_law} is \emph{entropic} for the entropy $\eta$ if it satisfies the \emph{entropy inequality} \begin{align} \label{entropy_inequality_distributional} \partial_t \eta(u)+\partial_x q (u) \leq 0 \end{align} in a distributional sense, where $q$ is any corresponding entropy flux. Precisely, \begin{align} \int\limits_{0}^{\infty} \int\limits_{-\infty}^{\infty}[\partial_t \phi \eta(u) +\partial_x \phi q(u)] \,dxdt + \int\limits_{-\infty}^{\infty} \phi(x,0)\eta(u^0(x))\,dx\geq0 \label{entropy_inequality_integral_formulation} \end{align} for every nonnegative Lipschitz continuous test function $\phi:\mathbb{R}\times[0,\infty)\to\mathbb{R}$, with compact support. Kruzhkov \cite{MR0267257} proved existence and uniqueness for bounded weak solutions to \eqref{conservation_law} which are entropic for the large family of entropies $\{\eta_k\}_{k\in\mathbb{R}}$, where \begin{align} \eta_k(u)\coloneqq\abs{u-k}. \label{kruzkov_entropies} \end{align} For a bounded and measurable solution to \eqref{conservation_law}, being entropic for each of the $\eta_k$ is equivalent to being entropic for every convex entropy \cite[Proposition 2.3.4]{serre_book}. See Bolley, Brenier and Loeper \cite{brenier_optimal_transport76} for an extension of Kruzhkov's theory, based on the Wasserstein distance. Ole\u{\i}nik discovered \emph{``condition E,''} and proved existence and uniqueness for bounded weak solutions to \eqref{conservation_law} which satisfy it \cite{MR0094541}. A solution $u$ to \eqref{conservation_law} satisfies condition E if \begin{align} \label{condition_e} \begin{cases} \mbox{There exists a constant $C>0$ such that}\\ \hspace{1.3in}u(x+z,t)-u(x,t)\leq \frac{C}{t} z\\ \mbox{for all $t>0$, almost every $z>0$, and almost every $x\in\mathbb{R}$.} \end{cases} \end{align} It is known that being entropic for each of the $\eta_k$ is equivalent to Ole\u{\i}nik's condition E when the conservation law \eqref{conservation_law} has a uniformly convex flux function $A$ \cite[p.~66 and p.~ 57]{serre_book}. When the solution to \eqref{conservation_law} is bounded and $A$ is strictly convex, we can assume $A$ is uniformly convex. \vskip0.3cm For general systems, using the $L^1$ theory, Bressan, Crasta, and Piccoli showed uniqueness in the class of solutions with small total variation in \cite{MR1686652}. It can be interesting to study the uniqueness of the same solutions in a larger class. For example, for the $2\times 2$ Euler system, existence of solutions with large data is known. But uniqueness is still open for such solutions. The present paper concerns the uniqueness of solutions to scalar conservation laws in one space dimension which are entropic for only one entropy. However we are careful in our theory to develop techniques that we believe will extend to the systems case. To attempt new progress on the theory of uniqueness, we take an entirely new approach. We use the method of relative entropy combined with the recent idea of stability up to a shift (first described by the second author in \cite{VASSEUR2008323}). Our methods are fundamentally $L^2$ theory. The method of relative entropy allows for stability estimates to be made between a smooth solution to a conservation law and a weak solution entropic for at least one convex entropy. The method of relative entropy is powerful, applying to the cases of scalar, systems, and multiple space dimensions. The proof of stability estimates between weak and strong solutions relies on the fact that classical solutions of a hyperbolic system of conservation laws satisfy the entropy inequality \eqref{entropy_inequality_distributional} as an exact equality. Stability breaks down when a discontinuity is introduced into the classical solution. In particular, the relative entropy method is much more involved when considering shocks. For a first result, see DiPerna \cite{MR523630} for uniqueness. In the case of stability, the theory of stability up to a shift allows for discontinuities to be introduced into the smooth solution that the method of relative entropy considers. In this way, the method of relative entropy can be used to make comparisons between a weak solution and piecewise-Lipschitz solutions. This is how the uniqueness result of the present paper is proven. Given the system \eqref{conservation_law} and entropy and entropy flux $\eta$ and $q$, respectively (or more generally, any hyperbolic system of conservation laws in multiple space dimensions endowed with any entropy), the method of relative entropy considers the quantity called the \emph{relative entropy}: \begin{align} \eta(a\|b)\coloneqq \eta(a)-\eta(b)-\eta'(b)(a-b) \end{align} for all $a,b\in\mathbb{R}$. We have also the associated \emph{relative entropy-flux}, \begin{align} q(a;b)\coloneqq q(a)-q(b)-\eta'(b)(A(a)-A(b)) \end{align} for all $a,b\in\mathbb{R}$. Both $\eta(a\|b)$ and $q(a;b)$ are locally quadratic in $a-b$. In particular, for all $a$ and $b$ in a fixed compact set, the strict convexity of $\eta\in C^2(\mathbb{R})$ gives \begin{align} \label{relative_entropy_controls_l2_4} c^(a-b)^2\leq \eta(a\|b) \leq c^{}(a-b)^2 \end{align} for constants $c^, c^{*}>0$. The method of relative entropy was invented by Dafermos \cite{doi:10.1080/01495737908962394,MR546634} and DiPerna \cite{MR523630} to prove weak-strong estimates: given a weak solution $u$ to \eqref{conservation_law}, entropic for $\eta$, and a classical solution $\bar{u}$, the method of relative entropy gives stability estimates on the growth in time of $\norm{u(\cdot,t)-\bar{u}(\cdot,t)}_{L^2}$ by considering the time derivative of $\int\eta(u\|\bar{u})\,dx$ and using the entropy inequality \eqref{entropy_inequality_distributional} (in particular, see \Cref{extended_entropy_inequality_lemma}). Due to \eqref{relative_entropy_controls_l2_4}, the quantity $\eta(u\|\bar{u})$ gives estimates of $L^2$-type, while being more amenable to study than the $L^2$ norm itself, due to the entropy inequality \eqref{entropy_inequality_distributional}. The relative entropy method is fundamentally an $L^2$ theory. The aforementioned recent insight of the second author has allowed for discontinuities to exist in the otherwise smooth solution $\bar{u}$ that the method of relative entropy considers, while maintaining stability. The key is that shocks have a contraction property in $L^2$ up to shift, even for large perturbations. For this, the discontinuities must not be allowed to move in time according to the conservation law, but instead their movement must be dictated by a special time-dependent function which shifts the solution. The first result in this program was by Leger \cite{Leger2011_original} for the scalar conservation law \eqref{conservation_law} for a strictly convex flux $A\in C^2(\mathbb{R})$. We now introduce the result of Leger. Let $u_L, u_R\in\mathbb{R}$ satisfy $u_L>u_R$. Let $\sigma$ satisfy $A(u_L)-A(u_R)=\sigma(u_L-u_R)$. Define \begin{align} \phi (x)\coloneqq \begin{cases} u_L & \text{if } x<0 \\ u_R & \text{if } x>0. \end{cases} \end{align} Let $u$ be any Kruzhkov solution. In this context, Leger proved the existence of a Lipschitz continuous function $h:[0,\infty)\to\mathbb{R}$ such that \begin{align} \norm{u(\cdot,t)-\phi(\cdot - h(t)-\sigma t)}_{L^2(\mathbb{R})}\leq\norm{u(\cdot,0)-\phi(\cdot)}_{L^2(\mathbb{R})} \end{align} for all $t\geq0$. Note that by shifting the position of the shock wave as a function of time, all $L^2$ growth in time is killed. The shift function $h$ depends on $u$. Leger gives control on $h$: $\abs{h(t)}\leq \lambda \norm{u(\cdot,0)-\phi(\cdot)}_{L^2(\mathbb{R})}\sqrt{t}$, for some constant $\lambda>0$. Leger only considered Kruzhkov solutions, but his methods are in fact very general and can be applied whenever a solution satisfies a strong trace property (see \Cref{strong_trace_condition} below) and is entropic for at least one strictly convex entropy $\eta\in C^2(\mathbb{R})$. The second author and coworkers have been actively developing the theory of contraction up to a shift function. Progress has been made on systems of conservation laws in one space dimension by introducing the notion of a-contraction up to shift \cite{MR3519973,MR3479527,MR3537479,serre_vasseur,Leger2011} and scalar viscous conservation laws in both one space dimension \cite{MR3592682} and multiple space dimensions \cite{multi_d_scalar_viscous_9122017}. For a more general overview of the theory of shifts and the relative entropy method in general, see \cite[Section 3-5]{MR3475284}. The theory of stability up to a shift has also been used to study asymptotic limits when the limit is discontinuous. See \cite{MR3333670} for the scalar case, and \cite{MR3421617} for the case of systems. There is a long history of using the relative entropy method to study the asymptotic limit. However, results on the asymptotic limit which do not use shifts have only been able to consider the case when the limit function is Lipschitz continuous (see \cite{MR1121850,MR1115587,MR1213991,MR2505730,MR2178222,MR2025302,MR1842343,MR1980855} and \cite{VASSEUR2008323} for a survey). Current work only allows for a single discontinuity to exist in the otherwise smooth solution considered by the method of relative entropy. A natural progression is to try to allow for arbitrarily many discontinuities in the smooth solution, so that we can use the method of relative entropy to compare not just a smooth solution and a weak solution, but ideally two weak solutions. In the present paper, we use the method of relative entropy to compare a weak solution to a solution with arbitrarily many discontinuities. Thus, the present paper expands the theory of stability up to a shift. The present paper lays out a new engine for proving uniqueness of solutions. In summary, the engine works by comparing two solutions using the method of relative entropy. One solution $u$ can be weak and entropic for just a single entropy. The second solution will have more regularity, but by allowing many discontinuities can still be from a dense class. By using the method of relative entropy to approximate the weak solution $u$ with a sequence of more regular solutions, we detect regularity in $u$. Hopefully, enough regularity in $u$ will ensure uniqueness. The uniqueness result in this paper is for the scalar conservation laws. But, much of our work should generalize to systems. The method of relative entropy works for systems, and also multiple space dimensions. The theory of stability up to a shift has been developed for systems. Many hyperbolic systems of conservation laws in one space dimension admit only one non-trivial entropy \cite[p.~238]{MR3475284}. In the present paper, we use only one entropy. For the scalar conservation laws, uniqueness of solutions entropic for a single entropy is not new. The first proof was given by Panov in 1994 \cite{panov_uniquness}, for the system \eqref{conservation_law} with any flux $A\in C^2(\mathbb{R})$ strictly convex, any entropy $\eta\in C^1(\mathbb{R})$ strictly convex, and $u^0\in L^{\infty}(\mathbb{R})$. A second proof was given approximately 10 years later by De Lellis, Otto and Westdickenberg \cite{delellis_uniquneness}. Their result is stronger than Panov's result: their proof allows for various right-hand sides in the entropy inequality, and can consider unbounded functions \cite[p.~688]{delellis_uniquneness}. However, both the proofs of Panov and De Lellis-Otto-Westdickenberg are fundamentally limited to the scalar conservation laws. Both proofs exploit the special connection between scalar conservation laws in one space dimension and Hamilton--Jacobi equations: the space derivative of the solution to a Hamilton--Jacobi equation is formally a solution to the associated scalar conservation law. It is well-known that the relation between scalar conservation laws in one space dimension and Hamilton--Jacobi equations breaks down in the more general case of hyperbolic systems of conservation laws in one space dimension (however, see \cite{jin_xin} for a formal connection between a general Hamilton--Jacobi equation in $n$ space dimensions and a \emph{weakly} hyperbolic system of conservation laws). \vskip0.3cm Before we can state precisely the uniqueness result proven in the present paper, let us introduce the strong trace property (originally introduced in \cite{Leger2011}): \begin{definition} \label{strong_trace_condition} Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$. Then $u$ verifies the \emph{strong trace property} if for any Lipschitz continuous function $h: [0,\infty)\to\mathbb{R}$, there exist $u_{+},u_{-}\in L^{\infty}([0,\infty))$ such that \begin{align} \lim_{n\to\infty} \int\limits_{0}^{T}\esssup_{y\in(0,\frac{1}{n})}\abs{u(h(t)+y,t)-u_{+}(t)}\,dt=\lim_{n\to\infty} \int\limits_{0}^{T}\esssup_{y\in(-\frac{1}{n},0)}\abs{u(h(t)+y,t)-u_{-}(t)}\,dt=0 \end{align} for all $T>0$. \end{definition} Note that any function $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ will satisfy the strong trace property if $u$ has a representative such that for any fixed $h$, the right and left limits \begin{align} \lim_{y\to0^{+}} u(h(t)+y,t) \hspace{.5in}\text{and}\hspace{.5in} \lim_{y\to0^{-}} u(h(t)+y,t) \end{align} exist for almost every t. In particular, any function $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ with a representative in $BV_{\text{loc}}$ will satisfy the strong trace property. But the strong trace property is weaker than $BV_{\text{loc}}$. The result we show in this paper is \begin{theorem}[Main theorem] \label{main_theorem} Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution with initial data $u^0\in L^{\infty}(\mathbb{R})$ to the scalar conservation law in one space dimension \eqref{conservation_law} with flux $A\in C^2(\mathbb{R})$ strictly convex. Assume $u$ satisfies the entropy inequality \eqref{entropy_inequality_distributional} for at least one strictly convex entropy $\eta\in C^2(\mathbb{R})$. Further, assume $u$ satisfies the strong trace property (\Cref{strong_trace_condition}). Then $u$ is the unique solution to \eqref{conservation_law} verifying \eqref{condition_e} and with initial data $u^0$. \end{theorem} We briefly outline the present paper and the proof of the main theorem, \Cref{main_theorem}. Let $u$ be any bounded weak solution to \eqref{conservation_law} with initial data $u^0\in L^{\infty}(\mathbb{R})$, entropic for a strictly convex entropy $\eta\in C^2(\mathbb{R})$ and satisfying the strong trace property (\Cref{strong_trace_condition}). In the scalar case, the special structure we are trying to detect is \eqref{condition_e}. To prove $u$ verifies \eqref{condition_e}, we use shift functions very similar to the ones constructed by Leger \cite{Leger2011_original} and in \cite{serre_vasseur} to give, at each fixed time $T$, a sequence of piecewise Lipschitz continuous functions $\{\psi_\epsilon\}_{\epsilon>0}$ defined on a subset of the real line which converge in $L^2$ to $u(\cdot,T)$ as $\epsilon\to0^{+}$. The $\psi_\epsilon$ will be constructed by gluing together at time $T$ various classical solutions to \eqref{conservation_law}, each of which satisfy \eqref{condition_e}. Thus, the $\psi_\epsilon$ will be shown to verify \eqref{condition_e}. By classical measure theory, the $\psi_\epsilon$ converge (up to a subsequence) to $u(\cdot,T)$ pointwise almost everywhere. Thus, $u$ will have the same structure as the $\psi_\epsilon$ and will satisfy \eqref{condition_e} for time $T$, where $T$ is arbitrary. This will complete the proof of \Cref{main_theorem}. In \Cref{Preliminaries}, we give some preliminary results and facts which we will need. Then, in \Cref{construction_of_shift} we give a construction of a shift function as suited to our needs in this paper, based on the construction in \cite{serre_vasseur}. With the shift construction out of the way, in \Cref{main_prop_section} we state and prove the main proposition (\Cref{main_lemma}) which gives the existence of the $\psi_\epsilon$. The proof of the main proposition is broken up into two lemmas (\Cref{psi_exists_with_limits} and \Cref{LIWAS_dense}). To conclude, in \Cref{main_prop_implies_main_theorem} we let $\epsilon\to0^{+}$ (up to a subsequence) to explain how the main proposition implies the main theorem (\Cref{main_theorem}). \section{Preliminaries}\label{Preliminaries} Throughout this paper, we will assume that there is a constant $B>0$ such that all of the weak solutions $w$ to \eqref{conservation_law} that we consider satisfy \begin{align} \label{define_B} \norm{w}_{L^{\infty}(\mathbb{R}\times[0,\infty))}\leq B. \end{align} For a function $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ that verifies the strong trace property (\Cref{strong_trace_condition}), and a Lipschitz continuous function $h:[0,\infty)\to\mathbb{R}$, we use the notation $u(h(t)\pm,t)$ to denote the values $u_{\pm}(t)$ at a time $t$ when $u$ has a left and right trace according to \Cref{strong_trace_condition}, where $u_{\pm}$ are as in the definition of strong trace. For time $t$ when $u$ has a strong trace, the values $u_{\pm}(t)$ are well-defined, and hence $u(h(t)\pm,t)$ are also well-defined. Most of the things in this preliminary section are well-known. But we include these facts here and their proofs so as to make sure we are not using more than we think. In \Cref{main_theorem} we do not assume that the solution $u$ to the scalar conservation law is Kruzhkov. \begin{lemma}[Properties of solutions with Lipschitz initial data]\label{lip_initial_data_properties} Let $v_i^0\in L^{\infty}(\mathbb{R})$ be nondecreasing Lipschitz continuous functions for $i=1,2$, verifying $v_1^0(x)\geq v_2^0(x)$ for all $x\in\mathbb{R}$. Let $v_i$ denote the unique solution to \eqref{conservation_law} satisfying \eqref{condition_e} and with initial data $v_i^0$. Then the following holds: \begin{enumerate}[label=(\alph)] \item \label{lip_initial_1} The $v_i$ are classical solutions to \eqref{conservation_law} on $\mathbb{R}\times[0,\infty)$. \item \label{lip_initial_2} The $v_i$ are given by the method of characteristics. In other words, for each $(x,t)\in\mathbb{R}\times[0,\infty)$, there exists $x^0\in\mathbb{R}$ such that $v_1(x,t)=v_1^0(x^0)$ and $x=x^0+tA'(v_1^0(x^0))$ (and similarly for $v_2$). \item \label{lip_initial_6} The $v_i$ satisfy \eqref{condition_e} with $C=1/ \inf A''$. \item \label{lip_initial_5} For $i=1,2$ and $t\geq0$, $v_i(\cdot,t)$ is a nondecreasing function in the $x$ variable. \item \label{lip_initial_3} $\norm{v_i^0}_{L^{\infty}(\mathbb{R})}=\norm{v_i}_{L^{\infty}(\mathbb{R}\times[0,\infty))}$ for $i=1,2$. \item \label{lip_initial_4} $v_1(x,t)\geq v_2(x,t)$ for all $(x,t)\in\mathbb{R}\times[0,\infty)$. \end{enumerate} \begin{proof} From \cite[p.~176-7]{dafermos_big_book}, we know that for each $i$ there exists a classical solution $v_i$ to \eqref{conservation_law} on $\mathbb{R}\times[0,\infty)$ with initial data $v_i^0$ which is given by the method of characteristics. For a fixed $i$, we now check by direct method that $v_i$ satisfies \eqref{condition_e}. By uniqueness of solutions satisfying \eqref{condition_e}, this will prove parts \ref{lip_initial_1} and \ref{lip_initial_2}. Fix $t,z>0$ and $x\in\mathbb{R}$. Because $v_i$ is given by the method of characteristics, there exists an $x^0$ and $\tilde{x}^0$ such that \begin{align} \begin{cases} v_i(x,t)=v_i^0(x^0) \\ x=x^0+tA'(v_i^0(x^0)) \end{cases} \end{align} and \begin{align} \begin{cases} v_i(x+z,t)=v_i^0(\tilde{x}^0) \\ x+z=\tilde{x}^0+tA'(v_i^0(\tilde{x}^0)). \end{cases} \end{align} We have \begin{equation} \begin{aligned}\label{characteristics1} v_i(x,t)=(A')^{-1}\bigg(\frac{x-{x}^0}{t}\bigg)\\ v_i(x+z,t)=(A')^{-1}\bigg(\frac{x+z-\tilde{x}^0}{t}\bigg)\\ \end{aligned} \end{equation} where the functional inverse $(A')^{-1}$ exists because $A$ is strictly convex. Because the characteristics do not cross, we have \begin{align}\label{characteristics4} 0 < \tilde{x}^0 -x^0=z+t(A'(v_i^0(x^0))-A'(v_i^0(\tilde{x}^0)))<z. \end{align} Then from \eqref{characteristics4}, \begin{align}\label{characteristics3} 0<\frac{x+z-\tilde{x}^0}{t}-\frac{x-{x}^0}{t}<\frac{z}{t}. \end{align} Finally, \begin{equation} \begin{aligned}\label{characteristics2} &\inf A'' \bigg[(A')^{-1}\bigg(\frac{x+z-\tilde{x}^0}{t}\bigg)-(A')^{-1}\bigg(\frac{x-{x}^0}{t}\bigg)\bigg] \\ &\leq A'((A')^{-1}\bigg(\frac{x-{x}^0}{t}\bigg))-A'((A')^{-1}\bigg(\frac{x+z-\tilde{x}^0}{t}\bigg))<\frac{z}{t}. \end{aligned} \end{equation} Lines \eqref{characteristics1}, \eqref{characteristics3} and \eqref{characteristics2} imply $v_i$ satisfies \eqref{condition_e}. In particular, note that we can take $C=1/ \inf A''$ in \eqref{condition_e}. This proves parts \ref{lip_initial_1}, \ref{lip_initial_2}, and \ref{lip_initial_6}. Part \ref{lip_initial_5} follows immediately from part \ref{lip_initial_2}. Similarly, part \ref{lip_initial_3} follows immediately from parts \ref{lip_initial_1} and \ref{lip_initial_2}. We now show how part \ref{lip_initial_4} follows from part \ref{lip_initial_2}. We argue by contradiction. Assume there exists $(x,t)\in\mathbb{R}\times[0,\infty)$ such that $v_1(x,t)<v_2(x,t)$. Then we have just proven that $v_1$ and $v_2$ are given by the method of characteristics. Thus there exists $x^0$ and $\tilde{x}^0$ such that \begin{equation} \begin{aligned}\label{characteristics_a1} x&=x^0+tA'(v_1^0(x^0))\\ &=\tilde{x}^0+tA'(v_2^0(\tilde{x}^0)) \end{aligned} \end{equation} and \begin{align} \label{characteristics_a2} v_1(x,t)=v_1^0(x^0)<v_2^0(\tilde{x}^0)=v_2(x,t). \end{align} Then from \eqref{characteristics_a1}, \begin{align} \label{characteristics_a3} x^0-\tilde{x}^0=t(A'(v_2^0(\tilde{x}^0))-A'(v_1^0(x^0))). \end{align} Then the right-hand side of \eqref{characteristics_a3} is nonnegative, which means $x^0\geq \tilde{x}^0$. Thus, \begin{align} v_1^0(x^0)\geq v_1^0(\tilde{x}^0)\geq v_2^0(\tilde{x}^0) \end{align} because $v_1^0(x)\geq v_2^0(x)$ for all $x\in\mathbb{R}$. However, this gives a contradiction with \eqref{characteristics_a2}. This proves part \ref{lip_initial_4}. \end{proof} \end{lemma} \begin{lemma}[An extension of the entropy inequality] \label{extended_entropy_inequality_lemma} Let $\bar{u}^0\in L^{\infty}(\mathbb{R})$ be a Lipschitz continuous and nondecreasing function. Let $\bar{u}$ be the unique solution to \eqref{conservation_law} with initial data $\bar{u}^0$ and which satisfies \eqref{condition_e}. Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution to \eqref{conservation_law} with initial data $u^0$. Assume that $u$ is entropic for the strictly convex entropy $\eta\in C^2(\mathbb{R})$. Then \begin{align} \label{relative_entropy_inequality_distributional} \partial_t \eta (u\|\bar{u})+\partial_x q(u;\bar{u}) \leq 0 \end{align} in the sense of distributions. In other words, the following holds for all nonnegative, Lipschitz continuous test functions $\phi$ on $\mathbb{R}\times [0,\infty)$ with compact support: \begin{align} \label{extended_entropy_inequality} \int\limits_{0}^{\infty} \int\limits_{-\infty}^{\infty}[\partial_t \phi \eta(u\|\bar{u})+\partial_x \phi q(u;\bar{u})] \,dxdt + \int\limits_{-\infty}^{\infty} \phi(x,0)\eta(u^0(x)\|\bar{u}^0(x))\,dx\geq0. \end{align} \begin{remark} The inequality \eqref{relative_entropy_inequality_distributional} extends the entropy inequality \eqref{entropy_inequality_distributional}. \end{remark} \begin{proof} By part \ref{lip_initial_1} of \Cref{lip_initial_data_properties}, $\bar{u}$ is a classical solution. Then, \Cref{extended_entropy_inequality_lemma} follows immediately from the following inequality: \begin{align} \label{fact_from_dafermos} \begin{split} \int\limits_{0}^{\infty} \int\limits_{-\infty}^{\infty}[\partial_t \phi \eta(u\|\bar{u})+\partial_x \phi q(u;\bar{u})] \,dxdt + \int\limits_{-\infty}^{\infty} \phi(x,0)\eta(u^0(x)\|\bar{u}^0(x))\,dx\geq \\ \int\limits_{0}^{\infty} \int\limits_{-\infty}^{\infty}\phi\partial_x \bar{u} \eta''(\bar{u})[A(u)-A(\bar{u})-A'(\bar{u})(u-\bar{u})]\,dxdt. \end{split} \end{align} Dafermos gives this inequality as one of the central steps in the proof of weak-strong stability. See \cite[p.~124, line (5.2.10)]{dafermos_big_book}. By part \ref{lip_initial_5} of \Cref{lip_initial_data_properties}, $\bar{u}$ is increasing in $x$. Furthermore, we have $\eta'',A''>0$. Thus the right-hand side of \eqref{fact_from_dafermos} is controlled from below by zero. \end{proof} \end{lemma} \begin{lemma}\label{rigorous_dissipation_rate_calc_middle_part} Let $[t^,t^{})$ be a bounded interval. Let $h_1(t), h_2(t): [t^,t^{*})\to \mathbb{R}$ be Lipschitz continuous functions, such that $h_2(t)-h_1(t)>0$ for all $t\in[t^,t^{*})$. Let $\bar{u}^0\in L^{\infty}(\mathbb{R})$ be a Lipschitz continuous and nondecreasing function. Let $\bar{u}$ be the unique solution to \eqref{conservation_law} with initial data $\bar{u}^0$ and which satisfies \eqref{condition_e}. Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution to \eqref{conservation_law} with initial data $u^0$. Assume $u$ is entropic for the strictly convex entropy $\eta\in C^2(\mathbb{R})$. Furthermore, assume that $u$ verifies the strong trace property (\Cref{strong_trace_condition}). Then for almost every $a,b\in[t^,t^{*})$ with $a<b$, \begin{equation} \begin{aligned}\label{rigorous_dissipation_rate_calc_middle_part_ineq} &\int\limits_{h_1(b)}^{h_2(b)}\eta (u(x,b)\|\bar{u}(x,b))\,dx-\int\limits_{h_1(a)}^{h_2(a)}\eta (u(x,a)\|\bar{u}(x,a))\,dx\leq\\ &\hspace{.2in}\int\limits_{a}^{b}\bigg[ q(u(h_1(t)+,t);\bar{u}(h_1(t),t))-q(u(h_2(t)-,t);\bar{u}(h_2(t),t))+\\&\hspace{.2in}\dot{h}_2(t)\eta(u(h_2(t)-,t)\|\bar{u}(h_2(t),t))-\dot{h}_1(t)\eta(u(h_1(t)+,t)\|\bar{u}(h_1(t),t))\bigg]\,dt. \end{aligned} \end{equation} If $t^=0$, then \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} holds for $a=0$ and almost every $b\in(t^,t^{})$. \begin{proof} For $0<\epsilon<\min\{b-a,t^{}-b\}$, define \begin{align} \chi_\epsilon (t)\coloneqq \begin{cases} 0 & \text{if } t<a \\ \frac{1}{\epsilon}(t-a) & \text{if } a\leq t<a+\epsilon \\ 1 & \text{if } a+\epsilon\leq t\leq b\\ -\frac{1}{\epsilon}(t-(b+\epsilon)) & \text{if } b<t \leq b+\epsilon\\ 0 & \text{if } b+\epsilon<t. \end{cases} \end{align} Let $\delta\coloneqq \inf_{t\in[a,b+\epsilon]}h_2(t)-h_1(t)$. Note $\delta>0$. Then for $0<\epsilon<\frac{\delta}{2}$, define \begin{align} \psi_{\epsilon}(x,t)\coloneqq \begin{cases} 0 & \text{if } x<h_1(t) \\ \frac{1}{\epsilon}(x-h_1(t)) & \text{if } h_1(t)<x<h_1(t)+\epsilon \\ 1 & \text{if } h_1(t)+\epsilon<x<h_2(t)-\epsilon\\ -\frac{1}{\epsilon}(x-h_2(t)) & \text{if } h_2(t)-\epsilon<x<h_2(t)\\ 0 & \text{if } h_2(t)<x. \end{cases} \end{align} The $\psi_{\epsilon}(x,t)$ and $\chi_{\epsilon} (t)$ are from \cite[p.~765]{Leger2011_original}. Use $\psi_{\epsilon}(x,t) \chi_{\epsilon}(t)$ as a test function for \eqref{extended_entropy_inequality}. The result is \begin{equation} \begin{aligned} \label{before_rearranged_dissipation_rate} &\frac{1}{\epsilon}\int\limits_{a}^{b}\bigg[\int\limits_{h_1(t)}^{h_1(t)+\epsilon} q(u;\bar{u})\,dx-\int\limits_{h_2(t)-\epsilon}^{h_2(t)} q(u;\bar{u})\,dx+\int\limits_{h_2(t)-\epsilon}^{h_2(t)} \dot{h}_2(t)\eta(u\|\bar{u})\,dx-\int\limits_{h_1(t)}^{h_1(t)+\epsilon}\dot{h}_1(t)\eta(u\|\bar{u})\,dx\bigg]\,dt \\ &\hspace{.2in}+ \frac{1}{\epsilon}\int\limits_{a}^{a+\epsilon} \int\limits_{h_1(t)}^{h_2(t)}\eta(u\|\bar{u}) \,dxdt -\frac{1}{\epsilon}\int\limits_{b}^{b+\epsilon} \int\limits_{h_1(t)}^{h_2(t)}\eta(u\|\bar{u}) \,dxdt+O(\epsilon)\geq0. \end{aligned} \end{equation} Let $\epsilon\to0^{+}$ in \eqref{before_rearranged_dissipation_rate}. Use the dominated convergence theorem, the Lebesgue differentiation theorem, and the fact that $u$ verifies the strong trace property (\Cref{strong_trace_condition}). This gives \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} for almost every $a,b\in[t^,t^{*})$ with $a<b$. When $t^=0$, we want to show that \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} holds for $a=t^=0$. This follows because we include the boundary term corresponding to $t=0$ in \eqref{entropy_inequality_integral_formulation}, and hence the boundary term corresponding to $t=0$ appears also in \eqref{extended_entropy_inequality}. We now show that when $t^=0$, \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} holds for $a=t^=0$. For $a=t^=0$ and $0<\epsilon<t^{*}-b$, define \begin{align} \chi_\epsilon^0 (t)\coloneqq \begin{cases} 1 & \text{if } 0\leq t< b\\ -\frac{1}{\epsilon}(t-(b+\epsilon)) & \text{if } b\leq t < b+\epsilon\\ 0 & \text{if } b+\epsilon \leq t. \end{cases} \end{align} The $\chi_\epsilon^0$ is borrowed from \cite[p.~124]{dafermos_big_book}. Test \eqref{extended_entropy_inequality} with $\psi_{\epsilon}(x,t) \chi_{\epsilon}^0(t)$. This gives \begin{equation} \begin{aligned} \label{before_rearranged_dissipation_rate_zero} &\frac{1}{\epsilon}\int\limits_{a}^{b}\bigg[\int\limits_{h_1(t)}^{h_1(t)+\epsilon} q(u;\bar{u})\,dx-\int\limits_{h_2(t)-\epsilon}^{h_2(t)} q(u;\bar{u})\,dx+\int\limits_{h_2(t)-\epsilon}^{h_2(t)} \dot{h}_2(t)\eta(u\|\bar{u})\,dx-\int\limits_{h_1(t)}^{h_1(t)+\epsilon}\dot{h}_1(t)\eta(u\|\bar{u})\,dx\bigg]\,dt \\ &\hspace{1in}+ \int\limits_{h_1(0)}^{h_2(0)}\eta(u^0(x)\|\bar{u}^0(x)) \,dx -\frac{1}{\epsilon}\int\limits_{b}^{b+\epsilon} \int\limits_{h_1(t)}^{h_2(t)}\eta(u\|\bar{u}) \,dxdt+O(\epsilon)\geq0. \end{aligned} \end{equation} Finally, let $\epsilon\to0^{+}$ in \eqref{before_rearranged_dissipation_rate_zero}. Once again, invoke the dominated convergence theorem, the Lebesgue differentiation theorem, and use that $u$ verifies the strong trace property (\Cref{strong_trace_condition}). We receive \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} for $a=0$ and for almost every $b\in(t^,t^{*})$. \end{proof} \end{lemma} \begin{lemma}\label{left_right_ap_limits} Let $\bar{u}^0\in L^{\infty}(\mathbb{R})$ be a Lipschitz continuous and nondecreasing function. Let $\bar{u}$ be the unique solution to \eqref{conservation_law} with initial data $\bar{u}^0$ and which satisfies \eqref{condition_e}. Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution to \eqref{conservation_law} with initial data $u^0$. Assume that $u$ is entropic for at least one strictly convex entropy $\eta\in C^2(\mathbb{R})$. Assume also that $u$ verifies the strong trace property (\Cref{strong_trace_condition}). Then for all $c,d\in\mathbb{R}$ verifying $c<d$, the approximate right- and left-hand limits \begin{align}\label{ap_right_left_limits_exist_44} &{\rm ap}\,\lim_{t\to {t_0}^{\pm}}\int\limits_{c}^{d}\eta (u(x,t)\|\bar{u}(x,t))\,dx \end{align} exist for all $t_0\in(0,\infty)$ and verify \begin{align} \label{left_and_right_limits_order} &{\rm ap}\,\lim_{t\to {t_0}^{-}}\int\limits_{c}^{d}\eta (u(x,t)\|\bar{u}(x,t))\,dx\geq {\rm ap}\,\lim_{t\to {t_0}^{+}}\int\limits_{c}^{d}\eta (u(x,t)\|\bar{u}(x,t))\,dx. \end{align} The approximate right-hand limit also exists for $t_0=0$ and verifies \begin{align} \label{right_limit_at_zero} \int\limits_{c}^{d}\eta (u^0(x)\|\bar{u}^0(x))\,dx\geq {\rm ap}\,\lim_{t\to {0}^{+}}\int\limits_{c}^{d}\eta (u(x,t)\|\bar{u}(x,t))\,dx. \end{align} \begin{proof} For some constant $C>0$ to be chosen momentarily, define the function $\Gamma:[0,\infty)\to\mathbb{R}$, \begin{align} \Gamma(t)\coloneqq \int\limits_{c}^{d}\eta (u(x,t)\|\bar{u}(x,t))\,dx -Ct. \end{align} Apply \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} to the case when $h_1(t)=c$ and $h_2(t)=d$ for all $t$. The integrand on the right-hand side of \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} is bounded. Thus, there exists some constant $C>0$ such that $\Gamma(t)\geq\Gamma(s)$ for almost every $t$ and $s$ verifying $t<s$. This means that there exists a function which agrees with $\Gamma$ almost everywhere and is non-increasing. This implies that $\Gamma$ has approximate left and right limits. In particular, we conclude that the approximate right- and left-hand limits \eqref{ap_right_left_limits_exist_44} exist for all $t_0\in(0,\infty)$ and verify \eqref{left_and_right_limits_order}. Note the approximate right-hand limit also exists for $t_0=0$ and because \eqref{rigorous_dissipation_rate_calc_middle_part_ineq} holds for the time $a=0$, the approximate right-hand limit verifies \eqref{right_limit_at_zero} at time zero. \end{proof} \end{lemma} \section{Construction of the shift} \label{construction_of_shift} In this section, we present a proof of \begin{proposition}[Existence of the shift function] \label{shift_theorem} Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution to \eqref{conservation_law}, entropic for at least one strictly convex entropy $\eta\in C^2(\mathbb{R})$. Assume also that $u$ verifies the strong trace property (\Cref{strong_trace_condition}). Let $\bar{u}^0_i\in L^{\infty}(\mathbb{R})$ be Lipschitz continuous and nondecreasing functions for $i=1,2$, verifying $\bar{u}^0_1(x)\geq\bar{u}_2^0(x)$ for all $x\in\mathbb{R}$. Let $\bar{u}_i\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be solutions to \eqref{conservation_law} verifying \eqref{condition_e} and with initial data $\bar{u}^0_i$, for $i=1,2$. Let $t^\geq0$ be some fixed time, and let $x_0\in\mathbb{R}$ be some fixed space coordinate. Then for $\epsilon>0$ there exists a Lipschitz continuous function $h_\epsilon:[t^,\infty)\to\mathbb{R}$ such that \begin{align} h_\epsilon(t^)=x_0, \hspace{.3in} \mbox{\upshape Lip}[h_\epsilon]\leq \sup_{\hspace{.02in}\abs{x}\leq\norm{u}_{L^{\infty}}} \abs{A'} \end{align} and \begin{align} q(u_{+};\bar{u}_2)-q(u_{-};\bar{u}_1)-\dot{h}_\epsilon(\eta(u_{+}\|\bar{u}_2)-\eta(u_{-}\|\bar{u}_1))\leq \epsilon \end{align} for almost every $t\in[t^,\infty)$, where $u_{\pm}=u(h_\epsilon(t)\pm,t)$ and $\bar{u}_i=\bar{u}_i(h_\epsilon(t),t)$. \end{proposition} In \cite{serre_vasseur}, the authors build a general framework for the construction of the shift functions necessary for $L^2$ stability in the general case of systems. They also apply their framework to the scalar case, recovering the result of Leger \cite{Leger2011_original}. We present here a construction of the shift function based on the work \cite{serre_vasseur}. We modify the construction slightly. In \cite{serre_vasseur}, the data $\bar{u}_i$ are constant. In contrast, we assume the $\bar{u}_i$ are Lipschitz continuous solutions to \eqref{conservation_law}. The presentation is also simplified due to our focus only on the scalar case. Lastly, in the general framework for systems in \cite{serre_vasseur}, there is a ``constraint'' labeled with equation number 7 \cite[p.~4]{serre_vasseur}. Although this constraint is used in \cite{serre_vasseur} for the scalar case in particular, we find that the proof for scalar in \cite{serre_vasseur} as-is does not require it. Lastly, the paper \cite{serre_vasseur} considers only Kruzhkov's solutions \cite[p.~9]{serre_vasseur} which are entropic for the family of entropies $\{\abs{u-k}\}_{k\in\mathbb{R}}$. We find that the proofs in \cite{serre_vasseur} hold without modification for solutions which are not necessarily of this type. \subsection{Lemmas necessary for the proof of \Cref{shift_theorem}} We need the following structural lemmas. \begin{lemma}[Structural lemma on entropic shocks] \label{entropic_shocks} Consider the equation \eqref{conservation_law}, endowed with a strictly convex entropy $\eta\in C^2(\mathbb{R})$, and an associated entropy flux $q$. Let $u_L,u_R,\sigma\in\mathbb{R}$ satisfy \begin{align} A(u_R)-A(u_L)=\sigma(u_R-u_L). \end{align} Then \begin{align} \label{entropic_shock} q(u_R)-q(u_L)\leq \sigma (\eta(u_R)-\eta(u_L)) \end{align} if and only if $u_L\geq u_R$. Which is to say, the shock $(u_L,u_R,\sigma)$ is entropic for the entropy $\eta$ if and only if $u_L\geq u_R$. \begin{remark} For the scalar conservation law \eqref{conservation_law} with flux $A$ strictly convex, we know that the ``physical'' or ``admissible'' shock wave solutions will be the ones with left- and right-hand states $u_L$ and $u_R$, respectively, which satisfy $u_L>u_R$. This is the \emph{Lax entropy condition} for the scalar case. It is immediate that shock solutions satisfying \eqref{condition_e} verify the Lax entropy condition. \Cref{entropic_shocks} says that shock solutions to the scalar conservation law \eqref{conservation_law} endowed with the entropy $\eta$, also verify the Lax entropy condition. \end{remark} \begin{proof} We only consider the case when $u_L\neq u_R$. Let \begin{align} \Lambda \coloneqq q(u_R)-q(u_L)-\sigma (\eta(u_R)-\eta(u_L))= \int\limits_{u_L}^{u_R} A'(u)\eta'(u)-\sigma\eta'(u)\,du. \end{align} We want to show $\Lambda \leq 0$ if and only if $u_L>u_R$. We can write \begin{align} \sigma=\frac{A(u_R)-A(u_L)}{u_R-u_L}=\frac{1}{u_R-u_L}\int\limits_{u_L}^{u_R} A'(v)\,dv. \end{align} Thus \begin{align} \Lambda &= \int\limits_{u_L}^{u_R} \eta'(u)\bigg(A'(u)- \frac{1}{u_R-u_L}\int\limits_{u_L}^{u_R} A'(v)\,dv\bigg)\,du\\ &= \int\limits_{u_L}^{u_R} \eta'(u)\bigg(\frac{1}{u_R-u_L}\int\limits_{u_L}^{u_R} A'(u)-A'(v)\,dv\bigg)\,du\\ &= \frac{1}{u_R-u_L}\int\limits_{u_L}^{u_R}\int\limits_{u_L}^{u_R}\eta'(u)(A'(u)-A'(v))\,dvdu\\ &= \frac{1}{u_R-u_L}\bigg(\frac{1}{2}\iint\limits_{D}\eta'(u)(A'(u)-A'(v))\,dvdu-\frac{1}{2}\iint\limits_{D}\eta'(v)(A'(u)-A'(v))\,dvdu\bigg) \end{align} where $D\coloneqq I(u_L,u_R)\times I(u_L,u_R)$. We use $I(a,b)$ to denote the interval with endpoints $a$ and $b$. Finally, \begin{align} \Lambda=\frac{1}{2}\frac{1}{u_R-u_L}\iint\limits_{D} (\eta'(u)-\eta'(v))(A'(u)-A'(v))\,dvdu. \end{align} Then, by strict convexity of $A$ and $\eta$, the quantity \begin{align} \label{always_nonnegative} \iint\limits_{D} (\eta'(u)-\eta'(v))(A'(u)-A'(v))\,dvdu \end{align} is always positive. Thus, the sign of $\Lambda$ is given by the sign of $u_R-u_L$. This completes the proof. \end{proof} \end{lemma} \begin{lemma}[Structural lemma from \cite{serre_vasseur}] \label{d_sm_rh_lemma} Let $u_L,u_R,u_{-},u_{+}\in\mathbb{R}$ satisfy $u_L > u_R$ and $u_{-} > u_{+}$. Let $\eta\in C^2(\mathbb{R})$ be a strictly convex entropy, with associated entropy flux $q$. Define \begin{align} \sigma(u_{-},u_{+})\coloneqq \frac{A(u_{+})-A(u_{-})}{u_{+}-u_{-}}.\label{rh_velocity} \end{align} Then, \begin{align} q(u_{+};u_R)-q(u_{-};u_L)-\sigma(u_{-},u_{+})(\eta(u_{+}\|u_R)-\eta(u_{-}\|u_L)) \leq 0. \label{d_rh} \end{align} Moreover, if \begin{align} \eta(u\|u_L)=\eta(u\|u_R) \label{denom_zero} \end{align} for some $u\in\mathbb{R}$, then \begin{align} q(u;u_R)-q(u;u_L)< 0. \label{d_sm} \end{align} \end{lemma} \Cref{d_sm_rh_lemma} was proven in \cite{serre_vasseur}. For completeness, we give the proof of this lemma in the appendix (\Cref{d_sm_rh_lemma_proof}). \begin{lemma}[Structural lemma] \label{v_epsilon_bounded} Let $\eta\in C^2(\mathbb{R})$ be a strictly convex entropy for \eqref{conservation_law}. Let $q$ be the associated entropy flux. For $\epsilon>0$, define the function $V_\epsilon: \{(u,u_L,u_R)\in \mathbb{R}^3 \| u_L\geq u_R\}\to \mathbb{R}$: \begin{align} V_\epsilon(u,u_L,u_R)\coloneqq \begin{cases} \frac{[q(u;u_R)-q(u;u_L)-\epsilon]_+}{\eta(u\|u_R)-\eta(u\|u_L)} & \text{if } \eta(u\|u_R)\neq\eta(u\|u_L) \\ 0 & \text{if } \eta(u\|u_R)=\eta(u\|u_L), \end{cases} \end{align} where $[\hspace{.025in}\cdot\hspace{.025in}]_{+}\coloneqq \max(0,\cdot)$. Then, $V_\epsilon$ verifies \begin{align} \abs{V_\epsilon(u,u_L,u_R)} \leq \abs{A'(u)}. \label{final_part_v_epsilon_bounded} \end{align} \begin{proof} Let \[ V(u,u_L,u_R)\coloneqq \begin{cases} \frac{[q(u;u_R)-q(u;u_L)]_+}{\eta(u\|u_R)-\eta(u\|u_L)} & \text{if } \eta(u\|u_R)\neq\eta(u\|u_L) \\ 0 & \text{if } \eta(u\|u_R)=\eta(u\|u_L). \end{cases} \] In order to show \eqref{final_part_v_epsilon_bounded}, the idea is to show \begin{align} \abs{V(u,u_L,u_R)} \leq \abs{A'(u)}\label{v_control} \end{align} and then use the basic inequality \[\abs{V_\epsilon(u,u_L,u_R)}\leq \abs{V(u,u_L,u_R)}.\] The proof of \eqref{v_control} depends on controlling the quantity \begin{align}\label{quantity_we_want_to_control} \frac{q(u;u_R)-q(u;u_L)}{\eta(u\|u_R)-\eta(u\|u_L)} \end{align} for the $(u,u_L,u_R)$ values that make $q(u;u_R)-q(u;u_L)\geq0$. We have three cases where we get control on the quantity \eqref{quantity_we_want_to_control}: $u>u_L>u_R$, $u_L>u_R>u$, and $u_R\leq u\leq u_L$. We begin with some elementary facts which will be used repeatedly. Remark that \begin{align} \partial_b \eta(a\|b)=\eta''(b)(b-a) \mbox{\hspace{.3in}and\hspace{.3in}} \partial_b q(a;b)=\eta''(b)(A(b)-A(a)). \end{align} In particular, we can write for all $a,b,c\in\mathbb{R}$, \begin{align} q(a;b)-q(a;c)= \int\limits_{c}^{b}\eta''(v)(A(v)-A(a))\,dv= \int\limits_{c}^{b}\eta''(v)(v-a)\frac{A(v)-A(a)}{v-a}\,dv. \end{align} We can now begin the casework to prove \Cref{v_epsilon_bounded}. \emph{Case} $u>u_L>u_R$ We note that for all $u,u_L,u_R\in\mathbb{R}$ such that $u>u_L>u_R$, \begin{align} \eta(u\|u_R)-\eta(u\|u_L)=\int\limits_{u_L}^{u_R}\partial_v \eta(u\|v)\,dv=\int\limits_{u_L}^{u_R}\eta''(v)(v-u)\,dv>0 \end{align} by strict convexity of $\eta$. We conclude \begin{align} \label{sign_denom_1} \eta(u\|u_R)-\eta(u\|u_L)>0. \end{align} For all $u,u_L,u_R\in\mathbb{R}$ such that $u>u_L>u_R$ and $\eta(u\|u_R)-\eta(u\|u_L)\neq0$, we compute \begin{align} \frac{q(u;u_R)-q(u;u_L)}{\eta(u\|u_R)-\eta(u\|u_L)} &= \frac{\int\limits_{u_L}^{u_R}\eta''(v)(v-u)\frac{A(v)-A(u)}{v-u}\,dv}{\eta(u\|u_R)-\eta(u\|u_L)} \label{refer_back_1} \shortintertext{By the mean value theorem and strict convexity of $A$, $\frac{A(v)-A(u)}{v-u}<A'(u)$ for $v\in[u_R,u_L]$. Thus, recalling also the strict convexity of $\eta$ and \eqref{sign_denom_1}, we continue from \eqref{refer_back_1} to get} &< \frac{A'(u)\int\limits_{u_L}^{u_R}\eta''(v)(v-u)\,dv}{\eta(u\|u_R)-\eta(u\|u_L)} =A'(u). \end{align} In summary, \begin{align}\label{bound_v_case1} \frac{q(u;u_R)-q(u;u_L)}{\eta(u\|u_R)-\eta(u\|u_L)}<A'(u). \end{align} \emph{Case} $u_L>u_R>u$ Our method is analogous to the case $u>u_L>u_R$ above. Remark that for all $u,u_L,u_R\in\mathbb{R}$ such that $u_L>u_R>u$, \begin{align} \eta(u\|u_R)-\eta(u\|u_L)=\int\limits_{u_L}^{u_R}\partial_v \eta(u\|v)\,dv=\int\limits_{u_L}^{u_R}\eta''(v)(v-u)\,dv<0 \end{align} by strict convexity of $\eta$. We conclude \begin{align} \label{sign_denom_2} \eta(u\|u_R)-\eta(u\|u_L)<0. \end{align} For all $u,u_L,u_R\in\mathbb{R}$ such that $u_L>u_R>u$ and $\eta(u\|u_R)-\eta(u\|u_L)\neq0$, we calculate \begin{align} \frac{q(u;u_R)-q(u;u_L)}{\eta(u\|u_R)-\eta(u\|u_L)} &= \frac{\int\limits_{u_L}^{u_R}\eta''(v)(v-u)\frac{A(v)-A(u)}{v-u}\,dv}{\eta(u\|u_R)-\eta(u\|u_L)}\label{refer_back_2} \shortintertext{By the mean value theorem and strict convexity of $A$, $\frac{A(v)-A(u)}{v-u}>A'(u)$ for $v\in[u_R,u_L]$. Thus, recalling also the strict convexity of $\eta$ and \eqref{sign_denom_2}, we continue from \eqref{refer_back_2} to get} &> \frac{A'(u)\int\limits_{u_L}^{u_R}\eta''(v)(v-u)\,dv}{\eta(u\|u_R)-\eta(u\|u_L)} =A'(u). \end{align} To summarize, \begin{align}\label{bound_v_case2} \frac{q(u;u_R)-q(u;u_L)}{\eta(u\|u_R)-\eta(u\|u_L)}>A'(u). \end{align} \emph{Case} $u_R\leq u\leq u_L$ This case is slightly different than the two cases above. For all $u,u_L,u_R\in\mathbb{R}$ such that $u_R\leq u\leq u_L$ and $\eta(u\|u_R)-\eta(u\|u_L)\neq0$, \begin{align} \frac{q(u;u_R)-q(u;u_L)}{\abs{\eta(u\|u_R)-\eta(u\|u_L)}} &= \frac{\int\limits_{u_L}^{u_R}\eta''(v)(v-u)\frac{A(v)-A(u)}{v-u}\,dv}{\abs{\eta(u\|u_R)-\eta(u\|u_L)}}\\ &= \frac{\int\limits_{u_L}^{u}\eta''(v)(v-u)\frac{A(v)-A(u)}{v-u}\,dv+\int\limits_{u}^{u_R}\eta''(v)(v-u)\frac{A(v)-A(u)}{v-u}\,dv}{\abs{\eta(u\|u_R)-\eta(u\|u_L)}} \shortintertext{By the mean value theorem and strict convexity of $A$, $\frac{A(v)-A(u)}{v-u}>A'(u)$ for $v\in(u,u_L]$, and $\frac{A(v)-A(u)}{v-u}<A'(u)$ for $v\in[u_R,u)$. Thus, recalling also the strict convexity of $\eta$,} &< \frac{A'(u)\int\limits_{u_L}^{u}\eta''(v)(v-u)\,dv+A'(u)\int\limits_{u}^{u_R}\eta''(v)(v-u)\,dv}{\abs{\eta(u\|u_R)-\eta(u\|u_L)}}\\ &= A'(u)\sgn(\eta(u\|u_R)-\eta(u\|u_L)). \end{align} We receive, \begin{align}\label{bound_v_case3} \frac{q(u;u_R)-q(u;u_L)}{\abs{\eta(u\|u_R)-\eta(u\|u_L)}}<A'(u)\sgn(\eta(u\|u_R)-\eta(u\|u_L)). \end{align} We combine all the cases $u>u_L>u_R$, $u_L>u_R>u$, and $u_R\leq u\leq u_L$, and in particular \eqref{sign_denom_1}, \eqref{bound_v_case1}, \eqref{sign_denom_2}, \eqref{bound_v_case2}, and \eqref{bound_v_case3}. We keep in mind that we only consider $(u,u_L,u_R)$ values that make $q(u;u_R)-q(u;u_L)\geq0$. In conclusion, \[\abs{V_\epsilon(u,u_L,u_R)}\leq \abs{V(u,u_L,u_R)} \leq \abs{A'(u)}.\] \end{proof} \end{lemma} \subsection{Proof of \Cref{shift_theorem}} This proof is based on the work \cite[p.~7-8]{serre_vasseur}. We modify the proof to consider $\bar{u}_i$ which are non-constant. Define $V_\epsilon: \{(u,u_L,u_R)\in \mathbb{R}^3 \| u_L\geq u_R\}\to \mathbb{R}$: \begin{align} V_\epsilon(u,u_L,u_R)\coloneqq \begin{cases} \frac{[q(u;u_R)-q(u;u_L)-\epsilon]_+}{\eta(u\|u_R)-\eta(u\|u_L)} & \text{if } \eta(u\|u_R)\neq\eta(u\|u_L) \\ 0 & \text{if } \eta(u\|u_R)=\eta(u\|u_L), \end{cases} \end{align} where $[\hspace{.025in}\cdot\hspace{.025in}]_{+}\coloneqq \max(0,\cdot)$. The function $V_\epsilon$ is continuous. For $(u,u_L,u_R)$ such that $\eta(u\|u_R)\neq\eta(u\|u_L)$, $V_\epsilon$ is clearly continuous. By \eqref{d_sm}, $V_\epsilon=0$ on some neighborhood of $\{(u,u_L,u_R)\in \mathbb{R}^3 \| \eta(u\|u_R)=\eta(u\|u_L)\}$. Thus, $V_\epsilon$ is continuous. Remark that by part \ref{lip_initial_1} of \Cref{lip_initial_data_properties}, the $\bar{u}_i$ are classical solutions, and by part \ref{lip_initial_4} of \Cref{lip_initial_data_properties}, $\bar{u}_1(x,t)\geq\bar{u}_2(x,t)$ for all $(x,t)\in\mathbb{R}\times[0,\infty)$. We construct a solution to \begin{align} \label{filippov} \begin{cases} \dot{h}_\epsilon=V_\epsilon(u(h_\epsilon(t),t),\bar{u}_1(h_\epsilon(t),t),\bar{u}_2(h_\epsilon(t),t))\\ h_\epsilon(t^)=x_0 \end{cases} \end{align} in the Filippov sense. We use the following lemma: \begin{lemma} \label{Filippov20} There exists a Lipschitz function $h_\epsilon:[t^,\infty)\to\mathbb{R}$ such that \begin{align} &\hspace{-1.5in}h_\epsilon(t^)=x_0,\label{Filippov1}\\ &\hspace{-1.5in}\norm{\dot{h}_\epsilon}_{L^{\infty}}\leq\norm{V_\epsilon}_{L^{\infty}},\label{Filippov2}\\ \shortintertext{and} \dot{h}_\epsilon(t)\in I[V_\epsilon(u_{-},\bar{u}_1,\bar{u}_2),V_\epsilon(u_{+},\bar{u}_1,\bar{u}_2)],\label{Filippov3} \end{align} for almost every $t>0$, where $u_{\pm}=u(h_\epsilon(t)\pm,t)$, and $\bar{u}_i=\bar{u}_i(h_\epsilon(t),t)$ for $i=1,2$. Here, $I[a,b]$ denotes the closed interval with endpoints $a$ and $b$. Moreover, for almost every $t>0$, \begin{align} A(u_{+})-A(u_{-})=\dot{h}_\epsilon(u_{+}-u_{-}),\label{Filippov4}\\ q(u_{+})-q(u_{-})\leq\dot{h}_\epsilon (\eta(u_{+})-\eta(u_{-})),\label{Filippov5} \end{align} which means that for almost every $t>0$, either the shock $(u_{-},u_{+},\dot{h}_\epsilon)$ is an entropic discontinuity for the entropy $\eta$ or $u_{-}=u_{+}$. \end{lemma} The proof of \eqref{Filippov1}, \eqref{Filippov2} and \eqref{Filippov3} is nearly identical to the proof of Proposition 1 in \cite{Leger2011}. For completeness, we provide a proof of \eqref{Filippov1}, \eqref{Filippov2} and \eqref{Filippov3} in the appendix (\Cref{proof_of_Filippov}). The properties \eqref{Filippov4} and \eqref{Filippov5} in fact hold for any Lipschitz function $h: [0,\infty)\to\mathbb{R}$. These properties are well-known in the $BV$ case. When $u$ only satisfies the strong trace property (\Cref{strong_trace_condition}), \eqref{Filippov4} and \eqref{Filippov5} are given in Lemma 6 in \cite{Leger2011}. We do not include a proof of \eqref{Filippov4} and \eqref{Filippov5} here; a proof is given in the appendix in \cite{Leger2011}. We are now ready to prove \Cref{shift_theorem}. \begin{proof}[Proof of \Cref{shift_theorem}] For almost every $t$ such that $u_{-}=u_{+}$, we borrow notation from \cite{serre_vasseur} and write $u_{\pm}\coloneqq u_{-}=u_{+}$, and then by \Cref{Filippov20} we have $\dot{h}_\epsilon=V_\epsilon(u_{\pm},\bar{u}_1,\bar{u}_2)$. This gives, \begin{align} q(u_{\pm};\bar{u}_2)-q(u_{\pm};\bar{u}_1)-\dot{h}_\epsilon(\eta(u_{\pm}\|\bar{u}_2)-\eta(u_{\pm}\|\bar{u}_1)) \leq \epsilon. \end{align} For almost every $t$ such that $u_{-}\neq u_{+}$, \Cref{Filippov20} says $\dot{h}_\epsilon=\sigma(u_{-},u_{+})$, where $\sigma(u_{-},u_{+})=(A(u_{+})-A(u_{-}))/(u_{+}-u_{-})$. Then, \begin{align} &q(u_{+};\bar{u}_2)-q(u_{-};\bar{u}_1)-\dot{h}_\epsilon(\eta(u_{+}\|\bar{u}_2)-\eta(u_{-}\|\bar{u}_1)) =\\ &q(u_{+};u_R)-q(u_{-};u_L)-\sigma(u_{-},u_{+})(\eta(u_{+}\|u_R)-\eta(u_{-}\|u_L)), \\ \shortintertext{and then by \eqref{d_rh}, \Cref{entropic_shocks}, and \Cref{Filippov20} again,} &\leq 0. \end{align} Thus, for almost every $t\in[t^,\infty)$ \begin{align} q(u_{+};\bar{u}_2)-q(u_{-};\bar{u}_1)-\dot{h}_\epsilon(\eta(u_{+}\|\bar{u}_2)-\eta(u_{-}\|\bar{u}_1))\leq \epsilon. \end{align} Finally, by \Cref{v_epsilon_bounded} and \Cref{Filippov20}, \begin{align} \mbox{Lip}[h_\epsilon]\leq \sup_{\hspace{.02in}\abs{x}\leq\norm{u}_{L^{\infty}}} \abs{A'}. \end{align} \end{proof} \section{Main proposition} \label{main_prop_section} The proof of \Cref{main_theorem} will follow from \begin{proposition}[Main proposition] \label{main_lemma} Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution to \eqref{conservation_law}, with initial data $u^0$. Let $\eta\in C^2(\mathbb{R})$ be a strictly convex entropy. Assume that $u$ is entropic for the entropy $\eta$ and verifies the strong trace property (\Cref{strong_trace_condition}). Then for all $R, T, \epsilon>0$, there exists a function $\psi:[-R,R] \to\mathbb{R}$ verifying: \begin{enumerate}[label=(\alph)] \item \label{relative_entropy_stable} \begin{align} \int\limits_{\abs{x}\leq R}\eta(u(x,T)\|\psi(x))\,dx \leq \epsilon. \end{align} \item \label{psi_satisfies_condition_e} \begin{align} \psi(x+z)-\psi(x)\leq \frac{c}{T}z \end{align} for $x\in[-R,R]$ and $z>0$ with $x+z\in[-R,R]$ and where $c=1/\inf{A''}$. \item \label{psi_bounded_u} \begin{align} \norm{\psi}_{L^\infty([-R,R])}\leq \norm{u}_{L^\infty(\mathbb{R}\times[0,\infty))}. \end{align} \end{enumerate} \end{proposition} We decompose the proof of \Cref{main_lemma} into two lemmas. We utilize functions of the form \begin{align} \hspace{1in} \label{LIWAS} \begin{cases} \intertext{For $v\in L^{\infty}(\mathbb{R})$, there exists a finite set of $x_i$ with} \hspace{1in}-\infty=x_0<x_1<x_2<\cdots<x_N<x_{N+1}=\infty \intertext{such that $v$ is nondecreasing and Lipschitz continuous on $(x_i,x_{i+1})$} \intertext{for $i=0,\ldots,N$, and} \hspace{1.7in}\lim\limits_{\substack{x\to x_i \\ x<x_i}} v(x) \geq \lim\limits_{\substack{x\to x_i \\ x>x_i}} v(x) \\ \mbox{for $1\leq i \leq N$.} \end{cases} \end{align} \begin{lemma} \label{psi_exists_with_limits} Let $R,T>0$. Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution to \eqref{conservation_law} with initial data $u^0$. Assume $u$ is entropic for at least one strictly convex entropy $\eta\in C^2(\mathbb{R})$. Assume that $u$ verifies the strong trace property (\Cref{strong_trace_condition}). Choose $s>0$ to verify $\abs{q(a;b)}\leq s \eta(a\|b)$ for all $a,b\in[-B,B]$, where $B$ is defined in \eqref{define_B}. Then for all $N\in\mathbb{N}\cup\{0\}$, we have: For any $v_1^0,\ldots,v_{N+1}^0\in L^{\infty}(\mathbb{R})$ Lipschitz continuous nondecreasing functions verifying $v_i^0(x)\geq v_{i+1}^0(x)$ for all $1\leq i \leq N$ and $x\in\mathbb{R}$, and for any $t^\in[0,T]$ and real numbers $x_0,\ldots,x_{N+1}$ verifying $x_0=-R+(t^-T)s<x_1<\cdots<x_{N}<R-(t^-T)s=x_{N+1}$ the following holds: There exist $N+2$ real numbers $x_{0,T},\ldots,x_{N+1,T}$ such that $x_{0,T}=-R\leq x_{1,T}\leq \cdots\leq x_{N,T}\leq R=x_{N+1,T}$ and \begin{align} \label{l2_stability1} {\rm ap}\,\lim_{t\to {T}^{+}}\sum_{i=0}^{N} \int\limits_{x_{i,T}}^{x_{i+1,T}}\eta(u(x,t)\|v_{i+1}(x,t))\,dx \leq {\rm ap}\,\lim_{t\to {t^}^{+}}\sum_{i=0}^{N} \int\limits_{x_i}^{x_{i+1}}\eta(u(x,t)\|v_{i+1}(x,t))\,dx \end{align} where $v_i$ is the unique solution to \eqref{conservation_law} with initial data $v_i^0$ and verifying \eqref{condition_e}. \end{lemma} \begin{lemma}[Density of functions of form \eqref{LIWAS} in $L^2$] \label{LIWAS_dense} Let $M,\epsilon>0$. Then for all $f\in L^2([-M,M])$, there is a function $f_\epsilon: \mathbb{R}\to\mathbb{R}$ of the form \eqref{LIWAS} such that \begin{align} \norm{f-f_\epsilon}_{L^2([-M,M])} < \epsilon, \\ \norm{f_\epsilon}_{L^\infty(\mathbb{R})} \leq \norm{f}_{L^\infty([-M,M])},\label{l_infinity_control_on_LIWAS_dense} \end{align} and all of the discontinuities of $f_\epsilon$ are contained in $(-M,M)$. \end{lemma} \subsection{\Cref{psi_exists_with_limits} and \Cref{LIWAS_dense} imply main proposition} As in \Cref{psi_exists_with_limits}, we choose $s>0$ such that $\abs{q(a;b)}\leq s \eta(a\|b)$ for all $a,b\in[-B,B]$, where $B$ is defined in \eqref{define_B}. We also choose $c^{}>0$ such that \begin{align} \label{control_rel_entropy_by_above} \eta(a\|b)\leq c^{}(a-b)^2 \end{align} for all $a,b\in[-B,B]$. By \Cref{LIWAS_dense}, there exists a function $v^0\in L^{\infty}(\mathbb{R})$ of the form \eqref{LIWAS} such that \begin{align} \label{l2_distance_LIWAS_guy} \norm{u^0-v^0}_{L^2([-R-sT,R+sT])} < \sqrt{\frac{\epsilon}{c^{*}}} \end{align} and if there is at least one discontinuity in $v^0$, the discontinuities are at points $x_1<x_2<\cdots<x_N$ for some $N\in\mathbb{N}$, and where $x_i\in(-R-sT,R+sT)$ for all $1\leq i \leq N$. If $v^0$ contains at least one discontinuity, define the functions $v_i^0 : \mathbb{R}\to\mathbb{R}$ for $1\leq i \leq N+1$ as follows: \[ v_1^0 (x)\coloneqq \begin{cases} v^0(x) & \text{if } x<x_1 \\ \operatorname{sup}_{x_1< y < x}\max(v^0(x_1-),v^0(y)) & \text{if } x_1<x \end{cases} \] For $2\leq i \leq N$, \[ v_i^0 (x)\coloneqq \begin{cases} \operatorname{inf}_{x < y < x_{i-1}}\min(v^0(x_{i-1}+),v_{i-1}^0(y)) & \text{if } x<x_{i-1} \\ v^0(x) & \text{if } x_{i-1}<x<x_{i} \\ \operatorname{sup}_{x_{i}< y< x}\max(v^0(x_{i}-),v^0(y)) & \text{if } x_{i}<x \end{cases} \] And \[ v_{N+1}^0 (x)\coloneqq \begin{cases} \operatorname{inf}_{x < y < x_{N}}\min(v^0(x_{N}+),v_{N}^0(y)) & \text{if } x<x_{N} \\ v^0(x) & \text{if } x_{N}<x. \end{cases} \] If $v^0$ has no discontinuities, then $N=0$ and we define $v^0_1\coloneqq v^0$. By construction, the $v^0_i$ are Lipschitz continuous, nondecreasing in $x$, and verify $v^0_i(x)\geq v^0_{i+1}(x)$ for all $x\in\mathbb{R}$ and $1\leq i \leq N$. We also have \begin{align}\label{sup_norm_control_on_v0i} \norm{v^0_i}_{L^\infty(\mathbb{R})}\leq\norm{v^0}_{L^\infty(\mathbb{R})} \end{align} for $1\leq i \leq N+1$. Let $v_i$ denote the unique solution to \eqref{conservation_law} with initial data $v^0_i$ and which satisfies \eqref{condition_e}. From \Cref{psi_exists_with_limits}, we have $N+2$ real numbers $x_{0,T},\ldots,x_{N+1,T}$ such that $x_{0,T}=-R\leq x_{1,T}\leq \cdots\leq x_{N,T}\leq R=x_{N+1,T}$ and \begin{align} \label{l2_stability_proof_of_main_prop} {\rm ap}\,\lim_{t\to {T}^{+}}\sum_{i=0}^{N} \int\limits_{x_{i,T}}^{x_{i+1,T}}\eta(u(x,t)\|v_{i+1}(x,t))\,dx \leq {\rm ap}\,\lim_{t\to {0}^{+}}\sum_{i=0}^{N} \int\limits_{x_i}^{x_{i+1}}\eta(u(x,t)\|v_{i+1}(x,t))\,dx \end{align} where $x_0\coloneqq-R-sT$ and $x_{N+1}\coloneqq R+sT$. We now control the right-hand side of \eqref{l2_stability_proof_of_main_prop}. Recalling \Cref{left_right_ap_limits}, we have \begin{equation} \begin{aligned}\label{control_rhs_l2_stability_proof_of_main_prop} {\rm ap}\,\lim_{t\to {0}^{+}}\sum_{i=0}^{N} \int\limits_{x_i}^{x_{i+1}}\eta(u(x,t)\|v_{i+1}(x,t))\,dx &\leq \sum_{i=0}^{N} \int\limits_{x_i}^{x_{i+1}}\eta(u^0(x)\|v_{i+1}^0(x))\,dx \shortintertext{Then, from the definition of the $v_i^0$,} &=\int\limits_{-R-sT}^{R+sT}\eta(u^0(x)\|v^0(x))\,dx \shortintertext{Using \eqref{control_rel_entropy_by_above}, \eqref{l_infinity_control_on_LIWAS_dense}, and that $\norm{u^0}_{L^\infty(\mathbb{R})}\leq \norm{u}_{L^{\infty}(\mathbb{R}\times[0,\infty))}$,} &\leq c^{} \int\limits_{-R-sT}^{R+sT}(u^0(x)-v^0(x))^2\,dx \shortintertext{Then, from \eqref{l2_distance_LIWAS_guy}} &< \epsilon. \end{aligned} \end{equation} On the other hand, by the convexity of $\eta$, \begin{align}\label{control_lhs_l2_stability_proof_of_main_prop} \sum_{i=0}^{N} \int\limits_{x_{i,T}}^{x_{i+1,T}}\eta(u(x,T)\|v_{i+1}(x,T))\,dx \leq {\rm ap}\,\lim_{t\to {T}^{+}}\sum_{i=0}^{N} \int\limits_{x_{i,T}}^{x_{i+1,T}}\eta(u(x,t)\|v_{i+1}(x,t))\,dx. \end{align} Combining \eqref{l2_stability_proof_of_main_prop}, \eqref{control_rhs_l2_stability_proof_of_main_prop} and \eqref{control_lhs_l2_stability_proof_of_main_prop}, we find \begin{align}\label{combine_rhs_lhs_l2_stability_proof_of_main_prop} \sum_{i=0}^{N} \int\limits_{x_{i,T}}^{x_{i+1,T}}\eta(u(x,T)\|v_{i+1}(x,T))\,dx < \epsilon. \end{align} We define $\psi:[-R,R] \to\mathbb{R}$, \begin{align} \psi(x)\coloneqq \begin{cases} v_1(x,T) & \text{if } -R < x< x_{1,T} \\ v_2(x,T) & \text{if } x_{1,T} < x< x_{2,T} \\ \hspace{.24in} \vdots \\ v_{N+1}(x,T) & \text{if } x_{N,T} < x< R. \end{cases} \end{align} By \eqref{combine_rhs_lhs_l2_stability_proof_of_main_prop}, $\psi$ satisfies part \ref{relative_entropy_stable} of \Cref{main_lemma}. We now show $\psi$ satisfies part \ref{psi_satisfies_condition_e} of \Cref{main_lemma}: this follows because each of the $v_i$ satisfy \eqref{condition_e}. In particular, \begin{align}\label{condition_e_for_v_i} v_i(x+z,T)-v_i(x,T)\leq \frac{C}{T} z \end{align} for all $z>0$ and all $x\in\mathbb{R}$. From part \ref{lip_initial_6} of \Cref{lip_initial_data_properties}, we can take $C=1/\inf A''$ in \eqref{condition_e_for_v_i}. Further, by part \ref{lip_initial_4} of \Cref{lip_initial_data_properties}, $v_i(x,T)\geq v_{i+1}(x,T)$ for all $x\in\mathbb{R}$ and $1\leq i \leq N$. This gives part \ref{psi_satisfies_condition_e} of \Cref{main_lemma}. Part \ref{psi_bounded_u} of \Cref{main_lemma} follows from \eqref{l_infinity_control_on_LIWAS_dense}, \eqref{sup_norm_control_on_v0i}, and part \ref{lip_initial_3} of \Cref{lip_initial_data_properties}. This completes the proof of \Cref{main_lemma}. \subsection{Proof of \Cref{psi_exists_with_limits}} We prove this lemma by strong induction on $N$. \uline{Base case} For $N=0$, let $v_1^0\in L^{\infty}(\mathbb{R})$ be any Lipschitz continuous nondecreasing function. Let $v_1$ be the unique solution to \eqref{conservation_law} with initial data $v_1^0$ and verifying \eqref{condition_e}. Let $h_0(t)\coloneqq-R+(t-T)s$ and $h_{1}(t)\coloneqq R-(t-T)s$. Then, from \Cref{rigorous_dissipation_rate_calc_middle_part}, \Cref{left_right_ap_limits}, and the dominated convergence theorem, \begin{equation} \begin{aligned}\label{dissipation_calculation_base} &{\rm ap}\,\lim_{t\to {T}^{+}} \int\limits_{h_{0}(T)}^{h_{1}(T)}\eta(u(x,t)\|v_1(x,t))\,dx- {\rm ap}\,\lim_{t\to {t^{}}^{+}}\int\limits_{h_{0}(t^{})}^{h_{1}(t^{})}\eta(u(x,t)\|v_{1}(x,t))\,dx\\ &\leq \int\limits_{t^{}}^{T}\bigg[ q(u(h_0(t)+,t);v_1(h_0(t),t))-q(u(h_{1}(t)-,t);v_1(h_{1}(t),t))\\ &\hspace{.2in}+\dot{h}_{1}(t)\eta(u(h_{1}(t)-,t)\|v_1(h_{1}(t),t))-\dot{h}_0(t)\eta(u(h_0(t)+,t)\|v_1(h_0(t),t))\bigg]\,dt\\ &=\int\limits_{t^{}}^{T}q(u(h_0(t)+,t);v_1(h_0(t),t))-s\eta(u(h_0(t)+,t)\|v_1(h_0(t),t))\,dt\\ &+\int\limits_{t^{}}^{T}-q(u(h_1(t)-,t);v_1(h_1(t),t))-s\eta(u(h_1(t)-,t)\|v_1(h_1(t),t))\,dt\\ &\leq 0 \end{aligned} \end{equation} by the definition of $h_0,h_2$ and $s$. We get, \begin{align} {\rm ap}\,\lim_{t\to {T}^{+}} \int\limits_{-R}^{R}\eta(u(x,t)\|v_1(x,t))\,dx \leq {\rm ap}\,\lim_{t\to {t^{}}^{+}}\int\limits_{-R+(t^-T)s}^{R-(t^-T)s}\eta(u(x,t)\|v_{1}(x,t))\,dx. \end{align} Thus \Cref{psi_exists_with_limits} holds for the base case $N=0$. \uline{Induction step} Suppose that $K\in\mathbb{N}\cup\{0\}$ is given such that \Cref{psi_exists_with_limits} holds for $N=0,1,\ldots, K$. We now prove that \Cref{psi_exists_with_limits} holds for $N=K+1$. Let $v_1^0,v_2,^0\ldots,v_{K+2}^0\in L^{\infty}(\mathbb{R})$ be any Lipschitz continuous nondecreasing functions, satisfying $v_i^0(x)\geq v_{i+1}^0(x)$ for all $1\leq i \leq K+1$ and $x\in\mathbb{R}$. For $1\leq i \leq K+2$, let $v_i$ be the unique solution to \eqref{conservation_law} with initial data $v_i^0$ and verifying \eqref{condition_e}. Let $t^\in[0,T]$ and $-R+(t^-T)s<x_1<\cdots<x_{K+1}<R-(t^-T)s$ be arbitrary. Let $\epsilon>0$. By \Cref{shift_theorem} we can construct Lipschitz continuous functions $h_{\epsilon,1},\ldots,\\h_{\epsilon,K+1}$ on the interval $[t^,T]$ such that for $1\leq i \leq K+1$, $h_{\epsilon,i}(t^)=x_i$ and \begin{align} \label{dissipation_rate3} q(u^{i}_{+};v_{i+1})-q(u^{i}_{-};v_i)-\dot{h}_{\epsilon,i}(\eta(u^{i}_{+}\|v_{i+1})-\eta(u^{i}_{-}\|v_i))\leq \frac{\epsilon}{T(K+1)} \end{align} for almost every $t\in[t^,T]$, where $u^{i}_{\pm}=u(h_{\epsilon,i}(t)\pm,t)$ and $v_l=v_l(h_{\epsilon,i}(t),t)$ for $l=i,i+1$. To simplify the exposition, denote \begin{align} h_{\epsilon,0}(t)\coloneqq -R+(t-T)s,\\ h_{\epsilon,K+2}(t)\coloneqq R-(t-T)s. \end{align} The Lipschitz constants of the $h_{\epsilon,i}$ are uniformly bounded in $\epsilon$ by \Cref{shift_theorem}. Thus, by Arzel\`a--Ascoli there exists a sequence $\{\epsilon_j\}_{j\in\mathbb{N}}$ such that $\epsilon_j\to0^{+}$ and for each $0\leq i \leq K+2$, $h_{\epsilon_j,i}$ converges uniformly on $[t^,T]$ to a Lipschitz function $h_{i}$. Then, let $t^{}$ be the first time that there exist two of the $h_{i}$ for $0\leq i \leq K+2$ that are equal to each other. If such a time does not exist, let $t^{}\coloneqq T$. We now show: \begin{claim} \begin{align} \label{relative_entropy_decreasing_claim1} {\rm ap}\,\lim_{t\to {t^{}}^{+}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(t^{})}^{h_{i+1}(t^{})}\eta(u(x,t)\|v_{i+1}(x,t))\,dx &\leq {\rm ap}\,\lim_{t\to {t^{}}^{+}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(t^{})}^{h_{i+1}(t^{})}\eta(u(x,t)\|v_{i+1}(x,t))\,dx. \end{align} \end{claim} \begin{claimproof} Due to the uniform convergence of the $h_{\epsilon_j,i}$ as $j\to\infty$, for each $\tau\in[t^,t^{})$, there exists $J_\tau>0$ large enough such that $h_{\epsilon_j,i+1}(t)- h_{\epsilon_j,i}(t)>0$ for all $t\in[t^,\tau]$, $j>J_\tau$ and $0\leq i \leq K+1$. Then, for almost every $t$ and $\tau$ verifying $t^\leq t < \tau < t^{}$, we have from \Cref{rigorous_dissipation_rate_calc_middle_part} (for $j>J_\tau$): \begin{align} &\sum_{i=0}^{K+1} \int\limits_{h_{\epsilon_j,i}(\tau)}^{h_{\epsilon_j,i+1}(\tau)}\eta(u(x,\tau)\|v_{i+1}(x,\tau))\,dx-\sum_{i=0}^{K+1} \int\limits_{h_{\epsilon_j,i}(t)}^{h_{\epsilon_j,i+1}(t)}\eta(u(x,t)\|v_{i+1}(x,t))\,dx\\ & \leq \sum_{i=0}^{K+1} \int\limits_{t}^{\tau}\bigg[ q(u(h_{\epsilon_j,i}(r)+,r);v_{i+1}(h_{\epsilon_j,i}(r),r))-q(u(h_{\epsilon_j,i+1}(r)-,r);v_{i+1}(h_{\epsilon_j,i+1}(r),r))\\ &\hspace{1in}+\dot{h}_{\epsilon_j,i+1}(r)\eta(u(h_{\epsilon_j,i+1}(r)-,r)\|v_{i+1}(h_{\epsilon_j,i+1}(r),r))\\ &\hspace{1in}-\dot{h}_{\epsilon_j,i}(r)\eta(u(h_{\epsilon_j,i}(r)+,r)\|v_{i+1}(h_{\epsilon_j,i}(r),r))\bigg]\,dr \shortintertext{Then, we collect the terms corresponding to $h_{\epsilon_j,i}$ into one sum, and the terms corresponding to $h_{\epsilon_j,i+1}$ into another sum,} &=\sum_{i=0}^{K+1} \int\limits_{t}^{\tau}\bigg[ q(u(h_{\epsilon_j,i}(r)+,r);v_{i+1}(h_{\epsilon_j,i}(r),r))-\dot{h}_{\epsilon_j,i}(r)\eta(u(h_{\epsilon_j,i}(r)+,r)\|v_{i+1}(h_{\epsilon_j,i}(r),r))\bigg]\,dr\\ &\hspace{1in}+\sum_{i=0}^{K+1} \int\limits_{t}^{\tau}\bigg[-q(u(h_{\epsilon_j,i+1}(r)-,r);v_{i+1}(h_{\epsilon_j,i+1}(r),r))\\ &\hspace{1in}+\dot{h}_{\epsilon_j,i+1}(r)\eta(u(h_{\epsilon_j,i+1}(r)-,r)\|v_{i+1}(h_{\epsilon_j,i+1}(r),r))\bigg]\,dr \shortintertext{Next, we peel off the $i=0$ term from the first sum, and the $i=K+1$ term from the second sum,} &=\int\limits_{t}^{\tau}\bigg[ q(u(h_{\epsilon_j,0}(r)+,r);v_1(h_{\epsilon_j,0}(r),r))-\dot{h}_{\epsilon_j,0}(r)\eta(u(h_{\epsilon_j,0}(r)+,r)\|v_1(h_{\epsilon_j,0}(r),r))\bigg]\,dr\\ &+\sum_{i=1}^{K+1} \int\limits_{t}^{\tau}\bigg[ q(u(h_{\epsilon_j,i}(r)+,r);v_{i+1}(h_{\epsilon_j,i}(r),r))-\dot{h}_{\epsilon_j,i}(r)\eta(u(h_{\epsilon_j,i}(r)+,r)\|v_{i+1}(h_{\epsilon_j,i}(r),r))\bigg]\,dr\\ &\hspace{1in}+\sum_{i=0}^{K} \int\limits_{t}^{\tau}\bigg[-q(u(h_{\epsilon_j,i+1}(r)-,r);v_{i+1}(h_{\epsilon_j,i+1}(r),r))\\ &\hspace{1in}+\dot{h}_{\epsilon_j,i+1}(r)\eta(u(h_{\epsilon_j,i+1}(r)-,r)\|v_{i+1}(h_{\epsilon_j,i+1}(r),r))\bigg]\,dr\\ &\hspace{1in}+\int\limits_{t}^{\tau}\bigg[-q(u(h_{\epsilon_j,K+2}(r)-,r);v_{K+2}(h_{\epsilon_j,K+2}(r),r))\\ &\hspace{1in}+\dot{h}_{\epsilon_j,K+2}(r)\eta(u(h_{\epsilon_j,K+2}(r)-,r)\|v_{K+2}(h_{\epsilon_j,K+2}(r),r))\bigg]\,dr \shortintertext{We then reindex the second sum $\sum_{i=0}^{K}\int\limits_{t}^{\tau}[\cdots]\,dr$ to start at $i=1$, and combine it with the first sum $\sum_{i=1}^{K+1}\int\limits_{t}^{\tau}[\cdots]\,dr$,} &=\int\limits_{t}^{\tau}\bigg[ q(u(h_{\epsilon_j,0}(r)+,r);v_1(h_{\epsilon_j,0}(r),r))-\dot{h}_{\epsilon_j,0}(r)\eta(u(h_{\epsilon_j,0}(r)+,r)\|v_1(h_{\epsilon_j,0}(r),r))\bigg]\,dr\\ &+\sum_{i=1}^{K+1} \int\limits_{t}^{\tau}\bigg[ q(u(h_{\epsilon_j,i}(r)+,r);v_{i+1}(h_{\epsilon_j,i}(r),r))-q(u(h_{\epsilon_j,i}(r)-,r);v_{i}(h_{\epsilon_j,i}(r),r))\\ &-\dot{h}_{\epsilon_j,i}(r)(\eta(u(h_{\epsilon_j,i}(r)+,r)\|v_{i+1}(h_{\epsilon_j,i}(r),r))-\eta(u(h_{\epsilon_j,i}(r)-,r)\|v_{i}(h_{\epsilon_j,i}(r),r)))\bigg]\,dr\\ &\hspace{1in}+\int\limits_{t}^{\tau}\bigg[-q(u(h_{\epsilon_j,K+2}(r)-,r);v_{K+2}(h_{\epsilon_j,K+2}(r),r))\\ &\hspace{1in}+\dot{h}_{\epsilon_j,K+2}(r)\eta(u(h_{\epsilon_j,K+2}(r)-,r)\|v_{K+2}(h_{\epsilon_j,K+2}(r),r))\bigg]\,dr \\ &\leq \frac{\epsilon_j}{T(K+1)}(\tau-t)(K+1) < \epsilon_j \end{align} by \eqref{dissipation_rate3}, the definition of $s$, and noting that $\dot{h}_{\epsilon_j,0}=s$ and $\dot{h}_{\epsilon_j,K+2}=-s$. Thus, \begin{equation} \begin{aligned} \label{relative_entropy_decreasing_claim2} \sum_{i=0}^{K+1} \int\limits_{h_{\epsilon_j,i}(\tau)}^{h_{\epsilon_j,i+1}(\tau)}\eta(u(x,\tau)\|v_{i+1}(x,\tau))\,dx <\\ \sum_{i=0}^{K+1} \int\limits_{h_{\epsilon_j,i}(t)}^{h_{\epsilon_j,i+1}(t)}\eta(u(x,t)\|v_{i+1}(x,t))\,dx+ \epsilon_j. \end{aligned} \end{equation} Then, let $j\to\infty$ in \eqref{relative_entropy_decreasing_claim2} and use the dominated convergence theorem to get \begin{align} \label{relative_entropy_decreasing_claim55} \sum_{i=0}^{K+1} \int\limits_{h_{i}(\tau)}^{h_{i+1}(\tau)}\eta(u(x,\tau)\|v_{i+1}(x,\tau))\,dx &\leq \sum_{i=0}^{K+1} \int\limits_{h_{i}(t)}^{h_{i+1}(t)}\eta(u(x,t)\|v_{i+1}(x,t))\,dx \end{align} for almost every $t$ and $\tau$ verifying $t^\leq t < \tau<t^{}$. From \eqref{relative_entropy_decreasing_claim55}, we get \begin{align} \label{relative_entropy_decreasing_claim555} {\rm ap}\,\lim_{\tau\to {t^{}}^{-}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(t^{})}^{h_{i+1}(t^{})}\eta(u(x,\tau)\|v_{i+1}(x,\tau))\,dx\leq\\ {\rm ap}\,\lim_{t\to {t^{}}^{+}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(t^{})}^{h_{i+1}(t^{})}\eta(u(x,t)\|v_{i+1}(x,t))\,dx \end{align} where we have used \begin{align} \label{fixed_time_terminals_ap_lim_a} {\rm ap}\,\lim_{\tau\to {t^{}}^{-}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(\tau)}^{h_{i+1}(\tau)}\eta(u(x,\tau)\|v_{i+1}(x,\tau))\,dx=\\ {\rm ap}\,\lim_{\tau\to {t^{}}^{-}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(t^{})}^{h_{i+1}(t^{})}\eta(u(x,\tau)\|v_{i+1}(x,\tau))\,dx \end{align} and \begin{align} \label{fixed_time_terminals_ap_lim_b} {\rm ap}\,\lim_{t\to {t^{}}^{+}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(t)}^{h_{i+1}(t)}\eta(u(x,t)\|v_{i+1}(x,t))\,dx={\rm ap}\,\lim_{t\to {t^{}}^{+}}\sum_{i=0}^{K+1} \int\limits_{h_{i}(t^{})}^{h_{i+1}(t^{})}\eta(u(x,t)\|v_{i+1}(x,t))\,dx.\end{align} The approximate limits exist by \Cref{left_right_ap_limits}. Then \eqref{relative_entropy_decreasing_claim555} and \eqref{left_and_right_limits_order} give the claim \eqref{relative_entropy_decreasing_claim1}. \end{claimproof} If $t^{}=T$, then we have proven \Cref{psi_exists_with_limits} holds for $N=K+1$: define $x_{i,T}\coloneqq h_{i}(T)$ for $0\leq i \leq K+2$. By \eqref{relative_entropy_decreasing_claim1}, the $x_{0,T},x_{1,T},\ldots,x_{K+2,T}$ satisfy the conclusions of \Cref{psi_exists_with_limits}. Otherwise, $t^{}<T$ and we consider the $0\leq i,j\leq K+3$ such that the following holds: \begin{align} \begin{cases} \label{good_i_and_j} h_{i-1}(t^{})<h_{i}(t^{})<h_{j}(t^{})\\ \mbox{for $i<k<j$, $h_{i}(t^{})=h_{k}(t^{})$} \end{cases} \end{align} where $h_{-1}(t)\coloneqq-\infty$ and $h_{K+3}(t)\coloneqq+\infty$ for all $t$. Then, let $\{(i_n,j_n)\}_{n\in\{1,\ldots,L\}}$ for $L\in\mathbb{Z}_{+}$ be the set of $i$ and $j$ pairs which satisfy \eqref{good_i_and_j} (label the $i$ and $j$ pairs such that $i_n < i_{n+1}$ for all $n$). Note that each $i$ has only one corresponding $j$. Thus, due to at least two of the $h_{i}(t^{})$ equalling each other (for $i$ ranging over $0,\ldots,K+2$), $L\leq K+2$. By the induction hypothesis with $N=L-2$: there exist real numbers $\tilde{x}_{0,T},\ldots,\tilde{x}_{L-1,T}$ verifying $\tilde{x}_{0,T}=-R\leq\tilde{x}_{1,T}\leq \cdots\leq \tilde{x}_{L-2,T}\leq R=\tilde{x}_{L-1,T}$ and \begin{equation} \begin{aligned} \label{l2_stability_induction_hypothesis} {\rm ap}\,\lim_{t\to {T}^{+}}\sum_{l=0}^{L-2} \int\limits_{\tilde{x}_{l,T}}^{\tilde{x}_{l+1,T}}\eta(u(x,t)\|v_{i_{l+2}}(x,t))\,dx \leq \\ {\rm ap}\,\lim_{t\to {t^{}}^{+}}\sum_{l=0}^{L-2} \int\limits_{h_{i_{l+1}}(t^{})}^{h_{i_{l+2}}(t^{})}\eta(u(x,t)\|v_{i_{l+2}}(x,t))\,dx. \end{aligned} \end{equation} For each $n\in\{1,\ldots,L\}$, define \begin{align} x_{i,T}\coloneqq \tilde{x}_{n-1,T} \mbox{ for all $i_n\leq i<j_n$.} \end{align} Then by construction $x_{0,T},\ldots,x_{K+2,T}$ satisfy the conclusions of \Cref{psi_exists_with_limits} with $N=K+1$. In particular, \eqref{l2_stability1} follows from \eqref{relative_entropy_decreasing_claim1} and \eqref{l2_stability_induction_hypothesis}. Thus, by the principle of strong induction we have proven \Cref{psi_exists_with_limits} for all $N\in\mathbb{N}\cup\{0\}$. \subsection{Proof of \Cref{LIWAS_dense}} Step functions are dense in $L^2([-M,M])$. Thus, there exists a step function $s \in L^2([-M,M])$ such that $\norm{f-s}_{L^2([-M,M])} < \frac{\epsilon}{2}$ and $\norm{s}_{L^\infty([-M,M])} \leq \norm{f}_{L^\infty([-M,M])}$. We can write $s$ in the form \begin{align} s(x)=\bigg(\sum_{i=1}^{n_{+}} \alpha^{+}_i H(x-x^{+}_{i})\bigg)+\bigg(\sum_{i=1}^{n_{-}} \alpha^{-}_i H(x-x^{-}_{i})\bigg) \end{align} for some $n_{+},n_{-}\in\mathbb{N}$, $\{\alpha^{+}_i\}_{i=1}^{n_{+}}\subset(0,\infty)$, $\{\alpha^{-}_i\}_{i=1}^{n_{-}}\subset(-\infty,0)$, $\{x^{+}_i\}_{i=1}^{n_{+}}\subset(-M,M)$ and $\{x^{-}_i\}_{i=1}^{n_{-}}\subset(-M,M)$. $H$ is the Heaviside step function \begin{align} H(x)\coloneqq \begin{cases} 0 & \text{if } x<0\\ 1 & \text{if } x>0. \end{cases} \end{align} Define \begin{align} s^{+}(x)\coloneqq \sum_{i=1}^{n_{+}} \alpha^{+}_i H(x-x^{+}_{i}),\\ s^{-}(x)\coloneqq \sum_{i=1}^{n_{-}} \alpha^{-}_i H(x-x^{-}_{i}). \end{align} We can then write $s=s^{+}+s^{-}$. Consider the standard mollifier $m: \mathbb{R}\to\mathbb{R}$, where $m$ is smooth and compactly supported, $m\geq0$, and $\int m(x)\,dx=1$. Let $\delta>0$. Define \begin{align} m_\delta(x)\coloneqq \frac{1}{\delta} m\bigg(\frac{x}{\delta}\bigg). \end{align} Define \begin{align} f_\delta (x)\coloneqq \begin{cases} \displaystyle\lim_{y\to -M^{+}}[(s^{+} m_\delta)(y)+s^{-}(y)] & \text{if } x\leq-M\\ (s^{+}* m_\delta)(x)+s^{-}(x) & \text{if } -M<x<M\\ \displaystyle\lim_{y\to M^{-}}[(s^{+}* m_\delta)(y)+s^{-}(y)] & \text{if } x\geq M. \end{cases} \end{align} Note that $f_\delta$ is of the form \eqref{LIWAS}, $\norm{f_\delta}_{L^\infty(\mathbb{R})} \leq \norm{f}_{L^\infty([-M,M])}$ when $\delta < \inf_{i,j}\|x^{+}_i-x^{-}_j\|$, and all of the discontinuities in $f_\delta$ are in the interval $(-M,M)$ because $\{x^{-}_i\}_{i=1}^{n_{-}}\subset(-M,M)$. From the Minkowski inequality, \begin{equation} \begin{aligned} \label{triangle_ineq_LIWAS} \norm{f-f_\delta}_{L^2([-M,M])}&\leq \norm{f-s}_{L^2([-M,M])} + \norm{s-f_\delta}_{L^2([-M,M])}\\ &< \frac{\epsilon}{2} + \norm{(s^{+}* m_\delta)-s^{+}}_{L^2([-M,M])}. \end{aligned} \end{equation} Choose $\delta$ even smaller such that $\norm{(s^{+}* m_\delta)-s^{+}}_{L^2([-M,M])}< \frac{\epsilon}{2}$. This completes the proof. \section{Main proposition implies main theorem}\label{main_prop_implies_main_theorem} Let $u\in L^{\infty}(\mathbb{R}\times[0,\infty))$ be a weak solution to \eqref{conservation_law}, with initial data $u^0$. Let $\eta\in C^2(\mathbb{R})$ be a strictly convex entropy. Assume that $u$ is entropic for the entropy $\eta$ and verifies the strong trace property (\Cref{strong_trace_condition}). We will show that $u$ satisfies \eqref{condition_e}. First, by strict convexity of $\eta$ we can choose a constant $c^>0$ such that \begin{align}\label{control_rel_entropy_below1} c^{}(a-b)^2\leq \eta(a\|b) \end{align} for all $a,b\in[-B,B]$, where $B$ is defined in \eqref{define_B}. Let $R,T>0$. From \Cref{main_lemma}, for all $\epsilon>0$, there exists $\psi_\epsilon: [-R,R]\to\mathbb{R}$ such that \begin{align}\label{relative_entropy_stable_epsilon} \int\limits_{\abs{x}\leq R}\eta(u(x,T)\|\psi_\epsilon(x))\,dx \leq c^* \epsilon^2 \end{align} and \begin{align}\label{psi_satisfies_condition_e_epsilon} \psi_\epsilon(x+z)-\psi_\epsilon(x)\leq \frac{c}{T}z \end{align} for $x\in[-R,R]$ and $z>0$ with $x+z\in[-R,R]$. Here $c=1/\inf{A''}$. We have also \begin{align}\label{psi_bounded_u_epsilon} \norm{\psi_\epsilon}_{L^\infty([-R,R])}\leq \norm{u}_{L^\infty(\mathbb{R}\times[0,\infty))}. \end{align} By \eqref{psi_bounded_u_epsilon}, $\norm{\psi_\epsilon}_{L^\infty([-R,R])}\leq B$. Likewise, we have $\norm{u(\cdot,T)}_{L^\infty(\mathbb{R})}\leq B$. Thus, from \eqref{control_rel_entropy_below1}, we get \begin{align}\label{l_2_control_psi_epsilon_u31231} c^(\psi_\epsilon(x)-u(x,T))^2\leq \eta(u(x,T)\|\psi_\epsilon(x)) \end{align} for almost every $x\in\ [-R,R]$. Then from \eqref{relative_entropy_stable_epsilon} and \eqref{l_2_control_psi_epsilon_u31231} we have \begin{align}\label{l_2_control_psi_epsilon_u} \norm{\psi_\epsilon(\cdot)-u(\cdot,T)}_{L^2([-R,R])}\leq \epsilon. \end{align} Thus, there exists a sequence $\{\epsilon_j\}_{j=1}^{\infty}$ with $\epsilon_j\to0^{+}$ such that \begin{align} \psi_{\epsilon_j}(x)\to u(x,T) \mbox{ as $j\to\infty$ for almost every x.} \end{align} Additionally, from \eqref{psi_satisfies_condition_e_epsilon}: \begin{align}\label{psi_j_satisfies_condition_e_epsilon} \psi_{\epsilon_j} (x+z)-\psi_{\epsilon_j}(x)\leq \frac{c}{T}z \end{align} for all $j\in\mathbb{N}$, $x\in[-R,R]$, and $z>0$ with $x+z\in[-R,R]$. We let $j\to\infty$ in \eqref{psi_j_satisfies_condition_e_epsilon} to get: \begin{align}\label{u_satisfies_condition_from_psi_j} u(x+z,T)-u(x,T)\leq \frac{c}{T}z \end{align} for almost every $x\in[-R,R]$, and almost every $z>0$ with $x+z\in[-R,R]$. Because $R,T>0$ are arbitrary, \eqref{u_satisfies_condition_from_psi_j} implies that $u$ satisfies \eqref{condition_e}. This concludes the proof of \Cref{main_theorem}. \section{Appendix} \subsection{Proof of \Cref{d_sm_rh_lemma}} \label{d_sm_rh_lemma_proof} Let $u_L,u_R,u_{-},u_{+}\in\mathbb{R}$. We then borrow the following notation from \cite{serre_vasseur}: If $F$ is a function of $u$, then we define \begin{align} F_{R}\coloneqq F(u_{R}), \hspace{.1in}F_{L}\coloneqq F(u_{L}), \hspace{.1in} [F] \coloneqq F_{R}-F_{L}, \hspace{.1in} F_{\pm}\coloneqq F(u_\pm). \end{align} \emph{Proof of \eqref{d_rh}} This proof is from \cite[p.~9-10]{serre_vasseur}. In \cite{serre_vasseur}, for the general systems case, the authors develop a condition which they label with the equation number 7 \cite[p.~4]{serre_vasseur}. In particular, they claim to use this condition to show \eqref{d_rh} for the scalar case. In fact, the condition is not necessary in the scalar case and their proof goes through unchanged without the condition. Denote \begin{align} D\coloneqq q(u_{+};u_R)-q(u_{-};u_L)-\sigma(u_{-},u_{+})(\eta(u_{+}\|u_R)-\eta(u_{-}\|u_L)). \end{align} Further, let $\sigma$ denote $\sigma(u_{-},u_{+})$. From Rankine-Hugoniot (as noted in \cite[p.~5]{serre_vasseur}), \begin{align} D=[\eta'(u)A(u)-q(u)]-\sigma[\eta'(u)u-\eta]+q_{+}-q_{-}-\sigma(\eta_{+}-\eta_{-})-[\eta'](A(u)-\sigma u)_{\pm}, \end{align} where $(A(u)-\sigma u)_{\pm}$ denotes that \begin{align} A(u_{+})-\sigma u_{+}=A(u_{-})-\sigma u_{-} \label{equality_RH1} \end{align} because of the Rankine-Hugoniot relation \eqref{rh_velocity}. From the fundamental theorem of calculus and integration by parts, \begin{align} D=\int\limits_{u_{+}}^{u_{-}} \eta''(u)(A(u)-\sigma u)\,du -\int\limits_{u_{+}}^{u_{-}} \eta''(u)\,du (A(u)-\sigma u)_{\pm}\\ -\int\limits_{u_{R}}^{u_{L}} \eta''(u)(A(u)-\sigma u)\,du+\int\limits_{u_{R}}^{u_{L}} \eta''(u)\,du (A(u)-\sigma u)_{\pm}. \end{align} We can then write \begin{align} D=\epsilon(I)B(I)+\epsilon(J)B(J) \end{align} where $I, J$ are disjoint intervals such that \begin{align} I\cup J =((u_{+},u_{-})\cup(u_R,u_L))\setminus ((u_{+},u_{-})\cap(u_R,u_L)). \end{align} We define the sign $\epsilon(I)$ to be $+1$ if $I\subset (u_{+},u_{-})$ and $-1$ otherwise. Finally, \begin{align} B(I)\coloneqq \int\limits_{I}\eta''(u)(A(u)-\sigma u)\,du - (A(u)-\sigma u)_{\pm} \int\limits_{I}\eta''(u)\,du. \end{align} The function $u\mapsto A(u)-\sigma u$ is strictly convex and \eqref{equality_RH1} holds. Thus, $B(I)<0$ if $I\subset (u_{+},u_{-})$ and positive otherwise. Thus, for all intervals $\epsilon(I)B(I)<0$. Thus $D<0$. \emph{Proof that \eqref{denom_zero} implies \eqref{d_sm}} This proof is from \cite[p.~9]{serre_vasseur}. Remark that the equality \begin{align} \eta(u\|u_L)=\eta(u\|u_R) \end{align} is equivalent to \cite[p.~4]{serre_vasseur} \begin{align} [\eta']u=[\eta'(u)u-\eta(u)]. \end{align} Remark also (as noted in \cite[p.~4]{serre_vasseur}) \begin{align} q(u;u_R)-q(u;u_L)=[\eta'A-q]-[\eta']A(u). \end{align} Then, \eqref{d_sm} is equivalent to (as noted in \cite[p.~9]{serre_vasseur}) \begin{align} \frac{\int\limits_{u_R}^{u_L} \eta''(u)A(u)\,du}{\int\limits_{u_R}^{u_L}\eta''(u)\,du} > A\Bigg(\frac{\int\limits_{u_R}^{u_L} \eta''(u)u\,du}{\int\limits_{u_R}^{u_L}\eta''(u)\,du} \Bigg). \label{equivalent1} \end{align} Finally, \eqref{equivalent1} is true by Jensen's inequality because $\eta''>0$ so $\eta''(u)\,du$ is a measure, and $A$ is strictly convex. \subsection{Proof of \Cref{Filippov20}}\label{proof_of_Filippov} The following proof of \eqref{Filippov1}, \eqref{Filippov2} and \eqref{Filippov3} is based on the proof of Proposition 1 in \cite{Leger2011} and the proof of Lemma 2.2 in \cite{serre_vasseur}. We do not prove \eqref{Filippov4} and \eqref{Filippov5}: these properties are in Lemma 6 in \cite{Leger2011}, and their proofs are in the appendix in \cite{Leger2011}. Define \begin{align} v_n(x,t)\coloneqq \int\limits_{0}^{1} V_\epsilon(u(x+\frac{y}{n},t),\bar{u}_1(x+\frac{y}{n},t),\bar{u}_2(x+\frac{y}{n},t))\,dy. \end{align} Let $h_{\epsilon,n}$ be the solution to the ODE: \begin{align} \begin{cases}\label{ode_for_h_epsilon_n} \dot{h}_{\epsilon,n}(t)=v_n(h_{\epsilon,n}(t),t), \text{ for } t>0\\ h_{\epsilon,n}(t^)=x_0. \end{cases} \end{align} Due to $V_\epsilon$ being continuous, $v_n$ is bounded uniformly in $n$ ($\hspace{.03in}\norm{v_n}_{L^{\infty}}\leq \norm{V_\epsilon}_{L^{\infty}}$), Lipschitz continuous in $x$, and measurable in $t$. Thus \eqref{ode_for_h_epsilon_n} has a unique solution in the sense of Carath\'eodory. Because $v_n$ is bounded uniformly in $n$, the $h_{\epsilon,n}$ are Lipschitz continuous in time, with the Lipschitz constant uniform in $n$. Thus, by Arzel\`a--Ascoli the $h_{\epsilon,n}$ converge in $C^0(0,T)$ for any fixed $T>0$ to a Lipschitz continuous function $h_{\epsilon}$ (passing to a subsequence if necessary). Note that $\dot{h}_{\epsilon,n}$ converges in $L^{\infty}$ weak* to $\dot{h}_{\epsilon}$. We define \begin{align} V_{\text{max}}(t)\coloneqq \max\{V_\epsilon(u_{-},\bar{u}_1,\bar{u}_2),V_\epsilon(u_{+},\bar{u}_1,\bar{u}_2)\},\\ V_{\text{min}}(t)\coloneqq \min\{V_\epsilon(u_{-},\bar{u}_1,\bar{u}_2),V_\epsilon(u_{+},\bar{u}_1,\bar{u}_2)\}, \end{align} where $u_{\pm}=u(h_\epsilon(t)\pm,t)$ and $\bar{u}_i=\bar{u}_i(h_\epsilon(t),t)$. To show \eqref{Filippov3}, we will first prove that for almost every $t>0$ \begin{align} \lim_{n\to\infty}[\dot{h}_{\epsilon,n}(t)-V_{\text{max}}(t)]_{+}=0, \label{one_side_filippov}\\ \lim_{n\to\infty}[V_{\text{min}}(t)-\dot{h}_{\epsilon,n}(t)]_{+}=0,\label{other_side_filippov} \end{align} where $[\hspace{.025in}\cdot\hspace{.025in}]_{+}\coloneqq \max(0,\cdot)$. The proofs of \eqref{one_side_filippov} and \eqref{other_side_filippov} are similar. We will only show the first one. \begin{align} [\dot{h}_{\epsilon,n}(t)-&V_{\text{max}}(t)]_{+}\\ &=\Bigg[\int\limits_{0}^{1} V_\epsilon(u(h_{\epsilon,n}(t)+\frac{y}{n},t),\bar{u}_1(h_{\epsilon,n}(t)+\frac{y}{n},t),\bar{u}_2(h_{\epsilon,n}(t)+\frac{y}{n},t))\,dy-V_{\text{max}}(t)\Bigg]_{+}\\ &=\Bigg[\int\limits_{0}^{1} V_\epsilon(u(h_{\epsilon,n}(t)+\frac{y}{n},t),\bar{u}_1(h_{\epsilon,n}(t)+\frac{y}{n},t),\bar{u}_2(h_{\epsilon,n}(t)+\frac{y}{n},t))-V_{\text{max}}(t)\,dy\Bigg]_{+}\\ &\leq\int\limits_{0}^{1} [V_\epsilon(u(h_{\epsilon,n}(t)+\frac{y}{n},t),\bar{u}_1(h_{\epsilon,n}(t)+\frac{y}{n},t),\bar{u}_2(h_{\epsilon,n}(t)+\frac{y}{n},t))-V_{\text{max}}(t)]_{+}\,dy\\ &\leq \esssup_{y\in(0,\frac{1}{n})}[V_\epsilon(u(h_{\epsilon,n}(t)+y,t),\bar{u}_1(h_{\epsilon,n}(t)+y,t),\bar{u}_2(h_{\epsilon,n}(t)+y,t))-V_{\text{max}}(t)]_{+}\\ &\leq \esssup_{y\in(-\epsilon_n,\epsilon_n)}[V_\epsilon(u(h_{\epsilon}(t)+y,t),\bar{u}_1(h_{\epsilon}(t)+y,t),\bar{u}_2(h_{\epsilon}(t)+y,t))-V_{\text{max}}(t)]_{+},\label{final_line_for_V_max_bound_proof_of_Filippov} \end{align} where $\epsilon_n\coloneqq \abs{h_{\epsilon,n}(t)-h_{\epsilon}(t)}+\frac{1}{n}$. Note $\epsilon_n\to0^{+}$. Fix a $t\geq0$ such that we have a strong trace according to \Cref{strong_trace_condition}. Then, by the continuity of $V_\epsilon$, \begin{align} \lim_{n\to\infty}\esssup_{y\in(0,\frac{1}{n})}[V_\epsilon(u(h_{\epsilon}(t)\pm y,t),\bar{u}_1(h_{\epsilon}(t)\pm y,t),\bar{u}_2(h_{\epsilon}(t)\pm y,t))-V_\epsilon(u_{\pm},\bar{u}_1,\bar{u}_2)]_{+}=0, \end{align} where $u_{\pm}=u(h_\epsilon(t)\pm,t)$ and $\bar{u}_i=\bar{u}_i(h_\epsilon(t),t)$. This implies \begin{align} \lim_{n\to\infty}\esssup_{y\in(0,\frac{1}{n})}[V_\epsilon(u(h_{\epsilon}(t)\pm y,t),\bar{u}_1(h_{\epsilon}(t)\pm y,t),\bar{u}_2(h_{\epsilon}(t)\pm y,t))-V_{\text{max}}(t)]_{+}=0.\label{thing_we_need_goes_to_zero_proof_of_Filippov} \end{align} We can control \eqref{final_line_for_V_max_bound_proof_of_Filippov} from above by \begin{equation} \begin{aligned}\label{final_line_for_V_max_bound_proof_of_Filippov_next_line} \esssup_{y\in(-\epsilon_n,0)}[V_\epsilon(u(h_{\epsilon}(t)+y,t),\bar{u}_1(h_{\epsilon}(t)+y,t),\bar{u}_2(h_{\epsilon}(t)+y,t))-V_{\text{max}}(t)]_{+}+\\ \esssup_{y\in(0,\epsilon_n)}[V_\epsilon(u(h_{\epsilon}(t)+y,t),\bar{u}_1(h_{\epsilon}(t)+y,t),\bar{u}_2(h_{\epsilon}(t)+y,t))-V_{\text{max}}(t)]_{+}. \end{aligned} \end{equation} Then, by \eqref{thing_we_need_goes_to_zero_proof_of_Filippov}, the quantity \eqref{final_line_for_V_max_bound_proof_of_Filippov_next_line} goes to $0$ as $n\to\infty$. This proves \eqref{one_side_filippov}. Recall that $\dot{h}_{\epsilon,n}$ converges in $L^{\infty}$ weak* to $\dot{h}_{\epsilon}$. Thus, because the function $[\hspace{.025in}\cdot\hspace{.025in}]_{+}$ is convex, \begin{align} \int\limits_{0}^{T} [\dot{h}_{\epsilon}(t)-V_{\text{max}}(t)]_{+} \,dt \leq \liminf_{n\to\infty} \int\limits_{0}^{T}[\dot{h}_{\epsilon,n}(t)-V_{\text{max}}(t)]_{+}\,dt. \end{align} By the dominated convergence theorem and \eqref{one_side_filippov}, \begin{align} \liminf_{n\to\infty} \int\limits_{0}^{T}[\dot{h}_{\epsilon,n}(t)-V_{\text{max}}(t)]_{+}\,dt=0. \end{align} This proves \begin{align} \int\limits_{0}^{T} [\dot{h}_{\epsilon}(t)-V_{\text{max}}(t)]_{+} \,dt=0. \end{align} A similar argument gives \begin{align} \int\limits_{0}^{T} [V_{\text{min}}(t)-\dot{h}_{\epsilon}(t)]_{+} \,dt=0. \end{align} This proves \eqref{Filippov3}. \bibliographystyle{plain}	{ "redpajama_set_name": "RedPajamaArXiv" }	8,390
__author__ = 'vikesh' import os import wolframalpha from flask import Flask, request, Response, redirect try: import config wol_id = config.wolframalpha['app_id'] except: wol_id = os.environ.get('APP_ID') if not wol_id: import sys print 'No config.py found exisiting...' sys.exit(0) app = Flask(__name__) client = wolframalpha.Client(wol_id) @app.route('/thel',methods=['post']) def thel(): ''' :Example: /thel current weather in mumbai? ''' text = request.values.get('text') try: res = client.query(text) except UnicodeEncodeError: return Response(('Sorry I did\'t get you. Would you please simplify your query?' '%s is not valid input.' % text), content_type='text\plain; charset=utf-8') resp_qs = ['Hi Top Answer for "%s"\n' % text] resp_qs.extend(next(res.results).text) return Response(''.join(resp_qs), content_type='text/plain; chatset=utf-8') @app.route('/') def hello(): return redirect('https://github.com/vicky002/slack-TheL') if __name__ == '__main__': port = int(os.environ.get('PORT',5000)) app.run(host='0.0.0.0', port=port)	{ "redpajama_set_name": "RedPajamaGithub" }	4,048
\section{Introduction} We are concerned with mesh sensitivity for the finite element (FE) solution of linear elliptic boundary value problems (BVPs). The finite element method is a well developed method that is widely used in scientific and engineering computation and has been studied extensively in the numerical analysis community. In particular, numerous error estimates have been established for the FE solution of elliptic problems; see, e.g., \cite{AO00,BS94,Cia78}. Those error estimates (cf. Proposition~\ref{pro:error-1} below) are typically established for a given type of mesh and show convergence behavior of the error as the mesh is being refined. They also show the {\it stable} dependence of the FE solution on the mesh in the sense that the FE solution remains in a neighborhood of the exact solution for all meshes of same type no matter how different they are. However, those error estimates do not tell if and how the FE solution changes continuously with the mesh. Although it is commonly believed and numerically verified that the FE solution depends continuously on the mesh at least for linear elliptic problems, little work has been done so far on the theoretical study of this mesh sensitivity issue. To our best knowledge, \cite{HH2016} is the only known work on this issue where a bound for the change of the linear FE solution resulting from the mesh deformation has been obtained for one-dimensional elliptic boundary value problems. The objective of this work is to present an analysis of the mesh sensitivity of the FE solution to a BVP of genera linear elliptic partial differential equations (PDEs) defined on a polygonal/polyhedral domain in $\mathbb{R}^d$ ($d \ge 1$). We consider an arbitrary unstructured simplicial mesh and a general order FE approximation. We use an approach similar to gradient methods for optimal control (see, e.g. \cite{BH1975}) and sensitivity analysis in shape optimization (e.g., see \cite{HM2003,SZ92}) where a small deformation in the mesh is introduced and then an FE formulation is derived and bounds are established for the change in the FE solution resulting from the mesh deformation. The main results are stated in Theorems~\ref{thm:fem-1} and \ref{thm:fem-2}. An outline of the paper is as follows. The BVP under consideration and its FE formulation are described in Section~\ref{SEC:FE}. In Section~\ref{SEC:mesh-deform}, the mesh deformation is introduced and changes in mesh quantities and functions resulting from the mesh deformation are discussed. The mesh sensitivity of the FE solution is analyzed in Section~\ref{SEC:sensitivity}. Numerical results are presented in Section~\ref{SEC:numerics} for an example with smooth and nonsmooth velocity fields. Finally, conclusions are drown in Section~\ref{SEC:conclusions}. \section{Finite element formulation} \label{SEC:FE} We consider the boundary value problem of a general linear elliptic PDE as \begin{eqnarray} && - \nabla (a \nabla u) + \V{b}\cdot \nabla u + c u = f,\quad \mbox{ in } \Omega \label{bvp-1} \\ && u = 0,\qquad \mbox{ on } \partial \Omega \label{bc-1} \end{eqnarray} where $\Omega$ is a polygonal/polyhedral domain in $\mathbb{R}^d$ ($d \ge 1$) and the coefficients $a(\V{x})$, $\V{b}(\V{x})$, $c(\V{x})$, and $f(\V{x})$ are given functions satisfying \begin{align} & a,\; \V{b},\; c \in W^{1,\infty}(\Omega), \quad f \in W^{1,2}(\Omega), \\ & a(\V{x}) \ge a_0 > 0, \quad c(\V{x}) - \frac{1}{2} \nabla \cdot \V{b}(\V{x}) \ge 0, \quad \text{ in } \Omega . \end{align} The derivatives of the coefficients will be needed in the sensitivity analysis (cf. Section~\ref{SEC:sensitivity}). The coefficients can have lower regularity if only the convergence of the FE solution is concerned. Let $V =H_0^1(\Omega)$. The variational formulation of (\ref{bvp-1}) and (\ref{bc-1}) is to find $u \in V$ such that \begin{equation} \int_{\Omega} \left ( a \nabla u \cdot \nabla \psi + (\V{b}\cdot \nabla u)\psi + c u \psi\right ) d \V{x} = \int_{\Omega} f \psi d \V{x}, \quad \forall \psi \in V. \label{bvp-2} \end{equation} It can be shown that \begin{align} \int_{\Omega} \left ( a \nabla u \cdot \nabla u + (\V{b}\cdot \nabla u) u + c u^2\right ) d \V{x} \ge a_0 \\| \nabla u \\|_{L^2(\Omega)}^2, \quad \forall u \in V . \label{bvp-3} \end{align} We consider the FE solution of problem (\ref{bvp-2}). To this end, we assume that a simplicial mesh $\mathcal{T}_h$ has been given for $\Omega$. Let $K$ be the generic element of $\mathcal{T}_h$ and $h_K$ and $a_K$ be the diameter (defined as the length of the longest edge) and the minimum height of $K$, respectively. Here, a height of $K$ is defined as the distance from a vertex to its opposite facet. The mesh $\mathcal{T}_h$ is said to be regular if there exists a constant $\kappa > 0$ such that \begin{equation} \frac{h_K}{a_K}\le \kappa, \quad \forall K\in \mathcal{T}_h. \label{mesh-reg-1} \end{equation} Although it is common to assume that the mesh is regular in FE error analysis (cf. Proposition~\ref{pro:error-1}), we do not make such an assumption in our current mesh sensitivity analysis. Instead, we only require $a_K > 0$ for all $K \in \mathcal{T}_h$, which essentially says that all elements must not be degenerate or inverted. Edge matrices of mesh elements are a useful tool in our analysis. Denote the vertices of $K$ by $\V{x}_i^K, i = 0, ..., d$. An edge matrix of $K$ is defined as \[ E_K = [\V{x}_1^K-\V{x}_0^K, ...,\V{x}_d^K-\V{x}_0^K]. \] It is evident that this definition is not unique, depending on the ordering of the vertices. Nevertheless, many geometric properties of $K$, which are independent of the ordering of the vertices, can be computed using $E_K$. For example, the volume of $K$ can be calculated by $\|K\| = \det(E_K)/d!$. Moreover, it is known \cite{LH2017} that \begin{equation} E_K^{-T} = [\nabla \phi_1^K, ..., \nabla \phi_d^K], \label{E-basis-1} \end{equation} where $\phi_i^K$ is the linear Lagrange basis function associated with $\V{x}_i^K$. These basis functions satisfy $\sum_{i=0}^d \phi_i^K = 1$. It is also known that the $i$-th height of $K$ is equal to $1/\|\nabla \phi_i^K\|$ and thus, \begin{equation} a_K = \min_{i} \frac{1}{\|\nabla \phi_i^K\|}. \label{aK-1} \end{equation} We consider the FE space associated with $\mathcal{T}_h$ as \[ V_h=\{v\in H^1_0(\Omega)\cap C(\overline{\Omega});\; v\|_K\in P_r(K),~~\forall K\in \mathcal{T}_h\}, \] where $P_r(K)$ ($r \ge 0$) is the set of polynomials of degree no more than $r$ defined on $K$. Any function $v_h$ in $V_h$ can be expressed as \[ v_h = \sum_i v_i \psi_i(\V{x}), \] where $\{\psi_1, \psi_2, ...\}$ is a basis for $V_h$. We distinguish FE basis functions $\{\psi_1, \psi_2, ...\}$ from the linear Lagrange basis functions $\{ \phi_i, i = 1, 2, ...\}$ (with $\phi_i$ being associated with $\V{x}_i$) and emphasize that they can be different (even when $r = 1$). The FE solution of BVP (\ref{bvp-1}) and (\ref{bc-1}) is to find $u_h \in V_h$ such that \begin{equation} \int_{\Omega} \left ( a \nabla u_h \cdot \nabla \psi + (\V{b}\cdot \nabla u_h)\psi + c u_h \psi\right ) d \V{x} = \int_{\Omega} f \psi d \V{x}, \quad \forall \psi \in V_h. \label{fem-1} \end{equation} The following proposition is a standard error estimate that can be found in most FEM textbooks (e.g., see \cite{BS94}). \begin{pro} \label{pro:error-1} Assume that $u \in H^2(\Omega)$ and the mesh $\mathcal{T}_h$ is regular. Then, \begin{equation} \\| \nabla (u_h - u)\\|_{L^2(\Omega)} \le C h \\|\nabla^2 u \\|_{L^2(\Omega)} , \label{error-1-1} \end{equation} where $h = \max_{K \in \mathcal{T}_h} h_K$ and $C$ is a constant independent of $u$, $u_h$, and the mesh. \end{pro} The error estimate (\ref{error-1-1}) shows the stable dependence of the FE solution on the mesh. It shows that the FE solution remains in a neighborhood of the exact solution for all regular meshes with maximum element diameter $h$ no matter how different they are. However, the estimate does not tell if and how the FE solution changes continuously with the mesh. \section{Mesh deformation} \label{SEC:mesh-deform} For the mesh sensitivity analysis of the FE solution we use an approach similar to gradient methods for optimal control (see, e.g. \cite{BH1975}) and sensitivity analysis in shape optimization (e.g., see \cite{HM2003,SZ92}). In this approach, a small deformation in the mesh is introduced and then an FE formulation is derived and bounds are established for the change in the FE solution resulting from the mesh deformation. In this section, we focus on the mesh deformation and changes in mesh qualities and functions resulting from the mesh deformation. We will discuss the mesh sensitivity of the FE solution in the next section. We assume that a smooth vector field $\dot{X}=\dot{X}(\V{x})$ is given on $\Omega$ and satisfies \[ \\| \dot{X} \\|_{L^\infty(\Omega)} < \infty, \quad \\| \nabla \dot{X} \\|_{L^\infty(\Omega)} < \infty . \] We consider the deformation of the mesh $\mathcal{T}_h$ by keeping its connectivity and moving its interior vertices according to \begin{equation} \V{x}_i(t) = \V{x}_i(0) + t \dot{\V{x}}_i, \quad 0 \le t < \delta, \quad i = 1, 2, ... \label{xt-1} \end{equation} where $\delta$ is a small positive number and the nodal velocities are defined as $\dot{\V{x}}_i = \dot{X}(\V{x}_i(0))$ (that are considered constant in time). We denote the time-dependent mesh by $\mathcal{T}_h(t)$. Here, we fix the boundary vertices for notational simplicity. The analysis applies without major modifications if the boundary points are allowed to move. We also note that the linearization of any smooth mesh deformation can be cast in the form of (\ref{xt-1}). Thus, (\ref{xt-1}) is sufficiently general. \begin{figure}[thb] \centering \begin{tikzpicture}[scale = 1] \draw [thick] (0,0) -- (2,0) -- (0.8, 2) -- cycle; \draw [thick,->] (2.2,0.75) -- (4.2,0.75); \draw [above] (3.2,1) node {$\V{y} = F_{K(t)}(\V{x})$}; \draw [thick] (4.5,0) -- (6.5,0.2) -- (5.5, 2) -- cycle; \draw [above] (1,0.5) node {$K$}; \draw [below] (1,0.7) node {$(\V{x})$}; \draw [above] (5.5,0.5) node {$K(t)$}; \draw [below] (5.5,0.7) node {$(\V{y})$}; \end{tikzpicture} \caption{Affine mapping $F_{K(t)}$ from $K(0)=K$ to $K(t)$.} \label{fig:mapping-1} \end{figure} To understand effects of the mesh deformation on mesh quantities and functions, we consider the affine mapping $F_{K(t)}$ from $K(0) = K \in \mathcal{T}_h $ to $K(t) \in \mathcal{T}_h(t)$ (cf. Fig.~\ref{fig:mapping-1}). To avoid notational confusion, we use coordinates $\V{x}$ and $\V{y}$ for $K(0)$ and $K(t)$, respectively. Moreover, we express quantities and functions in $\V{y}$ by adding ``\,$\tilde{\mbox{ }}$\," on the top of the names. For example, $\nabla$ denotes the gradient operator with respect to $\V{x}$ while the gradient operator with respect to $\V{y}$ is written as $\tilde{\nabla}$. We consider mesh deformation effects by first transforming functions/mesh quantities from $\V{y}$ to $\V{x}$ and then differentiating them with respect to $t$ while keeping $\V{x}$ fixed. The time differentiation is similar to material differentiation in fluid dynamics. We denote corresponding derivatives by the symbol ``\,$\dot{\mbox{ }}$\,". We first consider time derivatives for the Jacobian matrix and determinant. Denote the Jacobian matrix of $F_{K(t)}$ by $\mathbb{J} = \frac{\partial F_{K(t)}}{\partial \V{x}}$ and the Jacobian determinant by $J = \det(\mathbb{J})$. We can express $\mathbb{J}$ in terms of the edge matrices of $K(0)$ and $K(t)$. Since $\V{y} = F_{K(t)}(\V{x})$ is affine, it can be expressed as \begin{equation} \V{y} = F_{K(t)}(\V{x}) = \V{x}_0^K(t) + \mathbb{J} \; (\V{x}-\V{x}_0^K(0)), \quad \forall \V{x} \in K(0). \label{FK-1} \end{equation} By taking $\V{x} = \V{x}_i^K(0)$ and $\V{y} = \V{x}_i^K(t)$ ($i = 1, ..., d$) sequentially, we get \[ [\V{x}_1^K(t) - \V{x}_0^K(t), ..., \V{x}_d^K(t) - \V{x}_0^K(t)] = \mathbb{J}\; [\V{x}_1^K(0) - \V{x}_0^K(0), ..., \V{x}_d^K(0) - \V{x}_0^K(0)], \] which gives \begin{equation} \mathbb{J} = E_{K(t)} E_{K(0)}^{-1}. \label{J-1} \end{equation} From this, it is evident that $\mathbb{J} \|_{t = 0} = I$ (the $d\times d$ identity matrix). Differentiating (\ref{FK-1}) with respect to $t$ gives \begin{equation} \dot{F}_{K(t)} = \dot{\V{x}}_0^K + \dot{\mathbb{J}}\, (\V{x}-\V{x}_0^K(0)) = \dot{\V{x}}_0^K + \dot{E}_{K(t)} E_{K(0)}^{-1} (\V{x}-\V{x}_0^K(0)). \label{FK-3} \end{equation} where we have used \begin{equation} \dot{\mathbb{J}} = \dot{E}_{K(t)} E_{K(0)}^{-1}, \quad \dot{E}_{K(t)} = [\dot{\V{x}}_1^K - \dot{\V{x}}_0^K, ..., \dot{\V{x}}_d^K - \dot{\V{x}}_0^K]. \label{Jdot-1} \end{equation} On the other hand, $\V{y} = F_{K(t)}(\V{x})$ can be expressed in terms of linear basis functions as \begin{equation} \label{FK-0} F_{K(t)} = \sum_{i=0}^d \V{x}_i^K(t) \phi_i^K(\V{x}). \end{equation} Differentiating this with respect to $t$ yields \begin{equation} \dot{F}_{K(t)} = \sum_{i=0}^d \dot{\V{x}}_i^K \phi_i^K(\V{x}). \label{FKdot-1} \end{equation} \begin{lem} \label{lem:FKdot-0} There hold \begin{align} & \dot{F}_{K(t)} = \dot{X}_h \|_K, \label{FKdot-2} \\ & \nabla \cdot \dot{F}_{K(t)} = \text{tr} (\dot{\mathbb{J}}) = \text{tr} (\dot{E}_{K(t)} E_{K(0)}^{-1}) = \nabla \cdot \dot{X}_h \|_K, \label{FKdot-3} \end{align} where $\dot{X}_h$ is a piecewise linear velocity field defined as \begin{equation} \label{Xt-0} \dot{X}_h = \sum_i \dot{\V{x}}_i \phi_i(\V{x}) . \end{equation} \end{lem} \begin{proof} The equality (\ref{FKdot-2}) follows from (\ref{FKdot-1}) and (\ref{Xt-0}). Applying the divergence operator to (\ref{FK-3}) and by direct calculation, we get \[ \nabla \cdot \dot{F}_{K(t)} = \text{tr} (\dot{\mathbb{J}}) = \text{tr} (\dot{E}_{K(t)} E_{K(0)}^{-1}) . \] Applying the divergence operator to (\ref{FKdot-1}), we get \[ \nabla \cdot \dot{F}_{K(t)} = \nabla \cdot \dot{X}_h \|_K . \] Combining the above results we obtain (\ref{FKdot-3}). \end{proof} \begin{lem} \label{lem:Jdot-1} There hold \begin{equation} \dot{\mathbb{J}}\|_{t = 0} = \dot{E}_{K(0)} E_{K(0)}^{-1},\quad \dot{(\mathbb{J}^{-1})}\|_{t = 0} = - \dot{E}_{K(0)} E_{K(0)}^{-1} ,\quad \dot{J}\|_{t=0} = \nabla \cdot \dot{X}_h\|_K . \label{der-1} \end{equation} \end{lem} \begin{proof} The first equation in (\ref{der-1}) follows from (\ref{Jdot-1}). By differentiating the identity $\mathbb{J} \mathbb{J}^{-1} = I$, we get \[ \dot{(\mathbb{J}^{-1})} = - \mathbb{J}^{-1} \dot{\mathbb{J}} \mathbb{J}^{-1} . \] Taking $t = 0$ and using the first equation in (\ref{der-1}) and the fact that $\mathbb{J}^{-1} \|_{t = 0} = I$, we obtain the second equation in (\ref{der-1}). Recalling from $J = \det(\mathbb{J})$ and using the derivative formula for matrix determinants, we get \[ \dot{J} = \det(\mathbb{J}) \text{tr} (\mathbb{J}^{-1} \dot{\mathbb{J}}). \] Taking $t=0$ and using (\ref{FKdot-3}) we obtain the third equation in (\ref{der-1}). \end{proof} \begin{lem} \label{lem:Xh-1} There hold \begin{align} & \\| \dot{X}_h \\|_{L^\infty(\Omega)} \le \\| \dot{X} \\|_{L^\infty(\Omega)} , \label{Xdot-1} \\ & \\| \nabla \cdot \dot{X}_h\\|_{L^\infty(\Omega)} \le d \\| \nabla \dot{X} \\|_{L^\infty(\Omega)} \max_K \frac{h_K}{a_K} . \label{Xdot-2} \end{align} \end{lem} \begin{proof} The inequality (\ref{Xdot-1}) is evident from the definition of $\dot{X}_h$ in (\ref{Xt-0}). On $K$, we can write $\nabla \cdot \dot{X}_h$ as \begin{equation} \nabla \cdot \dot{X}_h = \sum_{i=0}^d \dot{\V{x}}_i^K \cdot \nabla \phi_i^K(\V{x}) = \sum_{i=1}^d (\dot{\V{x}}_i^K-\dot{\V{x}}_0^K) \cdot \nabla \phi_i^K(\V{x}) . \label{Xh-3} \end{equation} From \[ \dot{\V{x}}_i^K-\dot{\V{x}}_0^K = \dot{X}(\V{x}_i^K)-\dot{X}(\V{x}_0^K) = \int_0^1 \nabla \dot{X}(\V{x}_0^K + t (\V{x}_i^K-\V{x}_0^K)) \cdot (\V{x}_i^K-\V{x}_0^K) d t, \] we have \begin{equation} \| \dot{\V{x}}_i^K-\dot{\V{x}}_0^K \| \le \\| \nabla \dot{X} \\|_{L^\infty(\Omega)} h_K . \label{Xdot-3} \end{equation} The inequality (\ref{Xdot-2}) follows from (\ref{aK-1}), (\ref{Xh-3}), and the above inequality. \end{proof} \begin{rem} \label{rem:Xdot-1} In (\ref{Xdot-2}) we have assumed that the mesh velocity field is smooth. If the mesh velocity field is not smooth, from (\ref{Xh-3}) we have \begin{equation} \\| \nabla \cdot \dot{X}_h\\|_{L^\infty(\Omega)} \le \frac{d+1}{\min\limits_K a_K} \\| \dot{X} \\|_{L^\infty(\Omega)} . \label{Xdot-4} \end{equation} \end{rem} \begin{lem} \label{lem:EK-1} There hold \begin{align} & \\|E_{K}^{-1}\\|_2 \le \frac{\sqrt{d}}{a_K}, \label{EK-3} \\ & \\| \dot{E}_{K}\\|_2 \le \sqrt{d} \, h_K \\| \nabla \dot{X} \\|_{L^\infty(\Omega)} , \label{EK-4} \\ & \\| \dot{E}_{K}\\|_2 \\|E_{K}^{-1}\\|_2 \le \frac{d\, h_K}{a_K} \\| \nabla \dot{X} \\|_{L^\infty(\Omega)} , \label{EK-2} \end{align} where $\\| \cdot \\|_2$ denotes the 2-norm for matrices. \end{lem} \begin{proof} From (\ref{E-basis-1}), we have \[ \\|E_{K(0)}^{-1}\\|_2^2 \le \\|E_{K(0)}^{-1}\\|_F^2 = \sum_{i=1}^d \|\nabla \phi_i^K\|^2 \le \frac{d}{a_K^2} , \] which gives (\ref{EK-3}). Here, $\\| \cdot \\|_F$ is the Frobenius norm. Moreover, from (\ref{Xdot-3}) we have \[ \\| \dot{E}_{K(0)}\\|_2^2 \le \\| \dot{E}_{K(0)}\\|_F^2 = \sum_{i=1}^d \|\dot{\V{x}}_i^K-\dot{\V{x}}_0^K\|^2 \le d \\| \nabla \dot{X} \\|_{L^\infty(\Omega)}^2 h_K^2 , \] which gives (\ref{EK-4}) (with $K(0)$ being replaced by $K$). The inequality (\ref{EK-2}) follows from (\ref{EK-3}) and (\ref{EK-4}). \end{proof} \begin{rem} \label{rem:EK-1} As in Remark~\ref{rem:Xdot-1}, if the mesh velocity field is not smooth, (\ref{EK-4}) and (\ref{EK-2}) can be replaced by \begin{align} & \\| \dot{E}_{K}\\|_2 \le \sqrt{2 d} \, \\| \dot{X} \\|_{L^\infty(\Omega)} , \label{EK-5} \\ & \\| \dot{E}_{K}\\|_2 \\|E_{K}^{-1}\\|_2 \le \frac{d\, \sqrt{2}}{a_K} \\| \dot{X} \\|_{L^\infty(\Omega)} . \label{EK-6} \end{align} \end{rem} In FE computation, a basis function $\tilde{\psi}$ on $K(t)$ is typically defined as the composite function of a basis function $\psi$ on $K(0)$ with the affine mapping $F_{K(t)}$, i.e., \begin{equation} \tilde{\psi}(\V{y}, t) = \psi(F_{K(t)}^{-1}(\V{y})),\quad \tilde{\psi}(F_{K(t)}(\V{x}), t) = \psi(\V{x}) . \label{phi-1} \end{equation} Taking gradient of both sides of the second equation with respect to $\V{x}$, we get \begin{equation} \tilde{\nabla} \tilde{\psi} (F_{K(t)}(\V{x}), t) = \mathbb{J}^{-T} \nabla \psi (\V{x}) . \label{phi-2} \end{equation} \begin{lem} \label{lem:phi-1} Assume that basis functions $\tilde{\psi}$ on $K(t)$ and $\psi$ on $K(0)$ are related through (\ref{phi-1}). Then, \begin{align} & \dot{\tilde{\psi}}\|_{t = 0} = 0, \quad \dot{} \quad \label{phi-3-0} \\ & \dot{(\tilde{\nabla} \tilde{\psi})}\; \; \|_{t=0} = - E_{K(0)}^{-T} \dot{E}_{K(0)}^T \nabla \psi . \label{phi-3} \end{align} \end{lem} \begin{proof} The equality (\ref{phi-3-0}) follows from differentiating the second equation in (\ref{phi-1}) with respect to time. The equality (\ref{phi-3}) can be obtained by differentiating (\ref{phi-2}) with respect to time, taking $t = 0$, and using Lemma~\ref{lem:Jdot-1}. \end{proof} \begin{lem} \label{lem:fun-1} Consider a function $f = f(\V{x})$ defined on $K$ and let $\tilde{f} = \tilde{f}(\V{y},t) = f(F_{K(t)}(\V{x}))$. Then, \begin{equation} \dot{\tilde{f}}\|_{t=0} = \nabla f \cdot \dot{X}_h \|_K . \label{fun-1} \end{equation} \end{lem} \begin{proof} Differentiating $\tilde{f} = \tilde{f}(\V{y},t) = f(F_{K(t)}(\V{x}))$ with respect to $t$, we get \[ \dot{\tilde{f}} = \nabla f (F_{K(t)}(\V{x})) \cdot \dot{F}_{K(t)}(\V{x}). \] Taking $t = 0$ and using Lemma~\ref{lem:FKdot-0}, we obtain (\ref{fun-1}). \end{proof} \begin{lem} \label{lem:FE-1} Consider an FE approximation $v_h = \sum_{i} v_i \psi_i(\V{x})$ and its perturbation $\tilde{v}_h(\V{y},t) = \sum_{i} v_i(t) \tilde{\psi}_i(\V{y},t)$. Then, \begin{align} & \dot{\tilde{v}}_h \|_{K, t = 0} = \dot{v}_h \|_K, \quad \forall K \in \mathcal{T}_h \label{der-4} \\ & \dot{(\tilde{\nabla} \tilde{v}_h)}\;\; \|_{K, t=0} = - E_{K}^{-T} \dot{E}_{K}^T \nabla v_h \|_K + \nabla \dot{v}_h\|_K, \quad \forall K \in \mathcal{T}_h \label{der-5} \end{align} where $\dot{v}_h = \sum_i \dot{v}_i(0) \psi_i(\V{x})$. \end{lem} \begin{proof} Restricted on $K(t)$, using (\ref{phi-1}) we can rewrite $\tilde{v}_h$ into \begin{equation} \tilde{v}_h(F_{K(t)}(\V{x}),t) = \sum_{i} v_i(t) \psi_i(\V{x}) \|_{K(0)}. \label{vh-1} \end{equation} Differentiating this with respect to $t$, we get \[ \dot{\tilde{v}}_h(F_{K(t)}(\V{x}),t) = \sum_{i} \dot{v}_i(t) \psi_i(\V{x}) \|_{K(0)}. \] Taking $t= 0$, we get (\ref{der-4}). Applying $\nabla \cdot$ to (\ref{vh-1}) and using the chain rule, we get \[ \tilde{\nabla} \tilde{v}_h(F_{K(t)}(\V{x}),t) = \mathbb{J}^{-T} \sum_{i} v_i(t) \nabla \psi_i(\V{x}) \|_{K(0)} . \] Differentiating this with respect to $t$, we obtain \[ \dot{(\tilde{\nabla} \tilde{v}_h)} \; \; = (\dot{(\mathbb{J}^{-1})})^T \sum_{i} v_i(t) \nabla \psi_i(\V{x}) \|_{K(0)} + \mathbb{J}^{-T} \sum_{i} \dot{v}_i(t) \nabla \psi_i(\V{x}) \|_{K(0)} . \] Taking $t = 0$ and using Lemma~\ref{lem:Jdot-1}, we obtain (\ref{der-5}). \end{proof} \section{Mesh sensitivity analysis for the finite element solution} \label{SEC:sensitivity} In this section we analyze the mesh sensitivity for the FE solution $u_h = \sum_{i} u_i \psi_i(\V{x})$ satisfying (\ref{fem-1}). On the deformed mesh $\mathcal{T}_h(t)$, the perturbed FE solution can be expressed as $\tilde{u}_h = \sum_{i} u_i(t) \tilde{\psi}_i(\V{y},t)$. We assume that $\tilde{u}_h$ is differential. From Lemma~\ref{lem:FE-1}, the material derivative of $\tilde{u}_h$ at $t=0$ is given by $\dot{\tilde{u}}_h \|_{t=0} = \sum_{i} \dot{u}_i(0) \psi_i(\V{x})$. We denote this by $\dot{u}_h$, i.e., $\dot{u}_h = \sum_{i} \dot{u}_i(0) \psi_i(\V{x})$. This derivative measures the change in $u_h$ with mesh deformation. In this section, we first derive the FE formulation for $\dot{u}_h$ and then establish a bound for $\\|\nabla \dot{u}_h\\|_{L^2(\Omega)}$. \begin{thm} \label{thm:fem-1} The material derivative $\dot{u}_h$ satisfies \begin{align} & \int_{\Omega} \left ( {a} \nabla \dot{u}_h \cdot \nabla \psi + ({\V{b}} \cdot \nabla \dot{u}_h){\psi} + {c} \dot{u}_h {\psi}\right ) d \V{x} \nonumber \\ & = \sum_{K} \int_{K} \left ( {a} \nabla u_h \cdot (\dot{E}_{K}E_{K}^{-1}+ E_{K}^{-T} \dot{E}_{K}^{T}) \cdot \nabla \psi + ({\V{b}} \cdot E_{K}^{-T} \dot{E}_{K}^{T} \cdot \nabla u_h){\psi} \right ) d \V{x} \nonumber \\ & \quad \quad - \int_{\Omega} \left ( (\nabla a \cdot \dot{X}_h) (\nabla {u}_h \cdot \nabla {\psi}) + a ( \nabla {u}_h \cdot \nabla {\psi}) (\nabla \cdot \dot{X}_h) \right ) d \V{x} \nonumber \\ & \quad \quad - \int_{\Omega} \left ( (\dot{X}_h \cdot \nabla \V{b} \cdot \nabla {u}_h) {\psi} + ({\V{b}} \cdot \nabla {u}_h){\psi} (\nabla \cdot \dot{X}_h) \right ) d \V{x} \nonumber \\ & \quad \quad + \int_{\Omega} \left ( c \psi (\nabla u_h \cdot \dot{X}_h) + {c} {u}_h (\nabla \psi \cdot \dot{X}_h) \right ) d \V{x} - \int_{\Omega} f (\nabla \psi \cdot \dot{X}_h) d \V{x} , \qquad \forall \psi \in V_h . \label{fem-4} \end{align} \end{thm} \begin{proof} We start with rewriting (\ref{fem-1}) into \begin{equation} \sum_{K} \int_{K} \left ( a \nabla u_h \cdot \nabla \psi + (\V{b}\cdot \nabla u_h)\psi + c u_h \psi\right ) d \V{x} = \sum_{K} \int_{K} f \psi d \V{x}, \quad \forall \psi \in V_h . \label{fem-2} \end{equation} On the deformed mesh $\mathcal{T}_h(t)$, the perturbed FE solution can be expressed as $\tilde{u}_h = \sum_{i} u_i(t) \tilde{\psi}_i(\V{y},t)$. Moreover, (\ref{fem-2}) becomes \begin{equation} \sum_{K(t)} \int_{K(t)} \left ( \tilde{a} \tilde{\nabla} \tilde{u}_h \cdot \tilde{\nabla} \tilde{\psi} + (\tilde{\V{b}}\cdot \tilde{\nabla} \tilde{u}_h)\tilde{\psi} + \tilde{c} \tilde{u}_h \tilde{\psi}\right ) d \V{y} = \sum_{K(t)} \int_{K(t)} f(\V{y}) \tilde{\psi} d \V{y}, \quad \forall \psi \in V_h(t) \label{fem-3} \end{equation} where $V_h(t) = \text{span}\{ \tilde{\psi}_1, \tilde{\psi}_2, ... \}$. Transforming all integrals into $K(0)$, we obtain \begin{align} & \sum_{K(0)} \int_{K(0)} \left ( \tilde{a} \tilde{\nabla} \tilde{u}_h \cdot \tilde{\nabla} \tilde{\psi} + (\tilde{\V{b}} \cdot \tilde{\nabla} \tilde{u}_h)\tilde{\psi} + \tilde{c} \tilde{u}_h \tilde{\psi}\right ) J d \V{x} = \sum_{K(0)} \int_{K(0)} f(F_{K(t)}(\V{x})) \tilde{\psi} J d \V{x} . \end{align} Differentiating both sides of the above equation with respect to $t$ while keeping $\V{x}$ fixed, we have \begin{align} & \sum_{K(0)} \int_{K(0)} \left ( \tilde{a} \dot{(\tilde{\nabla} \tilde{u}_h)}\;\; \cdot \tilde{\nabla} \tilde{\psi} + (\tilde{\V{b}} \cdot \dot{(\tilde{\nabla} \tilde{u}_h)}\;\;)\tilde{\psi} + \tilde{c} \dot{\tilde{u}}_h \tilde{\psi}\right ) J d \V{x} \\ & = \quad - \sum_{K(0)} \int_{K(0)} \left ( \tilde{a} \tilde{\nabla} \tilde{u}_h \cdot \dot{(\tilde{\nabla} \tilde{\psi})}\;\; + (\tilde{\V{b}} \cdot \tilde{\nabla} \tilde{u}_h)\dot{\tilde{\psi}} + \tilde{c} \tilde{u}_h \dot{\tilde{\psi}}\right ) J d \V{x} \\ & \quad \quad - \sum_{K(0)} \int_{K(0)} \left ( \dot{\tilde{a}} \tilde{\nabla} \tilde{u}_h \cdot \tilde{\nabla} \tilde{\psi} + (\dot{\tilde{\V{b}}} \cdot \tilde{\nabla} \tilde{u}_h)\tilde{\psi} + \dot{\tilde{c}} \tilde{u}_h \tilde{\psi}\right ) J d \V{x} \\ & \quad \quad - \sum_{K(0)} \int_{K(0)} \left ( \tilde{a} \tilde{\nabla} \tilde{u}_h \cdot \tilde{\nabla} \tilde{\psi} + (\tilde{\V{b}} \cdot \tilde{\nabla} \tilde{u}_h)\tilde{\psi} + \tilde{c} \tilde{u}_h \tilde{\psi}\right ) \dot{J} d \V{x} \\ & \quad \quad + \sum_{K(0)} \int_{K(0)} \left ( \dot{\tilde{f}} \tilde{\psi} J + \tilde{f} \dot{\tilde{\psi}} J + \tilde{f} \tilde{\psi} \dot{J} \right )d \V{x} . \end{align} Using Lemmas~\ref{lem:Jdot-1}, \ref{lem:phi-1}, \ref{lem:fun-1}, and \ref{lem:FE-1}, taking $t = 0$, and noticing that $J = 1$ and $\V{y} = \V{x}$ at $t = 0$, we get \begin{align} & \sum_{K(0)} \int_{K(0)} \left ( {a} (\nabla \dot{u}_h - E_{K(0)}^{-T} \dot{E}_{K(0)}^{T}\nabla u_h) \cdot \nabla \psi + ({\V{b}} \cdot (\nabla \dot{u}_h - E_{K(0)}^{-T} \dot{E}_{K(0)}^{T}\nabla u_h)){\psi} + {c} \dot{u}_h {\psi}\right ) d \V{x} \\ & = \quad - \sum_{K(0)} \int_{K(0)} \left ( {a} \nabla {u}_h \cdot (- E_{K(0)}^{-T} \dot{E}_{K(0)}^{T}\nabla \psi)\right ) d \V{x} \\ & \quad \quad - \sum_{K(0)} \int_{K(0)} \left ( (\nabla a \cdot \dot{X}_h) \nabla {u}_h \cdot \nabla {\psi} + (\dot{X}_h \cdot \nabla \V{b} \cdot \nabla {u}_h) {\psi} + (\nabla c\cdot \dot{X}_h) {u}_h {\psi}\right ) d \V{x} \\ & \quad \quad - \sum_{K(0)} \int_{K(0)} \left ( {a} \nabla {u}_h \cdot \nabla {\psi} + ({\V{b}} \cdot \nabla {u}_h){\psi} + {c} {u}_h {\psi}\right ) (\nabla \cdot \dot{X}_h) d \V{x} \\ & \quad \quad + \sum_{K(0)} \int_{K(0)} \left ( (\nabla f \cdot \dot{X}_h) \psi + f {\psi} (\nabla \cdot \dot{X}_h) \right )d \V{x} . \end{align} Noticing that $K(0) = K$, we can rewrite the above equation into \begin{align} & \int_{\Omega} \left ( {a} \nabla \dot{u}_h \cdot \nabla \psi + ({\V{b}} \cdot \nabla \dot{u}_h){\psi} + {c} \dot{u}_h {\psi}\right ) d \V{x} \\ & = \sum_{K} \int_{K} \left ( {a} \nabla u_h \cdot (\dot{E}_{K}E_{K}^{-1}+ E_{K}^{-T} \dot{E}_{K}^{T}) \cdot \nabla \psi + ({\V{b}} \cdot E_{K}^{-T} \dot{E}_{K}^{T} \cdot \nabla u_h){\psi} \right ) d \V{x} \\ & \quad \quad - \int_{\Omega} \left ( (\nabla a \cdot \dot{X}_h) \nabla {u}_h \cdot \nabla {\psi} + (\dot{X}_h \cdot \nabla \V{b} \cdot \nabla {u}_h) {\psi} + (\nabla c\cdot \dot{X}_h) {u}_h {\psi}\right ) d \V{x} \\ & \quad \quad - \int_{\Omega} \left ( {a} \nabla {u}_h \cdot \nabla {\psi} + ({\V{b}} \cdot \nabla {u}_h){\psi} + {c} {u}_h {\psi}\right ) (\nabla \cdot \dot{X}_h) d \V{x} \\ & \quad \quad + \int_{\Omega} \left ( (\nabla f \cdot \dot{X}_h) \psi + f {\psi} (\nabla \cdot \dot{X}_h) \right )d \V{x} . \end{align} This can be rewritten into \begin{align} & \int_{\Omega} \left ( {a} \nabla \dot{u}_h \cdot \nabla \psi + ({\V{b}} \cdot \nabla \dot{u}_h){\psi} + {c} \dot{u}_h {\psi}\right ) d \V{x} \\ & = \sum_{K} \int_{K} \left ( {a} \nabla u_h \cdot (\dot{E}_{K}E_{K}^{-1}+ E_{K}^{-T} \dot{E}_{K}^{T}) \cdot \nabla \psi + ({\V{b}} \cdot E_{K}^{-T} \dot{E}_{K}^{T} \cdot \nabla u_h){\psi} \right ) d \V{x} \\ & \quad \quad - \int_{\Omega} \left ( (\nabla a \cdot \dot{X}_h) (\nabla {u}_h \cdot \nabla {\psi}) + a ( \nabla {u}_h \cdot \nabla {\psi}) (\nabla \cdot \dot{X}_h) \right ) d \V{x} \\ & \quad \quad - \int_{\Omega} \left ( (\dot{X}_h \cdot \nabla \V{b} \cdot \nabla {u}_h) {\psi} + ({\V{b}} \cdot \nabla {u}_h){\psi} (\nabla \cdot \dot{X}_h) \right ) d \V{x} \\ & \quad \quad - \int_{\Omega} \left ( (\nabla c\cdot \dot{X}_h) {u}_h {\psi} + {c} {u}_h {\psi} (\nabla \cdot \dot{X}_h) \right ) d \V{x} \\ & \quad \quad + \int_{\Omega} \left ( (\nabla f \cdot \dot{X}_h) \psi + f {\psi} (\nabla \cdot \dot{X}_h) \right )d \V{x} . \end{align} Using the divergence theorem, we get \begin{align} & \int_{\Omega} \left ( {a} \nabla \dot{u}_h \cdot \nabla \psi + ({\V{b}} \cdot \nabla \dot{u}_h){\psi} + {c} \dot{u}_h {\psi}\right ) d \V{x} \nonumber \\ & = \sum_{K} \int_{K} \left ( {a} \nabla u_h \cdot (\dot{E}_{K}E_{K}^{-1}+ E_{K}^{-T} \dot{E}_{K}^{T}) \cdot \nabla \psi + ({\V{b}} \cdot E_{K}^{-T} \dot{E}_{K}^{T} \cdot \nabla u_h){\psi} \right ) d \V{x} \nonumber \\ & \quad \quad - \int_{\Omega} \left ( (\nabla a \cdot \dot{X}_h) (\nabla {u}_h \cdot \nabla {\psi}) + a ( \nabla {u}_h \cdot \nabla {\psi}) (\nabla \cdot \dot{X}_h) \right ) d \V{x} \nonumber \\ & \quad \quad - \int_{\Omega} \left ( (\dot{X}_h \cdot \nabla \V{b} \cdot \nabla {u}_h) {\psi} + ({\V{b}} \cdot \nabla {u}_h){\psi} (\nabla \cdot \dot{X}_h) \right ) d \V{x} \nonumber \\ & \quad \quad - \int_{\partial \Omega} c u_h \psi \dot{X}_h\cdot \V{n} d S + \int_{\Omega} \left ( c \psi (\nabla u_h \cdot \dot{X}_h) + {c} {u}_h (\nabla \psi \cdot \dot{X}_h) \right ) d \V{x} \nonumber \\ & \quad \quad + \int_{\partial \Omega} f \psi \dot{X}_h\cdot \V{n} d S - \int_{\Omega} f (\nabla \psi \cdot \dot{X}_h) d \V{x} , \label{fem-7} \end{align} where $\V{n}$ is the outward unit normal of $\partial \Omega$. We obtain (\ref{fem-4}) by noticing that the surface integrals vanish since $\psi = 0$ on $\partial \Omega$. \end{proof} \begin{thm} \label{thm:fem-2} Assume that $\mathcal{T}_h$ is a simplicial mesh with the minimum element height $\min_K a_K > 0$. Then, the material derivative of the FE solution to BVP (\ref{bvp-1}) and (\ref{bc-1}) due to mesh deformation is bounded by \begin{align} a_0 \\| \nabla \dot{u}_h \\|_{L^2(\Omega)} & \le \\| f \\|_{L^2(\Omega)}\left ( 1 + C_{\Omega} \\|\nabla a \\|_{L^\infty(\Omega)} + C_{\Omega}^2 \\|\nabla \V{b} \\|_{L^\infty(\Omega)} + 2 C_{\Omega}^2 \\| c \\|_{L^\infty(\Omega)} \right ) \\| \dot{X} \\|_{L^\infty(\Omega)} \nonumber \\ & \quad + \\| f \\|_{L^2(\Omega)}\left ( 3 d C_{\Omega} \\| a \\|_{L^\infty(\Omega)} + 2 d C_{\Omega}^2 \\| \V{b} \\|_{L^\infty(\Omega)}\right ) \\| \nabla \dot{X} \\|_{L^\infty(\Omega)} \max_K \frac{h_K}{a_K} , \label{fem-5} \end{align} where $C_{\Omega}$ is the constant appearing in Poincar\'{e}'s inequality that depends only on $\Omega$. \end{thm} \begin{proof} Recall that $\dot{u}_h \equiv \sum_i \dot{u}_i(0) \psi_i$ belongs to $V_h$. Taking $\psi = \dot{u}_h$ in (\ref{fem-4}) and using the Cauchy-Schwarz inequality, Poincar\'{e}'s inequality, and (\ref{bvp-3}), we get \begin{align} a_0 \\| \nabla \dot{u}_h \\|_{L^2(\Omega)} & \le \\| a \\|_{L^\infty(\Omega)} \\| \nabla u_h\\|_{L^2(\Omega)} \left ( \\| \nabla \cdot \dot{X} \\|_{L^\infty(\Omega)} + 2 \max_K \\| \dot{E}_K E_K^{-1} \\|_2 \right ) \\ & \quad + \\| \nabla a \\|_{L^\infty(\Omega)} \\| \nabla u_h\\|_{L^2(\Omega)} \\| \dot{X}_h \\|_{L^\infty(\Omega)} \\ & \quad + C_{\Omega} \\| \V{b} \\|_{L^\infty(\Omega)} \\| \nabla u_h\\|_{L^2(\Omega)} \left ( \\| \nabla \cdot \dot{X} \\|_{L^\infty(\Omega)} + \max_K \\| \dot{E}_K E_K^{-1} \\|_2 \right ) \\ & \quad + C_{\Omega} \\| \nabla \V{b} \\|_{L^\infty(\Omega)} \\| \nabla u_h\\|_{L^2(\Omega)} \\| \dot{X}_h \\|_{L^\infty(\Omega)} + 2 C_{\Omega} \\| c \\|_{L^\infty(\Omega)} \\| \nabla u_h\\|_{L^2(\Omega)} \\| \dot{X}_h \\|_{L^\infty(\Omega)} \\ & \quad + \\| f \\|_{L^2(\Omega)} \\| \dot{X}_h \\|_{L^\infty(\Omega)} . \end{align} Taking $\psi = u_h$ in (\ref{fem-1}) and using the Cauchy-Schwarz inequality and Poincar\'{e}'s inequality, we get \[ \\| \nabla u_h \\|_{L^2(\Omega)} \le C_{\Omega} \\| f \\|_{L^2(\Omega)} . \] Combining the above results and using Lemmas~\ref{lem:Xh-1} and \ref{lem:EK-1}, we obtain (\ref{fem-5}). \end{proof} We emphasize that the bound in (\ref{fem-5}) has been obtained without assuming that the mesh is regular. In fact, the mesh can be arbitrary, isotropic or anisotropic, uniform or nonuniform, as long as it is simplicial and has a positive minimum element height. For meshes with large aspect ratio, the factor $\max_K h_K/a_K$ is large and the bound in (\ref{fem-5}) is more sensitive to $\\| \nabla \dot{X}\\|_{L^2(\Omega)}$. Moreover, the bound shows that the size and gradient of the mesh velocity field can have effects on the FE solution. The former is insensitive to the shape of mesh elements whereas the effects from the gradient of the mesh velocity are proportional to the maximum element aspect ratio $\max_K h_K/a_K$. Furthermore, (\ref{fem-5}) is homogeneous about time derivatives. Thus, $\dot{X}$ can be viewed as mesh displacement instead of mesh velocity. The bound shows that the change in the FE solution is small when $\dot{X}$ and $\nabla \dot{X}$ are small, implying a continuous dependence of the FE solution on the mesh. \begin{rem} \label{rem:fem-1} If the mesh velocity field is not smooth, from Remarks~\ref{rem:Xdot-1} and (\ref{rem:EK-1}) and inequality (\ref{fem-7}) we can see that (\ref{fem-5}) can be replaced by \begin{align} a_0 \\| \nabla \dot{u}_h \\|_{L^2(\Omega)} & \le \\| f \\|_{L^2(\Omega)}\left ( 1 + C_{\Omega} \\|\nabla a \\|_{L^\infty(\Omega)} + C_{\Omega}^2 \\|\nabla \V{b} \\|_{L^\infty(\Omega)} + 2 C_{\Omega}^2 \\| c \\|_{L^\infty(\Omega)} \right ) \\| \dot{X} \\|_{L^\infty(\Omega)} \nonumber \\ & \quad + \\| f \\|_{L^2(\Omega)}\left ( 3 d C_{\Omega} \\| a \\|_{L^\infty(\Omega)} + 2 d C_{\Omega}^2 \\| \V{b} \\|_{L^\infty(\Omega)}\right ) \\| \dot{X} \\|_{L^\infty(\Omega)} \frac{1}{\min\limits_K a_K} . \label{fem-6} \end{align} For a given mesh, $\min\limits_K a_K$ is fixed. Thus, (\ref{fem-6}) shows that the FE solution depends continuously on the mesh even in the situation where the mesh velocity field is not smooth. \end{rem} \section{A numerical example} \label{SEC:numerics} In this section we present some numerical results for a two-dimensional example in the form (\ref{bvp-1}) where $\Omega = (0,1)\times (0,1)$, $a = 1$, $\V{b} = (1,2)^T$, $c=0$, and $f$ is chosen such that the exact solution of the BVP is given by $u = \sin(2\pi x) \sin(3 \pi y)$. We use linear finite elements and choose the base mesh to be a triangular mesh by first partitioning $\Omega$ into $N\times N$ small rectangles and then dividing each small rectangle into four triangles using its diagonal lines. We first consider a smooth velocity field, \[ \dot{X} = \sin(\pi x) \sin(2 \pi y) \] and deform the mesh through (\ref{xt-1}). The results are listed in Table~\ref{tab:smooth}. We can see that $\\| \nabla \dot{u}_h\\|_{L^2(\Omega)}$ is a linear function of $t$. Moreover, the error is almost independent of the mesh size. This is consistent with (\ref{fem-5}) since the only mesh-dependent factor $\max_K h_K/a_K$ is constant when the mesh is being refined for the current situation. Next we consider a random mesh velocity field. In this case, we generate $\dot{X}_i = \dot{X}(\V{x}_i)$ ($i = 1, 2, ...$) using a uniformly distributed pseudorandom number generator and scale them to the range $(-1,1)$. Then we perturb the mesh through (\ref{xt-1}). In this case, the velocity field is not smooth. The results are listed in Table~\ref{tab:random} where $\\| \nabla \dot{u}_h\\|_{L^2(\Omega)}$ is shown as the average of the corresponding values obtained with twenty repeated runs for each pair of $t$ and the mesh size. From the table we can see that $\\| \nabla \dot{u}_h\\|_{L^2(\Omega)}$ is still linear about $t$. Interestingly, $\\| \nabla \dot{u}_h\\|_{L^2(\Omega)}$ is mesh-dependent: its values are about twice larger for the $80\times 80$ mesh than those for the $40\times 40$ mesh. This can be explained using (\ref{fem-6}) which contains a mesh-dependent factor $1/(\min_K a_K)$. This factor is about twice larger for the $80\times 80$ mesh than for the $40\times 40$ mesh. Moreover, we can see that $\\| \nabla \dot{u}_h\\|_{L^2(\Omega)}$ in Table~\ref{tab:random} are larger than those in Table~\ref{tab:smooth}, which reflects the nature of the bounds in (\ref{fem-5}) and (\ref{fem-6}) for smooth and nonsmooth velocity fields. \begin{table}[htb] \begin{center} \caption{$\\| \nabla \dot{u}_h\\|_{L^2(\Omega)}$ for smooth velocity field.} \vspace{2mm} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|}\hline \hline Mesh size & $t =$ 1e-6 & $t =$ 1e-5 & $t =$ 1e-4 & $t =$ 1e-3 & $t =$ 1e-2 \\ \hline $40\times 40$ & 3.759e-5 & 3.759e-4 & 3.759e-3 & 3.759e-2 & 3.760e-1 \\ $80\times 80$ & 3.770e-5 & 3.770e-4 & 3.770e-3 & 3.771e-2 & 3.772e-1 \\ \hline \hline \end{tabular} \label{tab:smooth} \end{center} \end{table} \begin{table}[htb] \begin{center} \caption{$\\| \nabla \dot{u}_h\\|_{L^2(\Omega)}$ (average) for random velocity field. The shown values are obtained as the average of the corresponding values obtained with twenty repeated runs for each pair of $t$ and the mesh size.} \vspace{2mm} \begin{tabular}{\|c\|c\|c\|c\|c\|}\hline \hline Mesh size & $t =$ 1e-6 & $t =$ 1e-5 & $t =$ 1e-4 & $t =$ 1e-3 \\ \hline $40\times 40$ & 3.641e-4 & 3.640e-3 & 3.633e-2 & 3.656e-1 \\ $80\times 80$ & 7.367e-4 & 7.341e-3 & 7.353e-2 & 7.432e-1 \\ \hline \hline \end{tabular} \label{tab:random} \end{center} \end{table} \section{Conclusions} \label{SEC:conclusions} We have presented an analysis on the mesh sensitivity for the finite element solution of a boundary value problem of linear elliptic partial differential equations. The main result is stated in Theorem~\ref{thm:fem-2} where a bound is obtained for the change in the finite element solution in terms of the mesh deformation and its gradient. A similar bound is given in (\ref{fem-6}) when the mesh velocity field is nonsmooth. These results show how the finite element solution depends continuously on the mesh. The results have been obtained in any dimension and for arbitrary unstructured simplicial meshes, general linear elliptic partial differential equations, and general finite element approximations.	{ "redpajama_set_name": "RedPajamaArXiv" }	3,768
Schuurs, ZP, Hammond, E, Elli, S, Rudd, TR, Mycroft-West, CJ, Lima, MA, Skidmore, MA, Karlsson, R, Chen, Y-H, Bagdonaite, I, Yang, Z, Ahmed, YA, Richard, DJ, Turnbull, J, Ferro, V, Coombe, DR and Gandhi, NS (2021) Evidence of a putative glycosaminoglycan binding site on the glycosylated SARS-CoV-2 spike protein N-terminal domain. Computational and Structural Biotechnology Journal, 19. 2806 - 2818. ISSN 2001-0370 Evidence of a putative glycosaminoglycan binding site on the glycosylated SARS-CoV-2 spike protein N-terminal domain.pdf - Published Version Official URL: https://www.sciencedirect.com/science/article/pii/... SARS-CoV-2 has rapidly spread throughout the world's population since its initial discovery in 2019. The virus infects cells via a glycosylated spike protein located on its surface. The protein primarily binds to the angiotensin-converting enzyme-2 (ACE2) receptor, using glycosaminoglycans (GAGs) as co-receptors. Here, we performed bioinformatics and molecular dynamics simulations of the spike protein to investigate the existence of additional GAG binding sites on the receptor-binding domain (RBD), separate from previously reported heparin-binding sites. A putative GAG binding site in the N-terminal domain (NTD) of the protein was identified, encompassing residues 245-246. We hypothesized that GAGs of a sufficient length might bridge the gap between this site and the PRRARS furin cleavage site, including the mutation S247R. Docking studies using GlycoTorch Vina and subsequent MD simulations of the spike trimer in the presence of dodecasaccharides of the GAGs heparin and heparan sulfate supported this possibility. The heparan sulfate chain bridged the gap, binding the furin cleavage site and S247R. In contrast, the heparin chain bound the furin cleavage site and surrounding glycosylation structures, but not S247R. These findings identify a site in the spike protein that favors heparan sulfate binding that may be particularly pertinent for a better understanding of the recent UK and South African strains. This will also assist in future targeted therapy programs that could include repurposing clinical heparan sulfate mimetics. This publication is available under a CC-BY license - https://creativecommons.org/licenses/by/4.0/ Heparan sulfate; SARS-CoV-2; COVID-19; Spike protein; Heparin; Coronavirus; Cosolvent MD simulations Faculty of Natural Sciences > School of Life Sciences	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	1,803
{"url":"https:\/\/physics.meta.stackexchange.com\/questions\/5948\/quickly-finding-policy-on-xxxxx","text":"# Quickly finding policy on Xxxxx\n\nI sometimes find myself searching for policy on something so that I can point a poster to the policy, or refer to it myself for guidance. Usually this is the homework policy, but sometimes it's something else such the policy on questions related to experimental physics. But I seem to re-invent the wheel of finding the policy each time I need it, and for me the search does not go smoothly.\n\nIs there some kind of central location for policies? Or can you provide advice on how to find these things quickly? It always seems to take me many clicks and many pages to find what I want.\n\n\u2022 You mean the help center? \u2013\u00a0Kyle Kanos Jul 8 '14 at 18:45\n\u2022 Hmm, there's also searching meta for the faq tag (e.g. by using the search bar on this site with [faq]). \u2013\u00a0user10851 Jul 8 '14 at 19:16","date":"2020-09-23 13:03:59","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\": 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\": 0.28319457173347473, \"perplexity\": 1012.4254710759163}, \"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-2020-40\/segments\/1600400210996.32\/warc\/CC-MAIN-20200923113029-20200923143029-00012.warc.gz\"}"}	null	null
{"url":"http:\/\/bjerck.net\/blog\/","text":"## blog @ bjerck.net\n\n#### Compiling an upstream kernel on ubuntu.\n\nby: tom\n##### Introduction\n\nAfter downloading the source from kernel.org or any of the mirrors, you have at least two options for how to compile the source for a ubuntu system. First you can just do it the traditional way and just use the Makefile that comes with the source directly. A second and more convenient and safer way to do this is to make a deb package, and let apt handle the installation and later eventually the removal of the kernel when it is no longer needed. Also making a deb will make sure that any installation scripts in the \/etc\/kernel directory is executed. These scripts currently updates the initramfs image, updates grub config and recompiles any third party modules. These scripts can not be part of the kernel package since they are dependent of the target system.\n\n##### Compiling using the make deb-pkg target in the Linux kernel source Makefile\n\nI didn't know this until recently, but the kernel Makefile has a deb-pkg target. It's the functionality used for compiling a new kernel documented in the Debian Kernel Handbook chapter 4. Se the scripts\/package\/builddeb file in the Linux source to see what this build target does. If you have a multi core cpu, you may use make's -j<num jobs> option, to speed things up.\n\n##### Compiling using kernel-package\n\nYou should install the following packages:\nkernel-package fakeroot build-essential\n\n###### Relevant documentation:\n\nman make-kpkg\nman kernel-package\nman make\nDebian Kernel Handbook\nwiki.ubuntu.com\/KernelTeam\/GitKernelBuild\n\/usr\/share\/doc\/kernel-package\/README.gz\n\/usr\/share\/doc\/kernel-package\/Kernel.htm\nman gpg\nman wget\nwww.kernel.org\/signature.html\nkernelnewbies.org\/FAQ\/KernelCompilation\n\n###### Versions\n\nThere are three version labels:\n\n1. The stable upstream kernel versions at kernel.org whose format are three numbers separated by dots, like 3.4.6.\n2. The deb package of the kernel has it's own version; usually composed of the upstream version numbers and local revision numbers.\n3. Then there is a third versioning labeling the locally configured and compiled kernel. This version number is the output of uname -r.\nThe last version label mentioned above is used to name the Linux kernel image file, and also to name the directory where it's modules are installed under \/lib\/modules. It's important that these names are unique for each new kernel installed on the system in order to have binary compatibility between the kernel image and it's modules and prevent previously installed kernels and modules from being overwritten.\n\nIn the following I'll download and unpack the source code for the 3.4.6 version of the upstream Linux kernel. I'll also download the gpg signature file and use it to verify the download. Using wget:\nwget \"https:\/\/www.kernel.org\/pub\/linux\/kernel\/v3.0\/linux-3.4.6.tar.xz\"\nwget \"https:\/\/www.kernel.org\/pub\/linux\/kernel\/v3.0\/linux-3.4.6.tar.sign\"\nThe compressed tar file needs to be decompressed before gpg verification:\nxz -d \"linux-3.4.6.tar.xz\"\nUse gpg to verify:\ngpg --verify \"linux-3.4.6.tar.sign\"\nIf you get an error message about a missing key, you need to import the signing key from a keyserver:\ngpg --recv-keys <key id>\nAnd then run the gpg verify command again. You should then get a message proclaming \"Good signature\". If you get a message stating \"Bad signature\" something is wrong. If this step goes well, continue and untar the archive:\ntar xf \"linux-3.4.6.tar\"\nThis should extract a directory named linux-3.4.6. Make this your working directory:\ncd linux-3.4.6\n\n###### Configure the kernel\n\nYou now need to configure the kernel. If you want base the new configuration on the old config that was used when building your currently running kernel, you just copy it from the \/boot directory like this:\ncp \/boot\/config-$(uname -r) .config The new kernel will likely have new configuration options not present in the config for the old kernel. Therefore you now need to run one of the several utilities supplied with the kernel source for editing the configuration. For a terminal user interface you can use menuconfig which you start by giving the command make menuconfig in the root directory of the source. ###### Use make-kpkg to compile the source created: 2012-07-20 15:38:14. Permalink #### Lenovo Thinkpad Edge S430 on Linux - First impressions by: tom A few days ago I received my new Thinkpad Edge S430, and so far I'm satisfied with it. I've upgraded to 16GB memory, and changed to a 128GB ssd from Samsung (830 series 2.5\"). If you plan on buying one of these laptops and install a ssd, you should know that it needs to have a maximum height of 7mm to fit. And remember to enable TRIM with the \"discard\" mount option in \/etc\/fstab. This was not done automaticalle by the Ubuntu installer. The system seems really fast and responsive with Ubuntu. (This is with 16GB memory and the Samsung ssd; I never ran Ubuntu with the original configuration.) Pros: \u2022 Low weight. (1.8kg) \u2022 Good screen resolution (1600x900) \u2022 Low price. \u2022 Anti Glare \u2022 Comfortable keyboard \u2022 Aesthetics Cons: \u2022 Battery won't last more that 3\u00bd - 4 hours with max backlight, general browsing and writing code in vim. And you will want to use max backlight in daylight. According to Lenovo the screen brightness is 250 nits. \u2022 TN LCD means low viewing angles. But what can you expect at this price? \u2022 The lid is a bit flexible. ###### Ubuntu 12.04 Installation of Ubuntu 12.04 went smoothly. Everything works; most notably WiFi, suspend and keyboard keys for backlight brightness, the thinklight, volume and media keys. However, I have not tested the dvd drive, fingerprint reader, thunderbolt, nor the hdmi port. There have been some stability issues where the system freezes completely every few hours, and there is nothing else to do than force a cold reboot using the power button. This seems to be an issue with Ivy Bridge: [IVB]12.04 64 bit System freezes (mouse, keyboard). A new kernel is in \"proposed\" which purportedly solves the freezes, and should be released within the next week. However I'm not patient enough to wait for that, so I've compiled and installed the latest stable release from kernel.org (3.4.6). This version is rock solid. ###### nVidia Optimus This laptop has a graphics chip from nVidia in addition to the integrated chip in the Intel Core i5 CPU. nVidia Optimus technology is used on Windows to switch this chip on and off automatically to save battery. nVidia has not supported this functionality on Linux, something everybody should know by now after a Finnish student asked Linus Torvalds about his views on nVidias lack of support to open source developers for this crucial functionality which doesn't exist in their binary drivers, and he replied that nVidia was the singel worst company they had ever worked with. Se Phoronix article and nVidias response. The BIOS has a choice between integrated graphics and nVidia Optimus. Since I don't want the nVidia chip to be switched on and draw battery, and because of the lack of Linux support I disabled the nVidia Optimus chip in BIOS before installing Ubuntu. There is of course the Bumblebee project: http:\/\/bumblebee-project.org\/. ###### Output from lscpi tom@edge:~$ lspci\n00:00.0 Host bridge: Intel Corporation Ivy Bridge DRAM Controller (rev 09)\n00:02.0 VGA compatible controller: Intel Corporation Ivy Bridge Graphics Controller (rev 09)\n00:14.0 USB controller: Intel Corporation Panther Point USB xHCI Host Controller (rev 04)\n00:16.0 Communication controller: Intel Corporation Panther Point MEI Controller #1 (rev 04)\n00:1a.0 USB controller: Intel Corporation Panther Point USB Enhanced Host Controller #2 (rev 04)\n00:1b.0 Audio device: Intel Corporation Panther Point High Definition Audio Controller (rev 04)\n00:1c.0 PCI bridge: Intel Corporation Panther Point PCI Express Root Port 1 (rev c4)\n00:1c.1 PCI bridge: Intel Corporation Panther Point PCI Express Root Port 2 (rev c4)\n00:1c.3 PCI bridge: Intel Corporation Panther Point PCI Express Root Port 4 (rev c4)\n00:1c.4 PCI bridge: Intel Corporation Panther Point PCI Express Root Port 5 (rev c4)\n00:1d.0 USB controller: Intel Corporation Panther Point USB Enhanced Host Controller #1 (rev 04)\n00:1f.0 ISA bridge: Intel Corporation Panther Point LPC Controller (rev 04)\n00:1f.2 SATA controller: Intel Corporation Panther Point 6 port SATA Controller [AHCI mode] (rev 04)\n00:1f.3 SMBus: Intel Corporation Panther Point SMBus Controller (rev 04)\n02:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. Device 5229 (rev 01)\n03:00.0 Network controller: Intel Corporation Centrino Wireless-N 2230 (rev c4)\n04:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111\/8168B PCI Express Gigabit Ethernet controller (rev 07)\n\n\nRelevant man pages are xinput, evdev and synaptics. I used xinput --list to find the names of the trackpad and trackpoint:\n\ntom@edge:~$xinput --list \u23a1 Virtual core pointer id=2 [master pointer (3)] \u239c \u21b3 Virtual core XTEST pointer id=4 [slave pointer (2)] \u239c \u21b3 SynPS\/2 Synaptics TouchPad id=11 [slave pointer (2)] \u239c \u21b3 TPPS\/2 IBM TrackPoint id=12 [slave pointer (2)] \u23a3 Virtual core keyboard id=3 [master keyboard (2)] \u21b3 Virtual core XTEST keyboard id=5 [slave keyboard (3)] \u21b3 Power Button id=6 [slave keyboard (3)] \u21b3 Video Bus id=7 [slave keyboard (3)] \u21b3 Power Button id=8 [slave keyboard (3)] \u21b3 Integrated Camera id=9 [slave keyboard (3)] \u21b3 AT Translated Set 2 keyboard id=10 [slave keyboard (3)] \u21b3 ThinkPad Extra Buttons id=13 [slave keyboard (3)] As can be seen, the relevant names are SynPS\/2 Synaptics TouchPad and TPPS\/2 IBM TrackPoint To turn the trackpad off, issue this command: xinput --set-prop --type=int \"SynPS\/2 Synaptics TouchPad\" \"Synaptics Off\" 1 By default the middle button of the trackpoint has a wheel emulation timeout of 200 milliseconds. This means that the driver issues a click event if the time between press and release is less than this value. If the time is larger, it issues scrolling events when used in combination with the trackpoint. This is really nice, but 200ms is too short for me, and I find it really difficult to issue clicks with this timeout. To increase this, issue the following command (on a single line): xinput --set-prop --type=int \"TPPS\/2 IBM TrackPoint\" \"Evdev Wheel Emulation Timeout\" 300 This sets the timeout to 300ms. Of course whenever the X server is restarted these commands must be issued again. One way to do this is to put these commands in ~\/.xsessionrc. Se man Xsession. Another thing you may want to change is the sensitivity and speed of the trackpoint. In order to do this you need to change the contents of two files in the \/sys virtual file system: \/sys\/devices\/platform\/i8042\/serio1\/serio2\/sensitivity \/sys\/devices\/platform\/i8042\/serio1\/serio2\/speed This can be done as root this way: echo -n 200 >\/sys\/devices\/platform\/i8042\/serio1\/serio2\/sensitivity echo -n 150 >\/sys\/devices\/platform\/i8042\/serio1\/serio2\/speed You may experiment with different numbers between 0 and 255. If you reboot you need to issue these commands again. I used to put these commands in \/etc\/rc.local, but this doesn't work any more. I've also written udev rules, but that didn't work either. What worked was to put these commands in an upstart script. I put the following upstart file in \/etc\/init: description \"Trackpoint settings.\" env TPDIR=\/sys\/devices\/platform\/i8042\/serio1\/serio2 start on virtual-filesystems task script while [ ! -f$TPDIR\/sensitivity ]; do\nsleep 1\ndone\necho -n 200 > $TPDIR\/sensitivity echo -n 150 >$TPDIR\/speed\nend script\n\n\n###### Update 2012-07-25\n\nI can confirm that with 3.4.6, the instabilities of the stock Ubuntu kernel are gone. A patched 3.2 kernel is released from Ubuntu, but I haven't tested it. However, if you use nfs, you are likely to hit this bug in the nfs idmap code. A temporary solution is to use the new idmap code as described in the mailing list. Also, if you use Compiz (which is the default on Ubuntu) and suspend, you are likely to run across this bug. The temporary solution is to choose Unity 2D from the login manager. But then you will likely experience tearing while watching videos. If you still want to use Compiz, it is a good idea to save any open files before going to suspend.","date":"2020-10-20 14:48:00","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\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3035162687301636, \"perplexity\": 12334.0055542424}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107872746.20\/warc\/CC-MAIN-20201020134010-20201020164010-00187.warc.gz\"}"}	null	null
\subsection{1. Specific features of equations descriptive of physical processes} Specific features of differential equations descriptive of physical processes and the types of their solutions can be demonstrated by the example of first-order partial differential equation using the properties of skew-symmetric differential forms. {\footnotesize [The method of investigating differential equations using skew-symmetric differential forms was developed by Cartan [1] in his analysis of the integrability of differential equations. Here we present this analysis to demonstrate specific features of differential equations and properties of solutions to these equations.]} Let $$ F(x^i,\,u,\,p_i)=0,\quad p_i\,=\,\partial u/\partial x^i \eqno(1)$$ be a first-order partial differential equation. Let us consider the functional relation $$ du\,=\,\theta\eqno(2)$$ where $\theta\,=\,p_i\,dx^i$ (the summation over repeated indices is implied). Here $\theta\,=\,p_i\,dx^i$ is a differential form of the first degree. The specific feature of functional relation (2) is that in the general case, when differential equation (1) describes any physical processes, this relation turns out to be nonidentical one. The left-hand side of this relation involves a differential, and the right-hand side includes the differential form $\theta\,=\,p_i\,dx^i$. For this relation be identical, the differential form $\theta\,=\,p_i\,dx^i$ must also be a differential (like the left-hand side of relation (2)), that is, it has to be a closed exterior differential form. To do this, it requires the commutator $K_{ij}=\partial p_j/\partial x^i-\partial p_i/\partial x^j$ of the differential form $\theta $ has to vanish. In the general case from equation (1) it does not follow (explicitly) that the derivatives $p_i\,=\,\partial u/\partial x^i $, which obey to the equation (and given boundary or initial conditions of the problem), make up a differential. For equations descriptive of any processes (without any supplementary conditions), the commutator $K_{ij}$ of the differential form $\theta $ is not equal to zero. The form $\theta\,=\,p_i\,dx^i$ turns out to be unclosed and is not a differential like the left-hand side of relation (2). Functional relation (2) appears to be nonidentical. The nonidentity of functional relation (2) points to a fact that without additional conditions the derivatives of original equation do not make up a differential. This means that the corresponding solution $u$ of the differential equation will not be a function of only variables $x^i$. The solution will depend on the commutator of the form $\theta $, that is, it will be a functional. To obtain a solution that is a function (i.e., the derivatives of this solution make up a differential), it is necessary to add the closure condition for the form $\theta\,=\,p_idx^i$ and for relevant dual form (in the present case the functional $F$ plays a role of a form dual to $\theta $) [1]: $$\cases {dF(x^i,\,u,\,p_i)\,=\,0\cr d(p_i\,dx^i)\,=\,0\cr}\eqno(3)$$ If we expand the differentials, we get a set of homogeneous equations with respect to $dx^i$ and $dp_i$ (in the $2n$-dimensional tangent space): $$\cases {\displaystyle \left ({{\partial F}\over {\partial x^i}}\,+\, {{\partial F}\over {\partial u}}\,p_i\right )\,dx^i\,+\, {{\partial F}\over {\partial p_i}}\,dp_i \,=\,0\cr dp_i\,dx^i\,-\,dx^i\,dp_i\,=\,0\cr} \eqno(4)$$ It is well-known that {\it vanishing the determinant} composed of coefficients at $dx^i$, $dp_i$ is a solvability condition of the system of homogeneous differential equations. This leads to relations: $$ {{dx^i}\over {\partial F/\partial p_i}}\,=\,{{-dp_i}\over {\partial F/\partial x^i+p_i\partial F/\partial u}} \eqno (5) $$ Relations (5) specify the integrating direction. namely, a pseudostructure, on which the form $\theta \,=\,p_i\,dx^i$ turns out to be closed one, i.e. it becomes a differential, and from relation (2) the identical relation is produced. One the pseudostructure, which is defined by relation(5), the derivatives of differential equation (1) constitute a differential $\delta u\,=\,p_idx^i\,=\,du$ (on the pseudostructure), and the means that the solution of equation (1) becomes a function. Solutions, namely, functions on the pseudostructures formed by the integrating directions, are the so-called generalized solutions. {\footnotesize [If we find the characteristics of equation (1), it appears that relations (5) are characteristic relations [2]. That is, the characteristics are examples of the pseudostructures on which the derivatives of the differential equation made up closed forms and the solutions prove to be functions (generalized solutions).]} If the requirements of closure of skew-symmetric form made up by the derivatives of differential equation and relevant dual form are not fulfilled, that is, the derivatives do not form a differential, the solution corresponding to such derivatives will depend on the differential form commutator formatted by derivatives. That means that the solution is a functional rather than a function. The first-order partial differential equation has been analyzed, and the functional relation with the form of the first degree has been considered. Similar functional properties have the solutions to all differential equations describing physical processes. And, if the order of the differential equation is $k$, the functional relation with the $k$-degree form corresponds to this equation. \bigskip Thus one can see that the differential equations describing any physical fields can have solutions of two types, namely, generalized solutions which depend on variables only, and the solutions which are functionals since they depend on the commutator made up by mixed derivatives. A specific feature of generalized solutions consists in the fact that they can be realized only under {\it degenerate transformations}. The relations (5) corresponding to generalized solutions had been obtained the condition of {\it vanishing the determinant} composed of coefficients at $dx^i$, $dp_i$ in the set of equations (4). This is a condition of {\it degenerate} transformation. (They are connected with symmetries of commutators of skew-symmetric forms.) {\footnotesize [It is clear that the degenerate transformation is a transition from tangent space to cotangent space (the Legendre transformations). The coordinates in relations (5) are not identical to the independent coordinates of the initial space on which equation (1) is defined.]} Since generalized solutions are possible only under realization of the conditions of degenerate transforms, they are discrete solutions (defined only on pseudostructures) and have discontinuities in the direction normal to pseudostructures. The solutions being functionals disclose the another peculiarity of the solutions of differential equations, namely, their instability. The dependence of the solution on the commutator may lead to instability. The instability develops when the integrability conditions are not realized and exact (generalized) solutions are not formatted. (Thus, the solutions to the equations of the elliptic type may be unstable.) {\footnotesize [One can see that the qualitative theory of differential equations that solves the problem of unstable solutions and integrability bases on the properties nonidentical functional relation.]} \subsection{2. Peculiarities of differential equations of mechanics and physics of material media } In analysis of partial differential equations the conjugacy of derivatives in different directions was studied (using the nonidentical functional relation). Under description of physical processes in material (continuous) media one obtains not one differential equation but a set of differential equations. And in this case it is necessary to investigate the conjugacy of not only derivatives in different directions but also the conjugacy (consistency) of the equations of this set. In this case from this set of equations one also obtains nonidentical relation that allows to study the conjugacy of equations and features of their solutions. {\footnotesize [The material (continuous) medium - material system - is a variety (infinite) of elements that have internal structure and interact among themselves. Thermodynamical, gasodynamical and cosmologic system, systems of elementary particles and others are examples of material system. (Physical vacuum can be considered as an analog of such material system.) Electrons, protons, neutrons, atoms, fluid particles and so on are examples of elements of material system.]} Equations of mechanics and physics of continuous media are equations that describe the conservation laws for energy, linear momentum, angular momentum and mass. Such conservation laws can be named as balance ones since they establish the balance between the variation of a physical quantity and corresponding external action. The equations of balance conservation laws are differential (or integral) equations that describe a variation of functions corresponding to physical quantities [3-5]. (The Navier-Stokes equations are an example [5].) (Mechanics and physics of continuous media treat the same equations. However an approach to solving these equations in mechanics and physics are different. Below it will be shown in what this difference manifest themselves.) It appears that, even without a knowledge of the concrete form of these equations, one can see specific features of these equations and their solutions using skew-symmetric differential forms. To do so it is necessary to study the conjugacy (consistency) of these equations. The functions for equations of material media sought are usually functions which relate to such physical quantities like a particle velocity (of elements), temperature or energy, pressure and density. Since these functions relate to one material system, it has to exist a connection between them. This connection is described by the state-function. Below it will be shown that the analysis of integrability and consistency of equations of balance conservation laws for material media reduces to a study the nonidentical relation for the state-function. Let us analyze the equations that describe the balance conservation laws for energy and linear momentum. We introduce two frames of reference: the first is an inertial one (this frame of reference is not connected with the material system), and the second is an accompanying one (this system is connected with the manifold built by the trajectories of the material system elements). The energy equation in the inertial frame of reference can be reduced to the form: $$ \frac{D\psi}{Dt}=A_1 $$ where $D/Dt$ is the total derivative with respect to time, $\psi $ is the functional of the state that specifies the material system, $A_1$ is the quantity that depends on specific features of the system and on external energy actions onto the system. {\footnotesize \{The action functional, entropy, wave function can be regarded as examples of the functional $\psi $. Thus, the equation for energy presented in terms of the action functional $S$ has a similar form: $DS/Dt\,=\,L$, where $\psi \,=\,S$, $A_1\,=\,L$ is the Lagrange function. In mechanics of continuous media the equation for energy of an ideal gas can be presented in the form [5]: $Ds/Dt\,=\,0$, where $s$ is entropy.\}} In the accompanying frame of reference the total derivative with respect to time is transformed into the derivative along the trajectory. Equation of energy is now written in the form $$ {{\partial \psi }\over {\partial \xi ^1}}\,=\,A_1 \eqno(6) $$ Here $\xi^1$ is the coordinate along the trajectory. In a similar manner, in the accompanying reference system the equation for linear momentum appears to be reduced to the equation of the form $$ {{\partial \psi}\over {\partial \xi^{\nu }}}\,=\,A_{\nu },\quad \nu \,=\,2,\,...\eqno(7) $$ where $\xi ^{\nu }$ are the coordinates in the direction normal to the trajectory, $A_{\nu }$ are the quantities that depend on the specific features of material system and on external force actions. Eqs. (6) and (7) can be convoluted into the relation $$ d\psi\,=\,A_{\mu }\,d\xi ^{\mu },\quad (\mu\,=\,1,\,\nu )\eqno(8) $$ where $d\psi $ is the differential expression $d\psi\,=\,(\partial \psi /\partial \xi ^{\mu })d\xi ^{\mu }$. Relation (8) can be written as $$ d\psi \,=\,\omega \eqno(9) $$ here $\omega \,=\,A_{\mu }\,d\xi ^{\mu }$ is the skew-symmetrical differential form of the first degree. Relation (9) has been obtained from the equation of the balance conservation laws for energy and linear momentum. In this relation the form $\omega $ is that of the first degree. If the equations of the balance conservation laws for angular momentum be added to the equations for energy and linear momentum, this form will be a form of the second degree. And in combination with the equation of the balance conservation law for mass this form will be a form of degree 3. In general case the evolutionary relation can be written as $$ d\psi \,=\,\omega^p \eqno(10) $$ where the form degree $p$ takes the values $p\,=\,0,1,2,3$. {\footnotesize (The relation for $p\,=\,0$ is an analog to that in the differential forms, and it was obtained from the interaction of energy and time.)} Since the balance conservation laws are evolutionary ones, the relations obtained are also evolutionary relations, and the skew-symmetric forms $\omega $ and $\omega^p $ are evolutionary ones. Relations obtained from the equation of the balance conservation laws, as well as functional relation (2), turn out to be nonidentical. To justify this we shall analyze relation (9). This relation proves to be nonidentical since the left-hand side of the relation is a differential, which is a closed skew-symmetric form, but the right-hand side of the relation involves the skew-symmetric differential form $\omega$, which is unclosed form. The commutator made up by the derivatives of coefficients $A_{\mu }$ the form $\omega $ itself is also nonzero, since the coefficients $A_{\mu }$ are of different nature, that is, some coefficients have been obtained from the energy equation and depend on the energetic actions, whereas the others have been obtained from the equation for linear momentum and depend on the force actions. In a similar manner one can prove the nonidentity of relation (10). Nonidentity of the evolutionary relation, as well as nonidentity on functional relation (2), means that initial equations of balance conservation laws are not conjugated, and hence they are not integrable. The solutions of these equations can be functional or generalized ones. In this case generalized solutions are obtained only under degenerated transformations. A type of solutions of the balance conservation law equations is essential to mechanics and physics of continuous media. And in physics and mechanics the interest is expressed in different types of solutions. In physics the interest is expressed in only generalized solutions that are invariant ones, and noninvariant solutions are ignored (even they have a physical meaning). In contrast to this, in mechanics of continuous media, where typically the equations are solved numerically, one searches for solutions that are functionals. In this case the question of searching for invariant solutions that are realized only under additional conditions is not posed. Such limited approach to solving the equations of material media has some negative points. The physical approach enables one to find possible invariant solutions, however in this approach there is no way of telling in what instant of time of evolutionary process one or another solution is obtained. This does not also discloses the causality of phenomenon described by these solutions. The approach exploited in mechanics of continuous media leads to difficulties in explaining such phenomena as origination any discrete formations (like a generation of waves or turbulent pulsations, birth of massless particles and so on), to which the invariant solutions are assigned. (The answer to the questions arisen in physics and mechanics while solving the equations describing material media can be found by analysis of the nonidentical evolutionary relation obtained from these equations.) The evolutionary relation obtained from equations of balance conservation laws for material systems (continuous media), in contrast to functional relation (2), carries not only mathematical but also large physical loading [6,7]. This is due to the fact that the evolutionary relation possesses the duality. On the one hand, this relation corresponds to material system, and on other, as it will be shown below, describes the mechanism of generating physical structures. This discloses the properties and peculiarities of the field-theory equations and their connection with the equations of balance conservation laws. \bigskip {\bf Physical significance of nonidentical evolutionary relation.} The evolutionary relation describes the evolutionary process in material system since this relation includes the state differential, which specifies the material system state. However, since this relation turns out to be not identical, from this relation one cannot get the differential $d\psi $. The absence of differential means that the system state is nonequilibrium. The evolutionary relation possesses one more peculiarity, namely, this relation is a selfvarying relation. (The evolutionary form entering into this relation is defined on the deforming manifold made up by trajectories of the material system elements. This means that the evolutionary form basis varies. In turn, this leads to variation of the evolutionary form, and the process of intervariation of the evolutionary form and the basis is repeated.) Selfvariation of the nonidentical evolutionary relation points to the fact that the nonequilibrium state of material system turns out to be selfvarying. {\footnotesize (It is evident that this selfvariation proceeds under the action of internal force whose quantity is described by the commutator of the unclosed evolutionary form $\omega^p $.)} State of material system changes but remains nonequilibrium during this process. Since the evolutionary form is unclosed, the evolutionary relation cannot be identical. This means that the nonequilibrium state of material system holds. But in this case it is possible a transition of material system to a locally equilibrium state. This follows from one more property of nonidentical evolutionary relation. Under selfvariation of the evolutionary relation it can be realized the conditions of degenerate transformation. And under degenerate transformation from the nonidentical relation it is obtained the identical relation. From identical relation one can define the state differential pointing to the equilibrium state of the system. However, such system state is realized only locally due to the fact that the state differential obtained is an interior one defined only on pseudostructure, that is specified by the conditions of degenerate transformation. And yet the total state of material system remains to be nonequilibrium because the evolutionary relation, which describes the material system state, remains nonidentical one. The conditions of degenerate transformation are connected with symmetries caused by degrees of freedom of material system. These are symmetries of the metric forms commutators of the manifold. {\footnotesize \{To the degenerate transformation it must correspond a vanishing of some functional expressions, such as Jacobians, determinants, the Poisson brackets, residues and others. Vanishing of these functional expressions is the closure condition for dual form. And it should be emphasize once more that the degenerate transformation is realized as a transition from the accompanying noninertial frame of reference to the locally inertial system. The evolutionary form and nonidentical evolutionary relation are defined in the noninertial frame of reference (deforming manifold). But the closed exterior form obtained and the identical relation are obtained with respect to the locally-inertial frame of reference (pseudostructure)\}}. Realization of the conditions of degenerate transformation is a vanishing of the commutator of manifold metric form, that is, a vanishing of the dual form commutator. And this leads to realization of pseudostructure and formatting the closed inexact form, whose closure conditions have the form $$d_\pi \omega^p=0, d_\pi{}^\omega^p=0 \eqno(12)$$ On the pseudostructure $\pi$ from evolutionary relation (10) it is obtained the relation $$ d_\pi\psi=\omega_\pi^p\eqno(13) $$ which proves to be an identical relation since the closed inexact form is a differential (interior on pseudostructure). The realization of the conditions of degenerate transformation and obtaining identical relation from nonidentical one has both mathematical and physical meaning. Firstly, this points to the fact that the solution of equations of balance conservation laws proves to be a generalized one. And secondly, from this relation one obtains the differential $d_\pi\psi $ and this points to the availability of the state-function (potential) and that the state of material system is in local equilibrium. \bigskip Relation (13) holds the duality. The left-hand side of relation (13) includes the differential, which specifies material system and whose availability points to the locally-equilibrium state of material system. And the right-hand side includes a closed inexact form, which is a characteristics of physical fields. The closure conditions (12) for exterior inexact form correspond to the conservation law, i.e. to a conservative on pseudostructure quantity, and describe a differential-geometrical structure. These are such structures (pseudostructures with conservative quantities) that are physical structures formatting physical fields[6]. The transition from nonidentical relation (10) obtained from the balance conservation laws to identical relation (13) means the following. Firstly, an emergency of the closed (on pseudostructure) inexact exterior form (right-hand side of relation (13)) points to an origination of the physical structure. And, secondly, an existence of the state differential (left-hand side of relation (13)) points to a transition of the material system from nonequilibrium state to the locally-equilibrium state. Thus one can see that the transition of material system from nonequilibrium state to locally-equilibrium state is accompanied by originating differential-geometrical structures, which are physical structures. Massless particles, charges, structures made up by eikonal surfaces and wave fronts, and so on are examples of physical structures. The duality of identical relation also explains the duality of nonidentical evolutionary relation. On the one hand, evoltionary relation describes the evolutionary process in material systems, and on the other describes the process of generating physical fields. Such duality, which establishes the connection between material systems and physical fields, discloses one more peculiarity of evolutionary processes in material media. The emergency of physical structures in the evolutionary process reveals in material system as an emergency of certain observable formations, which develop spontaneously. Such formations and their manifestations are fluctuations, turbulent pulsations, waves, vortices, and others. It appears that structures of physical fields and the formations of material systems observed are a manifestation of the same phenomena. The light is an example of such a duality. The light manifests itself in the form of a massless particle (photon) and of a wave. This duality also explains a distinction in studying the same phenomena in material systems and physical fields. As it had already noted, in the physics of continuous media (material systems) the interest is expressed in generalized solutions of equations of the balance conservation laws. These are solutions that describe the formations in material media observed. The investigation of relevant physical structures is carried out using the field-theory equations. \bigskip The unique properties of nonidentical evolutionary relation, which describes the connection between physical fields and material systems, discloses the connection of evolutionary relation with the field-theory equations. In fact, all equations of existing field theories are the analog to such relation or its differential or tensor representation. \subsection{3. Specific features of field-theory equations} The field-theory equations are equations that describe physical fields. Since physical fields are formatted by physical structures, which are described by closed exterior {\it inexact } forms and by closed dual forms (metric forms of manifold), is obvious that the field-theory equations or solutions to these equations have to be connected with closed exterior forms. Nonidentical relations for functionals like wave-function, action functional, entropy, and others, which are obtained from the equations for material media (and from which identical relations with closed forms describing physical fields are obtained), just disclose the specific features of the field-theory equations. The equations of mechanics, as well as the equations of continuous media physics, are partial differential equations for desired functions like a velocity of particles (elements), temperature, pressure and density, which correspond to physical quantities of material systems (continuous media). Such functions describe the character of varying physical quantities of material system. The functionals (and state-functions) like wave-function, action functional, entropy and others, which specify the state of material systems, and corresponding relations are used in mechanics and continuous media physics only for analysis of integrability of these equations. And in field theories such relations play a role of equations. Here it reveals the duality of these relations. In mechanics and continuous media physics these equations describe the state of material systems, whereas in field-theory they describe physical structures from which physical fields are formatted. {\footnotesize \{In differential equations of mathematical physics, which describe physical processes, the functions required are found by integrating derivatives obtained from the differential equation. And in field-theory equations the functions required follow not from derivatives, but from differentials of identical relations and they are exterior forms. That is, in mathematical physics one has to distinguish two types of differential equations, namely, the differential equations, which describe the variations of physical quantities, and the field-theory equations, which describe physical structures.\}} \bigskip It can be shown that all equations of existing field theories are in essence relations that connect skew-symmetric forms or their analogs (differential or tensor ones). And yet the nonidentical relations are treated as equations from which it can be found identical relation with include closed forms describing physical structures desired. Field equations (the equations of the Hamilton formalism) reduce to identical relation with exterior form of first degree, namely, to the Poincare invariant $$ds\,=-\,H\,dt\,+\,p_j\,dq_j\eqno(14)$$ {\footnotesize \{The field equation has the form [2] $${{\partial s}\over {\partial t}}+H \left(t,\,q_j,\,{{\partial s}\over {\partial q_j}} \right )\,=\,0,\quad {{\partial s}\over {\partial q_j}}\,=\,p_j \eqno(15)$$ here $s$ is a field function for the action functional $S\,=\,\int\,L\,dt$. Here $L$ is the Lagrangian function, $H(t,\,q_j,\,p_j)\,=\,p_j\,\dot q_j-L$ is the Hamilton function $p_j\,=\,\partial L/\partial \dot q_j$. These functions satisfy the relations: $${{dg_j}\over {dt}}\,=\,{{\partial H}\over {\partial p_j}}, \quad {{dp_j}\over {dt}}\,=\,-{{\partial H}\over {\partial g_j}}\eqno(16)$$ Relations (16), which present a set of the Hamilton equations, are the closure conditions for exterior and dual forms [7]. They are similar to relations (5).\}} The Schr\H{o}dinger equation in quantum mechanics is an analog to field equation, where the conjugated coordinates are replaced by operators. The Heisenberg equation corresponds to the closure condition of dual form of zero degree. Dirac's {\it bra-} and {\it cket}- vectors made up a closed exterior form of zero degree. It is evident that the relations with skew-symmetric differential forms of zero degree correspond to quantum mechanics. The properties of skew-symmetric differential forms of the second degree lie at the basis of the electromagnetic field equations. The Maxwell equations may be written as $d\theta^2=0$, $d^*\theta^2=0$, where $\theta^2= \frac{1}{2}F_{\mu\nu}dx^\mu dx^\nu$ (here $F_{\mu\nu}$ is the strength tensor). The Einstein equation is a relation in differential forms. This equation relates the differential of dual form of first degree (Einstein's tensor) and a closed form of second degree --the energy-momentum tensor. (It can be noted that, even Einstein's equation connects the closed forms of second degree, this equation is obtained from differential forms of third degree). The connection the field theory equations with skew-symmetric forms of appropriate degrees shows that there exists a commonness between field theories describing physical fields of different types. This can serve as an approach to constructing the unified field theory. This connection shows that it is possible to introduce a classification of physical fields according to the degree of skew-symmetric differential forms. From relations (10) and (13) one can see that relevant degree of skew-symmetric differential forms, which can serve as a parameter of unified field theory, is connected with the degree $p$ of evolutionary form in relation (10). It should be noted that the degree $p$ is connected with the number of interacting balance conservation laws. {\footnotesize \{The degree of closed forms also reflects a type of interaction [6]. Zero degree is assigned to a strong interaction, the first one does to a weak interaction, the second one does to electromagnetic interactions, and the third degree is assigned to gravitational field.\}} The connection of field-theory equations, which describe physical fields, with the equations for material media discloses the foundations of the general field theory. As an equation of general field theory it can serve the evolutionary relation (10), which is obtained the balance conservation laws for material media and has a double meaning. On the one hand, that, being a relation, specifies the type of solutions to equations of balance conservation laws and describes the state of material system (since it includes the state differential), and, from other hand, that can play a role of equations for description of physical fields (for finding the closed inexact forms, which describe the physical structures from which physical fields are made up). It is just a double meaning that discloses the connection of physical fields with material media (which is based on the conservation laws) and allows to understand on what the general field theory has to be based. \bigskip In conclusion it should be emphasized that the study of equations of mathematical physics appears to be possible due to unique properties of skew-symmetric differential forms. In this case, beside the exterior skew-symmetric differential forms, which are defined on differentiable manifolds, the skew-symmetric differential forms, which, unlike to the exterior forms, are defined on deforming (nondifferentiable) manifolds [7], were used.	{ "redpajama_set_name": "RedPajamaArXiv" }	2,149
\section{INTRODUCTION} \label{sect-intro} The properties of circumnuclear molecular gas (CMG) near an active galactic nucleus (AGN) can be influenced by the AGN itself. Some CMG observations of Seyfert galaxies with a spatial resolution around 100 to 300 pc have shown that the intensities of CO(2--1) line emission are about two times higher in temperature units than those of CO(1--0) emission \citep[e.g.,][]{mat04,hsi08}. This is different from the properties of molecular clouds in normal or star-forming galaxies, which usually show brightness temperatures in higher-J transitions similar to or lower than those in lower-J lines \citep[e.g.,][]{dev94,mau99,dum01,oka07,mat09}. From the LVG model, the high values of CO(2--1)/CO(1--0) arise when the emitting molecular gas is in a low opacity environment and the resulting molecular densities are larger than the critical densities of CO(1--0) and CO(2--1) transitions. A possible origin for such unusual behavior is that CMG near AGNs traces X-ray dominated regions due to the strong radiation \citep[e.g.,][]{mal96,koh01,use04,mei05,mei07}. Unlike far-UV photons, X-ray photons have greater penetration lengths and are more efficient in gas heating \citep{use04}. On the other hand, CMG can also be affected by mechanical processes, such as gas entrainment by jets \citep{mat07}. Star formation in galaxies is well correlated with gas surface density as embodied in the famous Schmidt-Kennicutt law \citep{sch59,ken98}. In the circumnuclear regions (CNRs) and inner structures of galaxies, the molecular gas usually has relatively higher densities and temperatures than the molecular gas in galaxy disks; in such environments, star formation is expected to be much more vigorous \citep{ken98,tan00}. \citet{kt07} suggest that the Schmidt-Kennicutt law might change slope when the averaged gas density is close to the line critical density. We note that the critical densities of CO(1--0) ($\sim10^3$~cm$^{-3}$) and CO(3--2) $\sim 3\times10^4$~cm$^{-3}$ are comparable with the densities of the CNRs of some nearby active galaxies, which are are around $300$ to $10^7$~cm$^{-3}$ \citep[e.g.,][]{mat04,sak07,per09}. It is unclear whether or how the Schmidt-Kennicutt law would vary in the CNRs of these galaxies when probed with CO(1--0) and CO(3--2) emission. Among nearby galaxies, NGC~1068 is the best-studied prototypical Seyfert. It has a distance of 14.4~Mpc \citep[][; $1\arcsec\sim70$~pc]{tul88} and is classified as (R)SA(rs)b in the RC3 catalog \citep{vau91}, with an inner bar in the central kpc and two tightly wound spiral arms starting from the tip of the bar \citep{sco88,thr89}. The inner 2 kpc is rich in star formation \citep{tel84}, and the active star forming regions are concentrated along the spiral arms \citep[e.g.,][]{tel88,dav98,ems06}. Molecular gas is also abundant along the spiral arms \citep[e.g.,][]{mye87,pla91,kan92,hel95}, and a weak offset ridge of emission along the leading side of the bar is also seen \citep{hel95,sch00}, which is a typical molecular gas distribution in barred galaxies. The nucleus of NGC~1068 shows radio jets \citep[e.g.,][]{wil83,gal96,gal04} and an ionization cone \citep[e.g.,][]{pog88,mac94}. The nuclear optical spectrum has type 2 characteristics, but the polarized spectrum shows type 1 features, reflecting the existence of an optically thick torus or disk around the central massive black hole \citep{ant85}. Indeed, a sub-parsec scale ionized gas disk perpendicular to the radio jets has been observed \citep{gal97,gal04}, with a molecular gas (maser) torus \citep{gre96} or disk \citep{gal01} located outside of the ionized gas disk. This maser torus/disk may be surrounded by a pc-scale warm dust torus \citep{jaf04}. Outside this structure, the existence of a warped molecular gas disk is suggested by interferometric CO(2--1) observations \citep{bake98,sch00}, but warm molecular gas kinematics dominated by irregular (infalling) motions have been observed in $2.12~\micron$ 1--0 S(1) molecular hydrogen emission \citep{mul09}. Interferometric observations of molecular gas emission with angular resolutions at $0.\arcsec$5--$2\arcsec$ also indicated non-circular motions in the central $\sim$100 pc \citep{krip11}. The intensity ratios of various molecular species exhibit peculiar values, including very high HCN/CO and HCN/HCO$^{+}$ ratios \citep{jac93,tac94,koh01,use04,krip08,per09}, suggesting that the molecular gas very close to the Seyfert 2 nucleus is irradiated by strong X-ray emission \citep{use04,koh05,koh08,gar10}. The central region of NGC~1068 has been mapped in different CO tranistion lines by both single-dish and interferometric observations. Among the former, it has been observed by \citet[CO(1--0), FCRAO)]{sco83}, \citet[CO(1--0), Nobeyama]{kan89}, \citet[CO(1--0) and CO(2--1), IRAM]{pla89}, \citet[CO(1--0), FCRAO]{you95}, \citet[CO(2--1) and CO(3--2), JCMT]{pap99}, and \citet[CO(1--0), CO(2--1), CO(3--2), and CO(4--3), IRAM and JCMT]{israel09}. With interferometers, it has been observed by \citet[CO(1--0), OVRO]{pla91}, \citet[CO(1--0), NMA]{kan92}, \citet[CO(1--0), BIMA]{hel95}, \citet[CO(1--0) and CO(2--1), IRAM]{sch00}, \citet[CO(1--0), BIMA]{hel03}, and \citet[CO(2-1) and CO(3-2), SMA/PdBI]{krip11}. In this paper, we present interferometric observations of the CO(3--2) line in the central 2 kpc of NGC~1068 using the Submillimeter Array (SMA). We also show interferometric observations of the CO(1--0) transition, with $uv$ sampling similar to that of the CO(3--2) data, from the Owens Valley Radio Observatory (OVRO) Millimeter Array. We describe our observations in Sect.~\ref{sect-obs} and show the overall molecular gas distributions in Sect.~\ref{sect-res-dist}. Line ratios, molecular gas masses, and kinematics are presented and discussed in Sect.~\ref{sect-res-ratio}, \ref{sect-res-mass}, and \ref{sect-res-rot}, respectively. The relation between line ratios and star formation is discussed in Sect.~\ref{sect-res-sfr}, and a summary is in Sect.~\ref{sect-sum}. \section{OBSERVATIONS AND DATA REDUCTION} \label{sect-obs} \subsection{SMA CO(3--2) Observations} \label{sect-obs-sma} We used the SMA to acquire three CO(3--2) datasets on August 13, 15, and 23, 2005. The zenith opacity at 225 GHz was about 0.07 in August 13 and 15 and 0.06 in August 23. Six out of eight 6-m antennas were used in the compact configuration. The receivers were tuned to the redshifted CO(3--2) line ($344.493$~GHz). Correlators were set to cover a velocity range of $\sim1700$~km~s$^{-1}$ ($\sim2$~GHz bandwidth) and configured to have a velocity resolution of $\sim0.7$~km~s$^{-1}$ ($=0.8125$~MHz frequency resolution). The SMA antenna primary beam has a half power beam width (HPBW) of $\sim36\arcsec$ ($\sim 2.5$~kpc in NGC~1068) at $345$~GHz. A 7-pointing mosaic (pointings separated by $18\arcsec$ and hexagonal in shape) was observed in order to image the inner $72\arcsec$ (5 kpc) of NGC~1068. The phase center was set at R.A.$ =2h42m40.798s $ and decl.$=-0d00m47.938s$ (J2000), corresponding to the AGN position \citep{mux96}. We performed the data reduction following the standard processes outlined in the SMA cookbook\footnote{\url{http://cfa-www.harvard.edu/~cqi/mircook.html}}. The visibilities were first calibrated with the IDL-based MIR package \citep{sco93} as modified for the SMA. We used $0423-013$ as the amplitude and phase calibrator to track phase and gain variations and used 3C~454.3 for bandpass and flux calibrations. We calibrated our observations by linearly interpolating the flux densities of the quasar 3C~454.3, which were $26.48\pm1.33$~Jy on August 12 and $18.90\pm0.95$~Jy on August 25, according to the SMA calibrator list\footnote{\url{http://sma1.sma.hawaii.edu/callist/callist.html}}. The expected flux densities of~3C 454.3 were about $25\pm 1.3$ (August 13-15), and $20\pm 1.0$ (August 23) during the observations. Mars and Uranus were also observed during the observations. However, Mars was resolved by the SMA observations, and Uranus was only observed on August 13 and 15. After performing the flux calibration with 3C~454.3, we compared the fluxes of Uranus and Mars to check the accuracy of our flux calibration. The uncertainty in our flux scale is estimated to be $\sim 10\%$. We combined the visibilities from all the mosaic observations and applied the mapping task INVERT in MIRIAD to produce ``dirty'' images with a velocity resolution of 10~km~s$^{-1}$. Primary beam correction was taken into account in the mosaic mode of the INVERT process. We used natural weighting in the mapping process in order to have the best sensitivity. We deconvolved the dirty images with the MOSSDI and MOSMEM packages in MIRIAD, and produced moment maps and spectra for further analysis. The resulting spatial resolution was $2\farcs29\times2\farcs00$ ($\sim160\times140~$pc$^2$) with a position angle of $-88\fdg8$. \subsection{OVRO CO(1--0) Observations} \label{sect-obs-ovro} We observed the CO(1--0) transition in NGC\,1068 (adopted pointing center: $\alpha_{\rm J2000} = $ 02:42:40.7 and $\delta_{\rm J2000} = $ $-$00:00:47.7) between April and September 1995 using the OVRO millimeter array \citep{padi91,scot93}. The array comprised six 10.4\,m antennas, which during our observations were deployed in three configurations providing a total of 40 distinct baselines. We configured the array's digital correlator \citep{padi93} to provide 112 contiguous frequency channels, each Hanning-smoothed to $4\,{\rm MHz}$ ($10.4\,{\rm km\,s^{-1}}$) resolution. We calibrated the data within the OVRO millimeter array database using the MMA package \citep{sco93}. Paired integrations on J0339$-$017, interleaved with observations of NGC\,1068 every 30-40 minutes, were used to correct for phase and gain variations; for passband calibration, we used 3C273, 3C454.3, and/or 0528+134. Our flux scale was determined by comparing J0339$-$017 with Uranus or Neptune in light of standard brightness temperature models for the latter \citep{muhl91,orto86}, or by bootstrapping from archived observations of the planets (and bright, frequently-observed quasars) obtained with similar elevations and coherences. From repeatibility of flux measurements on 1--2 week timescales, we estimate that the uncertainty in our flux scale is $\sim 10\%$. After editing for quality in the Difmap package \citep{shep97}, we were left with 11.4 hours of on-source data. To eliminate contamination by 3\,mm continuum emission associated with NGC\,1068's jet, we subtracted a $uv$-plane model based on the line-free channels at the ends of the recorded bandwidth. The line-free $uv$ data were then mapped using the IMAGR task in the NRAO AIPS package, with moderately robust weighting giving a synthesized beam of $3\farcs46 \times 2\farcs56$ (242\,pc $\times$ 179\,pc) at PA $67\fdg5$. For deconvolution, we adopted a single clean box for all channels that enclosed all of the emission in the zeroth moment map. A slightly different reduction of these data was first discussed by \citet{bake98}. Before analyzing moment maps and spectra, we applied a primary beam correction appropriate for the $\sim 56^{\prime\prime}$ OVRO HPBW. \section{CO Distribution} \label{sect-res-dist} In Figure~\ref{fig1}, we show the integrated intensity and velocity maps of the CO(3--2) line for the central region of NGC~1068. CO(3--2) emission appears in the nucleus and along the two spiral arms. The CO(3--2) distribution in the nucleus is elongated in an east-west direction, and the strongest peak in our data is located $\sim1\arcsec$ east of the nucleus. The distribution of the CO(3--2) emission is roughly consistent with that of previous CO(1--0) observations \citep[e.g.,][]{hel95,pla91,kan92,sch00} except that the nucleus is relatively brighter in the CO(3--2) image. The CO(3--2) intensity distributions in the two spiral arms are different. The southern arm is brighter than the northern arm. The second strongest peak in our map is located $\sim12\arcsec$ south of the nucleus, which is on the southern arm. The locations of these two strongest peaks in the map can explain the position of the peak intensity in an early CO(3--2) single-dish image \citep{pap99}, which showed a peak intensity offset south of the nucleus. The missing flux in the central $14\arcsec$ region is about 20\% when comparing our image with a JCMT observation of the central region of NGC~1068 \citep{israel09}. On the other hand, if we compare our data with an area-averaged CO(3-2) spectrum over a circular area of $30\arcsec$ radius \citep{pap99}, we find that our map only recovers $\sim50\%$ of the total flux. The missing flux might seem surprisingly large; however, \citet{pap99} have showed that the total CO(3--2) flux is actually dominated by a warm diffuse gas phase, which is highly excited and not virialized. Besides, similar extended structures of CO(3--2) emission are also observed in the central regions of other nearby AGNs, such as M51 \citep{mat04}. Most of this diffuse gas is likely resolved out and not detectable in our interferometer map. \begin{figure} \includegraphics[angle=-90,width=.5\textwidth]{fig1a.eps} \includegraphics[angle=-90,width=.5\textwidth]{fig1b.eps} \caption{$Left$: CO(3--2) integrated intensity (moment 0) map of the central region of NGC~1068. The grayscale range is shown in the wedge at right in units of Jy~beam$^{-1}$~km~s$^{-1}$. The contour levels are 10, 20, 30, 40, 50, 70, 90, 110, and 130 $\times\sigma$, where $1\sigma= 4.0$~Jy~beam$^{-1}$~km~s$^{-1}$. The synthesized beam is $2\farcs29\times2\farcs00$ ($\sim160\times140\rm~pc^2$) with a P.A.\ of $-88\fdg0$, which is shown in the bottom left corner. The cross indicates the galactic center determined from the 5~GHz and 22~GHz radio continuum data \citep{mux96}. $Right$: CO(3--2) intensity-weighted velocity field (moment 1) map. The grayscale range is shown in the wedge at right from -180 to 180 km~s$^{-1}$. The contour levels range from $-150$ to $+150$~km~s$^{-1}$ with a 30~km~s$^{-1}$ interval. \label{fig1}} \end{figure} \begin{figure} \includegraphics[angle=-90,width=.5\textwidth]{fig2a.eps} \includegraphics[angle=-90,width=.5\textwidth]{fig2b.eps} \caption{$Left$: CO(1--0) integrated intensity (moment 0) map of the central region of NGC~1068. The grayscale range is shown in the wedge at right in units of Jy~beam$^{-1}$~km~s$^{-1}$. The contour levels are multiplies of $5\sigma$, where $1\sigma= 0.81$~Jy~beam$^{-1}$~km~s$^{-1}$. The synthesized beam is $3\farcs46\times2\farcs56$ with a P.A.\ of $67\fdg5$, which is shown in the bottom left corner. The cross is the same as in Fig.~\ref{fig2}. $Right$: CO(1--0) intensity weighted velocity field (moment 1) map of the central region of NGC~1068. The grayscale range is shown in the wedge at the right, from -180 to 180 km~s$^{-1}$. The contour levels range from $-150$ to $+150$~km~s$^{-1}$ with a 30~km~s$^{-1}$ interval. \label{fig2}} \end{figure} The CO(1--0) integrated intensity and velocity maps are shown in Figure~\ref{fig2}. As in the CO(3--2) map, CO(1--0) emission is located in the nucleus and along the two spiral arms, but the strongest peak is located at the southern arm, and the peak intensity at the core region is relatively weak. Our CO(1--0) image is in general agreement with previous interferometric CO(1--0) observations \citep{pla91, kan92, sch00, hel03}. However, the BIMA image seems to show more emission at interarm regions. The difference between our results and the BIMA image might be due to the fact that the BIMA image have included modelled visibilities derived from single-dish NRAO observations and might contain large-scale emission that would be resolved out by our interferometric observations. The corresponding missing flux of CO(1--0) in the core region is about 20\% comparing with the BIMA results of \citet{hel03}. For the spiral regions, the situation is more complicated. We have estimated the missing fluxes for several selected spiral regions (see Figure 3) and obtained diverse results. For example, the missing flux for R4 is about 70\% and for R17 is about 38\%; however, we detect more flux than BIMA for R15, so there should be no missing flux for R15. We note that R4 is an inter-arm region, which is dominated by diffuse emission, while R15 is around a compact structure peak and R17 is at the boundary of an arm structure. These different results might be caused by different $uv$ coverage of the BIMA and our observations. The overall CO(1--0) velocity field is similar to that of CO(3--2). Both have a rotation axis with P.A. about $-30\degr$ and a rotation velocity of 120~km~s$^{-1}$ in the nucleus. In the spiral arms, both kinematic major axes run from east to west with a rotation velocity of 180~km~s$^{-1}$. \section{Discussion} \label{sect-dis} \subsection{CO Line Ratios} \label{sect-res-ratio} Before deriving line ratios at various regions, we first matched the $uv$ range between our CO(3--2) and CO(1--0) datasets. In Figure~\ref{fig3}, we overlay the CO(3--2) (solid contours) and CO(1--0) (dashed contours) intensity distributions. In this image, the shortest $uv$ length is set to 16~k$\lambda$, and the image resolutions are convolved into the same resolution ($3\farcs46\times2\farcs56$ with a P.A.\ of $67\fdg5$). As mentioned above, the overall distributions of both CO(3--2) and CO(1--0) are very similar, but some of the intensity peaks have shifted positions with respect to each other. The most obvious example is in the southern spiral arm, about $12\arcsec-14\arcsec$ south of the nucleus, where the CO(3--2) peak is located in the inner part of the spiral arm, but the CO(1--0) peak is shifted toward the outer edge of the spiral arm. \begin{figure} \includegraphics[angle=-90,width=.9\textwidth]{fig3.eps} \caption{ Integrated intensity maps of CO(3--2) (solid contours) and CO(1--0) (dashed contours), overlaid on the continuum-subtracted \emph{HST} F658N image (greyscale) of the central region of NGC~1068. Solid contour levels for CO(3--2) are 10, 30, 50, 70, 90, 110, and 130 $\times$ 5.3~Jy~beam$^{-1}$~km~s$^{-1}$, and dashed contour levels for CO(1--0) are 2, 4, 6, 8 $\times$ 5~Jy~beam$^{-1}$~km~s$^{-1}$. The CO(3--2) and CO(1--0) data are matched to the same $uv$ range and have the same synthesized beam size of $3\farcs46\times2\farcs56$ ($\sim242\times179\rm~pc^2$) with a P.A.\ of $67\fdg5$, which is shown in the bottom-left corner of the image. Cross is the same as in Figure~\ref{fig1}. We also plot 25 boxes (C1 and R1 -- R24) that are used to calculate the line ratios. \label{fig3}} \end{figure} We divide the central region of NGC~1068 into 25 regions covering the nucleus and spiral arms, as shown in Figure~\ref{fig3} (C1 and R1 -- R24). The size of each region is $4\arcsec\times5\arcsec$ except C1, which is $6\arcsec\times5\arcsec$. The intensity scale is then converted to a brightness temperature scale, and the CO(3--2)/CO(1--0) brightness temperature ratios, $R_{31}$, are derived for each region using the MIRIAD task MATHS. Area-averaged spectral peak and integrated brightness temperatures and $R_{31}$ for each region are shown in Table~\ref{tab-ratio}. The central core region C1 exhibits a very high integrated intensity ratio with $R_{31}=3.12\pm0.03$ and a spectra peak ratio of $2.83\pm0.10$. Figure~\ref{fig4} shows the spectra of C1. The CO(3--2)/CO(1--0) line ratio is slightly smaller than the results of \citet{krip11}, which are about 4--6. However, the observations of \citet{krip11} have a much higher angular resolution and might contain relatively more contribution from the circumnuclear region of the AGN, while our result is an average value over a much larger area and are likely to include emission from outside regions. In fact, if we only consider the line ratio within the brightest beam at the center, we would obtain a line ratio of $\sim4.6$, which is similar to the results of \citet{krip11}. On the other hand, the spiral arm regions R1 -- R24 have a wide range of integrated intensity ratios of 0.24 -- 2.34 and an average value of 0.75 with standard deviation 0.47. We note that the wide range of the intensity ratios might be caused by the different spatial distributions of the CO(1--0) and CO(3--2) emission due to varying physical conditions of the molecular gas. There is no obvious ratio difference between the northern and southern arms; the average ratio of the northern arm regions (R1 -- R4, R18 -- R24) is 0.8 and that of the southern arm regions (R5 -- R17) is 0.7. However, there is a difference between inner arm regions and outer arm regions; the average ratio of the inner arm regions (R1 -- R4, R9 -- R17, and R19 -- R22) is $0.88\pm0.14$, while the outer arm regions (R5 -- R8, R18, R23, and R24) show $0.42\pm0.05$. This difference indicates that there is a large-scale gradual decrease of the line intensity ratio from the inner radii to the outer radii. The radial change of molecular line ratios, and therefore of the physical conditions of the molecular gas, has also been detected in the Milky Way and other galaxies \citep{tur93,aal94,sak94,sak97,pet00,pag01,mat10}, suggesting that this trend is common. \begin{figure} \includegraphics[angle=0,width=\textwidth]{fig4.eps} \caption{Area-averaged spectra within the C1 region. The solid line shows the spectrum of CO(3--2), and the dash line shows the spectrum of CO(1--0). The uncertainties per channel are about 0.01~K for the CO(3--2) spectrum and 0.02~K for the CO(1--0) one. \label{fig4}} \end{figure} The line ratios $R_{31}$ in the spiral arm regions are similar to those observed in the centers of nearby normal and starburst galaxies. \citet{dev94} observed seven nearby starburst galaxies and found that the ratios are in the range of 0.5 -- 1.4 with an average ratio of $0.64\pm0.06$. A survey toward the centers of 28 nearby star-forming galaxies showed that most of their $R_{31}$ are within the range of 0.2 -- 0.7 \citep{mau99}. On the other hand, the value of $R_{31}$ in the nucleus is much larger than those observed in the nuclei of the nearby galaxies mentioned above, albeit similar to that observed in the nucleus of the Seyfert 2 galaxy M51. The inner $uv$-truncated $R_{31}$ of the central molecular core of M51 also has a very high value of $\sim5.2\pm1.7$ within a beam size of $3\farcs9\times2\farcs6$ or 160~pc $\times$ 110~pc \citep[missing flux corrected $R_{31}$ is $\geq 1.9\pm0.2$;][]{mat04}. If we only correct the 20\% missing flux of the CO(1--0) emission in the core region of NGC~1068 and ignore the missing flux of the CO(3--2) following the method of \citet{mat04}, the $R_{31}$ ratio of the core region will become 2.50 instead of 3.12. This suggests that the physical conditions of molecular gas around the Seyfert 2 AGN in NGC~1068 are very different from those in the centers of nearby star-forming and starburst galaxies, but might be similar to those around the Seyfert 2 nucleus of M51. \begin{deluxetable}{cccccccccc} \rotate \tabletypesize{\scriptsize} \tablecaption{Peak and integrated brightness temperature of CO(3--2) and CO(1--0), and the CO(3--2)/CO(1--0) line ratios, in different areas. \label{tab-ratio}} \tablehead{ \colhead{Area} & \colhead{Peak CO(3--2)} & \colhead{Peak CO(1--0)} & \colhead{Peak $R_{31}$} & \colhead{$\int T_{\rm B}(\rm{CO}$~$J = 3 - 2)~dv$} & \colhead{$\int T_{\rm B}(\rm{CO}$~$J = 1 - 0)~dv$} & \colhead{$R_{31}$} & \colhead{X offset} & \colhead{Y offset} \\ & \colhead{[K]} & \colhead{[K]} & & \colhead{[K~km~s$^{-1}$]} & \colhead{[K~km~s$^{-1}$]} & & \colhead{[arcsec]} & \colhead{[arcsec]} } \startdata C1 & $1.73\pm0.01$ & $0.61\pm0.03$ & $2.83\pm0.10$ & $ 438.5\pm 0.7$ & $ 140.6\pm 1.5$ & $ 3.12\pm0.03$ & -1 & 0\\ R1 & $1.80\pm0.03$ & $1.55\pm0.06$ & $1.16\pm0.05$ & $ 103.76\pm 0.97$ & $ 44.3\pm 1.8$ & $ 2.34\pm0.10$ & 0 & 14\\ R2 & $1.53\pm0.05$ & $1.90\pm0.08$ & $0.80\pm0.04$ & $ 104.97\pm 1.54$ & $ 133.4\pm 2.8$ & $ 0.79\pm0.02$ & 4 & 14\\ R3 & $1.29\pm0.06$ & $1.98\pm0.05$ & $0.65\pm0.03$ & $ 100.06\pm 2.07$ & $ 149.8\pm 1.8$ & $ 0.67\pm0.02$ & 8 & 12\\ R4 & $0.43\pm0.03$ & $0.73\pm0.04$ & $0.58\pm0.05$ & $ 6.26\pm 0.81$ & $ 26.6\pm 1.2$ & $ 0.24\pm0.03$ & 12 & 9\\ R5 & $0.78\pm0.04$ & $1.92\pm0.11$ & $0.41\pm0.03$ & $ 56.74\pm 1.39$ & $ 90.0\pm 3.8$ & $ 0.63\pm0.03$ & 16 & 6\\ R6 & $1.54\pm0.05$ & $3.65\pm0.15$ & $0.42\pm0.02$ & $ 78.95\pm 1.45$ & $ 131.1\pm 4.4$ & $ 0.60\pm0.02$ & 18 & 9\\ R7 & $0.71\pm0.04$ & $1.97\pm0.14$ & $0.36\pm0.03$ & $ 25.72\pm 1.23$ & $ 93.0\pm 4.4$ & $ 0.28\pm0.02$ & 18 & 14\\ R8 & $0.84\pm0.03$ & $1.80\pm0.08$ & $0.47\pm0.03$ & $ 28.85\pm 1.09$ & $ 94.4\pm 2.5$ & $ 0.31\pm0.01$ & 20 & 3\\ R9 & $1.35\pm0.03$ & $3.63\pm0.08$ & $0.37\pm0.01$ & $ 108.83\pm 1.11$ & $ 199.9\pm 2.9$ & $ 0.54\pm0.01$ & 16 & -1\\ R10 & $1.49\pm0.07$ & $2.20\pm0.08$ & $0.68\pm0.04$ & $ 62.03\pm 1.92$ & $ 111.0\pm 2.2$ & $ 0.56\pm0.02$ & 12 & -5\\ R11 & $1.49\pm0.07$ & $1.94\pm0.03$ & $0.77\pm0.04$ & $ 94.26\pm 2.63$ & $ 172.2\pm 1.1$ & $ 0.55\pm0.02$ & 8 & -8\\ R12 & $1.74\pm0.06$ & $3.02\pm0.04$ & $0.57\pm0.02$ & $ 112.76\pm 2.01$ & $ 176.3\pm 1.4$ & $ 0.64\pm0.01$ & 4 & -10\\ R13 & $1.93\pm0.09$ & $1.66\pm0.07$ & $1.16\pm0.07$ & $ 194.32\pm 3.64$ & $ 139.4\pm 2.8$ & $ 1.39\pm0.04$ & 0 & -12\\ R14 & $2.46\pm0.09$ & $3.40\pm0.03$ & $0.72\pm0.03$ & $ 252.08\pm 3.58$ & $ 192.5\pm 1.3$ & $ 1.31\pm0.02$ & -4 & -13\\ R15 & $2.68\pm0.08$ & $3.18\pm0.09$ & $0.84\pm0.04$ & $ 239.52\pm 3.24$ & $ 374.4\pm 3.4$ & $ 0.64\pm0.01$ & -8 & -11\\ R16 & $3.46\pm0.11$ & $4.80\pm0.18$ & $0.72\pm0.04$ & $ 211.39\pm 3.69$ & $ 301.1\pm 6.0$ & $ 0.70\pm0.02$ & -12 & -8\\ R17 & $1.67\pm0.01$ & $1.68\pm0.06$ & $0.99\pm0.04$ & $ 106.98\pm 0.22$ & $ 141.2\pm 2.3$ & $ 0.76\pm0.01$ & -16 & -5\\ R18 & $0.79\pm0.01$ & $2.47\pm0.07$ & $0.32\pm0.01$ & $ 56.68\pm 0.62$ & $ 159.4\pm 2.8$ & $ 0.36\pm0.01$ & -20 & 1\\ R19 & $1.28\pm0.05$ & $1.72\pm0.08$ & $0.75\pm0.04$ & $ 49.41\pm 1.41$ & $ 75.5\pm 2.4$ & $ 0.65\pm0.03$ & -16 & 6\\ R20 & $1.94\pm0.07$ & $3.19\pm0.06$ & $0.61\pm0.02$ & $ 115.44\pm 2.06$ & $ 168.9\pm 1.8$ & $ 0.68\pm0.01$ & -12 & 10\\ R21 & $1.83\pm0.05$ & $1.66\pm0.06$ & $1.10\pm0.05$ & $ 97.04\pm 1.48$ & $ 81.3\pm 1.8$ & $ 1.19\pm0.03$ & -8 & 13\\ R22 & $1.10\pm0.04$ & $0.47\pm0.04$ & $2.32\pm0.22$ & $ 62.59\pm 1.26$ & $ 46.3\pm 1.3$ & $ 1.35\pm0.05$ & -4 & 14\\ R23 & $1.02\pm0.05$ & $2.72\pm0.07$ & $0.38\pm0.02$ & $ 45.12\pm 1.70$ & $ 94.9\pm 2.1$ & $ 0.48\pm0.02$ & -24 & -4\\ R24 & $0.96\pm0.03$ & $2.95\pm0.16$ & $0.33\pm0.02$ & $ 39.06\pm 0.86$ & $ 126.5\pm 4.5$ & $ 0.31\pm0.01$ & -24 & -9\\ \enddata \tablecomments{Column (1): Region name. Column (2): Peak brightness temperature of CO(3--2). Column (3): Peak brightness temperature of CO(1--0). Column (4): Line ratio of peak brightness temperature of CO(3--2)/CO(1--0). Column (5): Integrated brightness temperature of CO(3--2). Column (6): Integrated brightness temperature of CO(1--0). Column (7): Line ratio of integrated brightness temperature of CO(3--2)/CO(1--0). The errors Column (8,9): Position offset of region center from the phase center. Errors were estimated from the statistical errors of the spectra; uncertainties of our flux calibration (about 10\% for both CO(1--0) and CO(3--2) lines) and missing fluxes are not included.} \end{deluxetable} \subsection{Molecular Gas Column Density and Mass} \label{sect-res-mass} Because CO(1--0) traces the bulk of the molecular gas, we can calculate the molecular gas column density and mass from the CO(1--0) integrated intensities. The column density of the molecular hydrogen can be estimated with a conversion factor, $X_{\rm CO}$, \begin{equation} N_{\rm H_2} = X_{\rm CO} \int T_{\rm B}({\rm CO}~J = 1 - 0)~dv ~{\rm [cm^{-2}]}, \end{equation} where $X_{\rm CO}\cong0.4\times10^{20}$~cm$^{-2}$~(K km s$^{-1}$)$^{-1}$ for circumnuclear molecular gas \citep{wil95,mau96,weis01,esp09} and $X_{\rm CO}\cong3\times10^{20}$~cm$^{-2}$~(K~km~s$^{-1}$)$^{-1}$ for spiral arm regions \citep{sol87,you91}. The column density $N_{\rm H_2}$ for each region is calculated using $\int T_{\rm B}({\rm CO}~J = 1 - 0)~dv$ shown in Table~\ref{tab-ratio}, and the derived values are shown in Table~\ref{tab-sfr}. We also derive the molecular gas mass, $M_{\rm H_2}$, which is displayed in Table~\ref{tab-sfr}. The total missing flux of CO(1--0) within the central circular area of $30\arcsec$ radius is about 35\% \citep{pap99}. However, the missing flux is dominated by extended structures, so the true missing fluxes in the core and the compact spiral structures should be smaller. For example, the missing flux of CO(1--0) in the core region is about 20\% as discussed in Sect.~\ref{sect-res-dist}. We thus expect that the errors of the column densities caused by the missing flux are less than 35\%. \begin{deluxetable}{ccccccccc} \tabletypesize{\scriptsize} \tablecaption{Molecular gas column density and mass, $8~\mu$m dust and H$\alpha$ + [N II] emission, and star formation rate surface density. \label{tab-sfr}} \tablehead{ \colhead{Area} & \colhead{$N_{\rm H_2}$} & \colhead{$M_{\rm H_2}$} & \colhead{$f_{\rm 8\micron, dust}$} & \colhead{$f_{\rm H\alpha+[N II]}$} & \colhead{$\Sigma_{\rm SFR}$} \\ & \colhead{$\times10^{21}$ [cm$^{-2}$]} & \colhead{$\times10^{6}$ [$M_\odot$]} & \colhead{[MJy~sr$^{-1}$]} & \colhead{$\times10^{4}$ [eps~pixel$^{-1}$]\tablenotemark{a}} & \colhead{[$M_\odot$~yr$^{-1}$~kpc$^{-2}$]} } \startdata C1 & $ 5.6\pm0.3$ & $ 12.3\pm0.8$ & $1.05\times10^{4}\pm6.4$ & 73.5 & $62.4\pm0.04$\\ R1 & $ 13.3\pm0.5$ & $ 19.5\pm0.8$ & $238\pm6.4$ & 1.51 & $1.41\pm0.04$\\ R2 & $ 40.0\pm0.8$ & $ 58.6\pm1.2$ & $260\pm6.4$ & $>1.87$\tablenotemark{b} & $1.54\pm0.04$\\ R3 & $ 44.9\pm0.5$ & $ 65.8\pm0.8$ & $337\pm6.4$ & $>2.57$\tablenotemark{b} & $2.00\pm0.04$\\ R4 & $ 8.0\pm0.4$ & $ 11.7\pm0.5$ & $241\pm6.4$ & 2.35 & $1.43\pm0.04$\\ R5 & $ 27.0\pm1.1$ & $ 39.5\pm1.7$ & $180\pm6.4$ & 1.08 & $1.07\pm0.04$\\ R6 & $ 39.3\pm1.3$ & $ 57.6\pm1.9$ & $166\pm6.4$ & $>0.81$\tablenotemark{b} & $0.98\pm0.04$\\ R7 & $ 27.9\pm1.3$ & $ 40.9\pm1.9$ & $107\pm6.4$ & ---\tablenotemark{b} & $0.63\pm0.04$\\ R8 & $ 28.3\pm0.8$ & $ 41.5\pm1.1$ & $ 89\pm6.4$ & $>0.72$\tablenotemark{b} & $0.52\pm0.04$\\ R9 & $ 60.0\pm0.9$ & $ 87.9\pm1.3$ & $163\pm6.4$ & $>0.65$\tablenotemark{b} & $0.97\pm0.04$\\ R10 & $ 33.3\pm0.7$ & $ 48.8\pm1.0$ & $158\pm6.4$ & 0.77 & $0.94\pm0.04$\\ R11 & $ 51.7\pm0.3$ & $ 75.7\pm0.5$ & $174\pm6.4$ & 0.89 & $1.03\pm0.04$\\ R12 & $ 52.9\pm0.4$ & $ 77.5\pm0.6$ & $226\pm6.4$ & 0.97 & $1.34\pm0.04$\\ R13 & $ 41.8\pm0.8$ & $ 61.3\pm1.2$ & $212\pm6.4$ & 0.76 & $1.26\pm0.04$\\ R14 & $ 57.8\pm0.4$ & $ 84.6\pm0.6$ & $236\pm6.4$ & 1.21 & $1.40\pm0.04$\\ R15 & $ 112.3\pm1.0$ & $164.5\pm1.5$ & $311\pm6.4$ & 2.17 & $1.84\pm0.04$\\ R16 & $ 90.3\pm1.8$ & $132.3\pm2.6$ & $381\pm6.4$ & 3.49 & $2.26\pm0.04$\\ R17 & $ 42.4\pm0.7$ & $ 62.1\pm1.0$ & $345\pm6.4$ & 2.81 & $2.05\pm0.04$\\ R18 & $ 47.8\pm0.8$ & $ 70.0\pm1.2$ & $172\pm6.4$ & 0.62 & $1.02\pm0.04$\\ R19 & $ 22.7\pm0.7$ & $ 33.2\pm1.0$ & $185\pm6.4$ & 1.15 & $1.10\pm0.04$\\ R20 & $ 50.7\pm0.6$ & $ 74.2\pm0.8$ & $207\pm6.4$ & 1.45 & $1.23\pm0.04$\\ R21 & $ 24.4\pm0.5$ & $ 35.7\pm0.8$ & $260\pm6.4$ & 1.33 & $1.54\pm0.04$\\ R22 & $ 13.9\pm0.4$ & $ 20.4\pm0.6$ & $231\pm6.4$ & 1.37 & $1.37\pm0.04$\\ R23 & $ 28.5\pm0.6$ & $ 41.7\pm0.9$ & $123\pm6.4$ & 0.27 & $0.73\pm0.04$\\ R24 & $ 38.0\pm1.4$ & $ 55.6\pm2.0$ & $105\pm6.4$ & 0.15 & $0.62\pm0.04$\\ \enddata \tablecomments{Column (1): Region names. Column (2): Column density. Column (3): Molecular gas mass. Column (4): Average intensity of dust emission. Column (5): Average intensity of $\rm H\alpha+[N II]$ emission. Column (6): Star formation rate surface density.} \tablenotetext{a}{Image units in electrons per second per pixels (eps~pixel$^{-1}$)} \tablenotetext{b}{These areas do not have complete H$\alpha$ emission information, since these regions are either located at the edge of or outside the \emph{HST} image.} \end{deluxetable} In the spiral arm regions, the box sizes correspond to $280\times350~$pc$^2$. The average values of $N_{\rm H_2}$ and $M_{\rm H_2}$ per region are $39.8\times10^{21}$~cm$^{-2}$ and $58.9\times10^6M_{\odot}$. The standard deviations of $N_{\rm H_2}$ and $M_{\rm H_2}$ are large inside the spiral arms, with a value of $24.0\times10^{21}$~cm$^{-2}$ and $34.8\times10^6M_{\odot}$, respectively, indicating that the molecular gas content within the spiral arms varies substantially from region to region. On the other hand, $N_{\rm H_2}$ and $M_{\rm H_2}$ for the central core C1 are estimated to be $5.6\times10^{21}$~cm$^{-2}$ and $12.3\times10^6$~M$_{\odot}$ (note that the area of C1 is $\sim$1.5 times larger than those in the spiral arms). This indicates that the column density and the mass of the central region are much smaller than the average values in the spiral arms. However, we note that the $R_{31}$ of C1 is very different from those of the spiral arm regions (see Sect.~\ref{sect-res-ratio}), and we have used different conversion factors in estimating the mass and column density \citep{weis01}. \citet{tac94} and \citet{stern94} showed that the $N_{\rm H_2}$ column density in the core region is about $4\times10^{22}$~ cm$^{-2}$ using the Galactic CO to $N_{\rm H_2}$ conversion factor. Our conversion factor for the core region is about $\frac{1}{6}$ of the Galactic conversion factor and we obtained a $N_{\rm H_2}$ column density of $5.6\times10^{21}$~cm$^{-2}$; in other words, the different results are mainly caused by the different conversion factors adopted. However, we note that a lower conversion factor is typically found in galaxy centers \citep{wil95,mau96,weis01,esp09}. In particular, the conversion factor in the center of NGC 1068 could be six times lower than the Galactic value \citep{use04,gar10}. In Figure~\ref{fig5}, we compare our CO(3--2) image with the X-ray image obtained with the \emph{Chandra X-ray Observatory} \citep{you01}. The X-ray image displays clear emission from the ionization cone emanating from the nucleus of NGC~1068; however, we note that there is an obvious dimmed area at the center of the image. This dimmed area matches well with the central core of our CO(3--2) image, suggesting that the obscuring material of the X-rays is closely related to the molecular gas. \begin{figure} \includegraphics[angle=-90,width=.9\textwidth]{fig5.eps} \caption{Comparison between our CO(3--2) image and the X-ray image taken with the \emph{Chandra X-ray Observatory} \citep{you01}. The contours show CO(3--2) emission, with levels 10, 20, 30, 40, 50, 70, 90, 110, and $130\times4.41$~Jy~beam$^{-1}$~km~s$^{-1}$. The gray scale shows the X-ray emission from 0 to 1000 counts pixel$^{-1}$($0.4-5.0$~keV). \label{fig5}} \end{figure} \subsection{CO(3--2) Rotation Curve} \label{sect-res-rot} The rotation curve in the nucleus of NGC~1068 is plotted in Figure~\ref{fig6}. The data points and the error bars were derived using the MIRIAD task VELFIT. We ignore points within $30\degr$ of the minor axis (P.A.~$=60\degr$), and we assume the inclination angle to be $45\degr$ following \citet{sch00}. Within the central $2\arcsec$ ($\sim140$~pc), the rotational velocity increases with radius (rigid rotation). The enclosed mass within the central $2\arcsec$ can be estimated using $M(r<2\arcsec) \sim rv^{2}/G$, and we find that the total mass within this area is $3.7\times10^{8}$~M$_{\odot}$, which is consistent with \citet{sch00}. The molecular gas mass in C1 is about $12.3\times10^6$~M$_{\odot}$, so that the gas-to-dynamical mass ratio is 3\%. This value is a factor of a few lower than those in star forming galaxies ($\sim10\%$) \citep{sak99,kod05}. For radii larger than $2\arcsec$, the rotation curve becomes slowly decreasing. This rotation curve is well fitted with the Brandt rotation curve \citep{bra60}: \begin{equation} V = \frac{V_{\rm max}\frac{R}{R_{\rm max}}} {\left(\frac{1}{3}+\frac{2}{3} \left(\frac{R}{R_{\rm max}}\right)^{n}\right)^{\frac{3}{2n}}} \end{equation} with $V_{\rm max}=116$~km~s$^{-1}$, $R_{\rm max}=2\farcs7$, and $n=6$ (Figure~\ref{fig6}). The mass from the Brandt model is: \begin{equation} M_{\rm tot} = \left(\frac{3}{2}\right)^{\frac{3}{n}} \frac{V^2_{\rm max}R_{\rm max}}{G}. \end{equation} The total mass inside 190~pc ($\sim2.7\arcsec$) radius is $7.2\times10^{8}$~M$_{\odot}$. \begin{figure} \includegraphics[angle=0,width=.9\textwidth]{fig6.eps} \caption{Rotation curve of CO(3--2) in the central region of NGC~1068. Crosses with error bars are the data points, and the solid curve is the fitted Brandt rotation curve (see the main text for details). \label{fig6}} \end{figure} \subsection{Relation between CO Line Ratio and Star Formation} \label{sect-res-sfr} As shown in Sect.~\ref{sect-res-ratio}, $R_{31}$ varies along the spiral arms. Since the CO(3--2) line is more closely related to star formation \citep{kom07}, and $R_{31}$ increases with increasing star formation efficiency \citep{mur07}, the variation in $R_{31}$ along the spiral arms of NGC~1068 may also be related to star formation. To study the cause of the variation, we compare $R_{31}$ with star formation surface density along the spiral arms. We use both infrared and optical observations as star formation tracers. The infrared data are obtained from the \emph{Spitzer Space Telescope} IRAC $3.6~\micron$ and $8~\micron$ images\footnote{\url{http://ssc.spitzer.caltech.edu/archanaly/archive.html}}. \citet{wu05} showed that dust emission at $8~\micron$ can be used as a star formation indicator. We estimate the dust emission of in the spiral arms of NGC~1068 using the observed \emph{Spitzer} IRAC $3.6~\micron$ and $8~\micron$ fluxes: \begin{equation} f_{8\micron}(\mathrm{dust}) = f_{8\micron} - \eta_{8} f_{3.6\micron}, \end{equation} where $\eta_{8}=0.232$ \citep{lei99,hel04}. The derived values are shown in Table~\ref{tab-sfr}. H$\alpha$ emission is also considered as a tracer of star formation. For NGC~1068, the wavelength of H$\alpha$ is shifted to 6587.8~\AA~ and those of [\ion{N}{2}] to 6573, 6603\AA. To obtain the H$\alpha$ emission, we retrieved the \emph{Hubble Space Telescope (HST)} F658N and F791W images from the STScI archive\footnote{\url{http://archive.stsci.edu}}. The F658N narrow-band filter has a central wavelength of 6590.8~\AA~ with a bandwidth of $\sim$30~\AA; therefore, the F658N image includes the H$\alpha$ line as well as [\ion{N}{2}] lines, which are shifted to 6573 and 6603\AA for NGC~1068. Since this image also includes continuum emission, we correct it using the nearby broad-band filter F791W. The F791W filter has a central wavelength of 7881~\AA~ with a bandwidth of $\sim$1231~\AA. The continuum subtraction can be performed using the following equation: \begin{equation} f_{\rm line} = \frac{f_{\rm NB}\Delta\lambda_{\rm BB} - f_{\rm BB}\Delta\lambda_{\rm NB}} {\Delta\lambda_{\rm BB} - \Delta\lambda_{\rm NB}}, \end{equation} where $f_{\rm line}$ is the continuum-subtracted line flux, $f_{\rm NB}$ and $f_{\rm BB}$ are the observed total fluxes in the narrow-band and broad-band filters, respectively, and $\Delta\lambda_{\rm NB}$ and $\Delta\lambda_{\rm BB}$ are the bandwidths of the narrow-band and broad-band filters, respectively. The resulting continuum-subtracted F658N image can be considered as the H$\alpha$ + [\ion{N}{2}] image. The derived values are shown in Table~\ref{tab-sfr}. We cannot separate the H$\alpha$ from the [\ion{N}{2}] lines and we cannot correct the H$\alpha$ + [\ion{N}{2}] image for internal extinction with available data. Therefore, we can not obtain any quantitive information from this image; we only use the continuum-subtracted emission as a possible indicator of relative star formation rates. We compare the \emph{HST} H$\alpha$ + [\ion{N}{2}] line intensity with the \emph{Spitzer} $8~\micron$ dust intensity. As can be seen in Figure~\ref{fig7}, the H$\alpha$ + [\ion{N}{2}] line intensity and the $8~\micron$ dust intensity are linearly correlated, indicating that the former image is effectively also a good star formation tracer. We note that the CO distribution also matches very well with the dust lanes along the spiral arms in the \emph{HST} H$\alpha$ + [\ion{N}{2}] line image (Figure~\ref{fig3}). The star forming regions are also located along the spiral arms but slightly shifted toward the outside of the spiral arms, especially in the southern spiral arm. This is similar to the results of previous observations for the spiral arms of the nearby galaxy M51 \citep{vog88,aal99,kod09}. \begin{figure} \includegraphics[angle=0,width=.9\textwidth]{fig7.eps} \caption{Comparison of the \emph{Spitzer} $8~\micron$ dust intensity with the \emph{HST} continuum-subtracted H$\alpha$ + [\ion{N}{2}] line intensity. The \emph{HST} line intensity is not flux-calibrated and is expressed in instrument units. The data points with upper arrows do not have complete H$\alpha$ emission information because they are located either at the edge of or outside the \emph{HST} image. \label{fig7}} \end{figure} \begin{figure} \includegraphics[angle=0,width=.9\textwidth]{fig8.eps} \caption{Comparison between the CO(3--2)/CO(1--0) integrated intensity ratio, $R_{31}$, and the star formation rate surface density ($\Sigma_{\rm SFR}$). The error bars have included the $\sim~10\%$ uncertainty of our flux calibrations for the CO(3--2) and CO(1--0) emission. \label{fig8}} \end{figure} \begin{figure} \includegraphics[angle=90,width=.9\textwidth]{fig9.eps} \caption{Relation between the surface density of molecular gas $\Sigma_{\rm gas}$ and $\Sigma_{\rm SFR}$ for normal galaxies (open squares), starburst galaxies (open circles), and the spiral arm regions of NGC~1068 (filled circles). The gas surface density is derived from CO(1--0) emission. The normal and starburst galaxy samples are obtained from \citet{ken98}. The solid triangle is the nuclear region of NGC 1068 (C1). The solid line represents the power-law fit to all data points. \label{fig9}} \end{figure} \begin{figure} \includegraphics[angle=0,width=.9\textwidth]{fig10.eps} \caption{Relation between integrated CO(3--2) and FIR luminosities. The filled circles represent the data of the spiral arm regions of NGC~1068, and the solid triangle is the nuclear region of NGC 1068 (C1). The open circles are the data obtained from \citet{mao10}. The solid line represents the linear fit to all data points. The outlier point with the lowest CO(3--2) emission is R4. \label{fig10}} \end{figure} We also compare molecular gas properties with the star formation rate surface density ($\Sigma_{\rm SFR}$) derived from the \emph{Spitzer} IRAC $8~\micron$ and $3.6~\micron$ data. The $\Sigma_{\rm SFR}$ is derived from the $8~\micron$ dust luminosity: \begin{equation} \frac{\Sigma_{\rm SFR}}{(M_\odot~{\rm yr^{-1}})} = \frac{\nu L_{\nu}[8\micron(dust)]}{1.57\times10^9~{L_{\odot}}} \end{equation} \citep{wu05}, where $L_{\nu}$ is the $8~\micron$ dust luminosity. The derived values are shown in Table~\ref{tab-sfr}. Figure~\ref{fig8} shows the comparison between $R_{31}$ and $\Sigma_{\rm SFR}$ of the spiral arm regions. The correlation coefficient $r$ between the $R_{31}$ and $\Sigma_{\rm SFR}$ is 0.33 with a probability $p=89\%$ of mutual correlation. We note that most of the $R_{31}$ of nearby star-forming galaxies are within 0.2--0.7 \citep{mau99}. For examples, \citet{mur07} found that the CO emission of the starburst galaxy M83 has $R_{31} < 1$ and show a good correlation between the $R_{31}$ and SFE of their data. If we consider the data with $R_{31} < 1$ in our results, we find that the correlation between the $R_{31}$ and $\Sigma_{\rm SFR}$ becomes highly significant with $r=0.63$ and $p=99\%$. This suggests that in normal star-forming regions the physical conditions of molecular gas indicated by $R_{31}$ are well correlated with dust emission. The data points where $R_{31} > 1.0$ are obviously out of the correlation, suggesting that the large $R_{31}$ might be caused by other reasons, such as a different heating mechanism for the molecular gas or different distributions of warm and cool molecular gas \citep[e.g.,][]{ho87}. Figure~\ref{fig9} shows the relation between the surface density of molecular gas $\Sigma_{\rm gas}$ and $\Sigma_{\rm SFR}$. The star formation rate surface densities of the spiral arm regions of NGC 1068 are much higher than those of normal galaxies and similar to those of starburst galaxies; however, they all seem to follow the same star formation law \citep{ken98}. This result strongly supports the idea that the spiral arms in the inner $\sim2$ kpc region are experiencing a starburst. On the other hand, the molecular gas at C1 is obviously offset from the Schmidt-Kennicutt law, suggesting that C1 is mainly affected by AGN activities instead of star formation. In Figure~\ref{fig10}, we present the Schmidt-Kennicutt law for the CO(3--2) emission. We used the FIR luminosity instead of the SFR in Figure~\ref{fig9} so that it is easier to compare with the results of \citet{mao10}. We first derive the star formation rate from the observed $8\micron$ dust luminosity \citep{wu05} and then use the SFR--$L_{FIR}$ relation, SFR$(M_\odot yr^{-1})\sim1.7\times10^{-10}(L_{FIR}/L_\odot)$ \citep{ken98}, to derive $L_{FIR}$. We note that this is effectively a Schmidt-Kennicutt law for CO(3--2) emission since $L_{FIR}$ is proportional to SFR, and the integrated CO(3--2) luminosity represents the molecular mass in relatively warm and/or dense regions. We find that most of our results follow the same relation obtained by \citet{mao10} for nearby galaxies. The only outlier point, which has the lowest CO(3--2) emission, is the interarm region R4. When we combine our results with \citet{mao10}, we find that the derived power-law index of the $L_{CO(3-2)}$--$L_{FIR}$ relation is $\sim0.9$. This value is smaller than the index of the traditional Schmidt-Kennicutt law, which is around 1.0 to 2.0, but is reasonable for the excitation conditions of warm and/or dense gas \citep{kt07,na08}. In Figure~\ref{fig9} and Figure~\ref{fig10}, the physical scale of the NGC~1068 data is about $280\times350~$pc$^2$, which is much smaller than those of \citet{ken98} and \citet{mao10}. We note that there is no correlation between $\Sigma_{\rm gas}$ and $\Sigma_{\rm SFR}$ in Figure~\ref{fig9} when only considering the NGC~1068 data. On the other hand, the CO(3--2) emission from the same NGC~1068 regions show a very good correlation with the infrared as shown in Figure~\ref{fig10}; the correlation coefficient $r$ is 0.789 with a probability $p = 99.9\%$. The power-law index of the $L_{CO(3-2)}$--$L_{FIR}$ relation for the NGC~1068 data alone (excluding R4) is about 0.49. This value is significantly lower than a typical power-law index of the Schmidt-Kennicutt law. In Figure~\ref{fig10}, the CO(3-2) to FIR luminosity distribution is generally consistent with previous studies, which have a power law index of the Schmidt-Kennicutt law $\sim$1.0. On the other hand, the Schmidt-Kennicutt law derived from the CO(1-0) data has a steeper slope than that from the CO(3-2) data (Figure~\ref{fig9}). Furtheremore, if we only consider our NGC 1068 data, we find a very flat power law index, which cannot be explained by the model of \citet{kt07} with a different critical density. One possibility is that the gas is under sub-thermal conditions, which would produce a flatter KS law as shown by \citet{na08}. This interpretation is also consistent with the relatively large line ratios of CO(3--2)/CO(1--0). We also note that most of our data are under the average value of the KS law, indicating that the SFE of the inner spiral regions of NGC 1068 is smaller than that in most of the sources in Figure~\ref{fig10}. There is also a radial variation of star formation activity in the spiral arm regions. The average star formation rate surface density of the inner arm regions (R1 -- R4, R9 -- R17, and R19 -- R22) is 1.45 $M_{\odot}$ yr$^{-1}$ kpc$^{-2}$, and the outer arm regions (R5 -- R8, R18, R23, and R24) is 0.80 $M_{\odot}$ yr$^{-1}$ kpc$^{-2}$. In other words, the radial variation of the physical conditions of the molecular gas mentioned in Sect.~\ref{sect-res-ratio} is correlated with the radial variation of the galaxy's star formation. \section{SUMMARY} \label{sect-sum} We have shown and compared the emission of different CO rotational transitions of the prototypical Seyfert 2 galaxy NGC~1068 observed with millimeter and submillimeter interferometers. The molecular gas in the central part of this galaxy is distributed in a central core and outer spiral arms. Both the CO(1--0) and CO(3--2) lines show similar distribution along the spiral arms, and most of the molecular gas mass is located in the spiral arms. However, the nucleus is rather different; the strongest CO(3--2) peak lies in the nucleus, but this is not true for CO(1--0). This is very similar to another Seyfert 2 galaxy M51, suggesting that the AGN is playing an important role in the different behaviors of these two CO transition lines. In the spiral arms, the CO(3--2)/CO(1--0) integrated intensity ratio is well correlated with the star formation rate surface density, indicating that the physical conditions of molecular gas are related to star formation. Both the CO(3--2)/(1--0) ratio and the star formation rate decrease with radius from the nucleus. \acknowledgements The authors thank an anonymous referee for important suggestions. MT and CYH acknowledge support from the National Science Council (NSC) of Taiwan through grant NSC 100-2119-M-008-011-MY3 and NSC 99-2112-M-008-014-MY3. SM acknowledges support from the NSC of Taiwan through grant NSC 97-2112-M-001--021-MY3. DE was supported by a Marie Curie International Fellowship within the 6$\rm^{th}$ European Community Framework Programme (MOIF-CT-2006-40298).	{ "redpajama_set_name": "RedPajamaArXiv" }	1,618
package ro.nextreports.designer; import java.io.FileInputStream; import java.io.FileNotFoundException; import java.util.EventObject; import java.util.List; import javax.swing.JLabel; import javax.swing.JTextField; import javax.swing.SwingUtilities; import ro.nextreports.designer.action.report.layout.cell.ClearCellAction; import ro.nextreports.designer.grid.DefaultGridCellEditor; import ro.nextreports.designer.querybuilder.BrowserDialog; import ro.nextreports.designer.querybuilder.BrowserPanel; import ro.nextreports.designer.util.I18NSupport; import ro.nextreports.designer.util.NextReportsUtil; import ro.nextreports.designer.util.Show; import ro.nextreports.engine.Report; import ro.nextreports.engine.ReportLayout; import ro.nextreports.engine.band.ReportBandElement; import ro.nextreports.engine.util.LoadReportException; import ro.nextreports.engine.util.ReportUtil; public class ReportCellEditor extends DefaultGridCellEditor { private BrowserPanel browser; private BrowserDialog dialog; private ReportBandElement bandElement; public ReportCellEditor() { super(new JTextField()); // not really relevant - sets a text field as the editing default. } @Override public boolean isCellEditable(EventObject event) { boolean isEditable = super.isCellEditable(event); if (isEditable) { editorComponent = new JLabel("...", JLabel.HORIZONTAL); delegate = new ReportDelegate(); } return isEditable; } class ReportDelegate extends EditorDelegate { ReportDelegate() { browser = new BrowserPanel(BrowserPanel.REPORT_BROWSER, false); dialog = new BrowserDialog(browser); dialog.pack(); dialog.setResizable(false); Show.centrateComponent(Globals.getMainFrame(), dialog); } public void setValue(Object value) { bandElement = (ReportBandElement) value; SwingUtilities.invokeLater(new Runnable() { public void run() { dialog.setVisible(true); if (dialog.okPressed()) { String reportPath = browser.getSelectedFilePath(); Report report = null; try { report = ReportUtil.loadReport(new FileInputStream(reportPath)); } catch (FileNotFoundException e) { // TODO Auto-generated catch block e.printStackTrace(); } catch (LoadReportException e) { // TODO Auto-generated catch block e.printStackTrace(); } List<String> names = NextReportsUtil.incompatibleParametersType(report); if (names.size() > 0) { cancelCellEditing(); new ClearCellAction().actionPerformed(null); Show.info(I18NSupport.getString("insert.report.action.incompatible", names)); } else { bandElement.setReport(report); stopCellEditing(); } } else { cancelCellEditing(); if (bandElement.getReport() == null) { new ClearCellAction().actionPerformed(null); } } } }); } public Object getCellEditorValue() { ReportLayout oldLayout = getOldLayout(); String reportPath = browser.getSelectedFilePath(); Report report = null; try { report = ReportUtil.loadReport(new FileInputStream(reportPath)); } catch (FileNotFoundException e) { // TODO Auto-generated catch block e.printStackTrace(); } catch (LoadReportException e) { // TODO Auto-generated catch block e.printStackTrace(); } bandElement.setReport(report); registerUndoRedo(oldLayout, I18NSupport.getString("edit.report"), I18NSupport.getString("edit.report.insert")); return bandElement; } } }	{ "redpajama_set_name": "RedPajamaGithub" }	4,333
It's created a buzz that can't be ignored. People are talking about it, discussion programmes are full of it and comedians are already making jokes about it. More importantly, it's also captured the imagination, taking people out of their homes to walk or cycle across cities and search through the countryside. With the Rio Olympics a few days away at the time of writing, are we talking about the sporting legacy of London 2012? Well no, we're actually referring to the latest interactive internet craze which is Pokémon Go. Initially released on July 6 in the USA, Australia and New Zealand such was the game's instant appeal that further rollouts were delayed until server capacity issues could be rectified. Since then the game's worldwide appeal has seen it breaking App store records with more than a hundred million downloads by the end of July. So what is it that has given rise to this instant appeal? Is it down to its simplicity or the challenge of the hunt; is it because it suits our urge to explore or is it because of the way in which virtual reality merges seamlessly with the world around us? Whatever the reason, Pokémon Go is a true example of innovation, combining existing technologies and branding in a way that not only captures the imagination but also encourages people to look afresh at the world around them. In fact, it's a recipe for success which many organisations would love to be able to copy. Not simply in terms of turnover, although that would be useful, but more for the way in which the game enthuses, engages and encourages. If businesses were able to boost employee engagement in the same way, how different would the world of work be? Just imagine what would happen if our employees bounded into work full of enthusiasm every day. How many problems would be solved, how many genuine and long lasting solutions would be created if our people simply lifted their eyes from the mundane and looked again at the world around them? When people interact then great things can happen. Admittedly, you may get a few problems along the way, but that is what engagement with innovation is all about, learning from mistakes and moving forward. And don't forget that true engagement not only benefit the business/employee relationship, it also boosts customer service and adds to the bottom line. You may not be able to achieve an overnight transformation; but if you start now, as you roll out your new approach across your organisation you have the chance to create a buzz, to create a level of excitement and engagement which can't be ignored.	{ "redpajama_set_name": "RedPajamaC4" }	3,540
Brighton Girls Prep School, BN1 Radinden Manor Road, Hove, East Sussex BN3 6NH 10 simple reasons to choose Brighton Girls Prep School Independent prep school and nursery in Hove. Girls' school. Academically selective intake. Medium sized school with an average of two classes per year. Prep school to Brighton and Hove High School, an all-through school to age 18. Fees range from £3,121 (Rec) to £3,513 per term (2020/21). 10 better reasons to choose Brighton Girls Prep School One of the best independent prep schools in Hove and one of the 3 best primary schools for 0.5 miles. On average, 12.2 pupils per class and 8.7 pupils per teacher, from Year 1. 80% of pupils go on to the senior school, a school with a track record for academic achievement. 10% go to Brighton College. School curriculum also includes PSHE, life skills, French. There are over 5 different academic clubs per year group per term. Specialist teachers for sport, music and languages throughout. School participates in the Eco Schools programme and has Green Flag Status. School offers over 5 different sports. Competitive teams in major sports from Year 3. Art, music and drama are taught as discrete curricular subjects to all pupils. Over half the pupils learn a musical instrument (from Year 3). 3 choirs, orchestra and several instrument ensembles. Over 5 different visual and performing arts clubs per year group per term. As the prep school to Brighton and Hove High School, pupils can expect access to specialised teaching and facilities, as well as eventual progression to the senior school. Find another school near Brighton Girls Prep School The Drive Prep School (KS2) (0.3 miles); Lancing Prep Hove (0.5 miles); The Montessori Place, Elementary (0.7 miles). Aldrington Church of England Primary School (1.1 miles); Westdene Primary School (2.1 miles); Our Lady of Lourdes Catholic Primary School, Rottingdean (5.7 miles). The Drive Prep School (0.3 miles); The Montessori Place (0.7 miles); Brighton Girls GDST (1.3 miles). Brighton Hove and Sussex Sixth Form College (0.6 miles); Blatchington Mill School and Sixth Form College (1.3 miles); City College Brighton and Hove (1.5 miles).	{ "redpajama_set_name": "RedPajamaCommonCrawl" }	7,108
\section{Introduction} The subject of Cremona transformations is a classical chapter of algebraic geometry. However, the Cremona group of the projective space $\pp^ n$ is well understood only for $n\leq 2$. In higher dimension the structure of this group is far from being clarified, and the classification of such maps, except for a few special classes, is poorly known. An important class of Cremona maps of $\pp^ n$ comes off the so-called {\em polar maps}, that is, rational maps whose coordinates are the partial derivatives of a homogeneous polynomial $f$ in the homogeneous coordinate ring $R:=k[x_0,\ldots,x_n]$ of $\pp^n$. Geometrically, the relevance of such a map is that its indeterminacy locus is the singular locus of the corresponding hypersurface $V(f)$. A homogeneous polynomial $f\in R$ whose polar map is a Cremona map is called {\em homaloidal} -- though more often this designation applies to the corresponding hypersurface rather than to $f$ itself. Unfortunately, there are scarcely any general methods to studying, much less recognizing, such polynomials. Results such as those in \cite{EKP} and \cite{KS}, although fascinating, are difficult to apply in practice and do not give a large picture. A more circumscribed environment consists of polynomials that are determinants of square matrices with homogeneous entries of the same degree. Alas, even for this class one lacks general methods, often happening that each such matrix requires a particular approach depending on its generic properties. One of the goals of this paper is to consider structured matrices whose entries are variables of a polynomial ring over a field. Even for those there seems to be no comprehensive study of the homaloidal behavior of the corresponding determinants. Still, one advantage of dealing with these matrices is that they are often $1$-generic in the sense of \cite[Definition-Proposition 1.1]{Eisenbud2}. This implies that their determinant is irreducible at the outset, thus allowing for a substantial class to search within. Our approach is algebraic throughout. Needless to justify, we will assume throughout that the base field has characteristic zero. A good deal of intuition about the results, if not the results themselves, gets lost in prime characteristic. Anyway, from the geometric point of view, the study of a polar map in characteristic zero drives primevally through the properties of the Hessian determinant $h(f)$ of $f$, the reason being the classically known criterion for the dominance of the polar map in terms of the non vanishing of the corresponding Hessian determinant. Although this criterion admits a vast generalization to arbitrary rational maps in terms of the Jacobian determinant of a basis of the corresponding linear system, it is in the polar case that the notion takes full role. Though the overall objective is to detect homaloidal determinants and their properties, we soon became aware of a richness of notions from commutative algebra that come alongside in a natural way. Often these notions and their use are crucial events for the geometric consequence. Thus, ideal theory in this paper is not just an aside, it is rather a live vein of the results. For example, a major underlying problem in this context is to understand the properties of the so-called {\em gradient ideal} (or {\em singular ideal}) of the polynomial $f$, that is, the ideal $J=J(f)\subset R$ generated by the partial derivatives of $f$. Regardless of whether $f$ is homaloidal, a particular question asks when $J$ has an irrelevant primary component, i.e., when it is not saturated. Of course, if Proj$(R/(f)$ is smooth its gradient ideal will itself be an irrelevant primary ideal (in addition, $f$ will not be homaloidal unless $\deg(f)\leq 2$). One can easily cook up families of irreducible plane curves whose gradient ideals have an irrelevant component. It is much harder to exhibit an irreducible polynomial in $f\in k[x_0,x_1,x_2]$ such that its gradient ideal has no irrelevant component (i.e., such that its gradient ideal is perfect of codimension $2$). This question would naturally drive us in the theory of free divisors (see \cite{LisbonProcDekker}, \cite{gradientsolo}, \cite{A-S}) which is slightly offshore our intention in this paper. Nevertheless, the question itself is relevant in the present context as well. The existence of tight lower bounds for the degrees of syzygies of the gradient ideal has been around in recent past, in connection with the so called inverse Poincar\'e problem -- see, e.g., \cite{BruMen}, where it is shown that there are no syzygies in degree less than $\deg(f)-n$ provided the singularities of $f$ are all normal crossings. Thus, in the context of this sort of singularities, only hypersurfaces $f$ of degree $\leq n+1$ qualify in order that the corresponding gradient ideal have any linear syzygies at all. This relates to the following problem which had been around as part of the folklore of homaloidal polynomials. \begin{Question}\label{degreebound}\rm Let $f\in k[\xx]=k[x_0,\ldots,x_n]$ denote a squarefree form. If $f$ is homaloidal when is $\deg(f)\leq n+1$? \end{Question} It has been shown in \cite{CRS} that there are irreducible homaloidal forms of degree arbitrarily larger than the number $n\geq 3$ of variables. The classes of examples found are the dual hypersurfaces of certain rational scroll surfaces. More recently, it has been announced in \cite[Section 1.3]{Huh2} the existence of many such examples rooted in Newton polytope theory as developed in \cite{Huh1}. However, for certain structured classes of homaloidal hypersurfaces it is reasonable to expect that the degree stays beneath the number of variables. Clearly, the notion of ``structured'' is quite foggy. One sort of modified question is open as far as we know: \begin{Question}\label{degreebound_determinantal}\rm Let $\mathcal{M}$ denote a square matrix over $k[\xx]=k[x_0,\ldots,x_n]$ whose entries are forms of equal degrees and such that $f:=\det(\mathcal{M})$ is an irreducible homaloidal polynomial. Is $\deg(f)\leq n+1$? \end{Question} It looks like the question is wide open even if the entries are linear. In \cite{CRS} a class of examples has been studied in which the answer to the last question is affirmative. One main motivation of this work is the hope to shed additional light into this facet of the theory. \medskip Here is a brief description of the contents of the paper. In Section 2 we initially review a few basic notions of ideal theory to be used throughout. These are mainly related to the syzygies of an ideal and its main associated algebras, stressing some or other notation that may not be universally accepted. Next there are a some preliminaries about homaloidal polynomials and displays some elementary examples. A discussion of birationality criteria is annotated for the reader's convenience. Our source for these criteria is generally \cite{AHA}, where the language looks quite encompassing both for the algebraist and the geometer. Special cases and more particular situations had been exposed before in various other sources (\cite{cremona}, \cite{bir2003}, \cite{CRS}). Sections 3 and 4 contain the core of the results and each is subdivided in several subsections. We focus on determinants of generic matrices of catalectic nature and their degenerations -- we use ``generic" both technically and informally, and in the latter case the emphasis is that the entries are variables, but in the degenerations one allows zeros instead of variables (but no arbitrary linear forms will be entries). This has the advantage of making the shape of the partial derivatives as close as possible to the submaximal minors. Section 3 deals with those classical matrices in their generic versions, where entries are always variables. We start with the completely generic and symmetric generic matrices, as a model to guide us through the other sorts of catalecticants. Most of what we will say in the fully generic case seems to be well-known, still our dealing recasts what is to be retained from an algebraic angle. Inspired by the behavior in this case and by the geometric notion of parabolic points, we reinstate a notion (or principle) of parabolism with a more algebraic content solely attached to the Hessian. For catalecticants the property of parabolism in this sense turns out to be quite surprising (to us) and even motivated our daring state a conjecture. Whatever direction the conjecture eventually takes up, the relationship to the ideal theoretic nature of the dual variety is curious and has some measure of puzzlement. Returning to the homaloidal search, the first consideration is the generic Hankel matrix of arbitrary size. Thus, let ${\cal H}_m$ stand for the generic $m\times m$ Hankel matrix and let $P\subset R$ denote its ideal of submaximal minors (i.e., $(m-1)$-minors). We prove that $P$ is the minimal primary component of the gradient ideal $J=J(f)\subset R$ of $f:=\det {\cal H}_m$. Although this assertion sounds naturally guessed, not so much its proof. We have used a bouquet of arguments, ranging from multiplicities to initial ideals to Pl\"ucker relations (straightening laws) of Hankel maximal minors via the Gruson--Peskine change of matrix trick. From this it is but one step to guess that the only remaining associated (necessarily embedded) prime of $R/J$ is the obvious candidate $Q:=I_{m-2}({\cal H}_m)$ -- defining the singular locus of the determinantal variety $V(P)$. This guess is equivalent to the expected equality $J:P=Q$. We chose to include the latter guess in a conjecture of much larger scope to the effect that $$JP^i:P^{i+1}=I_{m-2-i}({\cal H}_m),$$ for $0\leq i\leq m-2$. An almost immediate consequence of this conjectured statement is that $J$ is a minimal reduction of $P$ with reduction number $m-2$. This gives quite a spectacular relationship of algebraic content between these two ideals. Terse as it may look to a geometer, we expect it to pave the road to the next conjecture to the effect that the gradient ideal $J$ is an ideal of linear type, i.e., that its Rees algebra coincides with its symmetric algebra. Geometrically, one is saying that the defining equations of the blowup of $\pp^{2(m-1)}$ along the singular locus of the Hankel determinantal hypersurface can be taken to be of linear nature. If this conjecture proves to be affirmative then one can conclude, via the criterion stated in Proposition~\ref{polar_and_lineartype}, that the Hankel determinant is not homaloidal. For this one draws on the result proved in \cite[Theorem 3.3.5]{M_thesis} to the effect that the {\em linear syzygy rank} of $J$ is $3<2m-2$ for $m\geq 3$. These various aspects of the nature of the singular ideal give an idea of how much more difficult it is as compared with its nicely behaved minimal component $P$, the latter being linearly presented and thereby defining a birational map onto the image. So much for the Hankel matrix of arbitrary size. In the case where $m=3$ we are able to solve all previous conjectures in the affirmative. The most laborious is the one about the linear type property which, for this value of $m$, allows to apply some well established criteria in terms of Fitting ideals and Koszul homology. After this, we move on to other catalecticants of higher leap (step). The previous results in the Hankel case are not immediately adjustable to these matrices and, in fact, the theory takes a more sinuous route. First, the ideal of $2$-minors of the $2$-leap $3\times 3$ catalecticant is not prime, only radical. One of its minimal primes is generated by certain variables, while the other is the ideal of maximal minors of an associated catalecticant introduced as an analog of the classical Gruson--Peskine trick. This already tells us that the minimal part of the gradient ideal is reduced splitting into two components, thus indicating a measure of difficulty in this case as compared to the Hankel case. We next deal with the $3$-leap and $2$-leap $4\times 4$ catalecticants. To derive whether these are homaloidal or not requires quite a bit of juggling in-between theory and computation. In order to state what is expected in arbitrary dimension and leap, there will be unavoidably some uninspired guessing. On the bright side, some of the guessing is firmly based on the methods throughout, of which counting the rank of the linear syzygy part of the gradient ideal appeals to some muscle action. Looking from such an angle, we state the conjecture that only $m$-leap and $(m-1)$-leap $m\times m$ catalecticants will have enough linear syzygy rank. Unfortunately, we will not get any answers with the computer for $m\geq 5$ as even the computation of the syzygies of the partial derivatives drags along quite a bit. \smallskip In Section 4 our focus is on certain degenerations of the generic versions of the previous section. We avoid calling them specializations since the numerical invariants of the various ideals in consideration may change as do their properties of interest to this work. Actually, the only sort of degeneration introduced here is by means of replacing some variables (entries) in strategic positions by zeros. This idea was originally introduced in \cite[Section 4.1]{CRS} for Hankel matrices. A reminiscent mention of homaloidal systems of maximal minors of degenerations of $m\times (m+1)$ Hankel matrices had appeared in \cite[Section 3]{ST} as a limiting process to solve a problem related to the reciprocal transformation. The examples show that degenerating can both destroy or give rise to the property of homaloidness. This phenomenon is yet to be understood from the geometric point of view, but even the underlying algebra is not cleat either. By and large it is a provocative section which we found appropriate to enlighten ourselves if not the sympathetic reader. \medskip The paper includes several results with complete conceptual proofs. On the other hand, quite a few rely on a mix of conceptual and computational arguments, some of which are followed by various conjectured statements. It is perhaps fair to say that a facet of this work is to offer an open invitation for others to propose and carry completely conceptual proofs of some of the conjectured statements. \medskip A good deal of the material dealing with Hankel and sub-Hankel matrices was subject of the first author's PhD thesis under the second author's supervision. We wish to thank S. H. Hassanzadeh for his help and suggestions. \section{Preliminaries} \subsection{Tools from ideal theory} Let $R$ be a Noetherian ring and let $I\subset R$ be an ideal. Let ${\cal S}_R(I)\surjects {\cal R}_R(I)$ denote the structural graded $R$-algebra homomorphism from the symmetric algebra of $I$ to its {\sc Rees algebra} - the latter is the graded $R$-algebra that defines the blowup along the subscheme corresponding to the ideal $I$ (see \cite[\S 5.2] {Eisenbook}). We say that $I$ is of {\sc linear type\/} if this map is injective. An ideal $I\subset R$ of linear type satisfies the {\sc Artin--Nagata condition $G_{\infty}$\/} (see \cite{AN}) which states that the minimal number of generators of $I$ locally at any prime $p\in \spec{R/I}$ is at most the codimension of $p$. It is known that this condition is equivalent to a condition in terms of a free presentation $$R^m\stackrel{\varphi}{\lar} R^{n+1} \lar I \lar 0 $$ of $I$ and the Fitting ideals of $I$, namely: \begin{equation}\label{f1} \hht (I_t(\varphi))\geq {\rm rank}(\varphi)-t+2,\quad \mbox{\rm for}\quad 1\leq t\leq {\rm rank}(\varphi), \end{equation} where $I_t(\varphi)$ denotes the ideal generated by the $t\times t$ minors of a representative matrix of $\varphi$ and $\hht$ designates the {\sc height} of an ideal (see, e.g., \cite[\S 1.3]{Wolmbook}). Because of its formulation in terms of Fitting ideals, the condition has been dubbed as {\sc property} $(F_1)$. Suppose that $R$ is a standard graded over a field $k$ and $I$ is generated by forms of a given degree $s$. In this case, $I$ is more precisely given by means of a free graded presentation $$R(-(s+1))^{\ell}\oplus\sum_{j\geq 2} R(-(s+j)) \stackrel{\varphi}{\lar} R(-s)^{n+1}\rar I\rar 0$$ for suitable shifts $-(s+j)$ and rank $\ell\geq 0$. Of much interest in this work is the value of $\ell$, so let us state in which form. We call the image of $R(-(s+1))^{\ell}$ by $\varphi$ the {\sc linear part\/} of $\varphi$ -- often denoted $\varphi_1$. One says that the rank of $\varphi_1$ is the {\sc linear rank} of $\varphi$ (or of $I$ for that matter) and that $\varphi$ has {\sc maximal linear rank\/} provided its linear rank is $n$ ($={\rm rank}(\varphi)$). Clearly, the latter condition is trivially satisfied if the $\varphi_1=\varphi$, in which case $I$ is said to have {\sc linear presentation\/} (or is {\sc linearly presented\/}). A quite notable fact, whenever $R=k[\xx]=k[x_0,\ldots,x_n]$ is a standard graded polynomial ring over $k$, is the case where $I$ happens to be of linear type and generated by $r+1$ forms of the same degree. Then $I$ has maximal analytic spread and hence ($k$ infinite) is generated by analytically independent forms (with respect to the irrelevant maximal ideal) of the same degree. Then these forms are algebraically independent elements over $k$, hence define a dominant rational map $\p^n\dasharrow \p^n$. This will be a cornerstone of many an argument to follow. \subsection{Birationality criterion in terms of ideals} Let $f\in k[\xx]=k[x_0,\ldots,x_n]$ be a squarefree homogeneous polynomial of degree $d\geq 2$. Let $$I=\left(\frac{\partial f}{\partial x_0},\ldots,\frac{\partial f}{\partial x_n}\right) \subset k[\xx],$$ the so-called {\sc gradient\/} ideal of $f$ -- we refrain from using the terminology {\sc Jacobian\/} ideal since the latter usually refers to the residue ideal in the ring $k[\xx]/(f)$. The partial derivatives of $f$ can be looked upon as the coordinates of a rational map ${\cal P}_f\colon\pp^n\dasharrow \pp^n$. This map is called the {\sc polar map\/} defined by (or of) $f$. We note that the image of this map is the projective subvariety on the target whose homogeneous coordinate ring is given by the $k$-subalgebra $k[{\partial f}/{\partial x_0},\ldots,{\partial f}/{\partial x_n}]\subset k[\xx]$ (whose grading, for that purpose, is renormalized so as to have its generators of degree one). While this algebra describes the image of the map it falls short of giving the complete picture of the map itself. A thorough analysis of the polar map has been undertaken in \cite{CRS}. Here we focus on the algebraic properties of this map and algebraically structured examples. Note that the ideal $I$ has codimension at least two since $f$ is assumed to be squarefree. In particular, the partial derivatives give the unique representative of codimension $\geq 2$ of the polar map (cf. \cite[Proposition 1.2]{bir2003}). Thus, it makes sense to call the singular scheme of $f$ the (uniquely defined) {\sc base scheme\/} of the polar map. A complete understanding of the nature of the polar map hinges on describing its base scheme. If ${\cal P}_f$ is birational one says that $f$ is {\sc homaloidal}. One may expect that the finer properties of ${\cal P}_f$ are embodied in a rich interplay between the geometric side of the map and the properties of the gradient ideal $I$. In this vein, we will refrain from mechanically pass to the projective scheme defined by the gradient ideal $I$ since knowing whether $I$ is $(\xx)$-saturated may have direct bearing to the properties of the polar map. \medskip A general characteristic-free birationality criterion has been established in \cite{AHA} which depends on a unique numerical invariant. This invariant can be viewed as a replacement for the field (topological) degree of the given map and, for many purposes, it is more flexible and computationally effective. For polar maps such that the corresponding linear system generates an ideal of linear type, the following simplified criterion holds in all characteristics. \begin{Proposition}\label{polar_and_lineartype} {\rm \cite[Theorem 3.2 and Proposition 3.4]{AHA}} Let $f\in R:=k[\xx]=k[x_0,\ldots,x_n]$ denote a square-free homogeneous polynomial of degree $d\geq 3$ such that the image of the corresponding polar map has dimension $n$. Consider the following conditions statements: \begin{enumerate} \item[{\rm (a)}] The syzygy matrix of the gradient ideal of $f$ has maximal linear rank. \item[{\rm (b)}] $f$ is homaloidal. \end{enumerate} Then {\rm (a)} implies {\rm (b)}. If, moreover, the gradient ideal is of linear type then {\rm (a)} and {\rm (b)} are equivalent. \end{Proposition} As a reminder, an ideal of linear type generated in fixed degree is generated by algebraically independent forms, hence the rational map defined by these forms is dominant (onto $\pp^n$). As the converse is false, this is a source of difficulty to go from the geometric to the algebraic aspect. Additional impact to the algebraic side is a certain polynomial akin to the classical {\em principal curves} in plane Cremona map theory, being related to the so-called {\em Jacobian curve} of a homaloidal net. Its terminology has been formally introduced in \cite[Propositions 1.2 and 1.3]{Zaron} for the use in the theory of symbolic powers, while earlier appearances are in \cite[Proof of Theorem 3.1]{CremonaMexico} (see also \cite{CostaSimis}) for the case of maps defined by monomials. The object itself is part of the algebraic nature of a Cremona map, in that it ties the map and its inverse by means of an essentially unique polynomial. One interest in the present work is as to how this polynomial in the case of a homaloidal polynomial is further related to the polynomial itself. We briefly recall the definition. Let $k$ denote an arbitrary infinite field -- further assumed to be algebraically closed in a geometric discussion. Recall that, quite generally, a rational map $\mathfrak{F}:\pp^n\dasharrow \pp^n$ is defined by $n+1$ forms $\mathbf{f}=\{f_0,\ldots, f_n\} \subset R:=k[\xx]=k[x_0,\ldots,x_n]$ of the same degree $d\geq 1$, not all null. We often write $\mathfrak{F}=(f_0:\cdots :f_n)$ to underscore the projective setup and assume that $\gcd\{f_0,\cdots ,f_n\}=1$ (in the geometric terminology, the linear system defining $\mathfrak{F}$ ``has no fixed part''), in which case we call $d$ the {\em degree} of $\mathfrak{F}$. If $\mathfrak{F}$ is a Cremona map of $\pp^n$ then there is a rational map $\mathfrak{G}:\pp^{n}\dasharrow \pp^{n}$ based on a linear system spanned by forms $\mathbf{g}=\{g_0,\ldots, g_n\} \subset R$ of same degree satisfying the relation \begin{equation}\label{birational_rule} (\mathbf{g}_0(\mathbf{f}):\cdots :\mathbf{g}_n(\mathbf{f}))\equiv (x_0:\cdots :x_n). \end{equation} The congruence translates into the existence of a uniquely defined form $D\in R$ such that, using a short vector notation, $\mathbf{g}(\mathbf{f})=D\cdot (\xx)$. In \cite{Zaron} $D$ has been dubbed the {\em inversion factor} of the map $\mathcal{F}$ or, more precisely, its {\em source inversion factor}. Its degree is $\deg (\mathfrak{F})\deg(\mathfrak{G})-1$. For convenience, we quote the following basic result about the inversion factor: \begin{Proposition}\label{jac_vs_factor}{\rm (\cite[Proposition 1.3]{Zaron}, char$(k)=0)$} Let $\mathfrak{F}$ denote a Cremona map of $\pp^{n}$ defined by forms $\ff:\{f_0,\ldots,f_n\}\subset R$ without fixed part and let $\Theta(\ff)$ denote the Jacobian matrix of $\ff$. Then $\det \Theta(\ff)$ divides a power of the source inversion factor $E$ of $\mathfrak{F}$. In particular, if $\det \Theta(\ff)$ is reduced then it divides $E$. \end{Proposition} \section{Matrices. I: generic catalecticants and symmetric} Since we are mainly interested in the search of irreducible homaloidal polynomials, is is natural to start looking about in determinants of well-structured square matrices. All matrices in this section, except for the symmetric ones, are of the following type. Let $m\geq 2$ and $1\leq r\leq m$ be given integers. Let $R=k[x_0,\ldots,x_n]$ be a polynomial ring with $n+1=(m-1)(r+1)+1$ variables. The $r$-{\em leap} $m\times m$ {\em generic catalecticant} is the matrix \begin{equation}\label{gen_catalectic} \mathcal{C}_{m,r}= \begin{pmatrix} X_0 & X_1 & X_2 & \hdots & X_{m-1}\\ X_{r} & X_{r+1} & X_{r+2} & \hdots & X_{m+r-1}\\ X_{2r} & X_{2r+1} & X_{2r+2} & \hdots & X_{m+2r-1}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ X_{(m-1)r} & X_{(m-1)r+1} & X_{(m-1)r+2} & \hdots & X_{(m-1)r+(m-1)} \end{pmatrix} \end{equation} Note that the corresponding determinant will have low degree ($=n$) as compared to the dimension of the ring and still involve all variables. The extreme values $r=1$ and $r=m$ yield, respectively, the ordinary Hankel matrix and the generic matrix. An important result proved in \cite{Eisenbud2} is that the generic Hankel matrix of arbitrary size $m\times n$ is $1$-generic. Using this result, it has been proved in \cite{Zaron2} that all generic catalecticants of arbitrary size are $1$-generic. The advantage of this notion is that it implies, in particular, that the determinant of a square such a matrix is irreducible. Although the fully generic matrix is an extreme case of a catalecticant, not so the generic symmetric matrix. However, the two have similar behavior regarding the subject of this work, and hence will be treated together. \subsection{Generic and generic symmetric matrices}\label{symmetric} The main part of the following result is classically known in algebraic geometry. We restate it by emphasizing the algebraic side, along with a tiny addition. Recall that a homogeneous polynomial is called {\em totally Hessian} if up to a nonzero element of $k$ its Hessian determinat is a power of it. Thus, if $f\in k[x_0,\ldots,x_n]$ has degree $d\geq 1$, being totally Hessian means that $$H(f)=cf^{\frac{(d-2)(n+1)}{d}},$$ where $H(f)$ denotes the Hessian of $f$ and $c\in k\setminus \{0\}$. The totally Hessian situation is to be considered as an extremal condition, whereby one might expect that more commonly a certain power of $f$ divide the Hessian determinant. For reasons that will be explained soon, the maximal exponent for which a power of $f$ may happen to divide $H(f)$ is said to be the {\em expected multiplicity} if its value is $n-1-v(f)$, where $n$ is the dimension of the ambient projective space and $v(f)=\dim V(f)^$ (the dimension of the dual variety to $V(f)$). For further terminology we refer to \cite[Section 2.1]{CRS}, which contains a discussion of these notions referring back to B. Segre. \begin{Proposition}\label{genericdet} Let $\xx$ denote a generic or a generic symmetric $m\times m$ matrix over the field $k$ and let $f=\det\xx$. Then: \begin{enumerate} \item[{\rm (a)}] $f$ is a homaloidal polynomial and the polar map of $f$ is an involution up to a projective transformation. \item[{\rm (b)}] The source inversion factor of the polar map of $f$ coincides with the $(m-2)$th power of $f$. \item[{\rm (c)}] $f$ is totally Hessian$\,${\rm ;} in particular, the expected exponent of $f$ as a factor of $H(f)$ has the expected value if and only if $m=3$. \end{enumerate} \end{Proposition} \demo We first remark that in the generic case the partial derivatives of $f$ are the signed cofactors of the matrix $\xx$, which is immediate from the data. In the case of the generic symmetric matrix each $(m-1)$-minor appears twice, hence the partial derivatives relative to the variables off the main diagonal will be the corresponding cofactor multiplied by $2$. (a) The ideal of $k[\xx]$ generated by the cofactors is of linear type and linearly presented (cf. \cite{Huneke} for the generic case and \cite{Kotsev} for the generic symmetric case (cf. also \cite{Wolmbook}). Therefore, the assertion about homaloidness is a consequence of Proposition~\ref{polar_and_lineartype}. Since $f$ read in dual variables is the dual hypersurface to the variety defined by the cofactors ($(m-1)$-minors) then the inverse to the polar map of $f$ is of the same kind up to a linear change of coordinates. (Note that what is a true involution is the map defined by the cofactors, hence in the fully generic case the polar map is a true involution, while in the symmetric case it will be so up to suitable coefficient scrambling.) (b) We argue in the generic case, the symmetric case requiring small adjustments. Let $\underline{\Delta}$ denote the ordered list of the signed $(m-1)$-minors (i.e., the cofactors of the matrix). As noted above, it defines an involution, hence one has by (\ref{birational_rule}): \begin{equation}\label{involution_eq} \Delta_{x_i}(\underline{\Delta})=x_iD, \end{equation} for every $0\leq i\leq m^2$, where $\Delta_{x_i}$ denotes the (signed) cofactor of $x_i$ and $D$ is the inversion factor. Using Laplace to compute $f$ in the form $f=\sum_{j=0}^{m-1}x_j\Delta_{x_j},$ and applying (\ref{involution_eq}), one obtains one obtains \begin{equation}\label{basic_evaluation} f(\underline{\Delta})=\sum_{j=0}^{m-1}\Delta_{x_j}\Delta_{x_j}(\underline{\Delta})=\sum_{j=0}^{m-1}\Delta_{x_j}\,x_jD= D\,\sum_{j=0}^{m-1}x_j\Delta_{x_j}=D\,f. \end{equation} Then $D=f^{m-2}$ provided $f(\underline{\Delta})=f^{m-1}$. But the latter follows from Cauchy's formula for the cofactors which asserts that the determinant of the matrix of cofactors $C(\xx)$ (the so-called adjugate of $f=\det (\xx)$) is equal to $f^{m-1}$. Since $(\det (\xx)) (\underline{\Delta}) = \det( \xx (\underline{\Delta}) )=\det (C(\xx)^t)=\det (C(\xx))$, we are through (we thank Z. Ramos for pointing to us this passage). (c) For the assertion about being totally Hessian, we have to show that $H(f)=f^{m(m-2)}$. By Proposition~\ref{jac_vs_factor}, $H(f)$ divides a power of the inversion factor $D=f^{m-2}$, hence is itself a power of $f$ as $f$ is irreducible. A degree count gives that $H(f)=f^{m(m-2)}$. \smallskip As for the subsumed assertion, in the present situation, $n=m^2-1$. As remarked in the proof of (a), $V(f)^\subset \pp^{m^2-1}$ is the subvariety defined by the $(m-1)$-minors of the matrix (read in dual variables). But the latter has codimension $4$. therefore, we get $$n-1-v(f)=m^2-2-(m^2-1-4)=3.$$ On the other hand, since $f$ is totally Hessian, the true exponent of $f$ as a factor of $H(f)$ is $m(m-2)$. Clearly, $m(m-2)=3$ if and only if $m=3$. This proves the subsumed assertion. \qed \begin{Remark}\rm (1) A shorter elementary proof of part (a), more in the spirit of the linear algebra used in the proof of (b), has been communicated to us by F. Russo. (2) There has been interest in considering polynomials $f$ which are factors of their Hessian determinant with multiplicity higher that the expected one (see \cite[Section 2.1]{CRS} for appropriate references.) (3) It would be curious to decide if there exist strict linear specializations of the generic matrix -- that is to say, $n\times n$ matrices of linear forms in less than $m^2$ variables whose $(m-1)$-minors still generate a codimension $4$ Gorenstein ideal -- such that the dual variety to the $(m-1)$-minors is a homaloidal hypersurface. Though some of these will have a homaloidal determinant, the dual variety may even fail to be a hypersurface (see Remark~\ref{dual_variety}). A similar question can be asked in the case of specializations of the generic symmetric matrix. \end{Remark} \subsection{The Hessian principle: parabolism} The idea of this part is to relate more closely the two hypersurfaces $V(f)$ and $V(H(f))$, where $f\in R=k[x_0,\ldots, x_n]$ is the determinant of certain specializations of the generic and the generic symmetric matrices. Assuming that $f$ is irreducible, we follow the terminology established in \cite[Section 2.1]{CRS} taken from the references mentioned there. Accordingly, to say that the generic point of $V(f)$ is $h$-parabolic, for some integer $h> 0$, translates into $f$ being a factor of $H(f)$ with multiplicity at least $h$; then, necessarily, $h=n-1-\dim V(f)^$. Note that one cannot decide the non-vanishing of $H(f)$ via this definition, since if $H(f)=\{0\}$ then certainly $f$ is a factor of $H(f)$ with any multiplicity (cf. \cite[Corollary 4.4]{Zak} and the comments thereafter for this puzzling issue). Drawing upon this line of ideas, we will say that a homogeneous polynomial $f\in R$ is {\em parabolic} if $f$ is a factor of $H(f)$ with bounded multiplicity (not necessarily the expected multiplicity). This means, in particular, that $H(f)\neq 0$ and that the (true) multiplicity of $f$ as a factor of $H(f)$ is an upper bound for $n-1-\dim V(f)^$. (Calling parabolic even if the multiplicity is not the expected one seems like a good idea as the two properties can ba handled separately.) \begin{Conjecture}\rm Let $f$ denote the determinant of the $r$-leap $m\times m$ catalecticant (\ref{gen_catalectic}), with $1\leq r\leq m$. Then $f$ is parabolic and if $r\leq m-1$ then it has the expected multiplicity as a factor of $H(f)$. \end{Conjecture} As mentioned before, $f$ is irreducible since the generic catalecticant is $1$-generic (\cite[Proposition 2.1]{Zaron2}). Note that $r\leq m-1$ excludes the case of the fully generic matrix (though it could be included if $m=3$). Let us consider the case $r=m-1$, which is the case where one expects the Hessian determinant $H(f)$ to admit $f$ as factor of highest multiplicity among the class of $r$-leap catalecticants with $r\leq m-1$. As a slight, but beautiful, evidence to the conjecture, we can entirely describe the prime factorization of $H(f)$ for $m=3,4$: \begin{Example}\label{parabolic_exs}\rm $\bullet$ ($m=3$) One is looking at the matrix $$\mathcal{C}_{3,2}=\left( \begin{array}{ccc} x_0 & x_1 & x_2 \\ x_2 & x_3 & x_4 \\ x_4 & x_5 & x_6 \end{array} \right). $$ Here one has $H(f)=f\cdot g$, where $g$ is (up to a projective change of coordinates) the equation of the dual surface to the twisted cubic in the variables $x_0,x_2,x_4,x_6$. Note that the defining equations of the latter are the $2\times$2 minors of the first and third columns of $\mathcal{C}_{3,2}$. The projective change of coordinates is totally trivial, obtained by $x_0\mapsto 3x_0, x_6\mapsto 3x_6$ and fixing the remaining variables. Observe that the degrees match as the equation of the dual variety has degree $4$ and $H(f)$ has degree $7$. \smallskip The dual to the hypersurface $V(f)$ is an arithmetically Cohen--Macaulay variety of codimension $2$, a determinantal scheme defined by the maximal minors of a linear $4\times 3$ matrix. This scheme can be obtained by the method of \cite[Proposition 1.1 (ii)]{CRS} as the projection from $\pp^8={\rm Proj}(k[z_0,\ldots,z_8])$ to $\pp^6$ with projecting center cut by the linear forms $z_0,z_1,z_2-z_3,z_4,z_5-z_6,z_7,z_8$ (note that these define the specialization from the generic case to the present catalecticant case). The expected multiplicity of $f$ is $n-1-\dim V(f)^=5-4=1$, which coincides with the effective multiplicity above. \medskip $\bullet$ ($m=4$) One is looking at the matrix $$\mathcal{C}_{4,3}=\left( \begin{array}{cccc} x_0 & x_1 & x_2 & x_3\\ x_3 & x_4 & x_5 & x_6 \\ x_6 & x_7 & x_8 & x_9 \\ x_9 & x_{10} & x_{11} & x_{12} \end{array} \right). $$ Here one has $H(f)=f^5\cdot (\det \mathcal{H}_3)^2$, where $\mathcal{H}_3$ is the Hankel matrix $$\mathcal{H}_{3}=\left( \begin{array}{ccc} x_0 & x_3 & x_6 \\ x_3 & x_6 & x_9 \\ x_6 & x_9 & x_{12} \end{array} \right). $$ Note that the $2\times 4$ matrix corresponding to $\mathcal{H}_3$ by the classical principle (\cite[Lemme 2.3]{GP}) is the submatrix of $\mathcal{C}_{4,3}$ with first and fourth columns. Observe that the degrees match. Note, however, that this time around the dual variety of the normal quartic in variables $x_0,x_3,x_6,x_9,x_{12}$ is not coming into the picture -- it has the right matching degree (=$6$) but is irreducible. \smallskip The dual to the hypersurface $V(f)$ has codimension $6$. The expected multiplicity of $f$ is $n-1-\dim V(f)^=11-6=5$, which coincides with the effective multiplicity above. \end{Example} Let us now consider examples at the other extreme of the value of $r$, namely, for Hankel matrices. \begin{Example}\label{parabolic_hankel}\rm $\bullet$ ($m=3$) One is looking at the matrix $$\mathcal{H}_{3}=\left( \begin{array}{ccc} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \\ x_2 & x_3 & x_4 \end{array} \right). $$ The prime factorization of the Hessian determinant is $H(f)=f\cdot \partial$, where $\partial$ denotes the partial derivative $\partial f/\partial x_2$ up to a sign and a trivial projective change of coordinates. The dual variety $V(f)^$ is a well-known codimension $2$ subvariety of $(\pp^4)^$ defined by $7$ cubics, obtained either by suitable projection of the $2$-Veronese or else as image of the rational map defined by a linear system of quadrics of dimension $4$ in $3$ variables. Then, the expected multiplicity is $n-1-\dim V(f)^=3-2=1$, clearly coinciding with the effective multiplicity above. \smallskip $\bullet$ ($m=4$) One is looking at the matrix $$\mathcal{H}_{4}=\left( \begin{array}{cccc} x_0 & x_1 & x_2 & x_3\\ x_1 & x_2 & x_3 & x_4 \\ x_2 & x_3 & x_4 & x_5 \\ x_3 & x_4 & x_5 & x_6 \end{array} \right). $$ One finds that the effective multiplicity of $f$ is $2$. A reasonable bet is that the complementary factor is the square of $\partial f/\partial x_3$ up to an appropriate projective change of coordinates, in which case the prime factorization would be $H(f)=f^2\cdot (\partial f/\partial x_3)^2$ up to coordinate change. Unfortunately, we have not been able to decide if this is the case or if this factor of degree $6$ is irreducible. The dual variety of $V(f)$ is a codimension $3$ subvariety of $(\pp^6)^$ defined by quartics. As it comes out, the expected multiplicity $5-3=2$ coincides with the effective multiplicity. \end{Example} Finally, we consider one more case, where $r$ has an intermediate value so to say. \begin{Example}\label{parabolic_cat2}\rm $$\mathcal{C}_{4,2}=\left( \begin{array}{cccc} x_0 & x_1 & x_2 & x_3\\ x_2 & x_3 & x_4 & x_5 \\ x_4 & x_5 & x_6 & x_7 \\ x_6 & x_7 & x_8 & x_9 \end{array} \right). $$ The effective multiplicity of $f$ is $2$. The complementary factor (of degree $12$) is pretty unreachable for inspection, except that it belongs to the gradient ideal. It is plausible that, up to a coordinate change, it is a product of partial derivatives of $f$. The dual $V(f)^$ is a codimension $3$ arithmetically Cohen--Macaulay subvariety of $(\pp^9)^$ with linear resolution. By a well-known result (\cite[HuMi]), the expected multiplicity is $8-6=2$, which is the value of the effective one. \end{Example} \begin{Remark}\rm All the required computer calculation has been performed with {\em Macaulay} (\cite{Macaulay}). Cases beyond these will most probably get stalled due to the computation of determinants and dual varieties of hypersurfaces in large ambient spaces. A breach for the theory to come in is a specialization procedure from the fully generic case. Care must be taken, since although the specialization of the Hessian of the generic matrix contains as a submatrix the Hessian matrix of the specialized generic matrix (i.e., the catalecticant), the determinant of the latter won't be a factor of the former. As for the theoretic tool to get to the expected multiplicity at the moment we have no insight as far as algebraic methods are concerned. It is possible that some of the methods employed in \cite{CRS} might be used. \end{Remark} \subsection{Hankel matrices} Hankel matrices constitute the lower extreme of a catalecticant in terms of its leap. For this reason and also because they are the only symmetric catalecticants we deal with them first. In general, an obstruction for the cofactor technique, which has shown so successful in the fully generic and generic symmetric cases, is that the minimal number of generators of the ideal generated by the cofactors has to equal the number of variables retained in the specialized matrix. This immediately rules out the cofactor technique for $r$-leap catalecticant square matrices with $r\leq m-1$. To rephrase it, the problem is that the gradient of the generic determinant does not specialize to the gradient of the specialized determinant. The generic Hankel matrix of size $m\times m$ is the symmetric matrix $$\mathcal{H}_m:= \left( \begin{matrix} x_0&x_1&\ldots &x_{m-1}\\ x_1&x_2&\ldots &x_{m-2}\\ \vdots &\vdots &\ldots &\vdots \\ x_{m-1}&x_{m-2}&\ldots &x_{2m-2}\\ \end{matrix} \right) $$ Let $J \subset R=k[x_0,\ldots, x_{2m-2}]$ denote the gradient ideal of $\det \mathcal{H}_m$ and let $P:=I_{m-1}(\mathcal{H}_m)\subset R$ stand for the ideal of $(m-1)$-minors of the matrix. Our main goal in this subsection is to shows several properties relating these two ideals. Throughout the multiplicity (degree) of a graded residue ring $R/I$ is denoted $e(R/I)$. \begin{Lemma}\label{degree} Set $P:=I_{m-1}(\mathcal{H}_m)$. Then $P$ is a prime ideal and the multiplicity of $R/P$ is $$e(R/P)=\frac{1}{3!} (m-1)m(m+1)$$ \end{Lemma} \demo By \cite[Proposition 4.3]{Eisenbud2}, the ideal $P$ is prime and has maximal possible height. On the other hand, by the same principle of \cite[Lemme 2.3]{GP}, $P$ is the ideal of the maximal minors of the $(m-1)\times (m+1)$ Hankel matrix \begin{equation}\label{GP_trick} \left( \begin{matrix} x_0&x_1&\ldots &x_m\\ x_1&x_2&\ldots &x_{m-1}\\ \vdots &\vdots &\ldots &\vdots \\ x_{m-2}&x_{m-1}&\ldots &x_{2m-2}\\ \end{matrix} \right) \end{equation} Therefore, this ideal has height $m-(m-2)+1=3$. It follows that the Eagon--Northcott complex resolves $R/P$; since this complex is well-known to be a pure $(m-1)$-resolution one can apply the formula of \cite{HuMi} to get the desired expression. \qed \begin{Proposition}\label{codimension} The gradient ideal $J$ of $\det \mathcal{H}_m$ is a codimension $3$ ideal contained in $P$. \end{Proposition} \demo There are several ways to observe the containment $J\subset P$. We quote a more general fact which yields this result: \begin{Lemma}\label{partials_vs_cofactors}{\rm \cite{Golb}, \cite{M_thesis}} Let $\mathcal{M}$ denote a square matrix over $R=k[z_0,\ldots,z_s]$ satisfying the following requirements: \begin{itemize} \item Every entry of $M$ is either $0$ or $z_i$, for some $i=0,\ldots, s$ \item Any variable $z_i$ or $0$ appears at most once on every row or column. \end{itemize} Let $f:=\det(\mathcal{M})\in R$. Then, for each $i=0,\ldots, m$, the partial derivative $f_i$ of $f$ with respect to $z_i$ is the sum of the {\rm (}signed{\rm )} cofactors of the entry $z_i$ in all of its appearances on $\mathcal{M}$. \end{Lemma} \medskip We conclude the proof of the Lemma by showing that $J$ has codimension at least $3$. For this we consider the initial ideal of $J$ in the reverse lexicographic order. Using Lemma~\ref{partials_vs_cofactors} with $\mathcal{M}=\mathcal{H}_m$ and $R=k[x_0,\ldots,x_{2m-2}]$, direct inspection shows that ${\rm in}(f_0)=x_m^{m-1}, \,{\rm in}(f_{2m-2})=x_{m-2}^{m-1}, \, {\rm in}(f_{m-1})=mx_{m-1}^{m-1}$. Clearly then ${\rm in}(J)$ has codimension at least $3$. Therefore, $J$ has codimension at least $3$ as well. \qed \begin{Lemma}\label{multiplicity_inequality} Consider the subideal $J'=({\rm in}(f_0),\ldots, {\rm in}(f_{2m-2})\subset R$ of the initial ideal of $J$ in he reverse lexicographic order. Then $e(R/J')<2 e(R/P)$. \end{Lemma} \demo Let us compute $e(R/J')$ in a direct way. Namely, using Lemma~\ref{partials_vs_cofactors} once more with $\mathcal{M}=\mathcal{H}_m$ and $R=k[x_0,\ldots,x_{2m-2}]$, one sees that $J'$ is generated by the set of monomials $$G:=\{x_{m-2}^{m-1},\,x_{m-2}^{m-2}x_{m-1},\,\ldots,\,x_{m-1}^{m-1},\,x_{m-1}^{m-2}x_m,\,\ldots,\, x_{m}^{m-1}\}.$$ Writing $J''$ for the ideal of $S:=k[x_{m-2},x_{m-1},x_m]$ generated by these monomials one has $J'=J''R$, hence $e(R/J')=e(S/J'')$. Since $S/J''$ is a graded of finite length, $e(S/J'')=\dim_k(S/J'')$. We claim that \begin{equation}\label{mult_of_pre_initial} \dim_k(S/J'')=\frac{1}{6} (m-1)m(2m-1). \end{equation} To compute the latter, we argue that a basis thereof is formed by all monomials in $S$ of degrees $\leq m-2$ and in addition the monomials in degrees $m-1, m,\ldots, 2(m-2)=2m-4$ that are not multiples of the generators of $J''$. The first compound is given by the well-known number $$\sum_{0\leq d\leq m-2}{{d+2}\choose {2}}={{m+1}\choose {3}}.$$ For the second part, for every $d\in \{m-2, m-1,m,\ldots, 2m-4\}$, let ${\rm M}_d$ denote the set of monomials of degree $d$ that are divisible by both $x_{m-2}$ and $x_m$. The cardinality of the complementary set ${\rm M}_d\setminus (x_{m-1}){\rm M}_{d-1}$ can be seen to be $${{2m-d-2}\choose {2}}.$$ Adding up for $d=m-1,m,\ldots, 2m-4$ gives $$\sum_{2m-4\geq d\geq m-1} {{2m-d-2}\choose {2}}=\sum_{2\leq e\leq m-1}{{e}\choose {2}}={{m}\choose {3}}.$$ Summing up we have $$\dim_k(S/J'')={{m+1}\choose {3}}+{{m}\choose {3}}=\frac{1}{6} (m-1)m(2m-1),$$ as was to be shown. \medskip An alternative argument yielding this formula goes as follows: For lighter reading set $x=x_{m-2}, y=x_{m-1}, z=x_m $. We partition the basis of $(S/J'')$ into disjoint sets according to the powers of $y$. Let then $A_i$ denote the set of all basis elements with $y$-degree $i-1$, for $1\leq i \leq m-1.$ In other words, $A_i$ denotes the set of all monomials $x^\alpha ÿy^\beta z^\gamma$ such that $\beta=i-1$, $\alpha<m-i$ and $\gamma< m-i$. Indeed, if to the contrary $\alpha \geq m-i$ then the summation of the $y$-degree and the $x$-degree is $\geq i-1+m-i=m-1$, so the monomial would belong to $G.$ The same argument applies to show that, for any $i$, the highest possible value of $\gamma$ in $A_i$ is $m-i-1.$ By inspecting the nature of the monomials in $G$ it ensues that $\alpha$ and $\gamma$ run independently. It therefore follows that {\small \begin{eqnarray}\nonumber A_i&=&\{ y^{i-1}, y^{i-1}z, \ldots, y^{i-1}z^{m-i-1}, y^{i-1}x, y^{i-1}x z,\ldots , y^{i-1}x z^{m-i-1}, \ldots , y^{i-1}x^{m-i-1},\\ \nonumber &&y^{i-1}x^{m-i-1}z, \ldots , y^{i-1}x^{m-i-1}z^{m-i-1}\}, \end{eqnarray} where $1\leq i \leq m-1.$ } It follows that \begin{equation} \dim_k(S/J'')= \sum^{m-1}_{i=1}\|A_i\|=\sum^{m-1}_{i=1}(m-i)^2= \frac{1}{6} (m-1)m(2m-1). \end{equation} \medskip As a consequence of (\ref{mult_of_pre_initial}) one has $$e(R/J')= \frac{1}{6} (m-1)m(2m-1)< \frac{1}{3} (m-1)m(m+1)=2 e(R/P),$$ as required. \qed \medskip Next we introduce some ideas from the theory of Pl\"ucker relations. For this we recast the trick of passing to the matrix (\ref{GP_trick}) whose maximal minors are our $(m-1)$-minors in a suitable disposition. This allows to express some formulas related to the Pl\"ucker relations of these maximal minors. First, the set of maximal minors of the matrix (\ref{GP_trick}) are partially ordered in the usual way, using a well-known notation: \begin{equation}\label{lex ord minor} [i_1, \ldots , i_{n-1}]\leq [j_1,\ldots , j_{n-1}] \Leftrightarrow i_1\leq j_1, \ldots , i_{n-1}\leq j_{n-1}. \end{equation} As an illustration, in the case of $m=4$, we get \medskip \hspace{2.2in} \begin{tikzpicture} [scale=1.0,auto=center,every node/.style={circle,fill=grey}] \node (n1) at (2,0.5) {[123]}; \node (n2) at (2,2) {[124]}; \node (n3) at (1,3) {[125]}; \node (n4) at (3,3) {[134]}; \node (n5) at (2,4) {[135]}; \node (n6) at (4,4) {[234]}; \node (n7) at (1,5) {[145]}; \node (n8) at (3,5) {[235]}; \node (n9) at (2,6) {[245]}; \node (n10) at (2,7.5) {[345]}; \foreach \from/\to in {n1/n2,n2/n3,n2/n4,n3/n5,n4/n5,n4/n6,n5/n7,n5/n8,n6/n8,n7/n9,n8/n9,n9/n10} \draw (\from) -- (\to); \end{tikzpicture} \medskip In this diagram the minors horizontally aligned are incomparable. One important feature of the Pl\"ucker relations is that they can be used to get a straightening law for the minors, whereby a product of incomparable ones is a combination of products of minors each having a factor which is less than the starting minors. For the contents on this topic in general we refer to \cite[chapter 4]{BV} and for the present case of Hankel matrices our reference is \cite{Conca}. From the original square Hankel matrix $\mathcal{H}_m$, by \cite{Golb}, one has \begin{equation}\label{Golberg} f_i:=\partial(\det{\cal H}_m)/\partial x_i=\sum_{k+l=i+2} \Delta^k_l \end{equation} where $\Delta ^{k}_l$ is the $(m-1)$-minor of $\mathcal{H}$ obtained by omitting the $l$th row and $k$th column. Since $\mathcal{H}_m$ is symmetric, $\Delta^k_l =\Delta ^l_k$. On the other hand, from (\ref{GP_trick}), by \cite[Lemma2.3]{GP} and \cite[Corollary2.2]{Conca}, one has \begin{equation}\label{delta} \Delta ^j_i=\sum_{l=1}^i[1,\cdots,\widehat{l},\cdots,\widehat{(j+i+2-l)},\cdots,n+1] \end{equation} where $j\geq i.$ Collecting the information yields {\small \begin{equation}\label{star} f_j= \left\lbrace \begin{array}{c l} \sum_{i=0}^{j/2}(j+1-2i)[1,\cdots,\widehat{(i+1)},\cdots,\widehat{(j+2-i)},\cdots,n+1] & j<n,\\ \\ \sum_{i=1}^{n-j/2}(2n+1-j-2i)[1,\cdots,\widehat{(i+1+j-n)},\cdots,\widehat{(n+2-i)},\cdots,n+1] & j\geq n. \end{array} \right. \end{equation} } Note that, for any $j$, the minors appearing in the above expression of $f_j$ are incomparable. (In the above illustrative diagram minors horizontally aligned are present in one and the same partial derivative.) \begin{Proposition}\label{integrality} The radical of the gradient ideal $J$ of a Hankel determinant is the ideal $P$ generated by the $(m-1)$-minors of the matrix. \end{Proposition} \demo By symmetry, it suffices to consider the minors present in one of the partial derivatives $f_j$, for $j=0,\cdots,2n-2$. We proceed by descending induction. For $j=2n-2, 2n-3$ there is nothing to prove since the corresponding partial derivatives are themselves minors, to wit, $f_{2n-2}=[1,2,\ldots,n-1]$ and $f_{2n-3}=2[1,2,\ldots,n-2,n]$ respectively. We provide the next inductive step to make the argument clearer. The next minors in the diagram of partial order are exactly those present in the expression of $f_{2n-4}$ according to (\ref{star}). Consider the Pl\"ucker relation containing the term $[1,\ldots,n-3,n-1, n][1,\ldots,n-2,n+1]$: \begin{equation}\label{plucker} \begin{array}{lcl} [1,\ldots,n-3,n-1, n][1,\ldots,n-2,n+1] & = & 1/2[1,\ldots,n-3,n-1,n+1]f_{2n-3} \\ & - & f_{2n-2}[1,\ldots,n-3,n,n+1]. \end{array} \end{equation} Now, $[1,\ldots,n-3,n-1, n]$ satisfies the following equation: {\small \begin{equation}\label{xequation} X^2 -1/3 X ( 3[1,\ldots,n-3,n-1, n]+[1,\ldots,n-2,n+1] ) + 1/3 [1,\ldots,n-3,n-1, n][1,\ldots,n-2,n+1]=0. \end{equation} } By (\ref{star}) one has $f_{2n-4}=3[1,\ldots,n-3,n-1, n]+[1,\ldots,n-2,n+1]$, and by (\ref{plucker}) the constant coefficient of (\ref{xequation}) lies in $J$. Hence $[1,\ldots,n-3,n-1, n] \in \sqrt{J}.$ \smallskip Now suppose that $j<2n-4$ and that, for any $t$ such that $j<t\leq 2n-2$, any minor present in the expression of $f_t$ belongs to $\sqrt{J}$. Let $\Delta:=[1,\ldots,\widehat{i+1+j-n},\ldots,\widehat{n+2-i},\ldots,n+1]$ be a minor present in $f_j$, where $1 \leq i \leq [(n-j)/2]$ and $j\geq n$, and write $f_j=({\rm coef})\Delta+ \sum$, where $\sum$ denotes the complementary $k$-linear combination of minors as in (\ref{star}). Multiplying by $\Delta$ yields $f_j\Delta=({\rm coef})(\Delta)^2+\Delta\,\sum$. Making everything explicit via (\ref{star}), we see that $\Delta$ satisfies the following equation: \begin{equation}\label{equation} X^2 - \frac{1}{2n+1-j-2i} f_j\,X+\frac{1}{2n+1-j-2i} g=0, \end{equation} where {\small $$g=\sum_{i\neq l=1}^{[(n-j)/2]} [1,\ldots,\widehat{i+1+j-n},\ldots,\widehat{n+2-i},\ldots,n+1] [1,\ldots,\widehat{l+1+j-n},\ldots,\widehat{n+2-l},\ldots,n+1].$$ } Now, for any $i,l$ such that $1 \leq i < l \leq [(n-j)/2]$, one has the following Pl\"ucker relation: {\scriptsize \begin{eqnarray}\nonumber 0&=&[1,\ldots,\widehat{i+1+j-n},\ldots,n+2-l,\ldots,\widehat{n+2-i},\ldots,n+1] [1,\ldots,i+1+j-n,\ldots,\widehat{l+1+j-n},\ldots,\widehat{n+2-l},\ldots,n+1]\\\nonumber &-&{\bf [1,\ldots,{i+1+j-n},\ldots,\widehat{n+2-l},\ldots,\widehat{n+2-i},\ldots,n+1]} [1,\ldots,\widehat{i+1+j-n},\ldots,\widehat{l+1+j-n},\ldots,{n+2-l},\ldots,n+1]\\ \nonumber &+&[1,\ldots,\widehat{i+1+j-n},\ldots,\widehat{n+2-l},\ldots,{n+2-i},\ldots,n+1] {\bf [1,\ldots,i+1+j-n,\ldots,\widehat{l+1+j-n},\ldots,n+2-l,}\\ \nonumber &&{\bf \ldots,\widehat{n+2-i},\ldots,n+1]}\nonumber \end{eqnarray} } This is nearly a straightening relation as the minors in boldface are less than the starting minor $[1,\ldots,\widehat{i+1+j-n},\ldots,\widehat{n+2-i},\ldots,n+1]$ in the partial order -- but this is all we need. Since these two boldfaced minors are present in $f_t$ for some $t>j$, by the inductive hypothesis they belong to $\sqrt{J}$. Therefore $g \in \sqrt{J}$. It follows from (\ref{equation}) that $\Delta^2\in \sqrt{J}$, hence $\Delta\in \sqrt{J}$, as required. \qed \medskip Drawing upon the preceding results, one has: \begin{Theorem}\label{p_primary_part} The minimal component of the primary decomposition of $J$ in $R$ is $P$. \end{Theorem} \demo It suffices to prove that $P=\sqrt{J}$ and that $P$ is the $P$-primary part of $J$. The first assertion is the content of Proposition~\ref{integrality}. As for the second, let $\mathcal{P}$ denote the $P$-primary part of $J$. Since $\mathcal{P}$ is the contraction of $\mathcal{P}_P$, it suffices to show that $J_P=P_P$, i.e., that $\ell(J_P)=\ell(P_P)$. By the associativity formula of multiplicities, one has $e(R/J)\geq e(R/P)\ell (R_P/J_P)$, where $\ell$ denotes length. Since $J'=({\rm in}(f_0),\ldots, {\rm in}(f_{2m-2}))\subset {\rm in}(J)$ also has codimension $3$, using Lemma~\ref{multiplicity_inequality}, yields: $$e(R/P)\ell(R_P/J_P)\leq e(R/J)=e(R/{\rm in}(J))\leq e(R/J')<2e(R/P).$$ Clearly this is only the case if $\ell(R_P/J_P)=1$. \qed \begin{Corollary}\label{colon_by_P} One has $e(R/J)=e(R/P)\,${\rm ;} in particular, $J:P$ has codimension at least $4$. \end{Corollary} \demo The equality of multiplicities is clear from Proposition~\ref{p_primary_part} by the associativity formula. From the exact sequence $0\rar P/J\rar R/J\rar R/P\rar 0$ we then conclude that $$\dim P/J<\dim R/J=2m-1-3.$$ Since $J:P$ is the annihilator of $P/J$ we are through with the subsumed statement. \qed \bigskip We next state three open questions. Due to large computer verification, natural interspersing of the various properties so far and last, but not least, aesthetic formulation, we risk to call them conjectures. They are listed in increasing order of difficulty (to our best guessing). \begin{Conjecture}\label{reduction}\rm Let $\mathcal{H}_m$ denote the Hankel matrix of size $m\times m$. Let as above $J$ denote the gradient ideal of its determinant and $P=I_{m-1}(H)$. Then, for $0\leq i\leq m-2$ one has $$JP^i:P^{i+1}=I_{m-2-i}(H).$$ \end{Conjecture} \begin{Corollary} {\rm (of Conjecture~\ref{reduction})} \begin{enumerate} \item[{\rm (i)}] $Q:=I_{m-2}(H)$ is the only other associated prime of $R/J$ {\rm (}necessarily embedded{\rm )} \item[{\rm (ii)}] $JP^{m-3}:P^{m-2}=(x_0,\ldots,x_{2m-2})$ \item[{\rm (iii)}] $JP^{m-2}:P^{m-1}=(1)$, i.e., ${\rm red}_J(P)=m-2\,${\rm ;} in particular, $J$ is a minimal reduction of $P$. \item[{\rm (iv)}] The partial derivatives of $\det(H)$ are algebraically independent over $k$. \end{enumerate} \end{Corollary} \demo (Highlights) (i) Use Theorem~\ref{p_primary_part}. Note that this item is now equivalent to showing that $J:P=Q$. Indeed, let $J=P\cap {\cal N}$, stand for the primary decomposition of $J$, where ${\cal N}$ denotes the intersection of the remaining (necessarily embedded) primary components. Then $Q=J:P={\cal N}:P$, hence ${\cal N}$ is $Q$-primary. It follows that $Q$ is the only associated embedded prime of $R/J$. (ii) This guarantees that the reduction number is not smaller than $m-2$. In this regard we risk to say, moreover, that $\mu(P^{m-2})=\mu(JP^{m-3})+1$, so (ii) would correspond to having the entries in the (say) last coordinate of the syzygies of $P^{m-2}$ generate the maximal ideal. Thus, previous information about the syzygies of the powers of $P$ may be useful. (iii) Clearly, $J$ is a reduction of $P$. To see that it is a minimal reduction it suffices to see that it is minimally generated by $2m-1$ forms, while this is the analytic spread of $P$ because the dimension of the $k$-subalgebra of $R$ generated by the $(m-1)$-minors is $2m-1$. (iv) This follows since $J$ is a minimal reduction, hence is generated by analytically independent forms. But, since for forms of the same degree, ``analytical independence'' and ``algebraic independence'' are the same notion, we get that the partials themselves are algebraically independent. Equivalently, the inclusion of the algebras $k[J_{n-1}]\subset k[P_{n-1}]$ is a Noether normalization. \qed \begin{Conjecture}\label{linear_type}\rm $J$ is an ideal of of linear type. \end{Conjecture} \begin{Conjecture}\label{non-homaloidal}\rm The determinant of ${\cal H}_m$ is not homaloidal. \end{Conjecture} \begin{Remark}\rm (i) It is shown in \cite{M_thesis} that the syzygy matrix of $J$ has linear rank $3$. Together with an affirmative answer to Conjecture~\ref{linear_type} this would imply that $\det {\cal H}$ is {\em not} homaloidal (last of the three conjectures above). In the subsequent part we will prove both conjectures in the case $m=3$. (ii) Regarding assertion (iv) of the above conjectured corollary, it is indeed true that the Hankel partial derivatives are algebraically independent over $k$ - a proof is given in \cite[Proposition 3.3.11]{M_thesis} in terms of the non-vanishing of the corresponding Hessian determinant. In this regard it would be pertinent to give a proof of the parabolism of the Hankel determinant $f$, i.e., that $f$ divides the Hessian $H(f)$ and to find the exact multiplicity exponent. Note that this exponent is $1$ in the case of $m=3$ (Example~\ref{parabolic_hankel}). \end{Remark} \subsection{Cases study} This part is devoted to examining small sizes of catalecticants. The reason for this section is either because for small value of $m$ one has special properties inexistent for larger size, or else because some behavior is conjectured to hold in any size and yet we lack the tools to write a complete proof. \subsubsection{Hankel $3\times 3$} \smallskip This is the matrix $$\mathcal{H}_3=\left( \begin{array}{ccc} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \\ x_2 & x_3 & x_4 \\ \end{array} \right) $$ \begin{Proposition}\label{hankel3x3} Let ${\cal H}_3$ denote the generic $3\times 3$ Hankel matrix in the variables $x_0,\ldots,x_4$ and let $J\subset R:=k[x_0,\ldots,x_4]$ denote the gradient ideal of $\det {\cal H}_3$. \begin{enumerate} \item[{\rm (a)}] Let $P:=I_2({\cal H}_3)\subset R\,${\rm ;} then $P$ and $\fm:=(x_0,\ldots,x_4)$ are the only associated primes of $R/J$. \item[{\rm (b)}] $\det {\cal H}_3$ is not homaloidal. \end{enumerate} \end{Proposition} \demo Drawing on the previous results (Theorem~\ref{p_primary_part}), we only have to show that $\fm$ is an associated prime. For this it is enough to show that $\fm\subset J:P$. If the lower bound for $J:P$ from Corollary~\ref{colon_by_P} could be improved by $1$, we would be done since at any rate $J:P$ is contained in some associated prime of $R/J$. Instead we argue directly within the details of this case. We switch freely back and forth between the Hankel matrix and the associated $2\times 4$ Hankel matrix (\ref{GP_trick}), where the $2$-minors can be thought as maximal minors. By the shape of the generators of $J$, it suffices to show that, for $i=0,\ldots,4$, either $x_i\Delta_{23}\in J$ or $x_i\Delta_{14}\in J$. Taking the $3\times 2$ submatrix of the given $3\times 3$ Hankel matrix consisting of the last two columns, we immediately get that $x_3\Delta_{23}, x_4\Delta_{23}\in J$. By a similar token, the $2\times 3$ submatrix formed by the first two rows yield $x_0\Delta_{23}, x_1\Delta_{23}\in J$. To deal with $x_2$ choose $\Delta_{14}$ instead. This time around, consider the $2\times 3$ submatrix of the associated $2\times 4$ Hankel matrix (\ref{GP_trick}) consisting of the first and last two columns. Then $x_2\Delta_{14}\in J$, as required. \medskip (b) By (a), the saturated ideal $J:\fm^{\infty}$ coincides with $P$. Supposing that $J$ defines a Cremona map, one would have that the initial degree of $P/J$ is at least $2+1=3$ (\cite[Proposition 1.2]{PanRusso}). But this is nonsense since $P$ admits a generator of degree $2$ (minor) that does not belong to $J$. \qed \medskip Note that part (b) above proves Conjecture~\ref{non-homaloidal} for $m=3$, while the proof of part (a) showed that $J:P=\fm=I_1({\cal H}_3)$, which is half the statement of Conjecture~\ref{reduction}. To complete the other half, we need to show that $JP:P^2=(1)$. For convenience we isolate this result as a proposition. \begin{Proposition} Let ${\cal H}_3$ denote the generic $3\times 3$ Hankel matrix, let $J\subset R$ denote the gradient ideal of $\det {\cal H}_3$ and $P:=I_2({\cal H}_3)$. Then $P^2\subset JP\,${\rm ;} in particular, $J$ is a minimal reduction of $P$ with reduction number $1$. \end{Proposition} \demo Since $P=(J,\Delta)$, where $\Delta=\Delta_{23}=x_2^2-x_1x_3$, it suffices to show that $\Delta^2\in JP$. As in the proof of Theorem~\ref{p_primary_part} a direct calculation up to nonzero coefficients and adjusting signs yields the relation $$\Delta^2=f_2\Delta + \Delta_{14}\Delta_{23}.$$ Clearly, $f_2\Delta\in JP$. Now, using the Pl\"ucker relation $\Delta_{12}\Delta_{34}-\Delta_{13}\Delta_{24}+\Delta_{14}\Delta_{23}=0$,we get that $\Delta_{14}\Delta_{23}\in JP$ as well. \qed \medskip The conjecture about the linear type property is proved in the following theorem. \begin{Theorem} Let ${\cal H}_3$ denote the generic $3\times 3$ Hankel matrix and let $J\subset R$ denote the gradient ideal of $\det {\cal H}_3$. Then $J$ is an ideal of of linear type. \end{Theorem} \demo To show the linear type property we use the criterion of \cite[Theorem 9.1]{Trento}, informally known as the ``$(F_1)$ $+$ sliding depth'' criterion. Now, $(F_1)$ (also known as $(G_{\infty}$) is the following: \smallskip {\sc Claim}: $\mu(J_Q)\leq \hht Q$, for every prime ideal $Q$. \smallskip To prove this assertion we have to drill quite a bit through the syzygies of the generators of $J$. Since $P$ has the expected codimension ($=3$), its presentation matrix is linear, as a piece of the Eagon--Northcott complex for the maximal minors of the Hankel matrix \begin{equation}\label{GP3} \left( \begin{matrix} x_0&x_1&x_2&x_3\\ x_1&x_2 & x_3 &x_4\\ \end{matrix} \right) \end{equation} Noting that the syzygies derive from the Laplace relations coming from the four $2\times 3$ submatrices of (\ref{GP3}), and ordering the minors to adjust to the order in which they appear along the usual ordering of the partial derivatives, one obtains the following matrix: $$ \mathcal{S}=\left( \begin{matrix} 0&0&0&x_0&-x_1&0&x_1&x_2\\ 0&x_0 &0&0&x_2&x_1&0&-x_3\\ 0&-x_1&0&-x_2&0&-x_2&-x_3&0\\ x_0&0&x_1&0&-x_3&0&0&x_4\\ -x_1&0&-x_2&x_3&0&0&x_4&0\\ x_2 & x_3 & x_3 &0 &0 & x_4 & 0&0\\ \end{matrix} \right )$$ Let $\mathcal{S}_j\, (1\leq j\leq 8)$ denote the $j$th column of this matrix. A linear syzygy of the ordered partial derivatives comes by ``tweaking'' a $k$-linear combination of columns of $\mathcal{S}$ satisfying the following restrictions: first, the entry on the fourth coordinate is $3$ times the one on the third coordinate; second, the set of entries along the $3$rd row of the involved columns is the same as the set of entries along the $4$th row. After a detailed inspection of these restrictions, we get at least the following possibilities of pairs: $\{\mathcal{S}_2,\mathcal{S}_3\}$, $\{\mathcal{S}_4,\mathcal{S}_6\}$ and $\{\mathcal{S}_5,\mathcal{S}_7\}$ and the resulting syzygies being $3\mathcal{S}_3-\mathcal{S}_2, -3\mathcal{S}_5-\mathcal{S}_7, -\mathcal{S}_6+\mathcal{S}_4$. In this way the ordered partial derivatives admit the following linear syzygies \begin{equation}\label{linear_syzygies3} \left( \begin{matrix} 0 & -2x_0 & 4x_1\\ x_0 & -x_1 & 3x_2\\ 2x_1 & 0 & 2x_3\\ 3x_2 & x_3 & x_4\\ 4x_3 & 2x_4 & 0 \end{matrix} \right ). \end{equation} yielding a matrix of rank $3$. We now contend that, together with the Koszul syzygies, they suffice to check the stated property of $J$. For this, we note that this property is equivalent to a property of the Fitting ideals of $J$, and can be stated as follows: $$\hht I_t(\phi)\geq \rk (\phi)-t +2=4-t+2=6-t, \;\; {\rm for}\;\; 1\leq t\leq 4.$$ Obviously, it suffices to show these estimates for the above submatrix $\psi$. For $t=1,2$ the linear syzygies so far described immediately give the required inequalities. For $t=3,4$, we use the matrix of Koszul syzygies where one knows that the codimension of any of its Fitting ideals is $\geq \hht (J)=3$. So much for the property $(F_1)$. \smallskip The sliding depth condition reads as \smallskip {\sc Claim}: $\depth(H_i)_Q \geq \hht(Q) - \mu(J_Q) + i$, for every prime $Q \supset J$ with $0 \leq i \leq\mu(J_Q) -\hht(J_Q )$. Here $H_i$ denotes the $i$th Koszul homology module on the generators of $J$ \smallskip Note that if $Q\supset J$ then necessarily $Q\supset P$. We first assume that $Q=\fm$, in which case $\mu(J_{\fm})=\mu(J)=5$, $\hht(J_{\fm})=\hht(J)=3$, hence $\mu(J_{\fm})-\hht(J_{\fm})\leq 2$. In this case, our range for $i$ is $0\leq i\leq 2$. For $i=2$ the inequality is automatic since $H_2$ is Cohen--Macaulay and it is trivial if $i=0$. For $i=1$ we need to show that $H_1$ has depth at least $1$. This computation has been carried out with {\em Macaulay} by showing that the homological dimension of $H_1$ over $R$ is $4$. Next assume that $Q\subsetneq \fm$, hence $\mu(J_Q)\leq\hht(Q)\leq 4$ (by the first claim above). Clearly, $\hht(J_Q)=\hht(P_Q)=3$. Since $P$ is the minimal part of $J$ and $\fm$ is the only other associated prime of $R/J$, one has $J_Q=P_Q$. Therefore, $\mu(J_Q)-\hht(J_Q)=\mu(P_Q)-\hht(P_Q)=3-3=0$ since $R/P$ is an isolated singularity. Then $i=0$ is the only case. One has $\depth (H_0)_Q=\depth (R_Q/P_Q)=2 = \hht(Q)-\mu(J_Q) +0$. \smallskip Thus, $J$ is an ideal of linear type. \qed \begin{Remark}\rm The linear syzygies (\ref{linear_syzygies3}) in fact generate all linear syzygies of $J$ as is shown in \cite[Theorem 3.3.5]{M_thesis} in greater generality. Together with the linear type property, it implies via Proposition~\ref{polar_and_lineartype} that the Hankel determinant of size $3$ is not homaloidal. \end{Remark} \subsubsection{Catalecticant: $2$-leap $3\times 3$} In contrast with the Hankel case, we don't have as yet a general theory of the associated primes of the gradient ideal of the catalecticant determinant. In the case of a $3\times 3$ catalecticant, since the upper extreme case of the leap ($r=3$) is the generic $3\times 3$ matrix and the lower extreme is the Hankel matrix, both of which have been taken care of, we are left with the case $r=m-1=2$. This is the matrix $$\mathcal{C}=\mathcal{C}_{3,2}=\left( \begin{array}{ccc} x_0 & x_1 & x_2 \\ x_2 & x_3 & x_4 \\ x_4 & x_5 & x_6 \\ \end{array} \right) $$ We call attention to the sharp difference at the outset between this catalecticant case and the previous $3\times 3$ Hankel matrix: here the ideal of $2$-minors of $\mathcal{C}$ is radical, but not prime. Overall the structure of the primary components of $R/J$ is a lot more intricate as we now explain. \begin{Proposition}\label{2-cat3x3} Let $J\subset R:=k[x_0,\ldots,x_6]$ denote the gradient ideal of $\det \mathcal{C}$. Consider the associated $2$-leap catalecticant $$\widetilde{\mathcal{C}}:= \left( \begin{array}{ccccc} x_0 & x_1 & x_2 & x_3 & x_4 \\ x_2 & x_3 & x_4 & x_5 & x_6 \\ \end{array} \right) $$ \begin{enumerate} \item[{\rm (a)}] Let $I:=I_2(\mathcal{C})\subset R\,$ and $P:=I_2(\widetilde{\mathcal{C}})${\rm ;} then the minimal part of $J$ is the radical ideal $I=P\cap (I:P)=P\cap (x_0,x_2,x_4,x_6)$ and $Q:=(I:P)+P=(x_0,x_2,x_4,x_6, x_1x_5-x_3^2)$ is the only embedded prime of $R/J$. \item[{\rm (b)}] The generators of $J$ are algebraically independent over $k$ and the syzygy matrix of $J$ has maximal linear rank. In particular, $\det \mathcal{C}$ is homaloidal. \end{enumerate} \end{Proposition} \demo Since the data are so very explicit and of small size, one can set up a computation in {\sc Macaulay\/} \cite{Macaulay} to check all the assertions. We choose instead to play the theory against the background as much as possible before resorting to a computer calculation. (a) First, the present situation confronts two matrices by adapting a known general principle (\cite[Lemme 2.3]{GP}), which in this case tells that the $2$-minors of $\mathcal{C}$ are the $2$-minors of $\widetilde{\mathcal{C}}$ excluding the minor $x_1x_5-x_3^2$ of columns $2$ and $4$ of the latter. (Of course this can also be checked directly by inspection.) Thus, in terms of the stated notation, one has $P=(I,b)$, where $b=:x_1x_5-x_3^2$. Since $P$ is a prime ideal of codimension $4$, $I$ has codimension at least $3$. But $I\subset (x_0,x_2,x_4,x_6)$, the latter being a prime ideal of codimension $4$ and, clearly, a minimal prime thereof. Therefore, these two prime ideals are minimal primes of $R/I$. We claim that their intersection is contained in $I$. To see this, it suffices to show that $b\cdot(x_0,x_2,x_4,x_6)\subset I$ as $b$ is a non-zero-divisor modulo $(x_0,x_2,x_4,x_6)$. But this is the content of Laplace rule as applied to suitable $2\times 3$ submatrices of $\widetilde{\mathcal{C}}$ involving columns $2$ and $4$. (One can similarly show that $I:P=(x_0,x_2,x_4,x_6)$ as written on the statement, but we will have no use for it.) We now deal with the gradient ideal $J$. In particular, we can express the partial derivatives of $f:=\det \mathcal{C}$ in terms of the $2$-minors of $\widetilde{\mathcal{C}}$ rather than those of $\mathcal{C}$ itself, whereby $x_1x_5-x_3^2$ will not show. Letting $\Delta_{ij}$ denote the $2$-minor of $\widetilde{\mathcal{C}}$ with $i$th and $j$th columns ($i<j$), one has {\small \begin{equation}\label{gens_of_J_cat} \{f_0=\Delta_{45}, f_1=-\Delta_{35}, f_2= 2\Delta_{34}-\Delta_{25}, f_3=\Delta_{15}, f_4=2\Delta_{23}-\Delta_{14}, f_5=-\Delta_{13}, f_6=\Delta_{12}\} \end{equation} } We first check that $J$ has codimension $4$. Namely, the sequence $f_0,f_2, f_5,f_6$ is a regular sequence. In order to see this the easiest is perhaps to note that the three minors $f_0, f_5,f_6$ are known to form a regular sequence by specializing from the generic case (with $(x_2,x_3,x_4)$ being a minimal prime of minimal codimension). Then show that $f_2$ is a nonzerodivisor thereof -- to get around this point we resort to the computer since the structure of the associated primes of $R/(f_0,f_5,f_6)$ is rather involved. As a result, $P$ and $I:P$ are minimal primes of maximal dimension of $R/J$, hence $I=P\cap (I:P)$ contains the unmixed part $J^{\rm un}$ of $J$. In order to show that $J^{\rm un}=I$ one would first show that $R/J$ has no minimal primes other than $P$ and $I:P$, then that $J^{\rm un}$ is a radical ideal. The multiplicity of $R/J$ is at least $e(R/P)+e(R/(I:P))=5+1$, where the second of these is obvious and the first follows from the fact that the Eagon--Northcott resolution of $R/P$ is pure and $2$-linear. However, showing before hand that $e(R/J)\leq 6$ seems like a tough matter. We found no easy hand calculation for this part, not to mention the resources one had in the Hankel case that we lack here. As a dissonant note, here $I^2\subset J$ but $I^2\not\subset JI$ which dashes our hope for any reduction theory to come in. Resorting to {\em Macaulay} one easily computes the equality $J^{\rm un}=I$. As a result, $e(R/J)=6$ and hence, using the exact sequence $0\rar I/J\lar R/J \lar R/I \rar 0$, where $R/J$ and $R/I$ have same dimension and multiplicity, one derives that the annihilator $R/J:I$ has codimension at least $5$. This completes the proof that $R/J$ has no other minimal primes and any embedded prime has codimension at least $5$. At this point, an additional computation shows that $J:I=(x_0,x_2,x_4,x_6,x_1x_5-x_3^2)$ and this is then the only embedded prime of $R/J$. (b) When char$(k)=0$, the partial derivatives are algebraically independent over $k$ if and only if the Hessian determinant does not vanish (both express the condition that the polar map has image of maximal dimension). If one is in peace by accepting the result in Example~\ref{parabolic_exs}, the Hessian determinant is clearly nonzero. Since the non-vanishing is much less precise than that result, we might choose to proceed directly, as follows. First, a tedious but straightforward calculation gives the Hessian matrix: $$\left( \begin{matrix} 0 & 0 & 0 & x_6 & -x_5 & -x_4 & x_3\\ 0 & 0 & -x_6 & 0 & 2x_4 & 0 & -x_2\\ 0 & -x_6 & 2x_5 & -x_4 & -x_3 & 2x_2 & -x_1\\ x_6 & 0 & -x_4 & 0 & -x_2 & 0 & x_0\\ -x_5 & 2x_4 & -x_3 & -x_2 & 2x_1 & -x_0 & 0\\ -x_4 & 0 & 2x_2 & 0 & -x_0 & 0 & 0\\ x_3 & -x_2 & -x_1 & x_0 & 0 & 0 & 0 \end{matrix} \right) $$ Evaluating at $(0,0,1,0,0, 1, 1)\in \mathbb{Q}^7\subset k^7$ yields the numerical matrix $$ \left( \begin{matrix} 0 & 0 & 0 & 1 & -1 &0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 &-1\\ 0 & -1& 2 & 0 & 0 & 2 & 0\\ 1 & 0 & 0 & 0 & -1 & 0 & 0\\ -1 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 &0\\ 0 & -1& 0 & 0 & 0 & 0 & 0 \end{matrix} \right) $$ whose determinant is immediately computed by Laplace: its value is $8\in \mathbb{Q}$. Therefore the numerical matrix is non-singular and hence the Hessian determinant does not vanish. We next show that the linear rank of the partials is maximal, i.e., $6$. Together they imply by Proposition~\ref{polar_and_lineartype} that $f$ is homaloidal. Clearly, from the form of the partial derivatives in (\ref{gens_of_J_cat}), any linear syzygy of those is a linear syzygy of the ideal $I$ of $2$-minors of $\mathcal{C}$. More precisely, order the $2$-minors of so as to adjust to the order in which they appear along the partials in (\ref{gens_of_J_cat}). Then a linear syzygy $(l_0,\ldots,l_8)^t$ of this ordered minors gives rise to one of (\ref{gens_of_J_cat}) if and only $l_2=-2l_3$ and $l_5=-2l_6$ in which case the former will have coordinates $(l_0,-l_1,-2l_3,l_4,-2l_6,-l_7, -l_8)^t$. In order to search for such syzygies of $I$ one resorts to its presentation. We note that though $I$ is actually a nice Gorenstein ideal -- being a specialization of the generic case -- it is easier to work with $P$ since its presentation is a piece of the Eagon--Northcott complex, whereby the syzygies derive from the Laplace relations coming from the five $2\times 3$ submatrices of $\widetilde{\mathcal{C}}$. Thus, with the previous order of the generators of $I$, leaving out the syzygies of $P$ that involve the extra minor $x_1x_5-x_3^2$ with a nonzero coefficient, one obtains the following matrix: {\small $$ \left( \begin{array}{cccccccccccccccc} 0& 0& 0& 0& x_0 & 0& x_4 & 0& 0& 0& 0& x_2 & 0 & -x_2 & 0& x_1\\ 0& 0& 0& x_0 & 0& 0& -x_5 & x_1 & 0& 0& x_2 & 0& -x_3 & x_3 & 0& 0\\ 0& 0& x_0 & 0& 0& 0& x_6 & 0& x_2 & 0& 0& 0& 0 & -x_4 & x_1 & x_3\\ 0& x_0 & 0& 0& 0& 0& 0& -x_2 & 0& x_2 & 0& 0& x_4 & 0& 0& -x_3 \\ 0& -x_1 & 0& -x_2 & -x_3 & 0& 0& 0& 0& -x_3 & -x_4 & -x_5 & 0 & 0& 0& 0\\ x_0 & 0& 0& 0& 0& x_2 & 0& x_4 & 0& 0& 0& 0& -x_6 & 0& x_3 & x_5 \\ 0& 0& -x_2 & 0& x_4 & 0& 0& 0& -x_4 & 0& 0& x_6 & 0& 0& -x_3 & 0\\ -x_1 & 0& x_3 & x_4 & 0& -x_3 & 0& 0& x_5 & 0& x_6 & 0& 0 & 0& 0& 0\\ x_2 & x_4 & 0& 0& 0& x_4 & 0& 0& 0& x_6 & 0& 0& 0 &0& x_5 & 0 \end{array} \right) $$ } where the last two syzygies cannot be expressed as Laplace relations and compensate for the omission of the Laplace relations coming from the $2\times 3$ submatrices of $\widetilde{\mathcal{C}}$ fixing columns $2$ and $4$. Let $\mathcal{S}_j\, (1\leq j\leq 16)$ denote the $j$th column of this matrix. An additional restriction is that if a $k$-linear combination of a set of these columns is to produce a syzygy $(l_0,\ldots,l_8)^t$ such that $l_2=-2l_3$ and $l_5=-2l_6$ then the set of entries along the $3$rd row of the involved columns in the combination has to match the set of entries along the $4$th row, and the same regarding the $6$th and $7$th rows. By inspection, we get at least the following sets $$\{\mathcal{S}_2,\mathcal{S}_3,\mathcal{S}_6\}, \{\mathcal{S}_8,\mathcal{S}_9,\mathcal{S}_{10}\}, \{\mathcal{S}_{12},\mathcal{S}_{13},\mathcal{S}_{14}\}, \{\mathcal{S}_5,\mathcal{S}_8,\mathcal{S}_9,\mathcal{S}_{10}\}. $$ Each one of these sets gives rise to a linear syzygy of $I$ satisfying the required condition, from which we derive as explained the corresponding linear syzygy of $J$. This gives $4$ linear syzygies of $J$. Furthermore, we have two additional linear syzygies coming from the Laplace rule as applied to the generators $f_1,f_3,f_5$ formed with the corresponding $2\times 3$ submatrix of $\widetilde{\mathcal{C}}$. Therefore, the following linear syzygies are obtained: $$ \left( \begin{array}{cccccc} 0 & 0& 0 & 0 & x_2 & x_0 \\ x_0& x_2 & 0 & x_1 & 2/3x_3 & 2/3x_1 \\ 0 & 0& x_0 & -1/4x_2 & 2/3x_4 & 1/3x_2 \\ x_2 & x_4 & x_1 & 3/4x_3 & 1/3x_5 & 0\\ 0 & 0& 2x_2& -1/2x_4 & 1/3x_6 & -1/3x_4 \\ x_4 & x_6& 2x_3& 1/2x_5 & 0 & -2/3x_5 \\ 0 & 0& 3x_4& -3/4x_6 & 0 & -x_6 \end{array} \right) $$ Evaluating the upper $6\times 6$ submatrix at $(1,0,0,1,1,0,0)$ gives the following matrix $$ \left( \begin{array}{cccccc} 0 & 0 & 0 & 1& 0 & 0\\ 0 & 1 0 & 0 & 0 & 2/3\\ 0 & 0 & 1 0 & 0 & 2/3\\ 1 0 & 0 & 0 & 3/4 & 0\\ 0 & 0 & 0 & -1/3 & -1/2 & 0\\ 0 & 1 & 2 & 0 & 0 & 0 \end{array} \right) $$ whose determinant is easily calculated by Laplace and has value $-1$. Therefore, the rank of the above matrix of linear syzygies is $6$. \qed \begin{Remark}\label{dual_variety}\rm A computation with {\em Macaulay} (\cite{Macaulay}) actually shows that $J$ is of linear type and the inverse map has degree $6$ (maximum possible in this situation). Therefore, the inversion factor has degree $11$ and the Hessian determinant is a factor thereof, with complementary factor the equation $g$ of the dual variety to the twisted cubic as found in Example~\ref{parabolic_exs}. This fact clamors for a geometric explanation. \end{Remark} \begin{Question}\rm For what values of $1\leq r\leq m$ is the determinant $f_{m,r}$ of the generic $r$-leap catalecticant homaloidal? \end{Question} We believe that the answer is: $f_{m,r}$ is homaloidal if and only if $r=m$ or $r=m-1$. Next is some computational and theoretical evidence. \medskip \subsubsection {Additional cases} Consider the $3$-leap $4\times 4$ catalecticant \smallskip $$\mathcal{C}=\mathcal{C}_{4,3}=\left( \begin{array}{cccc} x_0 & x_1 & x_2 & x_3\\ x_3 & x_4 & x_5 & x_6\\ x_6 & x_7 & x_8 & x_9\\ x_9 & x_{10} & x_{11} & x_{12} \end{array}. \right) $$ We sketch the main features of this case, including the fact that $\det \mathcal{C}$ is homaloidal. Let $J\subset R:=k[x_0,\ldots,x_{12}]$ denote the gradient ideal of $\det \mathcal{C}$. Then, by an argument that mixes computational and conceptual results, the following hold. \begin{enumerate} \item[{\rm (a)}] (Ideal structure) Let $I:=I_3(\mathcal{C})\subset R\,$ and $P:=I_3(\widetilde{\mathcal{C}})$, where $$\widetilde{\mathcal{C}}:= \left( \begin{array}{ccccccc} x_0 & x_1 & x_2 & x_3 & x_4 & x_5 & x_6\\ x_3 & x_4 & x_5 & x_6 & x_7 & x_8 & x_9\\ x_6 & x_7 & x_8 & x_9 & x_{10} & x_{11} & x_{12} \end{array} \right). $$ Then the unmixed part of $J$ is the codimension $4$ prime ideal $I$ and moreover, $P$ is a codimension $5$ embedded prime of $R/J$ with $I=J:P$. The ideal $I$ has codimension $4$ since it specializes from the generic case -- and is, in fact, Gorenstein. The matrix $\widetilde{\mathcal{C}}$ is obtained by analogy to the Gruson--Peskine procedure, but of course $I$ and $P$ differ extensively in both in number of generators as in codimension; both ideals are however prime (cf. the case $m=3$, where $I$ was not prime) and are the associated primes of $R/J$. A computation with {\em Macaulay} (\cite{Macaulay}) gives that the unmixed part of $J$ is $I$ and the latter is $J:P$. \item[{\rm (b)}] (Structure of the partial derivatives) By inspection, $10$ out of the $13$ partial derivatives are $3$-minors of $\mathcal{C}$ up to signs. Of these, $8$ are up to signs the $3$-minors of the two submatrices of $\mathcal{C}$ consisting of columns $1,2,4$ and $1,3,4$, respectively. Each four generate a prime ideal of height $2$, hence the sum has height $\geq 3$. But then it has actually codimension $3$ since it is clearly contained in $I_2(\psi)$, where $\psi$ denotes the Hankel matrix formed with the first and the last columns of $\mathcal{C}$. Denote by $J'\subset J$ generated by these $8$ minors. A guess, inspired on the case of the general $4\times 4$ matrix, was that its unmixed part is $I_2(\psi)$. A computation with {\em Macaulay} confirmed this and, in addition, gave the equality $J':I_2(\psi)=I$, which was also expected to hold by specialization from the $4\times 4$ general matrix. \item[{\rm (c)}] (Linear syzygies) The syzygy matrix of $J$ has linear rank $11$. The above ideal $J'\subset J$ has $6$ independent linear syzygies by applying the Laplace trick. Extending these syzygies to syzygies of all partial derivatives by filling the remaining slots with zeros yields a submatrix of rank $6$ of the syzygy matrix of $J$. Five additional independent linear syzygies are found by the method of Proposition~\ref{2-cat3x3} and Proposition~\ref{hankel3x3}, bringing up a linear submatrix of rank $11$. Of course, this can also be computed with {\em Macaulay}. \item[{\rm (d)}] (Syzygetic part) The syzygetic part $\delta(J)$ has $4$ minimal generators (in standard degree $4$). The syzygetic part is the kernel $\delta(J)$ of the natural map $H_1\rar R^{13}/JR^{13}$ induced by applying $\otimes_RR/J$ to the presentation sequence $0\rar Z\lar R^{13}\lar J\rar 0$ of $J$; here $H_1$ denotes the first Koszul homology module on the partial derivatives. More explicitly, $\delta(J)=Z\cap JR^{13}/B$, where $B$ denotes the image of the Koszul map on those generators. Note that $H_1$ is a graded $R$-module with the standard grading of $R$ and that $\delta(J)$ inherits this grading. Write $\delta(J)=\oplus_{t\geq 1} \delta(J)_{t+3}$ By \cite[Section 1]{SV} one knows that the minimal number of generators of the $R$-module $\delta(J)_{t+3}$ coincides with the cardinality of a $k$-basis of $\mathcal{J}_{(t,2)}$ consisting of elements of bidegree $(t,2)$, where $\mathcal{J}$ denotes the bigraded presentation ideal of the Rees algebra of $J$ over $R$. In particular, the minimal generators of $\delta(J)$, being all of standard degree $4$, give a basis $Q$ of the $k$-vector space of Rees equations of bidegree $(1,2)$. \item[{\rm (e)}] (Conclusion) The partial derivatives are algebraically independent over $k$ and $\det \mathcal{C}$ is homaloidal. Consider the set consisting of the forms of $\mathcal{J}$ of bidegree $(1,1)$ (linear syzygies of $J$) and the set $Q$ above of forms of bidegree $(1,2)$. One would claim that the rank of the Jacobian matrix of this set with respect to the variables $\mathbf{x}$ is $12$. The idea is to look at this matrix in a suitable way to build up a non-zero $12\times 12$ minor thereof involving at least one of the last $4$ rows with quadratic entries in $\yy$. This is an additional technical challenge; alternatively, a computation with {\em Macaulay} gives the expected rank. By the general criterion of \cite[Theorem 2.18]{AHA}, $\det \mathcal{C}$ is homaloidal. \end{enumerate} \medskip For the $2$-leap $4\times 4$ catalecticant $$\mathcal{C}_{4,2}=\left( \begin{array}{cccc} x_0 & x_1 & x_2 & x_3\\ x_2 & x_3 & x_4 & x_5\\ x_4 & x_5 & x_6 & x_7\\ x_6 & x_{7} & x_{8} & x_{9} \end{array} \right) $$ we suspect that this case is not homaloidal. But alas, the sole evidence is computational: the linear rank is $6$, hence $3$ less than the maximum possible ($=9$). In addition, the Rees algebra has two generators of bidegree $(1,2)$ which come from computing the syzygetic part of the ideal. In principle, it could have additional equations of bidegree $(1,s)$ with $s>2$, although a direct (stalled) computation seems to give no new generators, so the Jacobian dual matrix would not have maximal rank. \begin{Question}\rm Is there an explicit formula for the linear rank of the gradient ideal in the case of catalecticants which depends solely on the size and leap? \end{Question} \section{Matrices. II: degenerations} As explained in the Introduction, this section is devoted to understanding the behavior of certain degenerations of the examples considered in the precious section. We chose to start with the Hankel degeneration since its homaloidal behavior is fully understood in \cite[Section 4.1]{CRS}, so we have nothing to add in this respect. Not so however its homological contents, so to say. In fact we will solve affirmatively the conjectural note stated in \cite[Remark 4.6 (c)]{CRS}, a step that, as mentioned there has strong implication to some of the technology developed. The actual remark there points to a reference that in fact never appeared and, somehow, can be thought of as being ``replaced'' by the present work. \subsection{Degeneration of Hankel matrices and their homology}\label{deg_of_cat} In this part we deal with the determinant of a so-called sub-Hankel matrix, considered in \cite[Section 4]{CRS}. Since a germ of these considerations existed prior to the latter paper, now used amongst the preliminaries of the present work, we will repeat some of its relevant aspects here. For consistency and appropriate referencing, we will keep the same notation as in \cite{CRS}. Let $x_0,\ldots,x_n$ be variables over a field $k$ and set $$ M^{(n)}=M^{(n)}(x_0,\ldots,x_n)= \left( \begin{matrix} x_0&x_1&x_2&\ldots &x_{n-2}&x_{n-1}\\ x_1&x_2&x_3&\ldots &x_{n-1}&x_n\\ x_2&x_3&x_4&\ldots &x_n&0\\ \vdots &\vdots &\vdots &\ldots&\vdots &\vdots \\ x_{n-2}&x_{n-1}&x_n&\ldots &0&0\\ x_{n-1}&x_n&0&\ldots &0&0\\ \end{matrix} \right ) $$ Note that the matrix has two tags: the upper index $(n)$ indicates the size of the matrix, while the variables enclosed in parentheses are the total set of variables used in the matrix. This detailed notation was introduced in \cite{CRS} as several of these matrices were considered with variable tags throughout. However, here we omit the list of variables if they are sufficiently clear from the context. This matrix will be called a {\sc generic sub--Hankel matrix}; more precisely, $M^{(n)}$ is the {\sc generic sub--Hankel matrix of order $n$ on the variables} $x_0,\ldots,x_n$. Throughout we fix a polynomial ring $R=k[\xx]=k[x_0,\ldots,x_n]$ over a field $k$ of characteristic zero. We will denote by $f^{(n)}(x_0,\ldots,x_n)$ the determinant of $M^{(n)}(x_0,\ldots,x_n)$ for any $r\geq 1$, and we set $f^{(0)}=1$. In \cite{CRS} the main objective was to prove that this determinant is homaloidal and explain the geometric contents of the corresponding polar map. Here we turn ourselves to the algebraic-homological behavior of the ideal $J\subset R$ generated by its partial derivatives. A common feature between the two approaches is a systematic use of a recurrence using the subideal $J_i\subset J$ generated by the first $i+1$ partial derivatives further divided by the gcd of these derivatives. We need the following results drawn upon \cite{CRS}. Throughout we set $f:=f^{(n)}$. \begin{Lemma}{\rm (\cite[Lemma 4.2]{CRS})}\label{prolegomena} If $n\geq 2$, then \begin{itemize} \item[{\rm (i)}] For $0\leq i\leq n-1$, one has \begin{equation}\label{g.c.d.s} \frac {\partial f } {\partial x_0}, \ldots, \frac{\partial f } {\partial x_i} \in k\left[x_{n-i},\ldots,x_{n}\right] \end{equation} and the g.c.d. of these partial derivatives is $x_n^{n-i-1}$ \item[{\rm (ii)}] For any $i\,$ in the range $1\leq i\leq n-1$, the following holds: \begin{equation}\label{basic_linear_relation} x_n\,\frac{\partial f }{\partial x_i}=-\sum_{k=0}^{i-1}\frac{2i-k}{i}\,\, x_{r-i+k}\, \frac{\partial f }{\partial x_k}. \end{equation} Moreover, \begin{equation}\label{perfect_linear_relation} x_n\, \frac{\partial f}{\partial x_n}=(n-1)x_0\,\frac{\partial f }{\partial x_0}+ (n-2)x_1\, \frac{\partial f}{\partial x_1}+\cdots + x_{n-2}\,\frac{\partial f} {\partial x_{n-2}} \end{equation} \end{itemize} \end{Lemma} \begin{Proposition}{\rm (\cite[Proposition 4.3 and its proof]{CRS})}\label{perfect_ideal} For every $1\leq i\leq n-1$, the ideal $J_i$ is perfect and linearly presented with recurrent Hilbert--Burch matrix of the form $$\varphi(J_i)=\left( \begin{array}{c@{\quad\vrule\quad}c} \raise5pt\hbox{$2\, x_{n-i}$}&\\ \raise5pt\hbox{${\frac{2i-1}{i}}\, x_{n-i+1}$}&\\ \vdots&\varphi(J_{i-1})\\ \raise5pt\hbox{$\frac{i+1}{i}\, x_{n-1}$}&\\ \multispan2\hrulefill\\ \lower3pt\hbox{$x_n$}&\lower3pt\hbox{{\boldmath $0$}} \end{array} \right) $$ \end{Proposition} \smallskip We next prove a few additional results not obtained in \cite{CRS}. \begin{Lemma}\label{multiplicities_annihilators} Keeping the above notation, set further $P=(x_{n-1},x_n)\subset R=k[\xx]$. Then \begin{enumerate} \item[{\rm (i)}] $P$ is the radical of $J$ and all the ideals $J_i, \, 1\leq i\leq n-1$ are $P$-primary. \item[{\rm (ii)}] The $R_P$-module $R_P/(J_{i})_P$ has length ${i+1} \choose {2}$. \item[{\rm (iii)}] The ideal of $R_P/(J_{n-1})_P$ generated by the forms $x_n,x_{n-1}^{n-1}$ has length ${n-1} \choose {2}$. \end{enumerate} \end{Lemma} \demo (i) This is clear from the form of these ideals: any prime ideal containing any of these has to contain $P$, which is clearly the unique minimal prime thereof. By Proposition~\ref{perfect_ideal}, each $J_i$ is perfect, hence $R/J_i$ is Cohen--Macaulay, thus implying that that $P$ is the only associated prime of $J_i$. (ii) By the primary case of the associativity formula for the multiplicities, one has $$e(R/J_{i})=\ell(R_P/(J_{i})_P)\, e(R/P)=\ell(R_P/(J_{i})_P),$$ since $P$ is generated by linear forms. On the other hand, from Proposition~\ref{perfect_ideal} one has the graded free resolution \begin{equation}\label{perfect_recurrence} 0\rar R(-(i+1))^{i}\rar R(-i)^{i+1}\rar R\rar R/J_{i}\rar 0 \end{equation} Applying the multiplicity formula for Cohen--Macaulay rings with pure resolution (\cite{HuMi}), one derives in this case $e(k[\xx]/J_{i})={{i+1} \choose {2}}$, as required. (iii) As pointed out previously, one has $(x_n, J_{n-1})= (x_n,x_{n-1}^{n-1})$. On the other hand, \begin{eqnarray} (x_n, J_{n-1})/J_{n-1}&\simeq &(x_n)/(x_n)\cap J_{n-1}\simeq \left(R/(J_{n-1}:x_n)\right)(1)\\ &=& \left(R/J_{n-2}\right) (1), \end{eqnarray} where the equality $(J_{n-1}:x_n)=J_{n-2}$ follows from Proposition~\ref{perfect_ideal}. Therefore $$\ell\left((x_n,x_{n-1}^{n-1})_P/(J_{n-1})_P\right)=\ell(R/J_{n-2})_P={{n-1} \choose {2}},$$ by part (ii). \qed \begin{Lemma}\label{equality} With the same notation, one has $$J/J_{n-1}\simeq \frac{R}{(x_n,x_{n-1}^{n-1})}\,(-(n-1)).$$ \end{Lemma} \demo The following isomorphisms of $R$-graded modules are immediate: \begin{equation}\label{colonisomorphism} \frac{J}{J_{n-1}}= \frac{\left(J_{n-1},\frac{\partial f}{\partial x_n}\right)}{J_{n-1}}\simeq \frac{\left(\frac{\partial f}{\partial x_n}\right)}{J_{n-1}\bigcap\left(\frac{\partial f}{\partial x_n}\right) }\simeq \frac{R}{\left(J_{n-1}\colon \frac{\partial f}{\partial x_n}\right)}\left(n-1\right). \end{equation} We claim that $\left(J_{n-1}\colon \frac{\partial f}{\partial x_n}\right)=(x_n, J_{n-1})$. Once this is proved, we will have $\left(J_{n-1}\colon \frac{\partial f}{\partial x_n}\right)=(x_n,x_{n-1}^{n-1})$ because it follows easily from the structure of $f$ and its derivatives that $(x_n, J_{n-1})=(x_n,x_{n-1}^{n-1})$. Now, by (\ref{perfect_linear_relation}), $(x_n, J_{n-1})\subset \left(J_{n-1}\colon \frac{\partial f}{\partial x_n}\right)$ as trivially $J_{n-1}\subset \left(J_{n-1}\colon \frac{\partial f}{\partial x_n}\right)$. For the reverse inclusion, we proceed as follows. Let $r\in (J_{n-1}:\partial f/\partial x_n)$ and write $P:=(x_{n-1},x_n)$. Since $J_{n-1}$ is a $P$-primary ideal and $\partial f/\partial x_n \not\in J_{n-1} $ then $r\in P=(x_n,x_{n-1})$. Next rewrite $r=r(x_0,\ldots,x_n)$ as $$r=x_n h(x_0,\ldots,x_n)+r'(x_0,\ldots,x_{n-1})\in P,$$ where $x_n$ divides no term on the second summand. Then $r'(x_0,\ldots,x_{n-1})\in (x_{n-1})$, so let $l\in \mathbb{N}$ be such that $r'(x_0,\ldots,x_{n-1})= x^l_{n-1}r''(x_0,\ldots,x_{n-1})$ and $x_{n-1}$ does not divide $r''(x_0,\ldots,x_{n-1})$. It follows that $r''(x_0,\ldots,x_{n-1})\not\in P$. Recall that $(x_n,J_{n-1})=(x_n,x_{n-1}^{n-1})$. Since $J_{n-1}$ is P--primary and $r''x^l_{n-1}\partial f/\partial x_n \in J_{n-1}$ then $x^l_{n-1}\partial f/\partial x_n \in J_{n-1}$. Thus, for the required reverse inclusion it is enough to show that $l\geq n-1$. Write \begin{equation}\label{formula 4} x^l_{n-1}\frac{\partial f}{\partial x_n} = \sum^{n-1}_{i=0} r_i\frac{\partial f}{\partial x_i}, \end{equation} where $r_i\in R.$ Writing $r_i=x_ng_{i}(x_0,\ldots,x_n)+g'_{i}(x_0,\ldots,x_{n-1})$ and drawing upon (\ref{basic_linear_relation}) yields \begin{eqnarray} x^l_{n-1}\partial f/\partial x_n &=& \sum^{n-1}_{i=0} r_i\partial f/\partial x_i =\sum^{n-1}_{i=0}(x_ng_{i}(x_0,\ldots,x_n)+g'_{i}(x_0,\ldots,x_{n-1}))\partial f/\partial x_i\\ &=&\sum^{n-1}_{i=0}g_{i}(x_0,\ldots,x_n)(-\sum^{i-1}_{k=0}{ \frac{2i-k}{i}x_{n-i+k} \frac{\partial f}{\partial x_k}) +\sum^{n-1}_{i=0}g'_{i}(x_0,\ldots,x_{n-1})\partial f/\partial x_i}. \end{eqnarray} By repeating this process for $g_i$ and so forth, after finitly many steps we may suppose that the coefficients of $\partial f/\partial x_i $ in (\ref{formula 4}) do not involve $x_n$. Multiplying both sides of (\ref{formula 4}) by $x_n$ and using (\ref{perfect_linear_relation}) yields a syzygy of the ideal $J_{n-1}$: \begin{equation}\label{partial} ((n-1)x^l_{n-1}x_0 -x_n r_0)\frac{\partial f} {\partial x_0} + \cdots +(x_{n-2} x^l_{n-1}-x_n r_{n-2})\frac{\partial}{\partial x_ {n-2}} - x_n r_{n-1}\frac{\partial f}{\partial x_{n-1}}=0. \end{equation} Thinking of this syzygy as a column vector $K$, we can write \begin{equation}\label{c} K=\alpha_1C_1+\cdots+\alpha_{n-1}C_{n-1} \end{equation} for suitable $\alpha_i\in k[x_0,\cdots,x_n]$, where $C_i$ denotes the $i$th column of the syzygy matrix of $J_{n-1}$ as in Proposition~\ref{perfect_ideal}: $$\begin{pmatrix} 2x_1 & 2x_2 &2x_3 &\dots &2x_{n-1}\\ \frac {2n-3}{n-1}x_2 &\frac {2n-5}{n-2}x_3& \frac{2n-7}{n-3}x_4 &\dots& x_n\\ \vdots & \vdots & \vdots & \dots &\vdots\\ \frac {n+2}{n-1}x_{n-3}& \frac{n}{n-2}x_{n-2}&\frac{n-2}{n-3}x_{n-1}&\dots & 0\\ \frac{n+1}{n-1}x_{n-2} & \frac{n-1}{n-2}x_{n-1} &x_n& \dots& 0\\ \frac{n}{n-1}x_{n-1}& x_n &0 & \dots & 0\\ x_n & 0 & 0 & \dots & 0 \end{pmatrix}_{n\times (n-1)} $$ This affords the following relations by looking at the last two rows: $$ x_{n-2} x^l_{n-1}-x_n r_{n-2}=-r_{n-1}\frac{n}{n-1}x_{n-1} + \alpha_2 x_n \quad \mbox{\rm and} \quad \alpha_1=-r_{n-1}.$$ Since we already assumed $r_i\in k[x_0,\ldots,x_{n-1}]$ for all $i$, then $\alpha_2=-r_{n-2}$. Therefore $$-r_{n-1}\frac{n}{n-1}x_{n-1}=x_{n-2} x^l_{n-1}.$$ In particular $l\geq 1$ and hence $ r_{n-1}=-\frac{n-1}{n} x_{n-2} x^{l-1}_{n-1}.$ \smallskip Inspecting the $(n-2)$'th row yields: $$2x^l_{n-1}x_{n-3}-x_n r_{n-3}=-\frac{n+1}{n-1} r_{n-1}x_{n-2}-\frac{n-1}{n-2}r_{n-2}x_{n-1}+\alpha_3 x_n.$$ Since $r_{n-1},r_{n-2}\in k[x_0,\ldots,x_{n-1}]$ then $\alpha_3=-r_{n-3}$. Substituting for $r_{n-1}$ obtains \begin{eqnarray} 2x^l_{n-1}x_{n-3}&=& -\frac{n+1}{n-1} r_{n-1}x_{n-2}-\frac{n-1}{n-2}r_{n-2}x_{n-1}\\ &=& \frac{n+1}{n} x^2_{n-2} x^{l-1}_{n-1}-\frac{n-1}{n-2}r_{n-2}x_{n-1} \end{eqnarray} Then necessarily $l\geq 2$ and furthermore \begin{equation}\label{sn} r_{n-2}=\frac {n-2}{n-1}(-2x^{l-1}_{n-1}x_{n-3}+\frac{n+1}{n} x^2_{n-2} x^{l-2}_{n-1})=x^{l-2}_{n-1}s_{n-2}, \end{equation} $\text{for some } s_{n-i}\in k[x_0,\ldots,x_{n-1}].$ Since $x_n$ appears exactly once on each row of the syzygy matrix below the first one, the argument inducts yielding that for every $1\leq i \leq n-2$ one has $\alpha_i=-r_{n-i}$ and $r_{n-i} = x^{l-i}_{n-1}s_{n-i}$ with $s_{n-i}\in k[x_0,\ldots,x_{n-1}]$ and $l\geq i$. In particular, $l\geq n-2$ and $r_2= x^{l-n+2}_{n-1}s_2$. \smallskip Finally, from the first row of $K$, we have: $$(n-1)x_0x^l_{n-1}-x_nr_0=-2x_1r_{n-1}-2x_2r_{n-2}-\ldots-2x_{n-2}r_2-2x_{n-1}r_1.$$ Hence $r_0=0$. Rearranging yields \begin{equation}\label{r1} -2x_1r_{n-1}-2x_2r_{n-2}-\ldots-2x_{n-1}r_1-(n-1)x_0x^l_{n-1}=2x_{n-2}r_2=2x_{n-2}x^{l-n+2}_{n-1}s_2. \end{equation} Since the left hand side is divisible by $x_{n-1}$ so is the right hand side. Thus $l-n+2>0$. In other words, $l\geq n-1,$ as desired. \qed \begin{Proposition}\label{resolution of J} Let $J$ denote the gradient ideal of the sub-Hankel determinant. Then the minimal graded resolution of $R/J$ has the form \begin{equation} 0\rar R(-(2n-1))\rar R(-n)^n\oplus R(-2(n-1))\stackrel{\phi}{\rar} R^{n+1}(-(n-1))\rar R. \end{equation} In particular, $R/J$ is strictly almost Cohen--Macaulay. \end{Proposition} \demo By Lemma~\ref{equality} we have a free minimal resolution \begin{equation} {\cal C}\colon\quad 0\rar R(-(2n-1))\rar R(-n)\oplus R(-2(n-1))\rar R(-(n-1)) \rar J/J_{n-1}\rar 0, \end{equation} On the other hand, (\ref{perfect_recurrence}) gives a resolution \begin{equation} {\cal J}_{n-1}\colon\quad 0\rar R(-n)^{n-1}\rar R(-(n-1))^n\rar R\rar R/J_{n-1}\rar 0 \end{equation} Since the inclusion $J/J_{n-1}\subset R/J_{n-1}$ induces a map of complexes ${\cal C}\rar {\cal J}_{n-1}$, the resulting mapping cone is a resolution of $R/J$ (\cite[Exercise A3.30]{Eisenbook}): \begin{equation}\label{resolution3} \quad 0\rar R(-(2n-1))\rar R(-n)^n\oplus R(-2(n-1))\stackrel{\phi}{\rar} R^{n+1}(-(n-1))\rar R\rar R/J\rar 0, \end{equation} (where the right end tail $R\oplus J/J_{n-1}\rar R/J_{n-1}\rar 0$ has been replaced by $R\rar R/J\rar 0$). Moreover, since for every relevant index $i$, the shifts of $({\cal J}_{n-1})_i$ are strictly smaller than those of $({\cal C})_i$, it follows by [loc. cit.] that (\ref{resolution3}) is minimal. In particular, one reads from it that the linear part $\phi_1$ has rank $n$, hence maximal. \qed \medskip Let us dwell a little more on the details of the mapping cone in the previous proof. It is of the form $$\begin{CD} 0 @>>> R^{n-1} @> \varphi_{n-1} >> R^n @> (\frac{\partial f}{\partial x_0},\ldots,\frac{\partial f}{\partial x_{n-1}}) >> R@>>> \frac {R}{J_{n-1}}@>>> 0\\ && @A {g_4}AA @A{g_3}AA @A{g_2}AA @A{g_1} AA && \\ 0 @>>> R @> (x^{n-1}_{n-1},-x_n)^t>> R^2 @>(x_n,x^{n-1}_{n-1})>> R @>>> J/J{n-1} @>>> 0 \end{CD}\ $$ \smallskip Note that $g_2$ is multiplication by $\partial f/ \partial x_n$ and the induced map $g_3$ is given by the following $n\times2$ matrix: $$g_3=\begin{pmatrix} (n-1)x_0 & r_0\\ (n-2)x_1 & r_1\\ \vdots\\ x_{n-2} & r_{n-2}\\ 0 & r_{n-1} \end{pmatrix}.$$ We next find out the entries of $g_4$. By the commutativity of the mapping cone diagram, we have $$\varphi_{n-1}\circ g_4=g_3 \circ \begin{pmatrix} x^{n-1}_{n-1}\\ -x_n \end{pmatrix}= \begin{pmatrix} (n-1)x_0 & r_0\\ (n-2)x_1 & r_1\\ \vdots\\ x_{n-2} & r_{n-2}\\ 0 & r_{n-1} \end{pmatrix}\begin{pmatrix} x^{n-1}_{n-1}\\ -x_n \end{pmatrix}=\begin{pmatrix} (n-1)x^{n-1}_{n-1}x_0 -x_n r_0 \\ (n-2)x^{n-1}_{n-1}x_1 - x_n r_1 \\ \vdots\\ x_{n-2} x^{n-1}_{n-1}-x_n r_{n-2}\\ -x_n r_{n-1} \end{pmatrix} $$ where the rightmost matrix is the syzygy $K$ in (\ref{partial}), viewed as a column vector, where $l=n-1$. By reasoning as in the argument that ensues (\ref{c}), one gets that the $i$th entry of $g_4$ is $- r_{n-i}$, $\,1\leq i\leq n-1$. As a result, the leftmost map in (\ref{resolution3}) is $\psi:=(x^{n-1}_{n-1}, -x_n, g_4^t)=(x^{n-1}_{n-1}, -x_n,-r_{n-1},\ldots,-r_1)$. \smallskip We make use of this in the following result, where $J$ is the gradient ideal of the sub-Hankel determinant. \begin{Theorem} The associated primes of $R/J$ are $(x_{n-1}, x_n)$ and $(x_{n-2},x_{n-1}, x_n)$. \end{Theorem} \demo Clearly, $P:=(x_{n-1}, x_n)$ is a minimal prime thereof. Since $P$ is also the radical of $J$ (Proposition~\ref{multiplicities_annihilators} (i)) then there are no additional minimal primes. To argue for embedded primes we proceed as follows. Let as above $\psi$ denote the tail map of (\ref{resolution3}). Since $R/J$ has homological dimension $3$, any $Q\in \Ass (R/J)$ has codimension at most $3$ and a prime $Q$ of codimension $3$ containing $J$ is an associated prime of $R/J$ if and only if $Q\supset I_1(\psi)$ (see, e.g., \cite[Corollary 20.14(a)]{Eisenbook} for the last part). As seen above, $I_1(\psi)=(x^{n-1}_{n-1}, x_n,-r_{n-1},\ldots,-r_1)$. On the other hand, we know from the proof of Proposition~\ref{equality} that $r_{n-i} = x_{n-1}^{n-i-1}s_{n-i}$, where $s_{n-i}\in k[x_0,\ldots,x_{n-1}]$ for all $1\leq i \leq n-1$. Additionally, the inductive argument in this proof and relation (\ref{sn}) also show that, for $1\leq i \leq n-2$, $s_{n-i}=x_{n-1}t_{n-i}+x_{n-i}^i $, where $t_{n-i}\in k[x_0,\ldots,x_{n-1}]$ . Therefore the condition for such a prime $Q$ to be an associated prime of $R/J$ is that it contain the ideal $(x_n, x_{n-1},x_{n-2}s_2)=(x_n, x_{n-1},x_{n-2}(x_{n-1}t_2+x_{n-2}^{n-2}))=(x_n, x_{n-1},x_{n-2}^{n-1} )$. It follows that $Q=(x_{n-2},x_{n-1}, x_n)$, as stated. \qed \begin{Corollary} Set $P:=\sqrt{J}=(x_{n-1}, x_n)$. Then the $P$-primary component of $J$ is $J_{n-2}$. \end{Corollary} \demo Since $J_{n-2}$ is a $P$-primary ideal (Proposition~\ref{multiplicities_annihilators} (i)), it is equivalent to show the equality $J_P={(J_{n-2})}_P$. Since $J\subset J_{n-2}$ we will be done by showing the equality of lengths $\emph{l}(\frac {R_P}{J_P})=\emph{l}( \frac {R_P}{{(J_{n-2})}_P})$. Now, with the present data, by the associativity formula one has $\emph{l}(\frac {R_P}{J_P})=e(R/J)$ -- the multiplicity of $R/J$. To compute the latter we deploy the numerator of the Hilbert series of $R/J$ in terms of the graded Betti numbers of $R/J$ as in (\ref{resolution3}); it obtains $$S(t):= 1-(n+1)t^{n-1} + t^{2n-2} + nt^n - t^{2n-1}.$$ Taking second derivatives, evaluating at $t=1$, etc., finally gives $e(R/J)= (n-1)(n-2)/2 = {n-1\choose 2}$. But the latter coincides with $\emph{l}(R_P / {(J_{n-2})}_P)$ by Lemma~\ref{multiplicities_annihilators} (ii). \qed \begin{Theorem} The gradient ideal $J$ of the sub-Hankel determinant is of linear type. \end{Theorem} \demo By definition, we have to show that the natural surjective $R$-homomorphism $\mathcal{S}_R(J)\surjects \mathcal{R}_R(J)$ from the symmetric algebra of $J$ to its Rees algebra is injective. One knows that this is the case if and only if $\mathcal{S}_R(J)$ is a domain. Set $\mathcal{S}_R(J)\simeq R[y_0,\cdots,y_n]/\mathcal{L}$, where $\mathcal{L}$ is the ideal generated by the $1$-forms coming from the syzygies of $J$. We will argue as follows: since $x_n$ belongs to the radical of $J$, $J_{x_n}$ is the unit ideal in $R_{x_n}$. Now, suppose one shows that $x_n$ is a non zero-divisor modulo $\mathcal{L}$. Then $\mathcal{L}_{x_n}$ is the defining ideal of the symmetric algebra of $J_{x_n}=R_{x_n}$, hence it is the zero ideal in a polynomial ring over a domain. In particular, it is a prime ideal, hence so must be $\mathcal{L}$. Therefore $\mathcal{S}_R(J)$ is a domain, thus the structural homomorphism is injective as observed. By a quirk, it will be easier to show first that $y_n$ is non-zero divisor modulo $\mathcal{L}$ and then that $(y_n,x_n)$ is a regular sequence modulo $\mathcal{L}$. In this case, since $\mathcal{S}_R(J)$ is a positively graded ring any permutation of a regular sequence is a regular sequence, hence $(x_n,y_n)$ is a regular sequence as well, in particular $x_n$ is a non-zero divisor over $\mathcal{S}_R(J)$. \textbf{Step 1.} $y_n$ is non-zero divisor modulo $\mathcal{L}$. Let $h=h(\textbf{x},\textbf{y})\in R[y_0,\ldots,y_n]$ be such that $y_n h(\textbf{x},\textbf{y})\in \mathcal{L} $. Say, $$y_n h=\sum_{i=1}^{n+1}h_i g_i,$$ for suitable $h_i=h_i(\xx,\yy) \in R[y_0,\ldots,y_n]$, where $$g_i=2x_{n-i}y_0 +\frac{2i-1}{i}x_{n-i+1}y_1+ \cdots +x_ny_i \;\;(1\leq i\leq n-1)$$ and $$g_n = (n-1)x_0y_0+(n-2)x_1y_1+\cdots +x_ny_n,\;\; g_{n+1}=\sum_{j=1}^{n-1} {r_jy_j} +x_{n-1}^{n-1}y_n$$ generate $\mathcal{L}$ (from (\ref{resolution3})), with $r_i$ being as in (\ref{partial}). Decompose further $h_i={h'_i}+y_n{h''_i}$, with ${h'_i}\in R[y_0,\ldots,y_{n-1}]. $ Then $$y_n h(\textbf{x},\textbf{y})=y_n\sum_{i=1}^{n-1}{h''_i}g_i + y_n{h''_n}g_n + {h'_n}x_ny_n + y_n{h''_{n+1}}g_{n+1} + {h'_{n+1}}x_{n-1}^{n-1}y_n.$$ Since in the right hand side the terms not divisible by $y_n$ must vanish, we get $$h=\biggl (\sum_{i=1}^{n-1}{h''_i} g_i + {h''_n}g_n + {h''_{n+1}}g_{n+1}\biggr) + {h'_n}x_n + {h'_{n+1}}x_{n-1}^{n-1}.$$ To show that $h\in \mathcal{L}$ it is then enough to check that $$ {h'_n}x_n+ {h'_{n+1}}x_{n-1}^{n-1}\in\mathcal{L}. $$ Now a form in $\mathcal{L}$ must vanish when evaluated at $y_i\mapsto \partial f/\partial x_i$ -- the generators of $J$. Letting $\partial\ff$ denote these partial derivatives we have $\frac{\partial f}{\partial x_n}h(\textbf{x},\partial\ff)=0$, hence $h(\textbf{x},\partial\ff)=0.$ Retrieving in terms of the expression of $h$ implies that $$ h'_n(\xx,\partial\ff)\,x_n+ h'_{n+1}(\xx,\partial\ff)\,x_{n-1}^{n-1}=0.$$ Since $h'_n ,h'_{n+1}\in R[y_0,\ldots,y_{n-1}]$, the form $h'_nx_n+ h'_{n+1}x_{n-1}^{n-1}$ belongs to the defining ideal of $\mathcal{R}_R(J_{n-1})$. By \cite[Proposition 4.3]{CRS}, $J_{n-1}$ is of linear type, hence $h'_nx_n+ h'_{n+1}x_{n-1}^{n-1}$ belongs to the defining ideal of $\mathcal{S}_R(J_{n-1})$, which is a subideal of $\mathcal{L}$ by the general theory developed in \cite{CRS}. This shows the contention. \medskip \textbf{Step 2.} $x_n$ is non-zero divisor modulo $(\mathcal{L},y_n)$. \smallskip One has $(\mathcal{L},y_n)=(g_1,\ldots,g_{n-1},g,h,y_n),$ where $g=(n-1)x_0y_0+(n-2)x_1y_1+\cdots+x_{n-2}y_{n-2}$ and $h=\sum_{j=1}^{n-1} {r_jy_j}.$ Then $$\frac{R[y_0,\ldots,y_n]}{(\mathcal{L},y_n)}\simeq \frac{R[y_0,\ldots,y_{n-1}]}{(g_1,\ldots,g_{n-1},g,h)}$$ hence we are to show that $x_n$ is a nonzerodivisor on the rightmost ring. Let then $\kappa=\kappa(\mathbf{x}, \mathbf{y}\setminus y_n)$ be a form in $R[y_0,\ldots,y_{n-1}]$ such that $x_n\kappa\in (g_1,\ldots,g_{n-1},g,h)$. Write $x_n\kappa=\sum_{j=1}^{n-1} \mu_jg_j+\mu g +\nu h$, for suitable forms $\mu_j,\mu,\nu\in R[y_0,\ldots,y_{n-1}]$. Evaluating $y_i\mapsto f_{x_i}:=\partial f/\partial x_i$ for $i=0,\cdots,n-1$, and taking in account the shape of $g$ and $h$, we have \begin{eqnarray} x_n\kappa(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})&=&\mu(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})g(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})\\ &+&\nu(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})h(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})\\ &=& -\mu(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})x_n\,f_{x_n} -\nu(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})x_{n-1}^{n-1}\,f_{x_n}. \end{eqnarray} On the other hand, by the shape of $f_{x_n}$, one has $\gcd(x_n, f_{x_n})=1$. It follows that $$\kappa(\mathbf{x},f_{x_0},\ldots, f_{x_{n-1}})=\delta\,f_{x_n},$$ for some $\delta\in R$. Pulling back to the $\mathbf{y}$-variables tells us that $\kappa -\delta\, y_n$ vanishes on the partial derivatives, and hence it belongs to affine ideal $\tilde{\mathcal{J}}$ of all polynomials vanishing on the partial derivatives. This ideal is prime because we can consider $\partial\ff:=(f_{x_0},\ldots , f_{x_n})$ as a point in $K^{n+1}$, where $K$ denotes the field of fractions of $R$ and consider the ideal of $K[\mathbf{y}]$ vanishing on $\partial\ff$ and then contract to $R[\mathbf{y}]$. We note that the Rees ideal $\mathcal{J}$ is the largest homogeneous ideal contained in $\tilde{\mathcal{J}}$. Now, multiplying $\kappa -\delta\, y_n$ by $x_n$ and using that $x_n\kappa\in \mathcal{L}\subset \mathcal{J} \subset \tilde{\mathcal{J}}$, it follows that $x_n \,\delta\, y_n\in \tilde{\mathcal{J}}$. Since this element is (trivially) homogeneous in $\mathbf{y}$, it must belong to the Rees ideal, hence $\delta=0$. Therefore, $\kappa\in \tilde{\mathcal{J}}$. But, since $\kappa$ is assumed to be homogeneous, it belongs to the Rees ideal. Thus, we have $\kappa\in \mathcal{J}\cap R[y_0,\ldots, y_{n-1}]$. But this means that $\kappa$ belongs to the Rees ideal $\mathcal{J}'$ of $J_{n-1}$. Since the latter is of linear type by we conclude as above that $\kappa\in \mathcal{L}$, as was to be shown. \qed \subsection{Degenerations of a catalecticant} The moral of this section is to search for predecessors of the sub-Hankel matrix, coming down all the way from the generic matrix. The case of the sub-Hankel determinant taught us that degenerating (with zeros) improves homaloidness and actually also the good algebraic properties of the polar map (such as the number of linear syzygies thereof). Guided by this case, we may expect a similar situation while taking degenerations of the generic, more generally, catalecticants. In what follows we will see that our hopes of analogy are a bit naive. \subsubsection{Degenerating the generic matrix}\label{deggeneric} We consider the simplest degeneration of the $m\times m$ generic matrix, consisting of replacing a variable by zero. Since in the generic matrix an entry has equal share as any other entry, we may assume that the resulting matrix has the shape: \begin{equation}\label{degener_gen} \mathcal{DG}=\mathcal{DG}_m:=\left( \begin{array}{cccc\|c} x_0 & x_1 & \ldots & x_{m-2} & \mathbf {x_{m-1}} \\ x_m & x_{m+1} & \ldots & x_{2m-2}& \mathbf{x_{2m-1}} \\ \vdots & \vdots & \vdots & \vdots & \vdots\\ x_{m(m-2)} & x_{m(m-2)+1} & \ldots & x_{m(m-1)-2} & \mathbf{x_{m(m-1)-1}}\\[5pt] \hline\\[-12pt] \mathbf{x_{m(m-1)}} & \mathbf{x_{m(m-1)+1}}& \ldots & \mathbf{x_{m^2-2}} & 0 \\ \end{array} \right) \end{equation} The reason for the boldfaced variables in a minute. We assume that $m\geq 3$. \begin{Proposition} Let $R=k[x_0,\ldots , x_{m^2-2}]$, let $f:=\det\mathcal{DG}\in R$ and let $J\subset R$ denote the gradient ideal of $f$. Then: \begin{enumerate} \item[{\rm (a)}] $f$ is a Gordan--Noether polynomial{\rm ;} in particular, its Hessian determinant vanishes. \item[{\rm (b)}] The image of the polar map of $f$ is the locus of the submaximal minors of the $(m-1)\times (m-1)$ square submatrix to the left and above the crossing lines{\rm ;} in other words, it is the cone over the Segre embedding $\pp^{m-2}\times \pp^{m-2}$. \item[{\rm (c)}] $J$ has maximal linear rank and the associated primes of $R/J$ are the Gorenstein ideals $I_{m-1}(\mathcal{DG})$ and the ideal generated by the boldfaced entries in {\rm (\ref{degener_gen})}. \end{enumerate} \end{Proposition} \demo (a) Expanding $f$ by Laplace along the last row, then expanding the corresponding (signed) cofactors along the last columns yields the following expression of $f$: \begin{equation} f=\bold\Delta\cdot \bold\Sigma ^T, \end{equation} where $\bold\Delta$ denotes the row of the signed $(m-2)$-minors of the $(m-1)\times (m-1)$ generic square submatrix $\mathcal{G}_{m-1}$ to the left and above the crossing lines in (\ref{degener_gen}), while $\bold\Sigma ^T$ stands for the transpose of the row vector whose entries are all the cross products of the entries of the last column and the last row (not counting the zero at the corner). Now, since the entries of $\bold\Sigma ^T$ are algebraically dependent if $m\geq 3$ and they share no variables with the entries of $\bold\Delta$ then $f$ is a particular case of a Gordan--Noether polynomial (see \cite[Section 2.3]{CRS}). Therefore, $f$ has vanishing Hessian. (b) Since $f$ has the structure of a Gordan--Noether polynomial, one knows that the defining equations of the polar variety (i.e., the image of the polar map) of $f$ are exactly the polynomial relations of the entries in $\bold\Sigma ^T$. But the latter are the parameters defining the Segre embedding of $\pp^{m-2}\times \pp^{m-2}$, whose defining equations are exactly the $(m-2)$-minors of the generic matrix $\mathcal{G}_{m-1}$ read in the dual variables. (c) We observe that the ideal $I_{m-1}(\mathcal{DG})$ is a Gorenstein ideal of codimension $4$ since it specializes from the full generic case, and as such it is a linearly presented ideal. Moreover, it is clear that since the gradient ideal of the fully generic predecessor of $\mathcal{DG}$ is its ideal of $(m-1)$-minors then one has $I_{m-1}(\mathcal{DG})=(J, \det \mathcal{G}_{m-1})$. From this follows easily that the number of $k$-linearly independent syzygies of $J$ falls by just a bit off the ones of $I_{m-1}(\mathcal{DG})$. But the latter is linearly presented with a number of syzygies $>> m^2-2$. Thus, $J$ has maximal linear rank $m^2-2$. By a codimension argument, $I_{m-1}(\mathcal{DG})$ is a minimal prime of $R/J$ (actually, it is the unmixed part of the gradient ideal). By the shape stressed in (\ref{degener_gen}) it is clear that the boldfaced variables conduct $\det \mathcal{G}_{m-1}$ into $J$ (for this use the Cramer--Hilbert--Burch relations along the obvious $(m-1)\times m$ and $m\times (m-1)$ submatrices of $\mathcal{DG}$). The other inclusion is easy as well. Therefore, $$J:I_{m-1}(\mathcal{DG})=J:\det \mathcal{G}_{m-1}=(\mbox{\rm boldfaced entries}),$$ which implies the assertion. \qed \subsubsection{Degeneration of a $2$-leap catalecticant} Consider the following $3\times 3$ matrix, obtained from the $2$-leap generic catalecticant by replacing the last entry with zero: $$\mathcal{SC}_3=\left( \begin{array}{ccc} x_0 & x_1 & x_2 \\ x_2 & x_3 & x_4 \\ x_4 & x_5 & 0 \\ \end{array} \right) $$ \begin{Proposition} Let $f:=\det \mathcal{SC}_3$ and let $J$ denote the gradient ideal of $f$. Then: \begin{enumerate} \item[{\rm (a)}] $f$ has non-vanishing Hessian determinant \item[{\rm (b)}] $J$ has maximal linear rank. \end{enumerate} In particular, $f$ is homaloidal. \end{Proposition} \demo For the proof we really need to resort to the computer: we cannot use Proposition~\ref{2-cat3x3} as guidance since even the ideal of $2$-minors does not specialize, its codimension now being $3$ and not Gorenstein (not even Cohen--Macaulay). Still, the shape of the Hessian matrix is nearly subdiagonal with $x_4$ along the main subdiagonal -- a situation pretty much like the one in the sub-Hankel case. This makes it easy to inspect the value of the Hessian determinant to conclude that it is indeed a power of $x_4$ up to a nonzero coefficient. This takes care of (a). For (b), we do not learn immediately from the syzygies of the $2$-minors even knowing that the latter are again linearly presented -- the reason being that only four of the partial derivatives are $2$-minors. Resorting to \cite{Macaulay} gives that $J$ has $7$ linear syzygies generating a submodule of rank $5=6-1$. \qed \begin{Remark}\rm Collecting the results so far, one has seen three sorts of behavior of a determinant subjected to degeneration: (1) It may not be homaloidal before degenerating and become homaloidal after (sub-Hankel); (2) it may be homaloidal before degeneration and stop being so afterwards (sub-Generic); and (3) it can be homaloidal before and continue being homaloidal after degeneration ($2$-leap sub-catalecticant.) \end{Remark}	{ "redpajama_set_name": "RedPajamaArXiv" }	4,949
\section{Introduction} For most microscopy applications, finding the right exposure and light intensity to be used involves a trade-off between maximizing the signal to noise ratio and minimizing undesired effects such as phototoxicity. As a consequence, researchers often have to cope with considerable amounts of noise. To mitigate this issue, denoising plays an essential role in many data analysis pipelines, enabling otherwise impossible experiments~\cite{belthangady2019applications}. Currently, deep learning based denoising, also known as content-aware image restoration (\mbox{\textsc{CARE}}\xspace)~\cite{weigert2018content}, achieves the highest quality results. \mbox{\textsc{CARE}}\xspace methods learn a mapping from noisy to clean images. Before being applied, they must be trained with pairs of corresponding noisy and clean training data. In practice, this dependence on training pairs can be a bottleneck. While noisy images can usually be produced in abundance, recording their clean counterparts is difficult or impossible. Over the last years, various solutions to the problem have been proposed. Lehtinen \emph{et al}\onedot showed that a network can be trained for denoising using only pairs of corresponding noisy images. This method is known as \mbox{\textsc{Noise2Noise}}\xspace~\cite{lehtinen2018noise2noise}. The first self-supervised approaches \mbox{\textsc{Noise2Void}}\xspace~\cite{krull2019noise2void} and \mbox{\textsc{Noise2Self}}\xspace~\cite{batson2019noise2self} were introduced soon after this. These methods can be trained on unpaired noisy image data. In fact, they can be trained on the very same data that is to be denoised in the first place. The underlying approach relies on the assumption that (given the true signal) the noise in an image is generated independently for each pixel, as is indeed the case for the dominant sources of noise in light microscopy (Poisson shot noise and Gaussian readout noise)~\cite{luisier2010image,zhang2019poisson}. Both methods employ so-called \emph{blind spot} training, in which random pixels are masked in the input image with the network trying to predict their value from the surrounding patch. Unfortunately, the original self-supervised methods typically produce visible high-frequency artifacts (see Figure~\ref{fig:schema}) and can often not reach the quality achieved by supervised \mbox{\textsc{CARE}}\xspace training. It is worth noting that the high-frequency artifacts produced by these self-supervised methods never occur in the real fluorescence signal. Since the image is diffraction-limited and oversampled, the true signal has to be smooth to some degree. Multiple extensions of \mbox{\textsc{Noise2Void}}\xspace and \mbox{\textsc{Noise2Self}}\xspace have been proposed~\cite{Krull:2020_PN2V,laine2019high,Prakash2019ppn2v,khademi2020self}. All of them improve results by explicitly modeling the noise distribution. Here, we propose an alternate and novel route to high-quality self-supervised denoising. Instead of making additional assumptions about the noise, we show that the result can be improved by including additional knowledge about the structure of our signal. We believe that our approach might ultimately complement existing methods that are based on noise modeling, to further improve denoising quality. We assume that the true signal is the product of a convolution of an unknown \emph{phantom image} and an approximately known point spread function (PSF) -- a common assumption in established deconvolution approaches~\cite{richardson1972bayesian}. We use a \mbox{\textsc{U-Net}}\xspace~\cite{ronneberger2015u} to predict the phantom image and then explicitly perform the convolution to produce the final denoised result (see Figure~\ref{fig:schema}). We follow~\cite{krull2019noise2void,batson2019noise2self} and use a blind spot masking scheme allowing us to train our network in an end-to-end fashion from unpaired noisy data. We demonstrate that our method achieves denoising quality close to supervised methods on a variety of real and publicly available datasets. Our approach is generally on-par with modern noise model based methods~\cite{Krull:2020_PN2V,prakash2020divnoising}, while relying on a much simpler pipeline. As a byproduct, our method outputs the predicted phantom image, which can be interpreted as a deconvolution result. While we focus on the denoising task in this paper, we find that we can produce visually convincing deconvolved images by including a positivity constraint for the deconvolved output. \section{Related work} \label{sec:relatedWork} In the following, we will discuss related work on self-supervised blind spot denoising and other unsupervised denoising methods. We will focus on deep learning-based methods and omit the more traditional approaches that directly operate on individual images without training. Finally, we will briefly discuss concurrent work that tries to jointly solve denoising and inverse problems such as deconvolution. \subsection{Self-Supervised Blind Spot Denoising} By now, there is a variety of different blind spot based methods. While the first self-supervised methods (\mbox{\textsc{Noise2Void}}\xspace and \mbox{\textsc{Noise2Self}}\xspace) use a masking scheme to implement blind spot training, Laine \emph{et al}\onedot~\cite{laine2019high} suggest an alternative approach. Instead of masking, the authors present a specific network architecture that directly implements the blind spot receptive field. Additionally, the authors proposed a way to improve denoising quality by including a simple pixel-wise Gaussian based noise model. In parallel, Krull \emph{et al}\onedot~\cite{Krull:2020_PN2V} introduced a similar noise model based technique for improving denoising quality, this time using the pixel masking approach. Instead of Gaussians, Krull~\emph{et al}\onedot use histogram-based noise models together with a sampling scheme. Follow-up work additionally introduces parametric noise models and demonstrates how they can be bootstrapped (estimated) directly from the raw data~\cite{Prakash2019ppn2v}. All mentioned methods improve denoising quality by modeling the imaging noise. We, In contrast, are the first to show how blind spot denoising can be improved by including additional knowledge of the signal itself, namely the fact that it is diffraction-limited and oversampled. While the blind spot architecture introduced in~\cite{laine2019high} is computationally cheaper than the masking scheme from \cite{krull2019noise2void,khademi2020self}, it is unfortunately incompatible with our setup (see Figure~\ref{fig:schema}). Applying a convolution after a blind spot network would break the blind spot structure of the overall architecture. We thus stick with the original masking scheme, which is architecture-independent and can directly be applied for end-to-end training. \subsection{Other Unsupervised Denoising Approaches} An important alternative route is based on the theoretical work known as \emph{Stein's unbiased risk estimator} (\textsc{SURE}\xspace)~\cite{stein1981estimation}. Given noisy observation, such as an image corrupted by additive Gaussian noise, Stein's 1981 theoretical work enables us to calculate the expected mean-squared error of an estimator that tries to predict the underlying signal without requiring access to the true signal. The approach was put to use for conventional (non-deep-learning-based) denoising in~\cite{ramani2008monte} and later applied to derive a loss function for neural networks~\cite{metzler2018unsupervised}. While it has been shown that the same principle can theoretically be applied for other noise models beyond additive Gaussian noise~\cite{raphan2007learning}, this has to our knowledge not yet been used to build a general unsupervised deep learning based denoiser. In a very recent work called \mbox{\textsc{DivNoising}}\xspace~\cite{prakash2020divnoising} unsupervised denoising was achieved by training a variational autoencoder (\mbox{\textsc{VAE}}\xspace)~\cite{KingmaW13} as a generative model of the data. Once trained, the \mbox{\textsc{VAE}}\xspace can produce samples from an approximate posterior of clean images given a noisy input, allowing the authors to provide multiple diverse solutions or to combine them to a single estimate. Like the previously discussed extensions of blind spot denoising~\cite{laine2019high,Krull:2020_PN2V,Prakash2019ppn2v,khademi2020self} all methods based on \textsc{SURE}\xspace as well as \mbox{\textsc{DivNoising}}\xspace rely on a known noise model or on estimating an approximation. We, in contrast, do not model the noise distribution in any way (except assuming it is zero centered and applied at the pixel level) and achieve improved results. A radically different path that does not rely on modeling the noise distribution was described by Ulyanov \emph{et al}\onedot~\cite{ulyanov2018deep}. This technique, known as \emph{deep image prior}, trains a network using a fixed pattern of random inputs and the noisy image as a target. If trained until convergence, the network will simply produce the noisy image as output. However, by stopping the training early (at an adequate time) this setup can produce high-quality denoising results. Like our self-supervised method, deep image prior does not require additional training data to be applied. However, it is fundamentally different in that it is trained and applied separately for each image that is to be denoised, while our method can, once it is trained, be readily applied to previously unseen data. \subsection{Concurrent Work on Denoising and Inverse Problems} Kobayashi \emph{et al}\onedot~\cite{kobayashi2020image} developed a similar approach in parallel to ours. They provide a mathematical framework on how inverse problems such as deconvolution can be tackled using a blind spot approach. However, while we use a comparable setup, our perspective is quite different. Instead of deconvolution, we focus on the benefits for the denoising task and show that the quality of the results on real data can be dramatically improved. Yet another alternative approach was developed by Hendriksen \emph{et al}\onedot~\cite{hendriksen2020noise2inverse}. However, this technique is limited to well-conditioned inverse problems like computer tomography reconstruction and is not directly applicable to the type of microscopy data we consider here. \section{Methods} \label{sec:methods} In the following, we first describe our model of the image formation process, which is the foundation of our method, and then formally describe the denoising task. Before finally describing our method for blind spot denoising with diffraction-limited data, we include a brief recap of the original \mbox{\textsc{Noise2Void}}\xspace method described in \cite{krull2019noise2void}. \subsection{Image Formation} \label{sec:imageFormation} We think of the observed noisy image $\mathbf{x}$ recorded by the microscope, as being created in a two-stage process. Light originates from the excited fluorophores in the sample. We will refer to the unknown distribution of excited fluorophores as the \emph{phantom image} and denote it as ${\mathbf{z}}$. The phantom image is mapped through the optics of the microscope to form a distorted image $\mathbf{s}$ on the detector, which we will refer to as \emph{signal}. We assume the signal is the result of a convolution $\mathbf{s} = {\mathbf{z}} * {\mathbf{h}}$ between the phantom image ${\mathbf{z}}$ and a known \textsc{PSF}\xspace ${\mathbf{h}}$~\cite{richardson1972bayesian}. Finally, the signal is subject to different forms of imaging noise, resulting in the noisy observation $\mathbf{x}$. We think of $\mathbf{x}$ as being drawn from a distribution $\mathbf{x} \sim \pnm{\mathbf{x}\|\mathbf{s}}$, which we call the \emph{noise model}. Assuming that (given a signal $\mathbf{s}$) the noise is occurring independently for each pixel, we can factorize the noise model as \begin{equation} \pnm{\mathbf{x}\|\mathbf{s}} = \prod_i^N \pnm{x_i, s_i}, \end{equation} where $\pnm{x_i, s_i}$ is the unknown probability distribution, describing how likely it is to measure the noisy value $x_i$ at pixel $i$ given an underlying signal $s_i$. Note that such a noise model that factorizes over pixels can describe the most dominant sources of noise in fluorescent microscopy, the Poisson shot noise and readout noise~\cite{foi2008practical,zhang2019poisson}. Here, the particular shape of the noise model does not have to be known. The only additional assumption we make (following the original \mbox{\textsc{Noise2Void}}\xspace~\cite{krull2019noise2void}) is that the added noise is centered around zero, that is the expected value of the noisy observations at a pixel is equal to the signal $\E{x_i}{ \pnm{x_i, s_i}}= s_i$. \subsection{Denoising Task} \label{sec:denoisingTask} Given an observed noisy image $\mathbf{x}$, the denoising task as we consider it in this paper is to find a suitable estimate $\hat{\mathbf{s}} \approx \mathbf{s}$. Note that this is different from the deconvolution task, attempting to find an estimate $\hat{\mathbf{z}} \approx {\mathbf{z}}$ for the original phantom image. \subsection{Blind Spot Denoising Recap} \label{sec:bsdRecap} In the originally proposed \mbox{\textsc{Noise2Void}}\xspace, the network is seen as implementing a function $\hat{s}_i = f(\img^\textsc{RF}_i;{\mathbf{\theta} })$, that predicts an estimate for each pixel's signal $\hat{s}_i$ from its surrounding patch $\img^\textsc{RF}_i$, which includes the noisy pixel values in a neighborhood around the pixel $i$ but excludes the value $x_i$ at the pixel itself. We use ${\mathbf{\theta} }$ to denote the network parameters. The authors of~\cite{krull2019noise2void} refer to $\img^\textsc{RF}_i$ as a \emph{blind spot receptive field}. It allows us to train the network using unpaired noisy training images $x$, with the training loss computed as a sum over pixels comparing the predicted results directly to the corresponding values of the noisy observation \begin{equation} \sum_{i} \left( \hat{s}_i - x_i \right)^2 . \label{eq:loss} \end{equation} Note that the blind spot receptive field is necessary for this construction, as a standard network, in which each pixel prediction is also based on the value at the pixel itself would simply learn the identity transformation when trained using the same image as input and as target. To implement a network with a blind spot receptive field \mbox{\textsc{Noise2Void}}\xspace uses a standard \mbox{\textsc{U-Net}}\xspace~\cite{ronneberger2015u} together with a masking scheme during training. The loss is only computed for a randomly selected subset of pixels $M$. These pixels are \emph{masked} in the input image, replacing their value with a random pixel value from a local neighborhood. A network trained in this way acts as if it had a blind spot receptive field, enabling the network to denoise images once it has been trained on unpaired noisy observations. \subsection{Blind Spot Denoising for Diffraction-Limited Data} \label{sec:ourMethod} While the self-supervised \mbox{\textsc{Noise2Void}}\xspace method~\cite{krull2019noise2void} can be readily applied to the data $\mathbf{x}$ with the goal of directly producing an estimate $\hat{\mathbf{s}} \approx \mathbf{s}$, this is a sub-optimal strategy in our setting. Considering the above-described process of image formation, we know that, since $\mathbf{s}$ is the result of a convolution with a \textsc{PSF}\xspace, high-frequencies must be drastically reduced or completely removed. It is thus extremely unlikely that the true signal would include high-frequency features as they are \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot visible in the \mbox{\textsc{Noise2Void}}\xspace result in Figure~\ref{fig:schema}. While a network might in principle learn this from data, we find that blind spot methods usually fail at this and produce high-frequency artifacts. To avoid this problem, we propose to add a convolution with the \textsc{PSF}\xspace after the \mbox{\textsc{U-Net}}\xspace (see Figure~\ref{fig:schema}). When we now interpret the final output after the convolution as an estimate of the signal $\hat{\mathbf{s}} \approx \mathbf{s}$, we can be sure that this output is consistent with our model of image formation and can \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot not contain unrealistic high-frequency artifacts. In addition, we can view the direct output before the convolution as an estimate of the phantom image $\hat{\mathbf{z}} \approx {\mathbf{z}}$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot an attempt at deconvolution. To train our model using unpaired noisy data, we adhere to the same masking scheme and training loss (Eq.~\ref{eq:loss}) as in \mbox{\textsc{Noise2Void}}\xspace. The only difference being that our signal is produced using the additional convolution, thus enforcing the adequate dampening of high-frequencies in the final denoising estimate. \subsection{A Positivity Constraint for the Deconvolved Image} \label{sec:posConstr} Considering that the predicted deconvolved phantom image $\hat{\mathbf{z}}$ describes the distribution of excited fluorophores in our sample (see Section~\ref{sec:imageFormation}), we know that it cannot take negative values. After all, a negative fluorophore concentration can never occur in a physical sample. We propose to enforce this constraint using an additional loss component, linearly punishing negative values. Together with the original \mbox{\textsc{Noise2Void}}\xspace loss our loss is computed as \begin{equation} \frac{1}{\|M\|} \sum_{i \in M} \left( \hat{s}_i - x_i \right)^2 + \lambda \frac{1}{N} \sum_{i=1}^N \max(0, -\hat{z}_i) \label{eq:lossFull}, \end{equation} where $N$ is the number of pixels and $\lambda$ is a hyperparameter controlling the influence of the positivity constraint. Note that the new positivity term can be evaluated at each pixel in the image, while the \mbox{\textsc{Noise2Void}}\xspace component can only be computed at the masked pixels. \section{Experiments and Results} \label{sec:experiments} In the following, we evaluate the denoising performance of our method comparing it to various baselines. Additionally, we investigate the effect of the positivity constraint (see Section~\ref{sec:posConstr}). Finally, we describe an experiment on the role of the \textsc{PSF}\xspace used for reconstruction. \subsection{Datasets} \label{sec:data} \miniheadline{Fluorescence Microscopy Data with Real Noise} We used 6 fluorescence microscopy datasets with real noise. The \textit{Convallaria}~\cite{Krull:2020_PN2V,Prakash2019ppn2v} and \textit{Mouse actin}~\cite{Krull:2020_PN2V,Prakash2019ppn2v} datasets each consist of a set of 100 noisy images of $1024 \times 1024$ pixels showing a static sample. The \textit{Mouse skull nuclei}~\cite{Krull:2020_PN2V,Prakash2019ppn2v} consist of a set of 200 images of $512 \times 512$ pixels. In all 3 datasets, the ground truth is derived by averaging all images. We use all 5 images in each dataset for validation and the rest for training. The authors of~\cite{Krull:2020_PN2V,Prakash2019ppn2v} define a region of each image that is to be used for testing, while the whole image can be used for training of self-supervised methods. We adhere to this procedure. We additionally use data from~\cite{zhou2020w2s}, which provides 3 channels with training and test sets each consisting of $80$ and $40$, respectively. We use 15\% of the training data for validation. Images are $512 \times 512$ pixels in size. Note that like~\cite{prakash2020divnoising} we use the raw data made available to us by the authors as the provided normalized data is not suitable for our purpose. The dataset provides 5 different versions of each image with different levels of noise. In this work, we use only the version with the minimum and maximum amount of noise. We will refer to them as \textit{W2S avg1} and \textit{W2S avg16} respectively, as they are created by averaging different numbers of raw images. \miniheadline{Fluorescence Microscopy Data with Synthetic Noise} Additionally, we use 2 fluorescence microscopy datasets from~\cite{buchholz2020denoiseg} and added synthetic noise. We will refer to them as \textit{Mouse (DenoiSeg)} and \textit{Flywing (DenoiSeg)}. While the original data contains almost no noise, we add pixel-wise Gaussian noise with standard deviation 20 and 70 for \textit{Mouse (DenoiSeg)} and \textit{Flywing (DenoiSeg)}, respectively. Both datasets are split into a training, validation, and test fraction. The \textit{Mouse} dataset, provides 908 images of $128 \times 128$ pixels for training, 160 images of the same size as a validation set, and 67 images of $256 \times 256$ as a test set. The \textit{Flywing} dataset, provides 1428 images size $128 \times 128$ as a training set, 252 images for validation (same size), and also 42 images size $512 \times 512$ as test set. As our method does not require ground truth, we follow \cite{prakash2020divnoising} and add the test fraction to the training data in order to achieve a fair comparison. \miniheadline{Synthetic Data} While the above-mentioned datasets are highly realistic, we do not know the true \textsc{PSF}\xspace that produced the images. To investigate the effect of a mismatch between the true \textsc{PSF}\xspace and the \textsc{PSF}\xspace used in the training of our method, we used the clean rendered text data from the book \emph{The beetle}~\cite{marsh2004beetle} previously introduced in~\cite{prakash2020divnoising}, synthetically convolved it using a Gaussian \textsc{PSF}\xspace with a standard deviation of 1 pixel width. Finally, we added pixel-wise Gaussian noise with a standard deviation of 100. The resulting data consists of 40800 small images of $128 \times 128$ pixels in size. We split off a validation fraction of 15\%. \subsection{Implementation Details and Training} \label{sec:implementation} Our implementation is based on the \emph{pytorch} \mbox{\textsc{Noise2Void}}\xspace implementation from~\cite{Krull:2020_PN2V}. We use the exact same network architecture, with the only difference being the added convolution with the \textsc{PSF}\xspace at the end of the network. In all our experiments, we use the same network parameters: A 3-depth \mbox{\textsc{U-Net}}\xspace with 1 input channel and 64 channels in the first layer. All networks were trained for 200 epochs, with 10 steps per epoch. We set the initial learning rate to 0.001 and used Adam optimizer, batch size = 1, virtual batch size = 20, and patch size = 100. We mask 3.125\% (the default) of pixels in each patch. We use the positivity constraint with $\lambda=1$ (see Section~\ref{sec:posConstr}). \subsection{Denoising Performance} \label{sec:denoisingPerformance} We report the results for all fluorescence microscopy datasets in Table~\ref{tab:results}. The performance we can achieve in our denoising task is measured quantitatively by calculation of the average peak signal-to-noise ratio (\textbf{PSNR}). Qualitative results can be found in Figure~\ref{fig:table}. We run our method using a Gaussian \textsc{PSF}\xspace with a standard deviation of 1 pixel width for all datasets. Figure~\ref{fig:table} shows examples of denoising results on different datasets. \figTable \tablePSNR To assess the denoising quality of our method we compare its results to various baselines. We compared our method to \mbox{\textsc{Noise2Void}}\xspace, noise model based self-supervised methods (\mbox{\textsc{PN2V}}\xspace~\cite{Krull:2020_PN2V}, \mbox{\textsc{DivNoising}}\xspace~\cite{prakash2020divnoising}), as well as the well-known supervised \mbox{\textsc{CARE}}\xspace~\cite{weigert2018content} approach. While we run \mbox{\textsc{Noise2Void}}\xspace ourselves, the PSNR values for all other methods were taken from \cite{prakash2020divnoising}. We created a simple additional baseline by convolving the \mbox{\textsc{Noise2Void}}\xspace result with the same \textsc{PSF}\xspace used in our own method. This baseline is referred to as \emph{N2V (conv.)}. \subsection{Effect of the Positivity Constraint} \label{sec:effectOfPosConstr} Here we want to discuss the effect of the positivity constraint (see Section~\ref{sec:posConstr}) on the denoising and deconvolution results. We compare our method without positivity constraint ($\lambda = 0$, see Eq.~\ref{eq:lossFull}) and with positivity constraint ($\lambda = 1$). Choosing different values for $\lambda$ did not have a noticeable effect. We find that the constraint does not provide a systematic advantage or disadvantage with respect to denoising quality (see Table~\ref{tab:results}). In Figure~\ref{fig:deconv} we compare the results visually. While it is difficult to make out any differences in the denoising results, we see a stunning visual improvement for the deconvolution result when the positivity constraint is used. While the deconvolution result without positivity constraint contains various artifacts such as random repeating structures and grid patterns, these problems largely disappear when the positivity constraint is used. We find it is an interesting observation that such different predicted phantom images can lead to virtually indistinguishable denoising results after convolution with the \textsc{PSF}\xspace, demonstrating how ill-posed the unsupervised deconvolution problem really is. \figDeconv \subsection{Effect of the Point Spread Function} \label{sec:effectOfPSF} Here we want to discuss an additional experiment on the role of the \textsc{PSF}\xspace used in the reconstruction and the effect of a mismatch with respect to the \textsc{PSF}\xspace that actually produced the data. We use our synthetic \emph{The beetle} dataset (see Section~\ref{sec:data}) that has been convolved with a Gaussian \textsc{PSF}\xspace with a standard deviation of $\sigma=1$ pixel width and was subject to Gaussian noise of standard deviation 100. We train our method on this data using different Gaussian \textsc{PSF}s\xspace with standard deviations between $\sigma=0$ and $\sigma=2$. We used an active positivity constraint with $\lambda=$ 1. The results of the experiment can be found in Figure~\ref{fig:psf}. We find that the true \textsc{PSF}\xspace of $\sigma=1$ gives the best results. While lower values lead to increased artifacts, similar to those produced by \mbox{\textsc{Noise2Void}}\xspace, larger values lead to an overly smooth result. \figPSF \section{Discussion and Outlook} \label{sec:Discussion} Here, we have proposed a novel way of improving self-supervised denoising for microscopy, making use of the fact that images are typically diffraction-limited. While our method can be easily applied, results are often on-par with more sophisticated second-generation self-supervised methods~\cite{Krull:2020_PN2V,prakash2020divnoising}. We believe that the simplicity and general applicability of our method will facilitate fast and widespread use in fluorescence microscopy where oversampled and diffraction-limited data is the default. While the standard deviation of the \textsc{PSF}\xspace is currently a parameter that has to be set by the user, we believe that future work can optimize it as a part of the training procedure. This would provide the user with an \emph{de facto} parameter-free turn-key system that could readily be applied to unpaired noisy raw data and achieve results very close to supervised training. In addition to providing a denoising result, our method outputs a deconvolved image as well. Even though deconvolution is not the focus of this work, we find that including a positivity constraint in our loss enables us to predict visually plausible results. However, the fact that dramatically different predicted deconvolved images give rise to virtually indistinguishable denoising results (see Figure~\ref{fig:deconv}) illustrates just how underconstrained the deconvolution task is. Hence, further regularization might be required to achieve deconvolution results of optimal quality. In concurrent work, Kobayashi \emph{et al}\onedot~\cite{kobayashi2020image} have generated deconvolution results in a similar fashion and achieved encouraging results in their evaluation. We expect that future work will quantify to what degree the positivity constraint and other regularization terms can further improve self-supervised deconvolution methods. We believe that the use of a convolution after the network output to account for diffraction-limited imaging will in the future be combined with noise model based techniques, such as the self-supervised~\cite{Krull:2020_PN2V,laine2019high} or with novel techniques like \mbox{\textsc{DivNoising}}\xspace. In the latter case, this might even enable us to produce diverse deconvolution results and allow us to tackle uncertainty introduced by the under-constrained nature of the deconvolution problem in a systematic way. \subsubsection{Code Availability.} \label{sec:code} Our code is available at \url{https://github.com/juglab/DecoNoising}. \subsubsection{Acknowledgments.} \label{sec:acknowledgments} We thank the Scientific Computing Facility at MPI-CBG for giving us access to their HPC cluster. \par\vfill\par \clearpage \bibliographystyle{splncs04}	{ "redpajama_set_name": "RedPajamaArXiv" }	6,920
{"url":"https:\/\/www.physicsforums.com\/threads\/triple-integral.454095\/","text":"# Triple Integral\n\n## Homework Statement\n\nEvaluate the triple integral.\n\u222b\u222b\u222bxyz dV, where T is the sold tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), and (1,0,1)\n\n## The Attempt at a Solution\n\nI'm having trouble finding the bounds. So far I'm integrating it in order of dzdydx with my x bounds as 0-1, my y bounds as 0-x, but I'm not sure how to find the z bounds.\n\nRelated Calculus and Beyond Homework Help News on Phys.org\nlanedance\nHomework Helper\nif you integrate over z first, it will be bounded by fucntion of both z & y, representing the top plane of the tetrathedron\n\nlanedance\nHomework Helper\n\n## Homework Statement\n\nEvaluate the triple integral.\n\u222b\u222b\u222bxyz dV, where T is the sold tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), and (1,0,1)\n\n## The Attempt at a Solution\n\nI'm having trouble finding the bounds. So far I'm integrating it in order of dzdydx with my x bounds as 0-1, my y bounds as 0-x, but I'm not sure how to find the z bounds.\nsimilarly your y bounds are not y=0 to y=x, but will be y=0 to y=1-x, to see this look at the line formed by the top surface of the tetrahedron in the xy plane\n\nsimilarly your y bounds are not y=0 to y=x, but will be y=0 to y=1-x, to see this look at the line formed by the top surface of the tetrahedron in the xy plane\nSorry, I'm still confused. How would I integrate over z first?\nAnd I am not seeing the line y=1-x formed. I plotted (0,0,0), (1,0,0), (1,0,1) and (1,1,0) onto a xy graph so the points would be (0,0), (1,0), and (1,1) giving me the line y=x or is that not how I approach it?\n\nlanedance\nHomework Helper\napologies you are correct, so the integral should be\n$$int^1_0 (int^0_x ( int_0^{f(x,y)} dz) dy ) dx$$\n\nnow you just need to find the upper bound for the first integral f(x,y) which is the upper bounding plane of the tetrahedron.","date":"2020-08-08 12:48:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.8031845688819885, \"perplexity\": 701.2872310975625}, \"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\/1596439737645.2\/warc\/CC-MAIN-20200808110257-20200808140257-00419.warc.gz\"}"}	null	null
Emma Gaala — фінський музичний гала-концерт, що щорічно організується Musiikkituottajat — IFPI Finland, на якому нагородами Emma Awards нагороджуються найвидатніші артисти та музичні професіонали року. Він проводиться щорічно з 1983 року, крім періоду з 1988 по 1990 рік. До 1991 року переможців обирали представники ÄKT. З 1992 року вони тільки відбирали кандидатів, з яких комісія музичних критиків обирала переможців. Критерії відбору включають цікаві музичні якості, музичний і комерційний успіх в індустрії звукозапису, а також популярність артиста, яка вимірюється різними способами. До 2022 року Nightwish отримали найбільше нагород Emma (всього 16). Примітки Музичні нагороди	{ "redpajama_set_name": "RedPajamaWikipedia" }	7,748
{-# LANGUAGE MultiParamTypeClasses #-} {-\| Module : Auto.Genome Description : The genetic algorithm implementation Copyright : (c) Bo Joel Svensson, 2015 Michael Vollmer, 2015 License : GPL-3 Maintainer : Stability : experimental Portability : A typeclass for basic operations on genomes. -} module Auto.Genome (Genome(..)) where import Control.Monad.Random -- \| A genome is parameterized by its representation and its -- "return value" (i.e. what type of value it represents). -- You can expect to mutate and cross over genomes. class Genome a b where -- \| Convert a value to a genome. toValue :: b -> a -- \| Randomly mutate a genome based on some probability. mutate :: (RandomGen g, Fractional f, Ord f, Random f) => g -> f -> b -> b -- \| Cross over two genomes. cross :: (RandomGen g) => g -> b -> b -> b	{ "redpajama_set_name": "RedPajamaGithub" }	9,587
\section{Introduction} A classical $6j$--symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of $SU(2)$, in other words by natural numbers. Its definition is roughly as follows. Let $V_a$ ($a=0,1,2,\ldots$) denote the $(a+1)$--dimensional irreducible representation. The $SU(2)$--invariant part of the triple tensor product $V_a \otimes V_b \otimes V_c$ is non-zero if and only if \begin{equation}\label{e:triang} a\leq b+c\qquad b\leq c+a \qquad c\leq a+b \qquad a+b+c\qua \hbox{is even}\end{equation} in which case we may pick, almost canonically, a basis vector $\epsilon^{abc}$ (details are given below). Suppose we have a tetrahedron, labelled so that the three labels around each face satisfy these conditions: we will call this an {\em admissible} labelling. Then we may associate to each face an $\epsilon$--tensor, and contract these four tensors together to obtain a scalar, the $6j$--symbol, denoted by a picture or a bracket symbol as in figure \ref{figtet}. \begin{figure}[h]\small \begin{equation}\vcenter{\hbox{\mbox{\input{tet.pst}}}}\quad \equiv \quad \sixj \end{equation} \caption{Pictorial representation\label{figtet}} \end{figure} This tetrahedral picture is traditionally used simply to express the {\em symmetry} of the $6j$--symbol, which is naturally invariant under the full tetrahedral group $S_4$. However, it has a deeper {\em geometric} significance. To an admissibly-labelled tetrahedron we may associate a metric tetrahedron $\tau$ whose side lengths are the six numbers $a,b,\ldots,f$. Its individual faces may be realised in Euclidean $2$--space, by the admissibility condition \eqref{e:triang}. As a whole, $\tau$ is either {\em Euclidean}, {\em Minkowskian} or {\em flat} (in other words has either a non-degenerate isometric embedding in Euclidean or Minkowskian $3$--space, or has an isometric embedding in Euclidean $2$--space), according to the sign of a certain polynomial in its edge-lengths. If $\tau$ is Euclidean, let $\theta_a, \theta_b, \ldots, \theta_f$ be its corresponding exterior dihedral angles and $V$ be its volume. \begin{theorem}[Asymptotic formula]\label{t:main} Suppose a tetrahedron is admissibly labelled by the numbers $a,b,c,d,e,f$. Let $k$ be a natural number. As $k \rightarrow \infty$, there is an asymptotic formula \begin{equation}\label{e:form} \ksixj \sim \begin{cases}{\displaystyle\sqrt{\frac{2}{3\pi Vk^3}}\cos{\left\{ \sum (ka+1) \frac{\theta_a}{2} + \frac{\pi}{4}\right\}}} &\hbox{{if $\tau$ is Euclidean}} \\ \hbox{{exponentially decaying}}&\hbox{{if $\tau$ is Minkowskian}}\end{cases} \end{equation} {\rm (}where the sum is over the six edges of the tetrahedron{\rm )}. \end{theorem} A (slightly different) version of this formula was conjectured in 1968 by the physicists Ponzano and Regge, building on heuristic work of Wigner; they produced much evidence to support it but did not prove it. It is the purpose of this paper to prove the above theorem using geometric quantization, and to explain the relation between $SU(2)$ representation theory and the geometry of ${\mathbb R}^3$. The formula has a lovely and peculiar consequence in elementary geometry. It is well-known that a generic tetrahedron is not {\em congruent} (by an orientation-preserving isometry of ${\mathbb R}^3$) to its mirror-image, but is {\em scissors-congruent} to it (in other words, the two tetrahedra are finitely equidecomposable). Inspired by the additional algebraic {\em Regge symmetry} of $6j$--symbols and the asymptotic formula above, one may derive from a generic tetrahedron a family of {\em twelve} non-congruent but scissors-congruent tetrahedra! Section 2 contains the algebraic and section 3 the differential-geometric preliminaries. Section 4 is a warm-up example, computing asymptotic rotation matrix elements for $SU(2)$ representations. It works in the same way as the eventual computation (in section 5) for the $6j$--symbol, but is much simpler and displays the method more clearly. Section 6 contains the geometric corollaries mentioned above and further notes on the Ponzano--Regge paper. Throughout the paper, the symbol ``$\sim$'' denotes an asymptotic formula, whereas ``$\approx$''denotes merely an approximation. \begin{ack} After having the basic ideas for this paper, I spent some time collaborating with John Barrett, trying to find a good method of doing the actual calculations required. Neither of us had much success during this period, and the details presented here were worked out by me later. (I feel quite embarrassed at ending up the sole author in this way.) I am especially grateful to John for many lengthy, interesting and helpful discussions on the subject, and also to J\o rgen Andersen, Johan Dupont, James Flude, Elmer Rees, Mike Singer and Vladimir Turaev for other valuable discussions. \end{ack} \section{Definition and interpretation of $6j$--symbols} \subsection{Combinatorial definition} The simplest definition is via Penrose's spin network calculus, which is related to Kauffman bracket skein theory at $A=\pm 1$. The details are in the book of Kauffman and Lins \cite{KL}. There is a topological invariant $\langle\,\rangle$ of planar links (systems of generically immersed curves) defined by sending a link $L$ to \begin{equation}\label{e:spinor}\langle L\rangle =(-2)^{\text{\em number of loops{\rm (}L{\rm )}}}(-1)^{\text{\em number of crossings{\rm (}L{\rm )}}}.\end{equation} It extends to an invariant of suitably-labelled trivalent graphs in $S^2$, for example the Mercedes (tetrahedron) and theta symbols shown in figure \ref{fskein}. \begin{figure}[ht] \begin{equation} \vcenter{\hbox{\mbox{\input{sphtet.pst}}}} \qquad \qquad \vcenter{\hbox{\mbox{\input{theta.pst}}}}\end{equation} \caption{Mercedes and theta graphs\label{fskein}} \end{figure} To define it, we replace each edge by a number of parallel strands equal to its label, and connect them up without crossings at the vertices (this imposes precisely the conditions \eqref{e:triang} on the the three incident labels). Then we replace this diagram by the set of all planar links obtained by inserting a permutation of the strands near the middle of each edge. Finally, evaluate each of these using \eqref{e:spinor}, add up their contributions, and divide by the number of such diagrams (the product of the factorials of the edge-labels). Explicit evaluations of these quantities are given in \cite{KL}. \begin{defn}\label{combdef} The $6j$--symbol shown in figure \ref{figtet} is defined as the spin-network evaluation of the above admissibly-labelled Mercedes symbol, divided by the product of the square-roots of the absolute values of the four theta symbols associated with its vertices. It is manifestly $S_4$--invariant. \end{defn} \begin{rem} It is important to note that the spin-network picture is {\em dual} to the one drawn in figure \ref{figtet}. There, the trilinear invariant spaces are associated with {\em faces} of the tetrahedron, whereas in the Mercedes symbol they are associated with {\em vertices}. \end{rem} Although this definition is the simplest, we will need a more algebraic version where the $6j$--symbol is exhibited as a hermitian pairing of two vectors. \subsection{Algebraic definition}\label{4def} Let $V_1$ be the fundamental $2$--dimensional representation of $SU(2)$, which we will consider as the space of linear homogeneous polynomials in coordinate functions $Z$ and $W$. Then the other irreducibles, the symmetric powers $V_a = S^aV$, $a=0, 1, 2, \ldots$, are the spaces of homogeneous polynomials of degree $a$. The dimension of $V_a$ is $a+1$; when $a$ is even, it is an irreducible representation of $SO(3)$. Making $Z$, $W$ orthonormal determines an invariant hermitian inner product $( - , - )$ on $V_1$, and induces inner products on the higher representations $V_a$, thought of as subspaces of the tensor powers of $V_1$. The fundamental representation has an invariant skew tensor $Z \otimes W - W \otimes Z$, which induces quaternionic or real structures on the $V_a$, according as $a$ is odd or even. The $SU(2)$--invariant part of the tensor product of two irreducibles $V_a \otimes V_b$ is zero unless $a=b$, when it is one-dimensional. Similarly, the invariant part of the triple tensor product $V_a \otimes V_b \otimes V_c$ of irreducibles is either empty or one-dimensional, according to the famous conditions \eqref{e:triang}. (The meaning of the parity condition is clear from the fact that the centre of $SU(2)$ is the cyclic group $\mathbb Z_2$. The other conditions, often written more compactly as $\vert a-b \vert \leq c \leq a+b$, are more surprising. Why the existence of a Euclidean triangle with the prescribed sides should have anything to do with this will be explained shortly.) We want to pick well-defined basis vectors $\epsilon^{aa}$ and $\epsilon^{abc}$ for these spaces of bilinear and trilinear invariants. Since each such space has a hermitian form and a real structure, we could just pick real unit vectors, but this would still leave a sign ambiguity. To fix this we may as well just write down the invariants concerned. Consider the vectors corresponding to the polynomials \[ (Z_1 W_2 - W_1 Z_2)^a \qquad (Z_1 W_2 - W_1 Z_2)^k(Z_1 W_3 - W_1 Z_3)^j(Z_2 W_3 - W_2 Z_3)^i \] on $\mathbb C^2\oplus \mathbb C^2$ and $\mathbb C^2\oplus \mathbb C^2\oplus \mathbb C^2$ respectively, where $i=(b+c-a)/2$, $j=(a+c-b)/2$, $k=(a+b-c)/2$. The required vectors are obtained from these by rescaling using positive real numbers, to obtain $\epsilon^{aa}$ with norm $\sqrt{a+1}$ and $\epsilon^{abc}$ with norm $1$. \begin{defn}\label{6jdef} Given six irreducibles $V_a, V_b, \ldots, V_f$, one can form $\epsilon^{abc} \otimes \epsilon^{cde} \otimes \epsilon^{efa} \otimes \epsilon^{fdb}$ (supposing these all exist) inside a 12--fold tensor product of irreducibles. One may always form $\epsilon^{aa} \otimes \epsilon^{bb} \otimes \cdots \otimes \epsilon^{ff}$, and permute the factors (without reversing the order of the paired factors) to match. Then the hermitian pairing of these two vectors (inside the 12--fold tensor product) defines the associated $6j$--symbol by \[ \sixj = (-1)^{\sum a} ( \overset{6}{\otimes} \epsilon^{aa}, \overset{4}{\otimes} \epsilon^{abc}) \] where $\sum a$ is simply the sum of the six labels. One should think of it as a function of six natural numbers $a,b, \ldots, f$, defined whenever the triples $(a,b,c), (c,d,e), (e,f,a), (f,d,b)$ satisfy the triangle and parity conditions \eqref{e:triang}, in other words when the associated tetrahedron labelling is admissible. It is standard to extend the definition to all such sextuples by setting the $6j$--symbol to zero elsewhere. \end{defn} \begin{lemma} These two definitions agree. \end{lemma} \begin{proof}[Proof {\rm (}Sketch{\rm )}] The Mercedes spin network evaluation used in definition \ref{combdef} can be reinterpreted as an explicit tensor contraction, using Penrose's diagrammatic tensor calculus (see \cite{KL}). The invariant $\langle L\rangle$ of a planar link may be evaluated by making the link Morse with respect to the vertical axis in ${\mathbb R}^2$, replacing cups and caps with $i$ times the standard skew tensor ($Z \otimes W - W \otimes Z \in \mathbb C \otimes \mathbb C$ and its dual, crossings with the flip tensor, and composing these morphisms to obtain a scalar. If we draw the Mercedes graph as in figure \ref{messy} and use this recipe to compute it, we see that it is given as the composition of a vector in $V_1^{\otimes 2 \sum a}$ (coming from the cups in the lower half of the diagram), a tensor product of twelve Young symmetrisers (coming from the flips associated to the crossings which are introduced where the labels are) and a vector in the dual of $V_1^{\otimes 2 \sum a}$ (coming from the caps). \begin{figure}[ht] \[ \vcenter{\hbox{\mbox{\input{pairing2.pst}}}} \] \caption{Writing the $6j$--symbol as a pairing\label{messy}} \end{figure} This can be interpreted as a bilinear pairing between a vector in the tensor product of twelve irreps and one in its dual, if we use the symmetrisers to project to these. Reinterpreting it as a hermitian pairing and including the normalisation factors gives the purely algebraic formulation of definition \ref{6jdef}. \end{proof} \begin{rem} This definition does not depend on the choice of hermitian form, coordinates, or real structure. It does depend on the sign conventions, but these can be seen to be sensible (in that the resulting $6j$--symbol is $S_4$--invariant) using the lemma. \end{rem} \begin{rem} A third way of defining the $6j$--symbols is to build a basis for $(V_a \otimes V_b \otimes V_c \otimes V_d)^{SU(2)}$ out of the trilinear invariants using an isomorphism such as \[ (V_a \otimes V_b \otimes V_c \otimes V_d)^{SU(2)} \cong \bigoplus_e (V_a \otimes V_b \otimes V_e)^{SU(2)} \otimes (V_e \otimes V_c \otimes V_d)^{SU(2)}\] where $e$ runs through all values such that $(a,b,e)$ and $(e,c,d)$ satisfy \eqref{e:triang}. There are three standard ways of doing this, corresponding to the three pairings of the four ``things'' $a,b,c,d$, and the change-of-basis matrix elements are (after mild renormalisation) the $6j$--symbols. Using this definition makes the Elliott--Biedenharn identity (pentagon identity) for $6j$--symbols very clear, but disguises their tetrahedral symmetry; therefore we will not consider this method here. See Varshalovich et al \cite{V} for this approach. Their definition coincides with the two given here, and with the one in Ponzano and Regge (though in these physics-oriented papers, half-integer spins are used). \end{rem} \subsection{Heuristic interpretation} The representation theory of $SU(2)$ is well-known to physicists as the theory of quantized angular momentum. The fundamental $2$--dimensional complex representation $V_1$ can be viewed as the space of states of spin of a spin--$\frac12$ particle. The other irreducibles, the symmetric powers $\{V_a=S^aV, a \in \mathbb N \}$, are state spaces for particles of higher spin; indeed, physicists label them by their associated spins $j=\frac12a$. Quantum and classical state-spaces are very different: the classical state of a spinning particle is described by an angular momentum vector in ${\mathbb R}^3$, whereas in the quantum theory, one should imagine the state vectors as wave-functions on ${\mathbb R}^3$, whose pointwise norms give {\em probability distributions} for the value of the angular momentum vector. However, when the spin is very large, the quantum and classical behaviour should begin to correspond. For example, the wave-functions representing states of a particle with large spin $j$ should be concentrated near the sphere of radius $j$ in ${\mathbb R}^3$. Many representation-theoretic quantities, most obviously square-norms of matrix elements of representations, can be interpreted as probability amplitudes for quantum-mechanical observations. Wigner \cite{Wig} explained the $6j$--symbol as follows. Suppose one has a system of four particles with spins $\frac12a, \frac12b, \frac12c, \frac12d$ and total spin $0$. Then the {\em square} of the $6j$--symbol is essentially the probability, given that the first two particles have total spin $\frac12e$, that the first and third combined have total spin $\frac12f$. (Compare with the third definition in the remark in subsection \ref{4def}.) He reasoned that for large spins, because of the concentration of the wave-functions, one can treat this statement as dealing with addition of vectors in ${\mathbb R}^3$. Suppose one has four vectors of lengths $\frac12a, \frac12b, \frac12c, \frac12d$ which form a closed quadrilateral. Then, given that one diagonal is $\frac12e$, what is the probability that the other is $\frac12f$? His analysis yielded the formula: \[ \sixj ^2 \approx \frac{1}{3 \pi V} \] where $V$ is the volume of the Euclidean tetrahedron whose edge-lengths are $a, b, \ldots, f$, supposing it exists. He emphasised that this is a dishonest approximation: the $6j$--symbols are wildly oscillatory functions of the dimensions, and his formula is only a local average over these oscillations, true in the same sense that one might write: \[ \hbox{For $\theta \neq 0$,}\ \cos^2(k\theta) \approx \frac12 \quad \hbox{as $k \rightarrow \infty$}.\] Ponzano and Regge improved his formula to one very similar to \eqref{e:form}, deducing the oscillating phase term from clever empirical analyses, and verified that as an approximation it is extremely accurate, even for small irreps. \section{Geometric quantization} \subsection{Borel--Weil--Bott} To rigorize Wigner's arguments, we need a concrete geometric realisation of the representations $V_a$. This is provided by the Borel--Weil--Bott theorem (see for example Segal \cite{CSM} or \cite{FH}): all finite-dimensional irreducible representations of semisimple Lie groups are realised as spaces of holomorphic sections of line bundles on compact complex manifolds, on which the groups act equivariantly. We only need the simplest case of this, namely that the irrep $V_a$ of $SU(2)$ is the space of holomorphic sections of the $a$th tensor power of the hyperplane bundle $\mathcal L$ on the Riemann sphere $\mathbb P^1$. If one thinks of these as functions on the dual tautological bundle, which is really just $\mathbb C^2$ blown up at the origin, they can be identified with spaces of homogeneous polynomial functions on $\mathbb C^2$ (ie in two variables) with the obvious $SU(2)$ action (or possibly the dual of the obvious one, depending on quite how carefully you considered what ``obvious'' meant!) Tensor products of such irreps are naturally spaces of holomorphic sections of the external tensor product of these line bundles over a product of Riemann spheres, for example \[ V_a \otimes V_b \otimes V_c = H^0(\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1, \mathcal L^a \boxtimes \mathcal L^b \boxtimes \mathcal L^c)\] with the diagonal action of $SU(2)$ on spheres and bundles. The calculation we are going to perform is a stationary phase integration, for which we need local differential-geometric information about these holomorphic sections. We will take the primarily symplectic point of view of Guillemin and Sternberg \cite{GS}, as well as using the main theorem of their paper (see below). Other relevant references are McDuff and Salamon \cite{MS} for general symplectic background and for symplectic reduction, Kirillov \cite{Kiri} for a concise explanation of geometric quantisation, and Mumford et al \cite{MFK} for the wider context of geometric invariant theory. \subsection{K\"ahler geometry} Suppose $M$ is a compact K\"ahler manifold of dimension $2n$. Thus, it has a complex structure $J$ acting on the real tangent spaces $T_pM$, a symplectic form $\omega$ and a Riemannian metric $B$ (we avoid the symbol $g$, which will denote a group element). The latter are $J$--invariant and compatible with each other according to the equation \[ B(X,Y)= \omega (X, JY), \] this being a positive-definite inner product. The Liouville volume form $\Omega = \omega^n/n!$ equals the Riemannian volume form. The hermitian metric on $T_pM$ (thought of as a complex space) is \[ H(X,Y)= B(X,Y) - i \omega(X,Y) \] which is linear in the first factor and antilinear in the second (the convention used throughout the paper). \subsection{Hamiltonian group action} Let $G$ be a compact group acting symplectically on $M$. We assume that the action is also Hamiltonian (ie that a moment map exists) and that it preserves the K\"ahler structure. This will certainly be true in the examples we will deal with. We let $\lie{g}$ be the Lie algebra of $G$. Then an element $\xi \in \lie{g}$ defines a Hamiltonian vector field $X_\xi$, a Hamiltonian $\mu(\xi)$ and thus a moment map $\mu\co M \rightarrow \coo{g}$ according to the conventions \[ d\mu(\xi) = \iota_{X_\xi}\omega \quad (= \omega(X_\xi, -)). \] I will later abuse notation slightly and write $\mu(X_\xi)$ instead of $\mu(\xi)$ when I want to emphasise the association of the moment map with a vector field corresponding to an infinitesimal action of $G$, rather than with an explicit Lie algebra element. \subsection{Equivariant hermitian holomorphic line bundle} If the symplectic form $\omega$ represents an integral cohomology class then there is a unique (we will assume $M$ is simply-connected) hermitian holomorphic line bundle over $M$, with metric $\langle -, -\rangle$, whose associated compatible connection has curvature form $F=(-2\pi i)\omega$ (so that $[\omega]$ is its first Chern class). We assume that $G$ acts equivariantly on $\mathcal L$, preserving its hermitian form. The space of holomorphic sections $V=H^0(M, \mathcal L)$ is finite-dimensional and has a natural left $G$--action defined by \[ (gs)(p) = g.s(g^{-1}p).\] $V$ becomes a unitary representation of $G$ when given the inner product \[ ( s_1, s_2 ) = \int_M \langle s_1, s_2 \rangle \Omega. \] The round bracket notation will be used to distinguish the {\em global} or {\em algebraic} hermitian forms from the {\em pointwise} form on the line bundle $\mathcal L$, which will be written with angle brackets. The infinitesimal action on sections is given by the formula \begin{equation}\label{e:quant} \xi s = \frac{d}{dt}(\exp(\xi t)s(\exp(-\xi t)p))= (-\nabla_{X_\xi} + 2\pi i \mu(\xi) )s \end{equation} This is the fundamental ``quantization formula'' of Kostant et al. \begin{rem} One has to be very careful with signs here, especially as there is such variation of convention in the literature. This formula is {\em minus} the Lie derivative $\mathcal L_{X_\xi}s$, because we are interested in the {\em left} action of $G$, and the Lie derivative is defined using the contravariant (right) action of $G$ on sections via pullback. There is an identical problem if one looks at the derivative of the left action of $G$ on vector fields, one has: \[ \frac{d}{dt}(\exp(\xi t)_Y)= - \mathcal L_{X_\xi}Y= - [X_\xi,Y]\] provided one uses the standard conventions on Lie derivative and bracket: \[ [ X, Y] = \mathcal L_XY, \qquad [X,Y]f = X(Yf)-Y(Xf) \] For further comments on sign conventions see McDuff--Salamon \cite{MS}, remark 3.3, though note that we do not here adopt their different Lie bracket convention. Anyway, it is a good exercise to check that the formula really does define a Lie algebra homomorphism: for this one also needs the standard conventions on curvature: \[ F(X,Y) = [ \nabla_X, \nabla_Y ] - \nabla_{[X,Y]} \] and on Poisson bracket: \[ \{f,g\}=- \omega(X_f, X_g)\] \end{rem} \subsection{Complexification} In the Borel--Weil--Bott setup, one actually has a complex group $G^{\mathbb C}$ acting on $\mathcal L$ and $M$. (Of course it does not preserve the hermitian structure on $\mathcal L$, but its maximal compact part $G$ does.) The action of $\clie{g}$ on sections of $\mathcal L$ is given by \begin{equation} (i\eta)s = i (\eta s) = (-i \nabla_{X_\eta} -2\pi \mu(\eta)) s = (-\nabla_{JX_\eta} -2\pi \mu(\eta)) s\end{equation} the last identity coming because $s$ is a holomorphic section, so is covariantly constant in the antiholomorphic directions in $TM \otimes \mathbb C$: \begin{equation} \nabla_{X+iJX}s=0\end{equation} The point of this identity is that it gives us information about the derivatives of an invariant section in directions orthogonal to the slice $\mu^{-1}(0)$. \subsection{Example}\label{exsph1} The $(k+1)$--dimensional irrep of $SU(2)$ is obtained by quantising $S^2$ with a round metric and with an equivariant hermitian line bundle $\mathcal L^{\otimes k}$ of curvature $k\omega$, where $\omega$ is the standard form with area 1. We will always view $S^2$ as being the unit sphere in ${\mathbb R}^3$. In cylindrical coordinates, its unit area form is then (``Archimedes' theorem'') \[ \omega = \frac{1}{4\pi}d\theta \wedge dz. \] Let $\xi$ be an element in the Lie algebra of the circle so that $e^\xi=1$. The moment map for the $1$--periodic rotation about the $z$--axis, generated by the vector field $X_\xi= 2\pi \partial/\partial \theta$ is $\mu\co S^2 \rightarrow \mathbb R$ given by $\mu=\frac12 kz$. (Here we identify the dual of the Lie algebra with $\mathbb R$ by letting $\xi$ be a unit basis vector.) Thus the image of the moment map is the interval $[-\frac{k}{2},\frac{k}{2}]$, and in accordance with the Duistermaat--Heckman theorem, the length of the interval equals the area of the sphere. \subsection{K\"ahler quotients} Given as above a K\"ahler manifold $M$ and the K\"ahler, Hamiltonian action of a compact group $G$, we may form the K\"ahler quotient $M_G$, which is just an enhanced version of the symplectic (Marsden--Weinstein) reduction. Let $M_0$ denote the slice $\mu^{-1}(0)$. The $G$--equivariance of the moment map $\mu\co M \rightarrow \coo{g}$ (coadjoint action on the right) means that $M_0$ is $G$--invariant, and consequently that $\omega(X_\xi, Y)= d\mu(\xi)(Y)=0$ for any $\xi \in \lie{g}$ and $Y \in T_{p}M_0$. Let us suppose that $G$ acts freely on $M_0$, since this will be enough for our purposes. We will use the symbol $\lie{g}p\equiv \{ X_\xi(p) : \xi \in \lie{g} \}$ to denote the tangent space to the $G$--orbit at $p$ and similarly, $i\lie{g}p$ will denote $\{ JX_\xi(p) : \xi \in \lie{g} \}$. In fact $\lie{g}p$ is the symplectic complement to $T_{p}M_0$ at $p$, and $i\lie{g}p$ is the (Riemannian) orthogonal complement to $T_{p}M_0$, because $B(Y, JX_\xi)=-\omega(Y, X_\xi)=0$ for any $Y \in T_{p}M_0$, and the dimensions add up. As a manifold, $M_G$ is just just the honest quotient $M_0/G$. It inherits an induced symplectic form $\omega_G$ whose pullback to $M_0$ is the restriction of that of $M$. Its tangent space $T_{[p]}M_G$ at a point $[p]$ (the orbit of a point $p\in M$) may be identified with its natural horizontal lift, namely the orthogonal complement of $\lie{g}{p} \subseteq T_{p}M_0$ at any lift $p$ of $[p]$. This space may also be described as the orthogonal complement of $\clie{g}{p} \subseteq T_{p}M$. As this subspace is complex, $T_{[p]}M_G$ inherits both a Riemannian metric and a complex structure by restriction. Hitchin proves in \cite{H} that these induced structures make $M_G$ into a K\"ahler manifold. Starting from $\mathcal L$ over $M$, we can also construct a hermitian holomorphic line bundle $\mathcal L_G$ over $M_G$ with curvature $-2\pi i \omega_G$ (in particular, the induced symplectic form is integral), as in \cite{GS}. The bundle and connection are such that their pullback to $M_0$ agrees with the restriction of $\mathcal L$. \subsection{Reduction commutes with quantization} Let $Q(M)$ denote the {\em quantization} $H^0(M, \mathcal L)$ associated to a K\"ahler manifold with equivariant hermitian line bundle $\mathcal L$ (which is suppressed in the notation). It is a representation of $G$, so we can consider the space of invariants $Q(M)^G \subseteq Q(M)$. (Whether we use $G$ or $G^{\mathbb C}$ here is of course irrelevant.) The main theorem in \cite{GS} is that there is an isomorphism $Q(M)^G \cong Q(M_G)$. There is obviously a restriction map from invariant sections over $M$ to sections over $M_G$, more or less by definition of $M_G$ and $\mathcal L_G$, so the task is to show injectivity and surjectivity. A vital ingredient in their proof of surjectivity is fact that norms of invariant sections achieve their maxima (in fact decay exponentially outside of) the slice $M_0$. We will rely on this fact too. Suppose $s$ is a holomorphic $G$--invariant section over $M$, and consider the real function $\\| s \\|^2$ on $M$. It is certainly $G$--invariant, but not $G^{\mathbb C}$--invariant. Following \cite{GS} we compute the derivative \begin{equation}\label{e:norm} (JX_\eta)\\| s \\|^2 = -4 \pi \mu(\eta) \\| s \\|^2 \end{equation} by using the quantization formula \eqref{e:quant} and the compatibility of hermitian metric and connection \[ X \\| s \\|^2 = \langle \nabla_X s, s \rangle + \langle s, \nabla_X s \rangle. \] Therefore, if $\gamma(t)$ is the flowline starting at $p \in M_0$ and generated by $JX_\eta$, \[ \frac{d}{dt} \\| s \\|^2_{\gamma(t)} = -4\pi \mu(\eta) \\| s \\|^2_{\gamma(t)} \] and combining with \[ \frac{d}{dt} \mu(\eta)_{\gamma(t)}= B(X_\eta, X_\eta) > 0\] we see that indeed the function $\\| s \\|^2_{\gamma(t)}$ has a single maximum at $t=0$, ie on $M_0$. \subsection{Refinement of the Guillemin--Sternberg theorem} We need an addition to the ``reduction commutes with quantization'' theorem. Any space $Q(M)$ is in a natural way a Hilbert space, with inner product defined by \[ (s_1,s_2) = \int_M \langle s_1, s_2 \rangle \Omega\] where $\langle - , - \rangle$ is its line bundle's hermitian form. One might imagine that the restriction isomorphism $Q(M)^G \cong Q(M_G)$ is an isometry, but in fact it is not. However, asymptotically it becomes an isometry if one redefines the measure on $M_G$, as will be shown below. First let us refine the observations about maxima of pointwise norms of sections given above. We can repeat the argument using the pointwise modulus of $\langle s_1, s_2 \rangle $, and establish that its maxima too are on $M_0$. Also, we can compute the second derivatives of the function $\langle s_1, s_2 \rangle $ in the $JX_\eta$ directions, which span the orthogonal complement of $TM_0$. Redoing the above calcuation yields, at $p \in M_0$, \[ (JX_\xi.JX_\eta \langle s_1, s_2 \rangle)_{p} = - 4 \pi JX_\xi(\mu(\eta) \langle s_1, s_2 \rangle)_{p} = - 4 \pi B_{p}(X_\eta, X_\xi)\langle s_1, s_2 \rangle_{p} \] because the moment map is zero at $p$. Parametrising a regular neighbourhood of $M_0$ as $M \times \exp(iU)$ for $U$ some small disc about the origin in $\lie{g}$, we see that to second order the function satisfies \begin{equation}\label{f:gauss} \langle s_1, s_2 \rangle_{(p, \xi)} \approx \langle s_1, s_2 \rangle_{p} e^{-2\pi B(X_\xi, X_\xi)} \quad \hbox{for}\ \xi \in U.\end{equation} To understand the asymptotics, we must first understand what is varying! Let $k$ be a natural number. Then one can consider $M$ with the new symplectic form $k\omega$; its Liouville form scales by $k^n$ (recall $\dim M=2n$), the moment map for its $G$--action scales by $k$, its Riemannian metric scales by $k$, and there is a new equivariant hermitian holomorphic bundle $\mathcal L^{\otimes k}$ over $M$ whose Chern form is $k\omega$. If $s$ is a $G$--invariant section of $\mathcal L$ then one can consider $s^k = s^{\otimes k}$, which is an invariant section of $\mathcal L^{\otimes k}$. We will always write the $k$ explicitly to indicate the scaling of forms, so that $B$, $\omega$, $\Omega$, $X_\xi$ and so on retain their original definitions. The new pointwise hermitian form satisfies \[ \langle s_1^k, s_2^k \rangle = \langle s_1, s_2 \rangle ^k.\] Thus, in view of \eqref{f:gauss}, as $k \rightarrow \infty$ the invariant section has pointwise norm concentrating more and more (like a Gaussian bump function) on the slice $M_0$. This localisation principle rigorises Wigner's ideas and forms the basis for the proof of the Ponzano--Regge formula. \begin{theorem}\label{t:norm} Let $\tilde s_1, \tilde s_2$ be $G$--invariant sections of $\mathcal L$ over $M$, and $s_1, s_2$ the induced sections of $\mathcal L_G$ over $M_G$. Let $\sigma\co M_G \rightarrow \mathbb R$ be the function which assigns to a point $[p]$ the Riemannian volume of the $G$--orbit in $M$ represented by $[p]$, and let $d=\dim(G)$. Then, as $k \rightarrow \infty$, there is an asymptotic formula \[(\tilde s_1^k, \tilde s_2^k) = \int_M \langle \tilde s_1^k, \tilde s_2^k \rangle (k^n\Omega) \sim \left(\frac{k}{2}\right)^{d/2} \int_{M_G} \langle s_1^k, s_2^k \rangle (\sigma k^{n-d}\Omega_G). \] \end{theorem} \begin{proof} From basic geometric invariant theory \cite{GS} we know $M_0/G=M_{ss}/G^{\mathbb C}$, where $M_{ss}=G^{\mathbb C} M_0$ is the set of semistable points, an open dense subset of $M$. We view the left hand integral as an integral over $M_{ss}$ and then integrate over the fibres of $\pi\co M_{ss} \rightarrow M_G$, which are $G^{\mathbb C}$--orbits. It helps to think of $\Omega$ as the Riemannian (coming from $B$) rather than the symplectic volume form. Suppose $[p]\in M_G$ and $p \in M_0$. The fibre can be parametrised via the map $G \times i\lie{g} \rightarrow \pi^{-1}([p])$ given by $(g, i\xi) \mapsto \exp(i\eta)gp.$ This is a diffeomorphism because of the Cartan decomposition of $G^{\mathbb C}$. (Recall we are assuming that $G$ acts freely on $M_0$.) Let $\psi$ be the pullback to $G \times i\lie{g}$ of the function $\log \langle \tilde s_1, \tilde s_2 \rangle$, so that $\log \langle \tilde s_1^k, \tilde s_2^k \rangle$ pulls back to $k\psi$. It is invariant in the $G$ directions but has Hessian form in the $i\lie{g}$ directions (at $0$) given by \[ (i\xi.i\eta.k\psi)_0 = -4\pi k B_{p}(X_\xi, X_\eta). \] The pullback Riemannian metric on $i\lie{g}$ is given by $\beta(i\xi, i\eta)= B(X_\xi, X_\eta)_{p}$. Consequently the integral of $\e^{k\psi}$ over $i\lie{g}$ is asymptotically given by \[ \langle \tilde s_1, \tilde s_2 \rangle_{p}^k \int_{i\lie{g}} e^{-2\pi k\beta(i\xi,i\xi)} d\vol_\beta = \langle \tilde s_1, \tilde s_2 \rangle_{p}^k \left(\frac{\pi}{2\pi k}\right)^{d/2}.\] (The symbol $\vol_\beta$ denotes the measure induced by the same metric $\beta$ as appears in the integrand; changing coordinates to an orthonormal basis, one obtains the a standard Gaussian integral, independent of $\beta$.) Finally we integrate over the $G$ orbit, picking up the factor $\vol(Gp)=\sigma([p])$. Substituting into the original left-hand side and separating the powers of $k$ in the correct way finishes the proof. \end{proof} \subsection{Example}\label{exsph2} Let us return to example \ref{exsph1} and check these formulae. If we have area form $2k\omega$ then sections of $\mathcal L^{\otimes 2k}$ are identified as homogeneous polynomials in $Z, W$ of degree $2k$, and under the circle action there is an invariant section which one can write as \[ s^k\co (Z,W) \mapsto (Z^k W^k) \in \mathbb C. \] In the complement of infinity, trivialise $\mathcal L^{\otimes 2k}$ using the nowhere-vanishing holomorphic section $b^{2k}$ corresponding to the homogeneous polynomial $W^{2k}$. Let $\zeta=Z/W$ be the coordinate on this chart. The pointwise norm of $b^{2k}$ is $(1+\vert \zeta \vert^2)^{-k}$, because a unit element of the tautological bundle above $\zeta$ is $(\zeta,1)/\sqrt{(1+\vert \zeta \vert^2)}$, which gets sent to $(1+\vert \zeta \vert^2)^{-k}$ by the section $b^{2k}$.) Thus the pointwise norm of $s^k$ is $\vert \zeta \vert^k/(1+\vert \zeta \vert^2)^k$. Under stereographic projection from the unit sphere in $\mathbb R^3$ \[ \zeta = \frac{x+iy}{1-z} \] we get $1/(1+\vert \zeta \vert^2) = (1-z)/2$ and $\vert \zeta \vert ^2/(1+\vert \zeta \vert^2) = (1+z)/2$. Thus the pointwise norm-squared of $s^k$ is $((1-z^2)/4)^k$. Its global norm-square is \begin{eqnarray} \\| s^k \\| ^2 &= &\int_{S^2} {\left(\frac{1-z^2}{4}\right)}^k 2k\frac{1}{4\pi}d\theta \wedge dz \\ &= & 2k \int_{-1}^1 {\left(\frac{1-z^2}{4}\right)}^k \frac12 dz\\ &=& 2k B(k+1, k+1) \end{eqnarray} by definition of the beta function $B$. Evaluating the beta function in terms of factorials gives \[ \\| s^k \\|^2 =\frac{2k}{2k+1} {\binom{2k}{k}}^{-1} \sim (\sqrt{\pi k})4^{-k}. \] Compare this with the computation of the norm on the reduced space, which is a single point: one finds the norm-square of $s^k$ at this point to be simply its value on the equator $z=0$ of the sphere, namely $4^{-k}$, so the ratio of the two is therefore $\sqrt{\pi k}$. If one applies theorem \ref{t:norm} with $\Omega = 2 \omega$ then one gets the same ratio: the scaling factor is $\sqrt{k/2}$ and the length of the equator (the factor $\sigma$ in the formula) is $2\pi \sqrt{2/4\pi}$, because the usual spherical {\em area} form has been divided by $4\pi$ and multiplied by $2$. \subsection{Orbit volumes} It is convenient here to note the formula for the volume of the $G$--orbit with respect to some basis (we will eventually have to roll up our sleeves and perform explicit calculations). \begin{lemma}\label{t:orbvol} If $\{\xi_i\}$ is a basis for and $\rho$ is an invariant metric on $\lie{g}$, then the volume of the $G$--orbit at $p\in M_0$ is \[ \vol(Gp) = \frac{\vol_{B_p}\{X_{\xi_i}\}}{\vol_{\rho}\{\xi_i\}} \vol_{\rho}(G).\] \end{lemma} \begin{proof} Pulling back the metric via the diffeomorphism $G \rightarrow Gp$, $g \mapsto g{p}$ gives a metric on $G$ of the form \[ \beta(g_\xi, g_\eta)=B_{g p}(X_\xi, X_\eta) = B_{p}(X_\xi, X_\eta)\] whose associated volume form differs from that of $\rho$ by the given factor. \end{proof} It is also worth giving a formula for the orbit volume when $G$ does not act freely on a space. Suppose as in the previous lemma that $\rho$ is an invariant metric, that the stabiliser at $p$ is $T$, whose Lie algebra is $\lie{t} \subseteq \lie{g}$. \begin{lemma}\label{t:rorbvol} If $\{\xi_i\}$ is set of vectors in $\lie{g}$ which, when projected onto into the orthogonal complement $\lie{t}^\perp \subseteq \lie{g}$, forms a basis $\{\hat\xi_i\}$ for that space, \[ \vol(Gp) = \frac{\vol_{B_p}\{X_{\xi_i}\}}{\vol_\rho\{\hat\xi_i\}} \vol_{\rho}(G/T).\] \end{lemma} \begin{proof} Repeat the earlier proof with $G/T$ mapping diffeomorphically to the orbit, and using the basis $\{\hat\xi_i\}$ for the tangent space of $G/T$. This gives a formula like the above except with $\vol_{B_p}\{X_{\hat\xi_i}\}$ on top. However, since the vectors $\hat \xi_i - \xi_i$ are in $\lie{t}$, they map to zero tangent vectors at $p$, and we can simply remove the hats. \end{proof} \subsection{Stationary phase formulae} The standard stationary phase formula is as follows: on a manifold $M^{2n}$ with volume form $\Omega$, for a smooth real function $f$ with isolated critical points $\{p\}$, one has \begin{equation} \int_M e^{ikf}\Omega \sim \left(\frac{2\pi}{k}\right)^{n} \sum_{p} \frac{e^{ikf(p)}.e^{\frac{i\pi}{4} \sgn(\Hess_{p}(f))}}{\sqrt{\Hess_{p}(f)}}.\end{equation} In our computation of we will actually have a {\em complex} function $\psi$, so it is probably easier to rewrite/generalise the above formula to such a situation as \begin{equation}\label{e:stat} \int_M e^{k\psi}\Omega \sim \left(\frac{2\pi}{k}\right)^{n} \sum_{p} \frac{e^{k\psi(p)}}{\sqrt{-\Hess_{p}(\psi)}}\end{equation} where the Hessian is now a complex number, and by the square root we mean the principal branch (the Hessian must not be real and positive). Of course, with $\psi = if$, $f$ real, this reduces to the previous version. \section{Warm-up example} In order to demonstrate more clearly the main points of the calculation to come in section 5, we will first work out a simpler case. Let $V_{2k}$ be an irreducible representation of $SO(3)$, and let $S^1_z$ be the circle subgroup fixing the $z$--axis in ${\mathbb R}^3$. With respect to this subgroup, $V_{2k}$ splits into one-dimensional weight spaces indexed by even weights from $-2k$ to $2k$. We may pick unit basis vectors inside these, uniquely up to a sign (by using the real structure of the representation). If $g \in SO(3)$ is a rotation, we may compute the matrix elements of $g$ with respect to such a basis by using the hermitian pairing. Most of these depend on the choices of sign, but the diagonal elements, those of the form $(v, gv)$, are independent. Below we will compute an asymptotic formula for such a matrix element, when $v$ is the zero-weight ($S^1$--invariant) vector. There are in fact explicit formulae for matrix elements given using Jacobi and Legendre polynomials, which are well-known in quantum mechanics. One can prove the theorem from these more easily, see for example Vilenkin and Klimyk \cite{VK}. Also, it could be computed explicitly from the example \ref{exsph2}. But this method demonstrates how to do the calculation without such explicit knowledge of the sections. If $v$ is a weight vector for $S^1_z$ then $gv$ is a weight vector for $gS^1_zg^{-1} = S^1_{gz}$, the subgroup fixing the rotated axis ``$gz$''. The elements $(v,gw)$ can also be thought of as elements of the matrix expressing one weight basis in terms of the other. \begin{theorem} As $k \rightarrow \infty$ there is an asymptotic formula \[ (v_0^{(k)}, gv_0^{(k)}) \sim \sqrt{\displaystyle\frac{2}{\pi k \sin \beta}} \cos \left\{(2k+1)\frac{\beta}{2} + \frac{\pi}{4}\right\} \] where $v_0^{(k)}$ is a unit zero-weight vector in $V_{2k}$ and $\beta>0$ the angle through which $g$ rotates the $z$--axis. \end{theorem} \begin{proof} As in example \ref{exsph1}, $V_{2k}$ is the space of holomorphic sections of $\mathcal L^{\otimes 2k}$ over $S^2$, with $SO(3)$ acting in the obvious way, symplectic form $2k\omega$, and the subgroup $S^1_z$ acting with moment map $kz$. The zero-slice is a circle, and the reduced space a single point. Therefore there is a one-dimensional space of invariant sections, and such a section will have maximal (and constant) modulus on the slice $z=0$. We choose a section $s$ of $\mathcal L^{\otimes 2}$ which is $S^1_z$--invariant and has peak modulus $1$ where $z=0$. (The phase doesn't matter, as noted above.) Then $s^{\otimes k}$ is a section of $\mathcal L^{\otimes 2k}$, invariant under $S^1_z$, and also with peak modulus 1 at $z=0$. It does not quite represent a choice of $v_0^{(k)}$ because its global norm is not 1 (we have fixed it locally instead). We must compute an asymptotic expression for \[ (v_0^{(k)}, gv_0^{(k)}) = \frac{(s^k, gs^k)}{(s^k,s^k)}.\] The denominator is asymptotically $\sqrt{\pi k}$, as we checked in example \ref{exsph2}. So the main work is computing the integral \[ (s^k, gs^k) = \int_{S^2} \langle s^{k}, gs^{k} \rangle 2k\omega = k\int_{S^2} \langle s, gs \rangle^k (2\omega)=k \int_M e^{k\psi}(2\omega) \] where $\psi = \log\langle s, gs \rangle$. Now $gs$ is an invariant section for $S^1_{gz}$, whose moment map is simply the ``$gz$ coordinate'', and whose zero-slice ``$gz=0$'' meets $z=0$ in two antipodal points $N$ and $S$. (They are both on the axis of $g$, and $N$ is the one about which $g$ is anticlockwise rotation.) Therefore, outside a neighbourhood of these two points, the modulus of the integrand is exponentially decaying, and the asymptotic contribution to the integral is just from $N$ and $S$. In fact these two points will also turn out to be the critical points of $\psi$, and we will evaluate the integral using the standard stationary phase procedure. Let us denote by $\mu, \nu$ the moment maps for $S^1_z$ and $S^1_{gz}$ acting on the sphere with symplectic form $2\omega$. Let $X,Y$ be the generating vector fields corresponding to these actions, as shown in figure \ref{sphere}. \begin{figure}[ht] \[ \vcenter{\hbox{\mbox{\input{sphere.pst}}}}\] \caption{Localisation regions and generating vectors\label{sphere}} \end{figure} The first derivatives of $\psi$ can be calculated as follows: start by computing \[ X \langle s, gs \rangle = \langle \nabla_{X}s, gs \rangle + \langle s, \nabla_{X} gs \rangle.\] The first term can be simplified to $2\pi i \mu \langle s, gs \rangle$ via the quantization formula \eqref{e:quant}, because $s$ is invariant under the group corresponding to $X$. The second term is quite so easy, but becomes simpler if we write \[X = p Y + q JY\] for some scalar functions $p,q$ (generically $Y, JY$ span the tangent space), and then expand again: \[ X\psi = 2\pi i \mu -2\pi i p \nu - 2 \pi q\nu.\] At either intersection point, the moment maps are both zero, so the whole thing vanishes. Similarly, the substitution \[ Y = p' X - q' JX \] gives \[ Y \psi = 2\pi i p'\mu +2\pi q' \mu - 2 \pi i \nu.\] Applying $X$ and $Y$ again to these formulae gives us the second derivatives at the critical points. The fact that ultimately we evaluate where the moment maps are zero shows we need only worry about the terms arising from the Leibniz rule in which the vector field differentiates them. These are evaluated using the following evaluations at $N$: \[ X\mu=0\qquad X\nu = 2\omega(Y,X)\qquad Y\mu = 2\omega(X,Y)\qquad Y\nu=0.\] In addition, at $N$ we have $p=p'= \cos \beta$ and $q=q'=-\sin \beta$, by inspection. Thus, with respect to the basis $\{X, Y\}$, we have the matrix of second derivatives \[ (-2\pi i)(-2\omega(X,Y)) \left( \begin{matrix} e^{i\beta}&1\\ 1&e^{i\beta}\end{matrix} \right). \] To obtain the Hessian, we have to divide the determinant of this matrix by $(2\omega(X,Y))^2$, to account for the basis $\{X, Y\}$ not being unimodular with respect to the volume form $2\omega$. Therefore the Hessian at $N$ is \[ (2\pi i)^2 e^{i\beta} 2i\sin \beta = -8\pi^2i\sin\beta e^{i\beta}. \] An easy check shows that the complex conjugate occurs at $S$. The $0$--order contributions $e^{k\psi(N)}, e^{k\psi(S)}$ must be computed. Both are of unit modulus, because of our convention on $s$. Their phases are easy to compute using the unitary equivariance of the bundle. Let $h$ be the rotation through $\pi$ lying in $S^1_z$. It exchanges $N$ and $S$, and preserves the section $s$. Thus \begin{align} e^{\psi(S)} & = \langle s,gs\rangle(S) = \langle h.h^{-1}s(hN), gh. h^{-1}s(hN) \rangle = \langle hs(N), ghs(N)\rangle \\ &= \langle s(N), h^{-1}ghs(N)\rangle.\end{align} Now $h^{-1}gh$ is clockwise rotation through $\beta$ at $N$, and therefore acts on the fibre of $\mathcal L^{\otimes 2}$, which is the tangent space at $N$, via $e^{-i\beta}$. Hence \[ e^{\psi(S)} = e^{i\beta}\ \text{and similarly}\ e^{\psi(N)}= e^{-i\beta} .\] Finally we can put this all together: \[ k\int_{S^2} e^{k\psi} (2\omega) = k \frac{2\pi}{k} \left\{ \frac{e^{-ik\beta}}{\sqrt{8\pi^2i\sin\beta e^{i\beta}}} + \frac{e^{ik\beta}}{\sqrt{-8\pi^2i\sin\beta e^{-i\beta}}}\right\}.\] Dividing by the normalising $(s,s)= \sqrt{\pi k}$ gives \[ \sqrt{\displaystyle\frac{2}{\pi k \sin \beta}} \cos \left\{(2k+1)\frac{\beta}{2} + \frac{\pi}{4}\right\}. \] \end{proof} \begin{rem} The answer consists of a modulus, coming from the modulus of the Hessian's determinant and the normalisation factors for the original sections, and a phase, coming partially from the phase of the Hessian and partially from the global phase shift (0--order terms). The $\pi/4$ is a standard stationary phase term. It will be possible to identify the terms in the $6j$--symbol formula similarly. \end{rem} \begin{rem} Other weight vectors behave similarly. If one repeats example \ref{exsph2}, one finds beta integrals with more `$(1-z)$'s than `$(1+z)$'s or vice versa, and the peak of the integrand also shifts to the correct position: a circle of constant latitude whose constant $z$ coordinate equals the weight divided by $k$. Similar formulae for general rotation matrix elements may be obtained. \end{rem} \section{Asymptotics of $6j$--symbols} \subsection{Geometry of the sphere}\label{51} The $6j$--symbol arises by pairing two $SU(2)$--invariant vectors in a 12--fold tensor product of irreducibles, according to definition \ref{6jdef}. Let us first consider the geometry of single irreducibles. Identify the Lie algebra of $SO(3)$ with ${\mathbb R}^3$ by using the standard vector product structure ``$\times$'', so that the standard basis vectors $e_i$ generate infinitesimal rotations $v \rightarrow e_i \times v$ in space. Define an invariant metric $\rho$ (the usual scalar product ``$.$'') by making them orthonormal. Using this scalar product we identify ${\mathbb R}^3$ with the coadjoint space also. This metric gives the flows generated by the `$e_i$'s period $2\pi$, and so gives the circle $T$ in $SO(3)$ generated by any of them length $2\pi$. We can think of $SO(3)$ as half of a 3--sphere whose great circle has length $4\pi$; such a sphere has radius $r=2$, and therefore volume $2\pi^2r^3 = 16 \pi^2$. Therefore $SO(3)$ has volume $8\pi^2$, and the quotient sphere $SO(3)/T$ (with its induced metric) has volume $4 \pi$. The irrep $V_a$ is the space of holomorphic sections of the bundle $\mathcal L_a$ on $S^2$. To write down explicit formulae for the various structures on the sphere it is better to view it as the sphere $S^2_a$ of radius $a$ in ${\mathbb R}^3$, instead of as the unit sphere. The symplectic form with area $a$ is given by \begin{equation} \omega_x(v,w) = \frac{1}{4\pi a^2} [x.v.w]\end{equation} where $x$ is a vector on the sphere, $v,w$ are tangent vectors at $x$ (orthogonal to it as vectors in ${\mathbb R}^3$) and the square brackets denote the triple product \[ [x.v.w] \equiv x . (v \times w). \] The complex structure at $x$ is the standard rotation \begin{equation} J_x(v) = \frac{1}{a} x \times v .\end{equation} The final piece of K\"ahler structure, the Riemannian metric, is then \begin{align} B_x(v,w)& = \omega_x(v,J_xw) = \frac{1}{4\pi a^3} (x \times v).(x \times w) = \frac{1}{4\pi a^3} (x^2 (v.w) - (x.v)(x.w))\\ & = \frac{1}{4\pi a} v.w.\end{align} This formula agrees with the fact that with the {\em natural} induced metric given by $B_x(v,w)=v.w$, the area of $S^2_a$ is $4 \pi a^2$. The group $G=SO(3)$ acts on the sphere, preserving its K\"ahler structure. This is a Hamiltonian action with moment map being simply inclusion $\mu_a\co S^2_a \hookrightarrow \mathbb R^3$. If $a$ is even then $G$ also acts on the corresponding hermitian line bundle, but if $a$ is odd one gets an action of the double cover $SU(2)$. (Since we need to work primarily with the geometric action, it is $SO(3)$ that wins the coveted title ``$G$''!) Products of K\"ahler manifolds have sum-of-pullback symplectic forms, and direct sums of complex structures. Their Liouville volume forms are wedge-products of pullbacks (there are no normalising factorials; this is one reason for the $n!$ in the definition of the Liouville form). The moment map for the diagonal action of $G$ on $S^2_a \times S^2_b \times S^2_c$ is therefore $\mu(x_1, x_2, x_3)=x_1+x_2+x_3 \in {\mathbb R}^3$. We know from the discussion earlier that the pointwise norm of an invariant section over this space will attain its maximum on the set $\mu=0$, which is in this case the ``locus of triangles'' $\{x_1+x_2+x_3=0\}$. $G$ acts freely and transitively on this space, except in the exceptional cases when one of $a,b,c$ is the sum of the other two. We can safely ignore this case, because it corresponds to a flat tetrahedron (about which the main theorem says nothing). We need to pick a section $s^{abc}$ corresponding to the invariant vector $\epsilon^{abc}$, whose normalisation was explained in section 2. There is a one-dimensional vector space of $SU(2)$--invariant sections of $\mathcal L^{\otimes a} \boxtimes \mathcal L^{\otimes b} \boxtimes \mathcal L^{\otimes c}$ over $S^2_a \times S^2_b \times S^2_c$. First we define $s^{abc}$ uniquely up to phase by setting its peak modulus to be 1. This section does {\em not} represent $\epsilon^{abc}$ exactly, because we fixed the norm {\em locally}, instead of globally. However, it is more convenient for calculations, and we can renormalise afterwards. Similarly, let $s^{aa}$ be a section over $S^2_a \times S^2_a$ with peak modulus 1. It will be concentrated near the zeroes of the moment map $\mu(x_1, x_2)=x_1+x_2$, namely the anti-diagonal. Note that $SO(3)$ acts transitively on the antidiagonal with circle stabilisers everywhere. To represent $\epsilon^{aa}$, this section would have to be renormalised so that its global section norm was $\sqrt{a+1}$. Fixing the phase of each section is slightly more subtle. The phase of the `$\epsilon$'s was fixed by using the spin-network normalisation. The analogue for sections is just the same, viewing the Riemann sphere as $\mathbb P^1$ with homogeneous coordinates $Z$ and $W$, and defining the sections just as before. The choice will not actually matter until subsection \ref{phase}. \subsection{A 24--dimensional manifold} Fix 6 natural numbers $a,b, \ldots f$ satisfying the appropriate admissibility conditions for existence of the $6j$--symbol. We will work on the following K\"ahler manifold $M$: \[ M= S^2_a \times S^2_b \times S^2_c \times S^2_c \times S^2_d \times S^2_e \times S^2_e \times S^2_f \times S^2_a \times S^2_f \times S^2_d \times S^2_b\] This is taken to lie inside $({\mathbb R}^3)^{12}$, and a point in it will be written as a vector $(x_1, x_2, \ldots, x_{12})$. There are three useful actions on $M$. First there is the diagonal action of $G$ on all 12 spheres. It has moment map $\phi\co M \rightarrow {\mathbb R}^3$ given by \[ \phi(x_1, x_2, \ldots, x_{12}) =\sum_1^{12} x_i. \] Algebraically, this generates the diagonal action of $G$ on the corresponding tensor product of 12 irreducible representations \[ V_a \otimes V_b \otimes V_c \otimes V_c \otimes \cdots \otimes V_b.\] Secondly, one has an action of $G^4=G \times G \times G \times G$, the first copy acting diagonally on the first three spheres, the next on the next three, and so on. The moment map for this action is $\mu\co M \rightarrow ({\mathbb R}^3)^4$, \[ (x_1, x_2, \ldots, x_{12})\mapsto (x_1+x_2+x_3, x_4+x_5+x_6, x_7+x_8+x_9, x_{10}+x_{11}+x_{12}).\] The section $\tilde s_\mu= s^{abc} \boxtimes s^{cde} \boxtimes s^{efa} \boxtimes s^{fdb}$ is a well-defined (given earlier conventions on $s^{abc}$) invariant section for this action with peak modulus 1. Thirdly, we let $G^6$ act on $M$, each copy acting diagonally on a pair of spheres of the same radius. The moment map (which shows precisely how this works) is $\nu\co M \rightarrow ({\mathbb R}^3)^6$, \[ \nu(x_1 , x_2, \ldots, x_{12})= (x_1+x_9, x_2+x_{12}, x_3+x_4, x_5+x_{11}, x_6+x_7, x_8+x_{10}).\] Then $\tilde s_\nu= s^{aa} \boxtimes s^{bb} \boxtimes \cdots \boxtimes s^{ff}$ (after a suitable permutation of its tensor factors, so that $s^{aa}$ lives over the first and ninth spheres, for example) is an invariant section for this action with peak modulus 1. \subsection{Proof of theorem \ref{t:main}} Recall from \ref{6jdef} the definition of the $6j$--symbol as a hermitian pairing of two vectors, and their normalisations. The corresponding geometric formula, in terms of the sections $\tilde s_\mu, \tilde s_\nu$ just defined is \begin{equation}\label{e:main} \ksixj = (-1)^{\sum a}\frac{(\tilde s_\mu^k, \tilde s_\nu^k)}{\\| \tilde s_\mu^k \\| \\| \tilde s_\nu^k\\|}\left(\prod (ka+1)\right) \end{equation} where the product on the right denotes simply $(ka+1)(kb+1) \cdots (kf+1)$.) Our convention on peak modulus 1 and phase of the sections mean that $(s^{abc})^{\otimes k}=s^{ka,kb,kc}$ and similarly $(s^{aa})^{\otimes k}=s^{ka,ka}$, so this formula is just the pairing of tensor products of these sections, with corrections for the global norms of $\tilde s_\mu$ and $\tilde s_\nu$ as explained at the end of subsection \ref{51}. To extract the asymptotic formula for the $6j$--symbol, we therefore need asymptotic formulae as $k \rightarrow \infty$ for the three integrals: \begin{eqnarray} I &= &(\tilde s_\mu^k, \tilde s_\nu^k) = \int_M \langle \tilde s_\mu^k, \tilde s_\nu^k \rangle k^{12}\Omega\\ I_\mu &=& (\tilde s_\mu^k, \tilde s_\mu^k) = \int_M \langle \tilde s_\mu^k, \tilde s_\mu^k \rangle k^{12}\Omega\\ I_\mu &= &(\tilde s_\nu^k, \tilde s_\nu^k) = \int_M \langle \tilde s_\nu^k, \tilde s_\nu^k \rangle k^{12}\Omega \end{eqnarray} (Note the explicit inclusion of all factors of $k$; everything else is unscaled.) We can in fact immediately write down asymptotic formulae for the correction integrals $I_\mu, I_\nu$ using theorem \ref{t:norm}, because the reductions $M_{G^4}$, $M_{G^6}$ are both single point spaces, and the sections have modulus $1$ over these points. \begin{eqnarray}\label{e:gnorm} I_\mu &\sim& \left(\frac{k}{2}\right)^6 \vol(\mu^{-1}(0))\\ \label{e:gnorm2}I_\nu &\sim& \left(\frac{k}{2}\right)^6 \vol(\nu^{-1}(0)) \end{eqnarray} (In the second case one must actually reconsider the proof of the theorem, because $G^6$ does not act freely on the set $\nu^{-1}(0)$, but there is no problem.) The remaining integral $I$ is evaluated by reduction to an integral over $M_G$ followed by the method of stationary phase, which fills the rest of this section. \subsection{Localisation of the integral $I$} As $k \rightarrow \infty$, the integrand decays exponentially outside the region where both moment maps $\mu, \nu$ are zero, because it is dominated by the pointwise norms of the invariant sections $\tilde s_\mu, \tilde s_\nu$. What then is the set $\mu^{-1}(0) \cap \nu^{-1}(0)$? At a point $(x_1, x_2, \ldots, x_{12})$, the condition $\nu=0$ requires that six of the $x_i$'s are simply negatives of the other six, and then $\mu=0$ forces the six remaining ones, say $(x_1, x_2, x_3, x_5, x_6, x_8)$, to form a tetrahedron, shown {\em schematically} in figure \ref{schem}. Recall that the lengths of the vectors are fixed integers $a,b,c,d,e,f$. \begin{figure}[ht] \[ \vcenter{\hbox{\mbox{\input{schtet.pst}}}}\] \caption{Schematic configuration of vectors\label{schem}} \end{figure} We have assumed that the numbers $a,b, \ldots, f$ satisfy the triangle inequalities in triples (otherwise the $6j$--symbol is simply zero), so the faces of this triangle can exist {\em individually} in ${\mathbb R}^3$. However, it is still quite possible that there is no Euclidean tetrahedron $\tau$ with sides $a,b, \ldots, f$. The sign of the Cayley polynomial $V^2(a^2, b^2, \ldots, f^2)$ (whose explicit form is irrelevant here) is the remaining piece of information needed to determine whether $\tau$ is Euclidean, flat or Minkowskian. In the last case, we see that $\mu^{-1}(0) \cap \nu^{-1}(0)=\emptyset$, and so have proved the second part of the main theorem: that if $\tau$ is Minkowskian then the $6j$--symbol is exponentially decaying as $k\rightarrow\infty$. Suppose on the other hand that $V^2$ is positive. Then we {\em can} find a set of six vectors $a,b,c,d,e,f$ in ${\mathbb R}^3$ forming a tetrahedron oriented as shown in figure \ref{f:vect}. Let $\tau$ denote both this tetrahedron and the corresponding point \[ (a,b,c,-c,d,e,-e,f,-a,-f,-d,-b) \in M \] and $\tau'$ be its mirror image (negate these 12 vectors). (Of course the whole tetrahedron is determined by just three of the vectors, say $a,c,e$.) It is clear that the localisation set $\mu^{-1}(0) \cap \nu^{-1}(0)$ will consist of exactly two $G$--orbits $G\tau$, $G\tau'$. \begin{figure}[h] \[ \vcenter{\hbox{\mbox{\input{tet2.pst}}}} \qquad \vcenter{\hbox{\mbox{\input{exptet.pst}}}}\] \caption{Actual configuration of vectors\label{f:vect}} \end{figure} \begin{rem} The symbols $a,b,c,d,e,f$ now denote {\em both} vectors and their integer lengths at the same time! This ought not be {\em too} confusing, as it should be clear from formulae what each symbol represents. \end{rem} Returning to the integral $I$, since both $\tilde s_\mu^k, \tilde s_\nu^k$ are invariant under the diagonal action of $G$, we can apply theorem \ref{t:norm} with respect to the diagonal action of $G$, and obtain an integral over an 18--dimensional manifold $M_G$: \begin{equation} \label{e:red} I = \int_M \langle \tilde s_\mu^k, \tilde s_\nu^k\rangle k^{12} \Omega = k^9\left(\frac{k}{2}\right)^{3/2} \int_{M_G} \langle s_\mu^k, s_\nu^k \rangle \sigma \Omega_G\end{equation} where in the right-hand integral, $s_\mu, s_\nu$ are the descendents of $\tilde s_\mu, \tilde s_\nu$, and the only thing depending on $k$ is the integrand, which is the $k$th power of something independent of $k$. The function $\sigma$ is the function on $M_G$ giving the volume of the corresponding $G$--orbit in $M$, and we view it as part of the measure in the integral. Let us define $\psi = \log \langle s_\mu, s_\nu \rangle$ on $M_G$ and $\tilde \psi = \log \langle \tilde s_\mu, \tilde s_\nu \rangle$ on $M$, so that the remaining problem is to compute \[ I' = \int_{M_G} e^{k\psi} \sigma \Omega_G.\] Since the modulus of $e^{k\tilde\psi}$ localises on the set $\mu^{-1}(0) \cap \nu^{-1}(0)= G\tau \cup G\tau' \subseteq M$, the above integrand localises to the two {\em points} $[\tau], [\tau']$. \subsection{Tangent spaces and stationary phase} The diagonal $G$--action does not commute with the other ones, so that $M_G$ will not have any kind of induced actions of $G^4$ or $G^6$, but we don't need this for the localisation calculation to go through. We always work on the upstairs space $M$ not $M_G$, precisely because the presence of the group actions defining the invariant sections being paired is so useful. The tangent space $T_{\tau}M$ is 24--dimensional, and contains two 12--dimensional subspaces $\ker d\mu$ and $\ker d\nu$ which meet in the 3--dimensional space $\lie{g}\tau$. (This degree of transversality can be checked explicitly from formulae below, but it should be clear from the fact that there are just two isolated critical points in $M_G$.) Together they span the 21--dimensional $T_{\tau}M_0$, which is orthogonal to $i\lie{g}\tau$. Projecting to 18--dimensional $T_{[\tau]} M_G$, we see two 9--dimensional subspaces we shall call $W_\mu$ and $W_\nu$ (the projections of $\ker d\mu$ and $\ker d\nu$) meeting transversely at the origin. We want to examine the behaviour of $\psi$ (its gradient and Hessian) at the point $[\tau] \in M_G$. Let us choose orthonormal bases $\{X_1, X_2, \ldots, X_9\}, \{Y_1, Y_2, \ldots, Y_9\}$ for the transverse 9--dimensional tangent spaces $W_\mu, W_\nu$ inside $T_{[\tau]}M_G$. Then we can need to compute quantities such as $X_i \psi$ and $X_i Y_j \psi$ (also at $[\tau]$, of course!). These can be computed by choosing {\em arbitrary} lifts of the vectors to $T_{\tau}M$ and applying them to the $G$--invariant function $\tilde \psi =\log \langle \tilde s_\mu, \tilde s_\nu \rangle$ on $M$. This is important, because it is very hard to write down any explicit {\em horizontal} lifts which would be needed to do computations directly in $T_{[\tau]}M_G$. So, to compute something like $X_i \psi$ one can choose any lift $\tilde X_i$ inside $\ker d\mu$ in $T_{\tau}M$, and write \begin{eqnarray} X_i \psi &=& \tilde X_i \tilde \psi = \tilde X_i \log \langle \tilde s_\mu, \tilde s_\nu \rangle\\ &= & \langle \tilde s_\mu, \tilde s_\nu \rangle^{-1}( \langle \nabla_{\tilde X_i} \tilde s_\mu, \tilde s_\nu \rangle + \langle \tilde s_\mu, \nabla_{\tilde X_i} \tilde s_\nu \rangle). \end{eqnarray} Now $\tilde X_i$ is a generator of the $G^4$ action under which $\tilde s_\mu$ is invariant, and therefore \[ \nabla_{\tilde X_i}\tilde s_\mu = 2\pi i \mu(\tilde X_i) \tilde s_\mu \] which vanishes at $\tau$. (Here $\mu(\tilde X_i)$ really denotes $\mu(\xi_i)$ for the Lie algebra element $\xi_i$ corresponding to $\tilde X_i$.) For the second term we must first express $X_i$ as a linear combination of the $Y_j$ and $JY_j$ (which span $T_{[\tau]}M_G$), then we can lift and use the quantization formula to compute. Therefore, introduce the $9 \times 9$ matrices $P_{ij}$ and $Q_{ij}$ according to \begin{align} X_i& = \sum P_{ik} Y_k + \sum Q_{ik} JY_k.\\ \intertext{Multiplying by $J$ we get} JX_i &= - \sum Q_{ik} Y_k + \sum P_{ik} JY_k . \end{align} By applying $\omega_G(X_j, -)$ and similar operators to these equations one obtains \[ P_{ij}= B_G(X_i, Y_j) \qquad Q_{ij}=-\omega_G(X_i, Y_j).\] These, together with the fact that the bases are orthonormal and span isotropic subspaces, determine completely matrices for $B$ and $\omega$ on $T_{[\tau]}(M_G)$. By a similar procedure one can invert the relations: \begin{eqnarray} Y_i &= &\sum P_{ki} X_k - \sum Q_{ki} JX_k \\ JY_i& = & \sum Q_{ki} X_k + \sum P_{ki} JX_k. \end{eqnarray} A final point is that since $\{X_i, JX_i\}$ and $\{Y_i, JY_i\}$ are both complex-oriented orthonormal bases for $T_{[\tau]}(M_G)$, the change of basis matrix is special orthogonal, and hence \[ P^TP+Q^TQ=1 \qquad QP^T=PQ^T \qquad Q^TP=P^TQ.\] Now we may rewrite the tangent vectors appropriately, lift everything to $T_{\tau}M$ and then apply them to $\tilde \psi$ via the fundamental formula (recall $\langle - , -\rangle$ is conjugate linear in the second factor). For example \begin{align} \tilde X_i \tilde \psi &= 2\pi i \mu(\tilde X_i) -2\pi i \sum P_{ik} \nu(\tilde Y_k) -2\pi \sum Q_{ik} \nu(\tilde Y_k).\\ \intertext{This right hand side vanishes at $\tilde\tau$, so indeed $ X_i \psi =0$ there. The companion formula is} \tilde Y_i \tilde \psi &= 2\pi i \sum P_{ki} \mu(\tilde X_k) +2\pi \sum Q_{ki} \mu(\tilde X_k) -2\pi i \nu(\tilde Y_i). \end{align} Together, these show that $\psi$ is stationary at $[\tau] \in M_G$, just as in the warm-up example. \subsection{Computation of the Hessian} Another application of the above formulae, remembering that \[ X\mu(Y)= d\mu(Y)(X)=\omega(Y,X)\] will obtain formulae for second derivatives such as \[ \tilde X_j \tilde X_i \tilde \psi = 2\pi i \omega_G(X_i, X_j) -2\pi i \sum P_{ik} \omega_G (Y_k, X_j) -2\pi \sum Q_{ik} \omega_G(Y_k, X_j)\] where everything in this formula is to be evaluated at $\tau$ (for example, the first term now dies), and we have used the defining identity $\omega(\tilde X, \tilde Y) = \omega_G(X, Y)$ to replace $\omega$ by $\omega_G$ and remove the tildes from the right-hand side. We can compute three similar formulae for the second derivatives and form the Hessian matrix for $\psi$ with respect to the basis $\{X_i, Y_i\}$ of $T_{[\tau]}M_G$ : \[ (-2 \pi i) \left( \begin{matrix} PQ^T-iQQ^T&Q\\ Q^T&P^TQ-iQ^TQ \end{matrix} \right) \] We can extract the matrix \[\left( \begin{matrix} Q^T&0\\ 0&Q \end{matrix} \right)\] form the right, and expand \[ \det \left( \begin{matrix} P-iQ&1\\ 1&P^T-iQ^T \end{matrix} \right)\] as \[ \det( (P-iQ)(P^T-iQ^T)-1) = \det (-2QQ^T-2iPQ^T) \] using properties of $P$ and $Q$ discussed earlier. Hence this temporary ``unnormalised Hessian'' of $\psi$ is: \[ (-2 \pi i)^{18}. (-2i)^9 .\det(P-iQ). \det(Q)^3\] The reason for separating the last two parts is that $\det Q$ is real, whereas $P-iQ$ represents the change of basis between $\{X_i\}$ and $\{Y_j\}$ as bases of $T_{[\tau]}M_G$ as a 9--dimensional {\em complex} vector space (ie, it is the matrix of the hermitian form, $(P-iQ)_{ij}= \overline{H_G(X_i, Y_j)}$), so is unitary and contributes just a phase as determinant. To normalise the Hessian we must compute the volume of the basis $\{X_i, Y_j\}$ with respect to the form $\sigma\omega_G^9/9!$ on $T_{[\tau]} M_G$. Expand, using the shuffle product, the expression $(\omega_G^9/9!)(X_1, X_2, \ldots, X_9, Y_1, Y_2, \ldots, Y_9)$: since the spaces spanned by the $X_i$ and by the $Y_j$ are isotropic, the terms appearing are simply all possible orderings of all possible products of 9 terms of the form $\omega(X_i, Y_j)$ (the $X$ before the $Y$). Reordering these cancels the denominator $9!$ and we obtain simply $\det(Q)$. So the determinant we actually computed was $(\vol(G\tau)\det(Q))^2$ times what it should have been when computed in a unimodular basis. Therefore \[ \Hess_{[\tau]}(\psi)= (-2 \pi i)^{18}. (-2i)^9. \det(P-iQ) .\det(Q).\vol(G\tau)^{-2}.\] This is the end of the general nonsense. To go any further we have to choose explicit bases, although not for $T_{[\tau]} M_G$, because of the difficulties already mentioned in writing down {\em any} vectors there. In the next two sections we will write down nice vectors ``upstairs'' in $T_\tau M$ and show how to lift the computations of $\det(P-iQ), \det(Q)$ into this space. \subsection{The modulus of the Hessian} We need to compute $\det(Q)$, where $Q_{ij}=-\omega_G(X_i, Y_j)$, and the $X_i$, $Y_j$ are orthonormal bases as chosen above. Let us start by introducing some useful vectors in $T_\tau M$, with which to compute ``upstairs''. We make an explicit choice of basis for each of the 12--dimensional spaces $\ker d\mu, \ker d\nu$ inside $T_{\tau}M$. Recalling that they intersect in the space $\lie{g}\tau$, we arrange for a suitable basis of this space to be easily obtained from each. Let $T^l_v$ be the infinitesimal rotation about the vector $v$, acting on the $l$th triple of vectors from $(x_1, x_2, \ldots, x_{12})$. For example, at any point $(x_1, x_2, \ldots, x_{12})$, we have \[ T^1_v=(v \times x_1, v \times x_2, v \times x_3, 0,0,0,0,0,0,0,0,0) .\] This vector clearly preserves the condition $x_1+x_2+x_3=0$, as well as the other three ${\mathbb R}^3$--coordinate parts of $\mu$. Recall that $a,c,e$ are three vectors defining the tetrahedron $ \tau$. Since $a,c,e$ are linearly independent, the vectors $T^1_a, T^1_c, T^1_e$ span the tangent space to $\{x_1+x_2+x_3=0\}$ inside the product of the first three spheres of $M$. Combining four such sets of vectors, we see that the 12 vectors $T^l_v$, where $l=1,2,3,4$ and $v$ is one of the three vectors $a,c$ or $e$, span the space $\ker d\mu$ at $\tau$. For convenience these vectors will also be numbered \[T_1, T_2, \ldots, T_{12} = T^1_a, T^1_c, T^1_e, T^2_a, \ldots, T^4_e.\] Note that although the formula defines a vector field everywhere on $M$, we only need the tangent vectors at two specific points, namely $\tau$ and $\tau'$. We can easily obtain a basis for the infinitesimal diagonal action of $G$ from these: \[R_a=T^1_a+T^2_a+T^3_a+T^4_a\] is the infinitesimal rotation of all 12 coordinates about $a$, and similarly we may define $R_c, R_e$, each a sum of four `$T$'s, which together span $\lie{g}\tau$. We will also denote these by \[ R_1, R_2, R_3 = R_a, R_c, R_e. \] Let $u$ denote an edge of the tetrahedron $\tau$, one of the vectors $a,b,c,d,e,f$. Let $U^u_w$ be the infinitesimal rotation about $w$ acting on the pair of spheres corresponding to $u$. For example, if $u=a$ then we have at $(x_1, x_2, \ldots, x_{12})$ \[ U^a_w=(w \times x_1, 0,0,0,0,0,0,0,w \times (-x_1),0,0,0) ,\] This vector preserves $x_1+x_9=0$ and hence $\nu$, and so do the other $U^u_w$. We want just two vectors $w_1, w_2$ such that $U^a_{w_1}, U^a_{w_2}$ span the tangent space to the orbit of $G$ acting on the first and ninth spheres in $M$ at $\tau$ (compare the previous case with the three `$T$'s.) Projecting into the first and ninth spheres, $\tau$ becomes $(a, -a)$ and $U^a_w$ becomes the tangent vector $(w \times a, w \times (-a))$. So all we need to do is pick $w_1, w_2$ such that $a, w_1, w_2$ are linearly independent. In this way we can construct 12 vectors spanning $\ker d\nu$ at $\tau$. Unfortunately there isn't a totally systematic way of deciding which two values of $w$ we should use, given $u$. We can at least choose them always to be two of the three vectors $a,c,e$, which forces for example the use of $U^a_c, U^a_e$ among our 12 vectors (because $U^a_a=0$). The twelve explicit choices are as follows: \[ U_1, U_2, \ldots, U_{12} = U^a_c, U^a_e, \quad U^b_a, U^b_e, \quad U^c_e, U^c_a, \quad U^d_c, U^d_a, \quad U^e_a, U^e_c, \quad U^f_c, U^f_e \] The same three diagonal generators $R_a, R_c, R_e$ can be expressed in terms of these vectors by observing that \[ R_a = U^a_a+U^b_a+U^c_a+U^d_a+U^e_a+U^f_a \] that $U^a_a=0$ and that $U^f_a$ (which is the only other not among our chosen $U_1, U_2, \ldots, U_{12}$) satisfies $U^f_a=-U^f_e$, because the fact that the three sides of the tetrahedron $\tau$ satisfy $-e+f-a=0$ implies \[ U^f_e+U^f_a=U^f_e-U^f_f+U^f_a=U^f_{e-f+a}= U^f_0 = 0.\] Similarly, one obtains $U^b_c=-U^b_a$ and $U^d_e=+U^d_c$, and hence: \begin{eqnarray} R_a &=& U_3+U_6+U_8+U_9-U_{12} \\ R_c &=& U_1-U_3+U_7+U_{10}+U_{11} \\ R_e &= &U_2+U_4+U_5+U_7+U_{12} \end{eqnarray} In the following calculation, a symbol such as $\det\omega(\{X_i\};\{Y_i\})$, where $\{X_i\}$ and $\{Y_i\}$ are some sets of vectors, will mean the determinant of the matrix whose entries are all evaluations of $\omega$ on pairs consisting of an element from the first set followed by one from the second set (arranged in the obvious way). In the case where the two sets of vectors are both bases of some fixed vector space, the symbol $\det(\{X_i\}/\{Y_i\})$ will be the determinant of the linear map taking $Y_i \mapsto X_i$. The grossly-abused subscript $i$ below stands for the complete list of such vectors (there are twelve `$T$'s, three `$R$'s, etc.) We regard all vectors as living in $T_\tau M$, in particular the original orthonormal bases are lifted horizontally into it. Let $\{e_1, e_2, e_3\}$ be some orthonormal basis of $\lie{g}\tau$. By extending the orthonormal sets of vectors and then changing bases inside the spaces $\ker d\mu$ and $\ker d\nu$ to bring in the `$T$'s and `$U$'s , we have \begin{align} \det(Q) &=-\det\omega(\{X_i\};\{Y_i\})\\ &= -\det\omega(\{X_i, e_i, Je_i\};\{Y_i, e_i, Je_i\})\\ &= -\det(\{T_i\}/\{X_i, e_i\})^{-1}\det(\{U_i\}/\{Y_i, e_i\})^{-1}\\ &\qquad\times \det\omega(\{T_i, Je_i\};\{U_i, Je_i\}) \end{align} The remaining $\det \omega$ term can be simplified further. Replace three of the `$T$'s ($T_{10}, T_{11}, T_{12}$) and three `$U$'s ($U_8, U_{11}, U_4$) by $R_1, R_2, R_3$. According to the earlier expressions for the $R_i$, each replacement is unimodular, and there is no sign picked up in reordering the `$U$'s to put $U_8, U_{11}, U_4$ (in that order) last. Then we change the `$R$'s back to `$e$'s and remove them. This gives: \begin{align} &\det\omega(\{T_i, Je_i\};\{U_i, Je_i\})\\ =&\det\omega(\{T_1, \ldots, T_9, R_i, Je_i\};\{U_1, U_2, U_3, U_5, U_6, U_7, U_9, U_{10}, U_{12}, R_i, Je_i\})\\ =& \det(\{R_i\}/\{e_i\})^2\\ &\times \det\omega(\{T_1, \ldots, T_9, e_i, Je_i\};\{U_1, U_2, U_3, U_5, U_6, U_7, U_9, U_{10}, U_{12}, e_i, Je_i\})\\ =& \det(\{R_i\}/\{e_i\})^2 \det\omega(\{T_1, \ldots, T_9\};\{U_1, U_2, U_3, U_5, U_6, U_7, U_9, U_{10}, U_{12}\}) \end{align} This remaining $9 \times 9$ determinant has to be done by explicit calculation. Fortunately the good choice of vectors helps enormously. The `$T$'s have only three non-zero coordinates (out of 12), the `$U$'s have only two, and these must overlap if there is to be a non-zero matrix entry. So a representative non-zero matrix element is something like \[ \omega(T^l_v, U^u_w)=\frac{1}{4\pi x^2} [x. (v \times x). (w \times x)] = \frac{1}{4\pi} [x. v .w]\] where $x$ is whichever of the `$x_i$'s corresponds to the overlap. (It will be plus or minus one of $a,b,c,d,e,f$, depending on whether the overlap of coordinates happens in the first or second of the two non-zero slots of the `$U$'--vector, respectively.) Writing down the matrix with rows corresponding to $T_1, T_2, \ldots, T_9$ and columns corresponding to $U^a_c, U^a_e , U^b_a, U^c_e, U^c_a, U^d_c, U^e_a, U^e_c, U^f_e$ gives \[ \frac{1}{4\pi}\left(\begin{matrix} 0 &0 &0 &[cae] &0 &0 &0 &0 &0\\ 0 &[ace] &[bca] &0 &0 &0 &0 &0 &0\cr [aec] &0 &[bea] &0 &[cea] &0 &0 &0 &0\\ 0 &0 &0 &-[cae] &0 &[dac] &0 &[eac] &0\\ 0 &0 &0 &0 &0 &0 &[eca] &0 &0\\ 0 &0 &0 &0 &-[cea] &[dec] &0 &0 &0\\ 0 &0 &0 &0 &0 &0 &0 &-[eac] &[fae]\\ 0 &-[ace] &0 &0 &0 &0 &-[eca] &0 &[fce]\\ -[aec] &0 &0 &0 &0 &0 &0 &0 &0 \end{matrix}\right) \] where $[ace]$ is just a shorthand for the vector triple product $[a.c.e]$. Substituting the relations $b=-a-c, d=c-e, f=a+e$ and extracting the factor of $[ace]$ gives \[ \frac{[ace]}{4\pi}\left(\begin{matrix} 0 &0 &0 &-1 &0 &0 &0 &0 &0 \\ 0 &1 &0 &0 &0 &0 &0 &0 &0 \\ -1 &0 &-1 &0 &1 &0 &0 &0 &0 \\ 0 &0 &0 &1 &0 &-1 &0 &1 &0 \\ 0 &0 &0 &0 &0 &0 &-1 &0 &0 \\ 0 &0 &0 &0 &-1 &0 &0 &0 &0 \\ 0 &0 &0 &0 &0 &0 &0 &-1 &0 \\ 0 &-1 &0 &0 &0 &0 &1 &0 &1 \\ 1 &0 &0 &0 &0 &0 &0 &0 &0 \end{matrix}\right) \] This matrix has determinant $[ace]^9/(4\pi)^9$. The various change-of-basis determinants may be evaluated in terms of orbit volumes. If we denote by $\sgn(\{X_i\}/\{Y_i\})$ the sign of the determinant of the appropriate transformation then \[ \det(\{T_i\}/\{X_i, e_i\}) = \sgn(\{T_i\}/\{X_i, e_i\})\vol_{B_\tau}\{T_i\}\] because $\{X_i, e_i\}$ is $B_\tau$--orthonormal. The volume is given by \[ \vol_{B_\tau}\{T_i\} = \frac{\vol(G^4\tau)}{\vol_\rho(G)^4}[ace]^4\] by lemma \ref{t:orbvol} and the fact that the 12 `$T$'s separate into four orthogonal triplets coming from the Lie algebra elements $a,c,e \in {\mathbb R}^3$. Similarly we have for the `$U$' case: \[ \det(\{U_i\}/\{Y_i, e_i\}) = \sgn(\{U_i\}/\{Y_i, e_i\})\vol_{B_\tau}\{U_i\}\] The volume term may be expressed via lemma \ref{t:rorbvol}. This will involve a product of six terms of the form $\vol_\rho\{\hat \xi_i\}$: for each choice of $u$, we have to calculate the area spanned by the vectors $w_1$ and $w_2$ once projected into the orthogonal complement of $u$. This is just $\vert [w_1. w_2.u]/u \vert$. Substituting the explicit choices we made gives \[ \vol_{B_\tau}\{U_i\} = \frac{\vol(G^6\tau)}{\vol_\rho(G/T)^6}\frac{[ace]^6}{\prod a} \] where the product on the right hand is simply $abcdef$. Yet another application of lemma \ref{t:orbvol} yields: \[\det(\{R_i\}/\{e_i\})^2 = \left(\frac{\vol(G\tau)}{\vol_\rho(G)}\right)^2[ace]^2 \] The sign terms here depend on the original choice of orthonormal bases, which we did not specify. So we do not yet know the actual sign of $\det(Q)$. Similar terms will appear in the computation of $\det(P-iQ)$, however, so that the product of the two terms does not depend on the original choice. So far we have \begin{equation}\begin{split}\det(Q)= &-\sgn(\{T_i\}/\{X_i, e_i\})\sgn(\{U_i\}/\{Y_i, e_i\})\left(\frac{[ace]}{4\pi}\right)^9 \\ &\quad\times \left(\frac{\vol(G^4\tau)}{\vol_\rho(G)^4}[ace]^4\right)^{-1}\left(\frac{\vol(G^6\tau)}{\vol_\rho(G/T)^6}\frac{[ace]^6}{\prod a}\right)^{-1} \left(\frac{\vol(G\tau)[ace]}{\vol_\rho(G)}\right)^2 \end{split}\end{equation} \subsection{Phase of the Hessian} We work out $\det(P-iQ)$ (defined by $((P-iQ)_{ij} = \overline{H_G(X_i, Y_j)}$) using similar techniques. We choose slightly different bases in $T_\tau M$ this time. For each face of the tetrahedron $\tau$, numbered by $l$ as earlier, choose three infinitesimal rotation vectors $T^l_v$ by letting $v$ be an exterior unit normal vector $v_l$ to the face or one of the two edges $x_{3l-2}, x_{3l-1}$ of that face. These occur in clockwise order, so that \[ x_{3l-2} \times x_{3l-1} = A_l v_l\] with $A_l$ twice the area of the $lth$ face. (Of course at the point $\tau$, we know that each $x_i$ is just one of the vectors $a,b,c,d,e,f$ or their negatives. However, it is easier to calculate without substituting these yet.) Pick a set of 12 vectors $U^u_w$ rather as before, except that given an edge $u$, we allow $w$ to be the exterior unit normal to either of the two faces incident at $u$. Order the two choices so that the first cross the second points along $u$ (in fact this corresponds to $v_i$ coming before $v_j$ iff $i<j$). Figure \ref{explode} shows these where these vectors are in ${\mathbb R}^3$, given the tetrahedron $\tau$. We will refer to the chosen vectors as $U_1', \ldots, U_{12}'$ and $T_1', T_2', \ldots, T_{12}'$. By a familiar change of basis procedure \begin{align} \det(P-iQ) &= \det H_G(\{Y_i\};\{X_i\}) \\ &=\det H(\{Y_i, e_i\};\{X_i, e_i\}) \\ &=\det(\{T_i'\}/\{X_i, e_i\})^{-1}\det(\{U_i'\}/\{Y_i, e_i\})^{-1}\det H(\{U_i'\};\{T_i'\}). \end{align} Since we are know the determinant is actually just a phase, we can throw away any positive real factors appearing during the computation. For example, the correcting determinants above may immediately be replace by correcting signs, because the difference (a volume) is positive. This principle also facilitates the direct computation of $ \det H(\{T_i'\};\{U_i'\})$ too. \begin{figure}[ht] \[\vcenter{\hbox{\mbox{\input{exptet2.pst}}}} \] \caption{The relevant vectors\label{explode}}\end{figure} Let us compute sample non-zero elements (once again, most of the matrix elements will be zero): \begin{align} H(U^u_w, T^l_v) &= \omega(T^l_v, JU^u_w + i U^u_w)\\ &= \frac{1}{4\pi x^3}[x .(v \times x) . (x \times (w \times x))] + \frac{i}{4\pi x^2}[x .(v \times x).(w \times x)] \\ & = \frac{1}{4\pi} (x(v.w)+i[xvw])\end{align} using the earlier notation for the triple product, and with $x$ being whichever of the `$x_i$'s corresponds to the overlap of the non-zero coordinates of $U^u_w, T^l_v$. There has been some simplification because $w.x=0$. Let us immediately forget about the $4\pi$ factors. If the $i$th and $j$th faces meet in an edge $u$, oriented along the direction of $v_i \times v_j$, then the exterior dihedral angle, written $\theta_u$ or $\theta_{ij}$, is defined (in the range $(0, \pi)$) by \[ u \sin (\theta_{ij}) = [u.v_i.v_j] \] \[ \cos(\theta_{ij}) = v_i.v_j. \] In performing the computation we run across three kinds of non-zero matrix elements. If $v=v_l$ is normal to the $l$th face, and $w=v_k$ then we obtain $\vert x\vert \e{k}{l} \equiv \vert x \vert e^{i\theta_{kl}}$. (If $k=l$ it is simply $\vert x \vert$.) If instead $v$ is a vector lying in the $l$th face, and $w=v_k$, then we distinguish according to whether $k=l$ or not. In the case of equality, we get case we get $0$, $iA_l$ or $-iA_l$ according to whether $v$ is $x$, its successor, or predecessor in the anticlockwise cyclic ordering around the face. If $k\neq l$ then we get $0$, $iA_l \e{k}{l}$ or $iA_l \e{k}{l}$, according to the same conditions. We can throw out the area factors and the eight powers of $i$ coming from these second and third cases, and end up with a matrix: \setcounter{MaxMatrixCols}{12} \footnotesize \[\left( \begin{matrix} a &a\e13 &b &b\e14 &c &c\e12 &0&0&0&0&0&0\\ 0 &0 &-1 &-\e14 &1 &\e12 &0&0&0&0&0&0\\ 1 &\e13 &0 &0 &-1 &-\e12 &0&0&0&0&0&0\\ 0&0&0&0 &c\e12 &c &d &d\e24 &e &e\e23 &0&0\\ 0&0&0&0 &0 &0 &-1 &-\e24 &1 &\e23 &0&0\\ 0&0&0&0 &\e12 &1 &0 &0 &-1 &-\e23 &0&0\\ a\e13 &a &0&0&0&0&0&0 &e\e23 &e &f &f\e34 \\ \e13 &1 &0&0&0&0&0&0 &0 &0 &-1 &-\e34 \\ -\e13 &-1 &0&0&0&0&0&0 &\e23 &1 &0 &0 \\ 0&0 &b\e14 &b &0&0 &d\e24 &d &0&0 &f\e34 &f \\ 0&0 &\e14 &1 &0&0 &-\e24 &-1 &0&0 &0 &0 \\ 0&0 &-\e14 &-1 &0&0 &0 &0 &0&0 &\e34 &1 \end{matrix} \right)\] \normalsize Easy row operations and then permutation of the rows reduces the determinant to minus that of the direct sum of the six $2\times 2$ blocks \[ \left(\begin{matrix}1&\e{i}{j}\\\e{i}{j}&1\end{matrix}\right) .\] The determinant of such a block is \[ -\e{i}{j}.2i \sin(\theta_{ij}) .\] Discarding the sines, which are positive and so only affect the modulus of the determinant, we find that its phase is \[ e^{i\sum \theta_a}. \] Let us collect up $\det(P-iQ)\det(Q)$ finally. The annoying sign terms may be combined into \[ \sgn(\{T_i\}/\{T_i'\}) \sgn(\{U_i\}/\{U_i'\}). \] The first term is a product of four signs arising from orientations of a three-dimensional vector space, and the second a product of six arising from two-dimensional spaces. All these signs are positive (an easy check). Thus we have for the Hessian: \begin{equation}\label{e:hess} \begin{split} \Hess_{[\tau]}(\psi) &= -i(2\pi)^{18}2^9 e^{i\sum \theta_a}\left(\frac{[ace]}{4\pi}\right)^9\\ &\qquad \times \left(\frac{\vol(G^4\tau)}{\vol_\rho(G)^4}[ace]^4\right)^{-1} \left(\frac{\vol(G^6\tau)}{\vol_\rho(G/T)^6}\frac{[ace]^6}{\prod a}\right)^{-1} \left(\frac{[ace]}{\vol_\rho(G)}\right)^2\\ & = -ie^{i\sum \theta_a}(2\pi)^9[ace]\left(\prod a\right)\left(\frac{\vol_\rho(G)^2\vol_\rho(G/T)^6}{\vol(G^4\tau)\vol(G^6\tau)} \right)\end{split}\end{equation} Looking at the argument again, it is easy to see that the Hessian at $[\tau']$ is the complex conjugate of this one. \subsection{The overall phase of the integrand}\label{phase} We need to account for the 0--order contributions $\psi([\tau']), \psi([\tau])$ of the integrand at the two critical points. Since the two sections being paired were fixed to have norm 1 along their critical regions, these 0--order contributions also have modulus 1. We start by calculating the phase {\em difference}. The chosen lifts $\tau, \tau '$ of these points lie on the slice $\mu=0$ inside $M$. Since this slice is just a product of four ``spaces of triangles'', each of which is a principal $G$--space, ($G=SO(3)$) there is a unique element of $G^4$ which translates $\tau$ to $\tau '$. In fact it is easy to describe such an element $g=(g_1, g_2, g_3, g_4)$ explicitly. The element $g_1$ must rotate the triangle $(a,b,c)$ (the projection of $\tau$ into the first three sphere factors of $M$) to its negative $(-a,-b,-c)$. Therefore it is the rotation of $\pi$ about the normal to the triangle's plane. The other $g_i$ are similarly half-turns normal to their respective triangular faces. We must compare the values of the pointwise pairing $\tilde s_\mu, \tilde s_\nu$ at $\tau, \tau '$. Define for each face a lift $\tilde g_i$ of $g_i$ into $SU(2)$ by lifting the path of anticlockwise rotations from $0$ to $\pi$. Together these form $\tilde g \in SU(2)^4$. Since $\tilde s_\mu$ is $SU(2)^4$--invariant: \[ \tilde s_\mu (\tau') = \tilde s_\mu (\tilde g\tau) = \tilde g \tilde g^{-1}(\tilde s_\mu(\tilde g \tau))= \tilde g ((\tilde g^{-1}\tilde s_\mu)(\tau)) = \tilde g (\tilde s_\mu(\tau)) \] By contrast, $\tilde s_\nu$ is not $SU(2)^4$--invariant, though it is $SU(2)^6$--invariant. We can write an equation like the above but we need to know what $(\tilde g^{-1}\tilde s_\nu)$ is to perform the last step. Now $\tilde s_\nu$ is a sextuple tensor product, and we can study the action of $\tilde g^{-1}$ on it by looking at the action on the six factors individually: \[\tilde g^{-1}\tilde s_\nu = (\tilde g_1^{-1}, \tilde g_3^{-1})s^{aa} \otimes (\tilde g_1^{-1}, \tilde g_4^{-1}) s^{bb} \otimes \cdots \otimes (\tilde g_3^{-1}, \tilde g_4^{-1})s^{ff}\] Using the diagonal invariance of each section of the form $s^{aa}$, we have identities like \[ (\tilde g_1^{-1}, \tilde g_3^{-1})s^{aa} = (1, \tilde g_3^{-1}\tilde g_1)s^{aa}.\] Now $\tilde g_1, \tilde g_3$ are lifts of rotations through $\pi$ about directions normal to the two faces of the tetrahedron meeting at side $a$, so the composite $\tilde g_3^{-1}\tilde g_1$ is a lift of an anticlockwise rotation through an angle equal to twice the exterior dihedral angle $\theta_a$ about the vector $a$, but it is slightly tricky to decide {\em which} lift. Let us denote by $\tilde r_a$ the lift of the path of anticlockwise rotations from $0$ to $2\theta_a$, and write $\tilde g_3^{-1}\tilde g_1= \delta_a \tilde r_a$, for some $\delta_a=\pm 1$. Then \begin{align} ((\tilde g_1^{-1}, \tilde g_3^{-1})s^{aa})(a, -a) & = (1, \delta_a\tilde r_a)(s^{aa}((1,\delta_a\tilde r_a^{-1})(a,-a)))\\ & = (1, \delta_a\tilde r_a)(s^{aa}(a,-a)).\end{align} The action of $\tilde r_a$ on the fibre of the bundle $\mathcal L^{\otimes a} \rightarrow \mathbb P^1$ at $-a$ is multiplication by $e^{-ia\theta_a}$ (remember that it acts as $e^{i\theta_a}$ on the fibre of the tangent bundle $\mathcal L^{\otimes 2}$ at $a$). The sign $\delta_a$ acts as its $a$th power $\delta_a^a$. So, using the invariance of the hermitian form, \begin{align} \langle \tilde s_\mu (\tau'), \tilde s_\nu(\tau') \rangle & = \langle \tilde g (\tilde s_\mu(\tau)), \tilde g (\left(\prod \delta_a^a\right) e^{-i\sum a\theta_a}s_\nu (\tau) )\rangle\\ & = \left(\prod \delta_a^a\right)e^{i\sum a\theta_a}\langle \tilde s_\mu(\tau), \tilde s_\nu (\tau)\rangle. \end{align} In fact this identity is independent of the choice of lifts $\tilde g_i$. For example, changing the lift of $\tilde g_1$ negates $\delta_a, \delta_b, \delta_c$, changing the right-hand side by $(-1)^{a+b+c}$, which is $+1$ because of the parity condition on $a,b,c$. Further, if we imagine varying the dihedral angles of the tetrahedron in the range $(0,\pi)$, all our chosen lifts are continuous and so we may evaluate the sign $\left(\prod \delta_a^a\right)$ by deformation to a flat one, for example where $\theta_a, \theta_b, \theta_c$ are $\pi$ and the others $0$. In this case, $\tilde g_2=\tilde g_3 =\tilde g_4 = - \tilde g_1$, and so all the $\delta_a$ turn out positive. Hence $\psi([\tau'])=e^{ik\sum a \theta_a}\psi([\tau])$. Because the $6j$--symbol itself is {\em real}, the two values of $\psi$ must be conjugate. Therefore $\psi([\tau'])=\pm e^{\frac12ik\sum a \theta_a}$, but we still have an annoying sign ambiguity. There are really three separate sign problems: how the sign depends on $k$ (for fixed $a,b, \ldots, f$); how it varies as we alter $a,b, \ldots, f$; and one overall choice of sign. One step is easy: the phase conventions on the sections used in the pairing implied for example that $s^{ka, ka} = (s^{aa})^{\otimes k}$ and that the integrand was a $k$th power of another function, the sign above must actually be $(-1)^k$ or or $1$. To do better requires a frustrating amount of work, which will only be sketched here. Recall that the signs in definition \ref{6jdef} were fixed by writing down explicit polynomial representatives of the trilinear and bilinear invariants. If we remove a suitable branch cut from each copy of the sphere in $M$, leaving a contractible manifold, we can extend this definition to allow real values of the variables $a,b, \ldots, f$, and extend $\psi([\tau])$ to a {\em real-analytic} function of these variables (at least locally). It is obtained by pairing two holomorphic sections of a trivial bundle with a hermitian structure which still satisfies the quantization formula \eqref{e:quant}. There is another way of computing the phase difference above, based on choosing two paths $\tau \rightarrow \tau'$ in $M$, one on which $\mu=0$ and one on which $\nu=0$, and computing the holonomy around the resulting loop. This can be done by computing the symplectic area of a bounded disc. To carry this out appropriately one must be very careful with {\em which} disc: once the symplectic form no longer has integral periods on $H_2(M)$, this matters. Further, the most obvious paths and disc intersect the branch cuts, so one must account for this too. Ultimately one obtains an analytic expression \[ \psi([\tau'])=\pm e^{\frac12ik\sum a \theta_a+i\sum a\pi}\] where the analyticity restricts this sign to a single {\em overall} ambiguity. (Note that at integral values of the lengths, we can see the sign $(-1)^{\sum a}$ appearing.) One could compute this sign using a single example, but as the reader will judge from this terse paragraph, the author is so bored with fixing signs that he no longer cares to! The experimental evidence in \cite{PR} confirms that the sign is positive. Therefore \begin{equation}\label{e:0order} \psi([\tau']=(-1)^{\sum a}e^{\frac12ik\sum a \theta_a} \quad \hbox{and} \quad \psi([\tau])=(-1)^{\sum a}e^{-\frac12ik\sum a\theta_a}.\end{equation} \subsection{Putting it all together} We combine the original integral definition \eqref{e:main} with the asymptotic normalisation factors \eqref{e:gnorm}, \eqref{e:gnorm2}, the reduction \eqref{e:red}, the stationary phase evaluation \eqref{e:stat} incorporating the Hessian \eqref{e:hess} and 0--order terms \eqref{e:0order}. \begin{equation}\begin{split} \ksixj &\sim \left(\prod\sqrt{ka+1}\right) \left(\frac{k}{2}\right)^{-6}(\vol(\mu^{-1}(0))\vol(\nu^{-1}(0)))^{-\frac12} k^9 \left(\frac{k}{2}\right)^{\frac32}\\ &\qquad\times \left(\frac{2\pi}{k}\right)^{9}\left\{\frac{e^{-\frac12ik\sum a \theta_a}}{\sqrt{-\Hess_{[\tau]}(\psi)}} +\frac{e^{\frac12ik\sum a \theta_a}}{\sqrt{-\Hess_{[\tau']}(\psi)}}\right\}.\end{split}\end{equation*} The terms from the Hessian and the normalisation involving \[\vol(\mu^{-1}(0))=\vol(G^4\tau)=\vol(G^4\tau')\ \hbox{and}\ \vol(\nu^{-1}(0))=\vol(G^6\tau)=\vol(G^6\tau')\] cancel. The normalisation factor $(\prod (ka+1))^{\frac12}$ cancels with the term in the Hessian involving $\prod a$, contributing asymptotically simply $k^3$. What remains is \[(2\pi)^{\frac92}2^{\frac{11}{2}}k^{-\frac32}[ace]^{-\frac12}\vol_\rho(G)^{-1}\vol_\rho(G/T)^{-3}\cos{\left\{ \sum (ka+1) \frac{\theta_a}{2} + \frac{\pi}{4}\right\}}.\] Substituting in the volumes $8\pi^2$ and $4\pi$ of $G$ and $G/T$ gives \[ \ksixj \sim \sqrt{\frac{2}{3\pi k^3V}} \cos{\left\{ \sum (ka+1) \frac{\theta_a}{2} + \frac{\pi}{4}\right\}} \] where $V=\frac16[ace]$ is the (scaling-independent) volume of $\tau$. This completes the proof of the theorem. \section{Further geometrical remarks} \subsection{Comparison with the Ponzano--Regge formula} It is important to note that the formula \eqref{e:form} is {\em not} the same as the original Ponzano--Regge formula. There are two main differences, apart from the trivial fact that they label their representations by half-integers instead of integers. Their claim, in our integer-labelling notation, is that for large $a,b,c,d,e,f$: \begin{equation}\label{e:formpr} \sixj \approx \begin{cases}{\displaystyle\sqrt{\frac{2}{3\pi V'}}\cos{\left\{ \sum (a+1)\frac{\theta_a'}{2} + \frac{\pi}{4}\right\}}} &\hbox{{if $\tau'$ is Euclidean,}} \\ \hbox{{exponentially decaying}}&\hbox{{if $\tau'$ is Minkowskian,}}\end{cases} \end{equation} where $\tau'$ is a tetrahedron whose edges are $a+1, b+1, \ldots, f+1$ and whose dihedral angles $\theta_a'$ and volume $V'$ are therefore slightly different from those of our $\tau$. This difference is worrying, as it is quite possible to find sextuples of integers such that $\tau'$ is Euclidean yet $\tau$ is Minkowskian, in which case the formulae seem to conflict: is the $6j$--symbol exponentially or polynomially decaying in this case? The second difference explains this. The Ponzano--Regge formula \eqref{e:formpr} is only claimed as an {\em approximation} for large irreducibles, rather than an asymptotic expansion in a strict sense as in theorem \ref{t:main}. Therefore the only meaningful comparison between the two formulae is to examine how their function behaves as we rescale $a,b,c,d,e,f$ by $k \rightarrow \infty$ in the precise sense of our theorem. Although for small $k$ it is possible that $\tau'$ might be Euclidean when $\tau$ is not, eventually the shift in edge-lengths becomes insignificant and either both are Euclidean or neither is. Therefore there is no inconsistency between cases in the two formulations. As for comparing the actual formulae, the asymptotic behaviour of the Ponzano--Regge function is \begin{equation}\label{e:formpr2} {\displaystyle\sqrt{\frac{2}{3\pi k^3V}}\cos{\left\{ \sum (ka+1) \frac{\theta_a'}{2} + \frac{\pi}{4}\right\}}}, \end{equation} because $V$ and $V'$ agree to leading order in $k$. The only problem is the dihedral angles relating to slightly different tetrahedra. Fortunately we may easily show that \[ e^{ik\sum (a+1) \theta_a'} \sim e^{ik\sum (a+1) \theta_a}\] by applying the {\em Schl\"afli identity} (see Milnor \cite{M2}), which says that the differential form $\sum a d\theta_a$ vanishes identically on the space of Euclidean tetrahedra. Therefore there is no inconsistency. \begin{rem} The case of a flat tetrahedron is not covered by either formula. \end{rem} \subsection{Regge symmetry and scissors congruence} Suppose one picks out a pair of opposite sides of the tetrahedron denoting the $6j$--symbol (as in figure \ref{figtet}), say $a,d$. Let $s$ be half the sum of the other four labels (twice their average). Define: \begin{equation}\label{f:regge} \begin{matrix} a'=a&\qquad&b'=s-b\\ d'=d&\qquad&c'=s-c\\ &\qquad&e'=s-e\\ &\qquad&f'=s-f \end{matrix} \end{equation} Regge discovered that the $6j$--symbols are invariant under this algebraic operation (the easiest way to see this is to look at the generating function for $6j$--symbols, \cite{V}): \[ \sixj = \sixjj\] We can also consider this as a {\em geometric} operation on a tetrahedron, altering its side lengths according to the above scheme. It is not meant to be obvious that the result of applying this to a Euclidean tetrahedron will return a Euclidean one! Regge and Ponzano considered the effect of this symmetry on the geometrical quantities occurring in their asymptotic formula, mainly as another check on its plausibility. They discovered that the volume and phase term associated to a Euclidean tetrahedron are indeed exactly invariant. This is amazing, given that it would be consistent with their appearance in an {\em asymptotic} expansion for them to change, but by lower-order contributions. Let us reconsider this surprising geometric symmetry. First note that the symmetry is an involution: if one thinks geometrically, it corresponds to reflecting the lengths of the four chosen sides about their common average. These involutions, together with the tetrahedral symmetries, form a group of 144 symmetries of the $6j$--symbol, isomorphic to $S_4 \times S_3$ (see \cite{V}). V.~G.~Turaev pointed out to me that the term $\sum l_i \theta_i$ in the phase part of the formula \eqref{e:form} for a tetrahedron $\tau$ is reminiscent of the Dehn invariant $\delta(\tau)$. Actually it would be fairer to say that it is the ``Hadwiger measure'' (or ``Steiner measure'') $\mu_1(\tau)$. Both invariants are connected with problems of equidissection of three-dimensional polyhedra. Two polyhedra are {\em scissors congruent} if one may be dissected into finitely-many subpolyhedra which may be reassembled to form the other. (Hilbert's third problem was to determine whether three-dimensional polyhedra with equal volumes were, as is the case in two dimensions, scissors-congruent. Dehn used his invariant to solve this problem in the negative.) The modern way of looking at the problem is to define a Grothendieck group of polyhedra $\mathcal P$. We take $\mathbb Z$--linear combinations of polyhedra in ${\mathbb R}^3$ with the relations: \begin{align} P \cup Q &= P + Q - P\cap Q \\ \label{e:deg}P&=0 \qquad \hbox{if $P$ is degenerate}\\ P&=Q \qquad \hbox{if $P$, $Q$ are congruent} \end{align} Volume is an obvious homomorphism $\mathcal P \rightarrow \mathbb R$. The Dehn invariant is a less obvious one $\mathcal P \rightarrow \mathbb R \otimes_{\mathbb Z} (\mathbb R/\pi \mathbb Z)$, defined for a polyhedron by summing, over its edges, their lengths tensor dihedral angles: \[ \delta(P) = \sum l_i \otimes \theta_i\] Sydler proved that these two invariants suffice to {\em classify} polyhedra up to scissors-congruence: two such are scissors-congruent {\em if and only if} they have the same volume and Dehn invariant. See Cartier \cite{C} for more details. If we look for homomorphisms $\mathcal P \rightarrow \mathbb R$ which are {\em continuous} under small perturbations of vertices of a polyhedron, then volume is the only one (up to scaling). However, if we remove the condition \eqref{e:deg} on degenerate polyhedra from the axioms defining $\mathcal P$, there is a four-dimensional vector space of continuous homomorphisms, spanned by the following {\em Hadwiger measures} (picked out as eigenvectors under dilation): \begin{align} \mu_3(P) &= \vol(P)\\ \mu_2(P)&= \textstyle{\frac12} \area(\partial P)\\ \mu_1(P) &= \textstyle{\sum} l_i \theta_i\\ \mu_0(P) &= \chi(P) \quad \hbox{(the Euler characteristic)} \end{align} See Milnor \cite{M} or Klain and Rota \cite{KR} for more on these beautiful functions. The relationship with the Regge symmetry of tetrahedra is as follows: \begin{theorem}\label{t:Dehn} The Regge symmetry \eqref{f:regge} takes Euclidean tetrahedra to Euclidean tetrahedra, preserving volume, Dehn invariant and Hadwiger measure $\mu_1$. {\rm (}Remark: simple examples show that Regge symmetry does not preserve the surface area measure $\mu_2$.{\rm )} \end{theorem} \begin{proof} The tour-de-brute-force of trigonometry in appendices B and D of \cite{PR} contains all the calculations necessary to prove this. They demonstrate that under the Regge symmetry, which is a rational linear transformation $A$ of the six edge lengths, the dihedral angles also transform according to $A$. The orthogonality of this matrix and the fact that we may view the Dehn invariant as being in $\mathbb R \otimes_{\mathbb Q} (\mathbb R/\pi \mathbb Z) \equiv \mathbb R \otimes_{\mathbb Z} (\mathbb R/\pi \mathbb Z)$ demonstrate its invariance, as well as that of $\mu_1$. The volume is checked by straightforward calculation using the Cayley determinant. \end{proof} \begin{coroll} The orbit under the group of 144 symmetries of a generic tetrahedron consists of {\em twelve} distinct congruence classes of tetrahedra, all of which are scissors-congruent to one another. \end{coroll} \begin{rem} The fact that a tetrahedron is scissors-congruent to its mirror-image was proved by Gerling in 1844 (see Neumann \cite{N}), using perpendicular barycentric subdivision about the circumcentre. One would expect that for the Regge symmetry, which is also a ``generic'' scissors congruence (as opposed to a ``random'' coincidence of volume and Dehn invariant for two specific tetrahedra), a similar general construction might be given. What is it? \end{rem} \subsection{Further questions} 1.\qua Ponzano and Regge give an explicit formula for the exponential decay of the $6j$--symbol, in the case when no Euclidean tetrahedron exists. Amazingly, it is an analytic continuation of the main formula, incorporating the volume of the {\em Minkowskian} tetrahedron which exists instead, and with the oscillatory phase term converted into a decaying hyperbolic function. Can this be extracted from a similar procedure? 2.\qua Can similar geometrically-meaningful formulae be obtained for general spin networks, the so-called $3nj$--symbols? 3.\qua The calculation in this paper is comparatively crude, since it computes a pairing of 12--linear invariants when one could really do with a pairing of 4--linear invariants (see the remark of section \ref{4def}). The space whose quantization gives quadrilinear invariants is 2--dimensional, in fact a sphere with a non-standard K\"ahler structure. Can one work directly on this space instead? (Possibly any advantage in dimensional reduction is lost when one needs to do explicit calculations, which end up like the ones here). 4.\qua Can similar formulae be obtained for other groups, and does their associated geometry have any physical meaning? The $6j$--symbols are scalars only for multiplicity-free groups such as $SU(2)$. In general they live in the tensor product of four trilinear invariant spaces, in which one would need preferred bases. 5.\qua Can one obtain similar formulae for quantum $6j$--symbols, which arise as pairings in the quantization of moduli spaces of flat connections on the 4--punctured sphere? \np	{ "redpajama_set_name": "RedPajamaArXiv" }	7,443
Баграт III од Грузије (око 960 - 7. мај 1014) члан грузијске династије Багратиони, краљ Абхазије од 978 (као Баграт II) и краљ Грузије од 1008. па надаље. Као и други грузијски монарси, познат је по својој монархијској титули mepe (მეფე). Он је ујединио ова два наслова династичке баштине, а освајањем и дипломацијом, додао још неке земље свом краљевству, те тиме ефикасно постао први краљ који је владао јединственим грузијским краљевством. Пре него је крунисан за краља, владао је као династ у Картлији од 976 до 978 године. Надгледао је изградњу катедрале Баграти у Кутаисију у западној Грузији. Катедрала је уврштена на УНЕСКО-в попис Светске културне баштине. Рођен је око 960. године као син Гургена, принца Картлије из династије Багратиони и његове жене Гурандукхт, кћери краља Гиоргија II од Абхазије. Као још малољетног, усвојио га је рођак Давид III Куропалат, који није имао деце. У то време, Давид је био најмоћнији владар на Кавказу. Абхашко краљевство било је под влашћу Теодозија III Слепог, немоћног краља који је био Багратов ујак. Краљевство ја запало у хаос и ратове међу феудалцима. Користећи ситуацију, принц Квирике II од Кахетије (939-976), која је тада најисточнија регија Грузије, упада у Картлију, дотада под влашћу абхашких краљева и опседа у стени уклесано упориште Уплисцихе. Јоане Марусхис, енергични гувернер (eristavi) Картлије, позвао је 976. Давида од Таоа да преузме контролу над покрајином или да га да Баграту у наследни посед. Давид је енергично одговорио и Кахетијци су се морали повући како би избегли сукоб. Литература Лордкипханиџе, М (1967), Georgia in the XI-XII centuries, Ganatleba, edited by George B. Hewitt. Такође расположиво на Рођени 960. Умрли 1014. Династија Багратиони Краљеви Грузије	{ "redpajama_set_name": "RedPajamaWikipedia" }	9,198

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